Tap Into Teen Minds https://tapintoteenminds.com Sparking Curiosity in Mathematics Through Tasks that Are Contextual, Visual and Concrete Thu, 15 Nov 2018 02:36:16 +0000 en-CA hourly 1 https://wordpress.org/?v=4.9.8 35226203 Exploring Proportional Relationships Developmentally https://tapintoteenminds.com/proportional-relationships-developmentally/ https://tapintoteenminds.com/proportional-relationships-developmentally/#respond Tue, 13 Nov 2018 02:51:31 +0000 https://tapintoteenminds.com/?p=53124 Another post from Tap Into Teen Minds. Enjoy!

Many educators underestimate the complexity of proportional relationships. Let's take one task and make small modifications to it so it is low floor and high ceiling; then, let's tackle it from a developmental continuum!

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Another post from Tap Into Teen Minds. Enjoy!

Over the past few years, I have been spending a lot of time thinking about proportional relationships. From unpacking the information in the Paying Attention to Proportional Reasoning document to meeting with a proportional reasoning symposium group through the Arizona Mathematics Project, my mind has been blown on a regular basis with how complex this seemingly basic and predictable mathematical concept actually is.

Having spent a ton of time teaching grade 9 courses in Ontario, the ideas we focus on all build on the learning of proportional relationships from previous years. However, because the content I was teaching was supposed to build on the ideas of proportional situations to move on to focus primarily on linear relationships, I found myself moving through the content quickly in order to get to direct and partial variation linear relationships graphically and algebraically.

I now realize that the vast majority of my students – including those I thought were “high achieving” – had an incomplete understanding of proportional relationships.

How do I know?

Because I had an incomplete understanding of proportional relationships myself.

There is so much goodness to unpack in the area of proportional relationships and I’m still finding myself going down rabbit holes with colleagues like Jon Orr, James Tanton, Dick Stanley, and Yvette Lehman to better understand how it all fits together.

To give you a bit of a backstory, most schools in my district have had student and teacher learning needs identified in fractions, measurement, and proportional reasoning, so the math team (Yvette Lehman, Andrea Lewis-Longmuir, Shelley Pike, Angeline Humber, and I) has been actively trying to create meaningful math professional development to highlight the very important connections between all three areas.

The task that we wanted teachers to engage in and then lead in their classrooms with students in Kindergarten through Grade 8 (modifying where appropriate) was a simple problem made a bit more curious centered around this question:

At a party, guests ate 14 packages of Reese’s Peanut Butter Cups.

There is 1 sixth cups of sugar in each package.

How much sugar is in 14 packages?

Reeses Peanut Butter Cups 1 sixth sugar question.001

This task was made much more curious by not rushing to this particular problem, but by starting with something a little more open and low floor. Check it out here.

For this particular question, we first withheld the information of how much sugar is in each package to have students make a prediction of how much sugar would be in 14 packs. That lowers the floor and also gives you some perspective about how much your students know about the sugar content in food, whether they have an understanding of units of measure for mass and much more.

When we engaged in this task as teachers, we anticipated certain student solutions, but most (including myself) were quite surprised at how few solution strategies students had when they did the task. Regardless of the grade level and how the task was modified to be more accessible for younger learners, there was a lot of procedures and not a whole lot of mathematical models demonstrated through the use of tools and representations.

Before moving on, I’d recommend that you stop and do the following:

  • solve this task in as many ways as you can using various concrete, visual, and symbolic representations;
  • consider what students must be able to do before they are able to be successful with this task; and,
  • think about how this task can be modified to ensure that all learners in your classroom have an entry point to this task.

Once you’re done doing that, come back and then have a look at some of the modifications that were made for a couple different grade levels as well as how students approached this task.

Modifications For A Grade 1 Classroom

After discussing possible modifications we could make to the task to help students in a grade 1 class access the task with the math learning team at an elementary school, we went to give it a try.

Here’s the modifications we landed on:

  • Change the quantity of packages from 14 to 5 packages of Reese’s Pieces.
  • Change the quantity of sugar and units of measure from 1 sixth cups of sugar per package to 1 half cups of sugar per package.

We also ensured that we had tangible materials for students to use. In this case, both real sugar and a non-standard “cup” that was transparent so the students could clearly see approximately half of each container without having to worry about the complexity and abstractness of standard “cups”. The best we could find in this case was 5 glass beakers with no standard units of measurement marked on the sides. We also had 5 printed images of Reese’s Peanut Butter Cup packages.

001 - Tangible Materials - Beakers and half cups of sugar copy

If you’re interested to see how two grade 1 students approached the task using this modification and tangible materials, here’s how it went:

What are your thoughts?

When we introduced this task to most educators, many believed that there would be no way for any younger students from say Kindergarten to Grade 2 to access this task. However, with slight modifications, we can get students thinking fractionally and start them on their proportional reasoning journey. These students were able to count how many half cups (or half beakers) of sugar there would be for 5 packages of Reese’s Peanut Butter Cups!

Some questions we could ask to stretch their thinking include:

  • How many half cups of sugar would we need for 6 packages of Reese’s Peanut Butter Cups?
  • How many half cups do you need to make a whole cup of sugar? (remove the other 3 packages and cups to focus their attention on just 2 cups)
  • Give the students a whole cup of sugar and ask them to pour it into as many half cups as they can. What happened? How many half cups are there?
  • What would happen if you pour 2 whole cups of sugar into as many half cups as you can?

In this particular case, these students were able to count how many halves, but were currently stuck at regrouping to determine how many whole cups there would be. This is something we can work with these two on by providing more experiences and by considering using different tangible materials (i.e.: whole apples and half apples, etc.).

Modifications For A Grade 4 Classroom

After seeing how much amazing thinking our friends in Grade 1 could do with this task, what do you think this task could look like in a Grade 4 classroom?

If the students in the class are comfortable with the fractional language of halves and fourths (not just quarters), it is realistic to think that students could work with this problem using the same situation of 1 sixth cups of sugar. However, working with 1 fourth cups of sugar could work to lower the floor.

With this group, we attempted the task “as is” to see what students could do and learn more about where we should go next.

The tangible materials and concrete manipulatives we had available were:

  • Printed images of Reese’s Peanut Butter Cup packages;
  • Connecting cubes;
  • Relational rods;
  • Square tiles;

I’ll recommend again that you stop to anticipate what student thinking might come out of this if you haven’t already. Maybe your thinking has changed since you saw the grade 1 students work on this problem.

Now that you’re back, let’s have a look at our first group of students who used the printed packs of Reese’s and single connecting cubes to represent 1 sixth cups of sugar and then simply counted how many sixths they had in total.

Counting With Manipulatives Linking Cubes copy

Another group of students were able to use fraction tiles to represent the number of packages and the number of 1 sixth cups of sugar. The tricky part here was that they were using these manipulatives without regard to area, which might not be the easiest way for students to show their thinking, however I wanted to help see their thinking through.

003 - Reeses Proportional Relationships - Counting with Manips copy

They were able to count and regroup or unitize to determine that there were 14 sixth cups of sugar and that was the same as 2 whole cups and 2 sixth cups of sugar.

Here’s these students sharing their thinking:

Another group of students had used connecting cubes after the teacher had prompted them to try and show their thinking concretely.

Linking Cubes 1 white out of 6 cubes copy

The white connecting cube represents the 1 sixth cups of sugar in each group of 6 cubes.

Many of the groups required teacher prompting and that suggested to me that modifying the task to a lower number of packages and maybe 1 fourth cups of sugar with tangible materials might have been useful for this group before getting to this point.

However, once students were able to represent the 14 sixth cups of sugar, many were able to regroup to unitize. Many teachers in later grades are frustrated when students cannot convert from improper fractions to mixed fractions and that is exactly what these students were doing with relative ease.

Some questions / next steps to stretch their thinking include:

  • Engaging in paper folding activities where students are taking whole pieces of paper and partitioning by making folds, cutting up the pieces and re-grouping;
  • Using other foods that are typically partitioned from wholes into parts like apples, cucumbers, cakes, brownies, etc.;
  • Asking students questions that promote fractional thinking through counting unit fractions to build fluency and flexibility;

Modifications For A Grade 7 Classroom

The next classroom we went into was a grade 7 classroom.

Based on what we’ve witnessed in grade 1 and grade 4, what sorts of modifications and/or materials would you have ready for these students?

Take a moment to reflect.

Maybe jot down your thinking on a scrap piece of paper and then come back.

Alright, now to give you the full story on this class, the teacher had left this task as a diagnostic while she was away at professional learning the previous week. She let us know that they had not done any work explicitly on fractions or proportional relationships, so this was definitely just to get a baseline of where students are and what they are bringing with them from previous years.

What we had found was that for the students who solved the problem correctly, most had relied on procedure and converting to a decimal to come to their solutions. Only two (2) others had relied on a concrete or visual representation to show their thinking. Many other students were trying to rely on a procedure symbolically, but when they applied their strategy incorrectly, they were unable to determine whether their answer made sense. This is very common in the intermediate/middle school classrooms I work with.

We decided to go back in and ask students to help us understand their thinking and give us an opportunity to press for understanding a wee bit.

This first student had a visual set / bar model and it was clear on his paper that he had been able to count and regroup / unitize to whole cups of sugar, so I wanted to ask him to take me through his thinking. I wasn’t quite sure why he had listed the numbers 1 through 14 each 3 times.

Bar and Set Model With Regrouping copy

So, let’s give a listen:

After prompting another group to show where they got their solution, they drew circle area models and used additive thinking to count up how many sixth cups of sugar there were:

Additive and Multiplicative Thinking copy

Many students immediately attempted converting from 1 sixth in standard notation to a decimal, but incorrectly. Many believed that 1 sixth is represented as 0.6 as a decimal.

With the following group, I asked them to show their thinking as they only had represented their solution symbolically as 14 x 0.6 = 8.4

Here’s what they did:

Visual Representation volume model and 1:6 as 0.6

After they drew out 14 packages, they just added 0.6 cups of sugar and were unable to make sense of the 8.4 they had come up with previously.

Another group had successfully found the total amount of sugar in 14 packages by converting to a decimal and multiplying by 14 symbolically.

Symbolic Representation Converting to Decimals Correctly

However, when I asked them to show their thinking using some sort of concrete or visual representation, they were unable to get started initially.

Another group had initially believed that 1 sixth cups of sugar was the same as 1.6 cups of sugar.

After encouraging them to represent their thinking, they were able to verbally describe the quantity of sugar as 14 sixth cups of sugar and regrouped / unitized successfully.

Additive and Multiplicative Thinking Using Visual and Symbolic With Decimal Conversion copy

Some questions / next steps to stretch their thinking include:

  • Asking students to show what 0.6 looks like using a linear, area, set and volume model (modelling will likely be required);
  • Asking questions similar to this that make use of linear, area, set and volume models to encourage students making use of those tools and representations;
  • Explicitly using fractional language that encourages the counting of unit fractions to build fluency and flexibility;

Download The Developmental Continuum Cheat Sheet!

You can learn more about Proportional Reasoning Developmentally by downloading the developmental continuum cheat sheet to save, print, and share!

What Is Your Biggest Take Away? What Questions Do You Still Have?

What was your biggest take away from this post?

What questions do you still have?

In order to truly learn, you must generate new knowledge and understanding through reflection. Without it, new ideas will wash away like footprints in the sand. I can’t wait to read your thoughts in the comments.

Also, keep an eye because we aren’t done with proportional relationships yet. There is so much to dig into and Jon Orr and I will be diving deeper in the coming months along the developmental continuum from counting to algebra!

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The 3 Act Math Beginner’s Guide https://tapintoteenminds.com/3act-guide/ https://tapintoteenminds.com/3act-guide/#respond Sun, 21 Oct 2018 15:15:02 +0000 https://tapintoteenminds.com/?p=53016 Another post from Tap Into Teen Minds. Enjoy!

Wondering how to use 3 act math tasks in your classroom? Let's dive into Dan Meyer's math class structure for building an effective problem solving experience for all learners!

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Sparking Curiosity With 3 Act Math Tasks

TITM - The 3 Act Math Beginner's Guide Featured Image

Looking to spice up your daily problem solving routine in your math class?

Serving up your math tasks in Dan Meyer’s 3 Acts of a Mathematical Story format are a great way to go!

When I was first introduced to 3 Act Math Tasks, it was when I heard Dan speak at an OAME Annual Conference. The idea of bringing the real world into my math classroom immediately resonated with me and I made a personal and professional commitment to start building these types of tasks into my lessons.

Although Dan and a huge number of others like Robert Kaplinsky, Graham Fletcher, Jon Orr, Dane Ehlert, and many others from the Math Twitter Blogosphere were openly sharing their tasks on the web, I found that the lessons didn’t always run so smoothly.

What I eventually realized is that while my immediate addiction to this type of task format was helpful for ensuring that I began the transformation that my lesson delivery desperately needed, I had failed to reflect on what elements were necessary to ensure my lessons ran without a hitch.

“While my immediate addiction to this type of task format was helpful for ensuring that I began the transformation that my lesson delivery desperately needed, I had failed to reflect on what elements were necessary to ensure my lessons ran without a hitch.”

In this post, I plan to share a few key tips that I wish I was more intentional about when I began delivering 3 act math tasks in my own classroom.

SPARK CURIOSITY

What caught your eye about the 3 act math task you’ve found?

What made you think it would be useful for your classroom?

Often times, we can be tricked into thinking kids will enjoy a problem because it is from the real world, it is relevant, or because there is a video they can watch. These are all misconceptions. Think about how the problem might make a student curious to engage and solve the problem.

In the Gummy Worms 3 Act Math Task, I’ve made an attempt to Spark Curiosity.

Take a look.

But what makes this a curious moment?

CREATE ANTICIPATION BY WITHHOLDING INFORMATION

One of the best ways to make a curious moment is to avoid giving students all of the information necessary to solve the problem upfront. Withhold counts, measurements, and even the question you intend to ask as a means to address the learning goal you’ve selected for that lesson.

When first exploring 3 act math tasks, it is easy to miss the fact that the first video, act 1, typically gives little information about the question we are asking or any measured quantities. Much like a well written movie script, the filmmaker is intentionally giving just enough information to capture the attention of the audience and will build the storyline slowly to keep that attention.

In act 1 of the Gummy Worms task, you’ll notice that there really wasn’t a whole lot of anything going on. Just a jar placed on counter leaving students to wonder: “what the heck is going on here?”

In order for a 3 act math task to be successful, we must build anticipation in our students by withholding information that you typically get upfront in a textbook or on a worksheet. Dan Meyer talks about this more here.

“In order for a 3 act math task to be successful, we must build anticipation in our students by withholding information that you typically get upfront in a textbook or on a worksheet.”

NOTICE AND WONDER

Once we have built anticipation through the withholding of information, we can now empower student voice by asking them what they notice and what they wonder about the image or video clip.

Some common ideas shared during the notice and wonder portion of the Gummy Worms task include:

  • There’s a jar with gummy worms in it.
  • What does the tag say?
  • Nice countertops!
  • How many gummy worms are in the jar?
  • Why is the jar so empty?
  • How many gummy worms would it take to fill the jar?
  • … and many others.

Be sure to leave this questioning open as asking for them to pay attention to only things related to mathematics may shut down some students, especially those who may not feel confident enough in their thinking. Some mistakes I’ve made in the first handful of years trying to use 3 act math tasks was asking questions like: “what is the question?” or “what math-related things do you notice and wonder?”

Questions like these sure did limit the conversation in my classroom to only include the more confident students and shut out those who aren’t so certain.

Once you’ve jotted down the noticings and wonderings of your students, it’s time to think about what questions you might consider solving. Of course you have planned out ahead of time which question you’d like to ensure that you answer to address a specific learning goal, however often times, students will have asked that question during the notice and wonder stage. If they don’t, that’s OK! It doesn’t always have to be the students who get to pick which questions we are going to solve and typically, the question I have in mind is a “gateway” question to give us enough information to answer other questions that have already been asked.

ESTIMATION

Did you know that the word “estimate” comes up in the Ontario curriculum 53 times while the word “calculate” only appears a total of 17 times?

Of course there are other words that imply “calculate” that we must consider, but my point is that estimation is an important part of building mathematics proficiency. Using the 3 act math structure to withhold information allows for a unique opportunity for students to engage in real, worthwhile estimations that we typically do not have when using traditional word problems from the textbook or a worksheet.

I can vividly remember myself as a student looking at questions like:

There are 7 rows of 6 cookies in the display case at the bakery.

a) Estimate how many cookies there are in total.
b) Calculate how many cookies there are in total.

After reading a question like this, I would routinely tackle part b) first to get an answer of 42 and then take 1 or 2 off to jot down my “estimate”.

This isn’t a great way to promote students wanting to take part estimation activities or develop their estimation skills.

With the 3 act math structure to approach problem solving, the anticipation we have built up through the withholding of information and engaging in the notice and wonder almost forces everyone to NEED to take part in making estimates.

This is where everyone puts some skin in the game and even a bit of friendly competition develops. Now everyone WANTS to know the answer to settle things.

A great strategy to get kids using their adaptive reasoning skills before just tossing out any old number for their estimate is to have students consider numbers that are “too low” and “too high”. This is a great way to help kids sort of “box in” a nice range for them to play in for their estimate. It might also get them thinking more deeply about the actual situation, slowly unravelling all of the details that might be in plain sight, but unnoticed until thinking about what is realistic.

Once students “box in” their reasonable estimate range, they can then go ahead and pick their “best guess”.

Allowing students to share their best guesses openly is a great way to set a range of estimates for the entire class. The beauty of this moment is that we can now look at the range of best guesses and come to a conclusion that we’ve got a pretty wide range of possibilities. This is where the mathematics begins to be a super helpful tool for us to narrow down that range to something much more precise.

“Allowing students to share their estimates openly is a great way to set a range of estimates for the entire class. It is then when we can highlight how we can use mathematics as a tool to narrow down our range to something much more precise.”

At this point, you’re in a great spot to ask students for what other information they’d like to know in order to “improve” their estimates. Asking them to explain their reasoning for requesting that information can give you a nice window into their mind to determine what sort of mathematical thinking and strategies they might be ready to use.

Wanting to focus more on estimation specifically? Be sure to check out Andrew Stadel’s Estimation 180.

New to Using 3 Act Math Tasks?

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!

PROMOTE STUDENT THINKING

I’ll typically share out more information at this point or, like in the Gummy Worms 3 act math task, I reveal the answer to this first question as a means to get us to the intended learning goal for that day. In this particular problem, after students witness me take 25 gummy worms out of the jar, I then put a handful back in and they are left to determine how many I must have put in the jar?

Gummy Worms 3 Act Math - Part-Part-Whole Model 07 - Sequel How many gummy worms were taken from the jar

Now, set them free on the task to solve it using any strategy they would like without pre-teaching them how they could/should or you would go about solving it.

This might be hard for some of us to swallow because we teachers always feel the need to ensure that kids can successfully attack the problem. Of course, we want to make sure we are giving tasks that they have an entry into, however we don’t want to be pre-teaching them or all of the curiosity, anticipation, and excitement is gone.

The most effective way to implement a 3 act math task or any other type of curious problem is to promote student thinking without explicitly pre-teaching the concept. If a lesson is taught before students have had an opportunity to solve a problem using their prior knowledge and through the inquiry process, this can immediately shut down some students who do not feel confident with the newly presented ideas and/or concepts.

“Avoid pre-teaching the rules, steps, and procedures in order to allow students to apply their prior knowledge and show you what they know before making connections during the consolidation.”

Not only does pre-teaching rob students of the opportunity to think, but it can often lead teachers to feel as though 3 act math tasks “aren’t working” because students don’t appear any more engaged or interested in the math lesson than before. While it might feel like we are helping students by teaching the steps and procedures that will help them successfully solve the problem, we are often times helping only those who have strong memorization skills while leaving the others behind. Pre-teaching may result in the abandonment of this problem solving structure before experiencing the huge benefits.

FUEL SENSE MAKING

Plan with intentionality to fuel sense making as you help push student thinking in the direction of the new learning. Making use of the 5 Practices for Orchestrating Productive Discussions as you anticipate, monitor, select, sequence, and connect the mathematical ideas you have planned with intentionality will be extremely important to maximize student learning.

Selected specific students to share out their useful mathematical models and strategies prior to you sharing additional models and strategies you would like to highlight through direct instruction.

Although you can use individual 3 act math tasks for many different learning goals, one that I tend to use the Gummy Worms task for is to explicitly introduce and/or re-visit the part-part-whole or “parts whole” model for addition and subtraction:

Gummy Worms 3 Act Math - Fuel Sense Making 2 - Adding and Subtracting With The Part-Part-Whole Model

So while I’ve allowed different students to share out different strategies based on the 5 Practices, I am taking some time to do some explicit teaching through direct instruction to help push student thinking forward in the area of addition and subtraction structures. Here are some other addition and subtraction structures you might consider checking out on my site Math Is Visual.

THE BIG REVEAL

Just like a great Hollywood movie, Act 3 is the conclusion of the storyline.

In math class, this is where we share what really happened in the real world. It’s great to have a video or image for this portion, but not always a requirement.

Keep in mind that some 3 act math tasks might have act 3 come at the end of the task, or if you think about the Gummy Worm problem, the first act 3 came pretty early on after the estimating. However, once students are engaged and can visualize the contextual situation in their mind, you’ve got them for all kinds of mathematical learning opportunities.

PURPOSEFUL PRACTICE

Practice is something that isn’t discussed a whole lot during workshops, keynotes, and presentations which can often lead us teachers to make interpretations as to why that is. While many teachers tend to think that if you don’t hear about it any more, then it must not be a thing or “allowed”, I’ll argue that there are many great things that happen in a traditional math classroom that are still really useful today. You might not hear about them much, but I think that is because most people are aware that they exist and thus they aren’t really exciting to talk about during presentations and workshops.

Practice for your students in math class is important.

However, the way we go about practicing is really important.

“Plan an opportunity for purposeful practice. While repetition is an important part of learning, focusing on too much repetition too quickly may lead to some students disengaging.”

Plan an opportunity for purposeful practice. Repetition when learning anything in life is really important, but focusing on too many reps too fast as well as focusing on speed can actually hurt more than help.

Think of ways that you can spread out your practice over time instead of all in one chunk. It’s also a huge bonus If you can make purposeful practice that is connected to the context/story from the 3 act math task, but this is not a requirement.

New to Using 3 Act Math Tasks?

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!

While I believe the ideas above would certainly help someone who is trying to implement 3 act math tasks in their classroom for the first time or to sharpen up on their current usage of this problem type, I’m pretty confident there are other important elements lurking in the background.

Can you help us identify some in the comments?

Also, if you’d like to take this learning on the run or share this with a colleague, be sure to download the printable 3 Act Math Tip Sheet here.

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Make Math Moments That Matter In Your Classroom https://tapintoteenminds.com/make-math-moments/ https://tapintoteenminds.com/make-math-moments/#respond Tue, 09 Oct 2018 07:00:27 +0000 https://tapintoteenminds.com/?p=52998 Another post from Tap Into Teen Minds. Enjoy!

Make Math Moments That Matter for YOUR students by learning how to Spark Curiosity, Fuel Sense Making, and Ignite Your Next Teacher Moves each and every day

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Teaching mathematics is complex.

First, there is the content; the “what” we are teaching. From as early as counting and quantity, students are wrestling with some pretty complex ideas including subitizing, abstraction, and unitizing which quickly develops into addition, subtraction, multiplication and division. Then, possibly without even knowing it, kids find themselves in a world of fractions, proportional relationships, algebra and beyond.

Then, there is the pedagogy; the “how” we are going to teach it. Should I use a 3-part math lesson? When should I be doing small group instruction? Should my math class always have kids working in collaborative groups or individually? How often should I give paper and pencil assessments? How long should my units of study be and should it be all one topic or should the math content spiral?

Regardless of whether you are brand new to teaching K-12 mathematics or a seasoned vet, these questions and many more probably still swarm inside your head on a regular basis.

There’s a lot to think about when teaching mathematics and it’s really hard if you try to do it alone. That’s why Jon Orr from MrOrr-IsAGeek.com and I started intentionally collaborating online a few years ago. We were frustrated with how hard we were working to deliver lessons that would have a positive impact on our students both in how they perceived math class and their understanding of the mathematics. At the time it seemed that one day, our lessons would be a huge hit, while on others they’d seem to flop hard.

We were left wondering how we could consistently Make Math Moments That Matter for our students and not just for that lesson or unit, but for weeks, months, or even years.

“We were left wondering how we could consistently Make Math Moments That Matter for our students; not just for that lesson or unit, but for weeks, months, and even years.”

We knew that those “hit” lessons had moments in them that didn’t happen by luck.

There had to be a way we can create these moments on a daily basis – but how?

Making Math Moments That Matter: A Framework

MMM Logo Circle Transparent.001

After dissecting our math lessons for over 3 years, we came to realize that we could classify the most important elements required to create memorable math moments in a 3-part framework we call “Making Math Moments That Matter”:

  • Spark Curiosity
  • Fuel Sense Making
  • Ignite Your Next Moves

This framework is not intended to be three specific steps that must be followed in order, but rather three overarching elements that must be present in some form to ensure your lessons make math moments that matter to your students.

Let’s dive into each part of the framework to unpack each element.

Spark Curiosity

Education is notorious for buzzwords and as I’ve explained in the past, the biggest buzzword of them all is engagement. So while we are all striving to engage our students regardless of the content area we teach in, the word engagement has led me to many dead ends when I wasn’t quite sure what type of engagement I was looking for.

If we look at the Google Dictionary definition of curiosity, we find:

  1. a strong desire to know or learn something.
  2. a strange or unusual object or fact.

Isn’t that first definition exactly what we’re hoping for in our math classroom?

We know that curiosity is a motivator for learning and it is this curiosity that fuels our natural tendency to make sense of the world around us.

“We know that curiosity is a motivator for learning and it is this curiosity that fuels our natural tendency to make sense of the world around us.”

So how do we spark curiosity in math class?

The best part is that there are infinitely many ways to spark curiosity on a daily basis in the classroom which means that each teacher can pull on their own personality and interests to build in this curiosity. While this sounds super awesome, the hard part is that this means the way in which you spark curiosity in your class may not necessarily look the same as a colleague down the hall.

While there is no exact one-size-fits-all solution, a great place to start is with lesson 1 in the 4-Part Video Lesson Series that Jon and I put together recently. While there are many different ways to spark curiosity, the video series dives into some of our favourite and most effective. We will also dig into other specific elements over the next while on our blogs.

Dive Into The FREE Make Math Moments That Matter Online Workshop!

START WITH VIDEO 1: How powerful math moments can change your class!

Fuel Sense Making

Both Jon and I have been fascinated with Dan Meyer’s 3 Act Math structure for delivering perplexing problems in math class in a similar fashion to that of a storyteller or screenwriter. Although the tasks themselves are really interesting, early in our implementation of these types of tasks, we often felt that we were getting kids engaged by sparking curiosity, but they still weren’t “getting it” at the end.

Over time we started to realize that although our classes were becoming more energetic and interesting, we hadn’t focused enough on how we were going to help kids make sense of the mathematics. The curiosity had sparked their attention and interest to learn, but we were leaving the opportunity to explicitly help them make connections and build their conceptual understanding on the table each and every day.

“Over time we started to realize that although our classes were becoming more energetic and interesting, we hadn’t focused enough on how we were going to help kids make sense of the mathematics.”

We now realize that we must have a deep conceptual understanding of the mathematics we are intentionally having our students engage in as well as the developmental continuum of how that concept and thinking develops in order to meet students where they are and help them get to the next stop in their learning journey.

This has been very hard, but extremely rewarding work that we are continuing to engage in to deepen our own mathematics content knowledge and understanding.

Ignite Your Next Moves

Finally, we realized that most of the important work we do as math teachers is in and around igniting our next teacher moves before, during, and after each lesson. It is this part of the framework that must be happening all of the time in order to be able to create math moments that matter with consistency and intentionality.

While this part of the framework is most important, it is also the most complex and difficult to unpack.

Some of the many before moves that must take place to plan for an effective math lesson include:

  • What is the intended learning for the lesson?
  • How can I help students to build a conceptual understanding before introducing procedures and algorithms?
  • What does this concept look like developmentally? Where did it “come from”? Where is it “going”?
  • How can I spark curiosity in order to build the student need/want to learn this concept?
  • … and many more.

We also need to be igniting our “during moves” by incorporating the 5 Practices for Orchestrating Productive Mathematical Discussions including:

  • Delivering a task that is open and accessible without pre-teaching steps, rules, and procedures.
  • Monitoring students as they work on the task.
  • Selecting student representations and strategies to highlight during consolidation.
  • Connecting the selected student representations and strategies to the mathematical concept and thinking you have planned to intentionally introduce/address.
  • … and many more.

And finally, we need to be reflecting on the “after moves” including:

  • Do we proceed as planned to the next lesson in my long range plan?
  • Should I pivot my plan by incorporating another lesson to help address any gaps or areas of concern?
  • Which tools and representations were students comfortable with during the lesson? Which were not present?
  • … and many more.

“In order to create math moments that matter for our students, we must Ignite Our Next Moves by intentionally planning our teacher moves before, during, and after each and every lesson.”

Let’s Learn To Make Math Moments That Matter Together!

While it is convenient to have a nice and tidy 3-part framework to help us as we plan to create math moments that matter on a consistent basis in our math classrooms, putting it into action will take time and effort before it will feel like a natural process or habit.

Don’t worry! Jon and I are here to help!

In the coming posts here on my blog and over on Jon’s blog MrOrr-IsAGeek.com we will be diving into each of these pieces in-depth!

Want to ensure you don’t miss out on any of the learning? Join our mailing list here to receive tools, tips, and resources each week!

Dive Into The FREE Make Math Moments That Matter Online Workshop!

VIDEO 1: How powerful math moments can change your class!

VIDEO 2: How avoiding rushing to the algorithm can build resilient problem solvers.

VIDEO 3: What red Dodge Chargers have in common with conceptual understanding in math class.

VIDEO 4: Helping teachers empower students to become resilient problem solvers.

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Make Room for Spatial Reasoning https://tapintoteenminds.com/make-room-for-spatial-reasoning/ https://tapintoteenminds.com/make-room-for-spatial-reasoning/#respond Tue, 02 Oct 2018 10:00:22 +0000 https://tapintoteenminds.com/?p=52791 Another post from Tap Into Teen Minds. Enjoy!

The research is so strong it is no longer worth asking the question of whether Spatial Reasoning is important for math class. Make room for it EVERYDAY!

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One afternoon, as I was visiting a friend of mine, his 2 year old daughter came out of her room with six stuffed dogs and set them all down in a row. She went back into her room and came out with a stack of bowls which she proceeded to place in front of each of her pups – it was time for their dinner. When she had distributed all of the bowls, she realized that two pups did not have a bowl so she went back into her room to retrieve exactly two more bowls.

This may seem like a pretty typical example of a child’s tea party, but when you consider the cognitive functions this two year old was employing, it becomes so much more. Luckily, the significance of what was happening in her curious mind was not lost on me and it made me wonder how deep her number sense ran.

The Power of Spatial Reasoning

What do you suppose was going through this child’s head when she went back for more bowls? It’s highly doubtful that at 2 years old she was thinking “Six dogs minus four bowls means I need to add two more bowls.” So how was she able to know exactly how many more bowls she needed?

The answer lies in her spatial awareness and spatial reasoning.

Although the context may vary wildly from child to child, young learners have “foundational experiences” (Clements & Sarama, 2004) like the one described above all the time and it demonstrates the role spatial reasoning plays in their cognitive development – but this anecdotal evidence is far from the full story.

The role spatial reasoning plays in a child’s understanding and success in other cognitive domains has been researched for decades and the correlation may be surprising to some. For instance, there is a strong correlation between early successes in spatial reasoning and success in STEAM careers later in life (Shumway, 2013). Furthermore, the more experiences a child has with spatial challenges the more their skills develop. This research suggest that a strong emphasis ought to be put on spatial reasoning and developmental opportunities in early grades rather than on rote or abstract processes such as algorithms and procedures. The decades of research connecting strong spatial sense and high levels of achievement in mathematics makes one wonder why we don’t pay more attention to building this extremely important skill in our young learners.

Spatial reasoning is not limited to geometry as some might believe. As we saw at the tea-party at the beginning of this discussion, spatial reasoning can be effectively applied to many other aspects of numeracy such as number sense, data management, measurement, and many others. As such, it is valuable to investigate new ways to promote spatial reasoning when approaching any branch of mathematics and avoid the temptation to jump straight to abstract representation. Instead, we ought to start with the concrete by leveraging and building upon young learners’ foundational experiences and only then fade this into visual representations and finally abstract representation which is often referred to as concreteness fading.

Technology’s Role in Spatial Reasoning

Technology can allow us to manipulate and investigate situations and problems in ways that would never be available through paper and pencil or traditional manipulatives. For example a +1 counter and -1 counter do not physically ‘cancel each other out’ on a student’s desk, but in a digital environment, putting them in contact with one another can cause a mysterious reaction that results in nothing. Although these digital manipulatives are not truly “concrete” in that they are not tangible, they can be useful to bridge the gap between the concrete and visual/pictorial stages of representation as well as the source of new and developing foundational experiences.

The implementation of the right digital resources can transform any lesson into a visual and memorable exploration that develops the spatial awareness of young learners. However, it is important to note that not all digital math resources provide this bridge between the concrete and visual that we are hoping for. I’d like to highlight a few digital tools that effectively leverage spatial reasoning to help young learners develop a deeper understanding of new concepts.

Zorbit’s Math Adventure

Zorbits Math Adventure LogoWhen students play Zorbit’s Math Adventure, they join Zorbit and his crew on a journey through the cosmos visiting planets that represent strands within our math curriculum. Each planet contains a cast of interesting characters that need Zorbit’s help and the students complete contextual activities that lead towards a solution to their new friends’ problem. Although the planets focus on different areas of the curriculum, each activity is designed to use familiar, but digitally supercharged, manipulatives to help students engage with the math concepts at hand.

On the Kindergarten-aligned planet of Pacifica, for example, students manipulate a variety of objects to measure fish which encourages students to understand that measurements are relative and not absolute – there is more than one way to measure a fish. The same fish can be 10 pearls long AND 5 seashells long AND 2 bolts long. What an amazing way to help students learn about the importance of “unit” for both measurement and proportional reasoning.

This is just one small example of the huge variety of activities from Zorbit’s Math Adventure. Not only does the game leverage spatial reasoning on 14 curriculum-aligned planets across grades K-3, it also features a library of lesson plans and teaching activities that build on activities within the game. These activities have been designed around materials found in many primary classrooms and are a great way to bring their in-game experiences into a more tangible environment or vice-versa.

This digital platform collects student performance data based on their activities within the game and features a suite of analytic and instructional tools for the teacher. These tools make Zorbit’s Math Adventure a very powerful platform for teaching math in the Kindergarten through grade 3 classroom.

Lucky for you, the good folks at Zorbit’s have been generous enough to provide anyone who uses this link to get an exclusive Zorbit’s Trial for your own classroom! If you’d like to try Zorbit’s Math Adventure in your classroom, visit this special link for your extended free trial.

Math is Visual

Math Is VIsual - Let's Teach It That WayOne of my sites, www.mathisvisual.com, features a library of videos that demonstrate mathematical ideas with animated manipulatives. These short animations are a great way to introduce new ideas by allowing students to investigate and hypothesize about the interplay between operations before being represented with numerals – an abstraction of the idea at hand. This evokes a strong number sense rather than simply the memorization of procedures and math-facts alone.

For full blown math tasks that make math visual, you might consider checking out 3 Act Tasks as well.

DragonBox Numbers

Dragonbox Numbers LogoDragonBox Numbers is a game that uses “Nooms” – little creatures whose height represent different numbers from 1 to 10. Within the game’s sandbox areas, ladder levels, and puzzle challenges, students can combine, slice, stack, compare, and sort the Nooms to build an appreciation of how numbers affect and relate to one another. It is through these contextual manipulations of the Nooms that math is made visual and interactive. For example, when combining Nooms, one will eat the other and grow proportionally to the sum of those two Nooms. This is a wacky yet concrete representation of how addition of simple numbers works.

My daughter loves tinkering with the different variations of the DragonBox Numbers game and I’m sure your young kids or students will love it too!

DragonBox has a few other games that expand these ideas into larger numbers and other mathematical domains including algebra and Euclidean geometry.

They can be purchased separately or in bundles from the App Store.

As a bonus, our friends at Dragonbox will give teachers access to Dragonbox Numbers for free!

Click here to register for FREE access to Dragonbox Numbers.

Math Visuals

Math Visuals Website LogoMath Visuals is another website with a growing collection of animations that illuminate key concepts from Kindergarten through Grade 5. The visuals give students an entry-point into difficult concepts by asking Annie Fetter’s signature (and my now favourite) questions: “What do you notice? What do you wonder?” to encourage mathematical discourse and encourage the use of spatial reasoning.

Visit the website here.

Effective Mathematics Instruction Leverages Spatial Reasoning

The power of spatial reasoning in a young learner’s education is amazing! Since the day they are born, children use their perception of the world around them to learn new things. Once they begin manipulating concrete objects in their surroundings, they are already fully immersed into a continuous state of exploration, experimentation, and tinkering.

So while I would never want to rush past this stage of students playing, tinkering and manipulating concrete objects, there will come a time when we should help nudge their thinking towards the second stage of concreteness fading by encouraging visualization. While the scope of the objects we can manipulate in this digital age may seem limitless, being intentional when selecting the appropriate digital resource for each child at their individual developmental stage is so important. Hopefully this post has given you a head start as we all work to become better at identifying which digital math tools leverage and help students build their spatial reasoning to deepen their understanding of mathematics.

So the next time you’re in the presence of young children at play, be on the lookout for instances of spatial reasoning taking place – both concretely and visually – right before your eyes!

I’d love to hear your experiences of spatial reasoning in action. Be sure to share in the comments!

References

Clements, D.H.; Sarama, J. (2004). Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education. Lawrence Erlbaum Associates Publishers, New Jersey.

Shumway, J. F. (2013). Building bridges to spatial reasoning. Teaching Children Mathematics, 20(1), 44–51.

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Memorization vs. Automaticity: Back to Basics or Beyond the Basics? https://tapintoteenminds.com/memorization-automaticity/ https://tapintoteenminds.com/memorization-automaticity/#comments Sat, 28 Jul 2018 17:03:59 +0000 https://tapintoteenminds.com/?p=52705 Another post from Tap Into Teen Minds. Enjoy!

We’ve all heard it before; kids don’t know their math facts! While this may be true for some, what is more important than memorizing alone is building a conceptual understanding.

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We’ve all heard it before; kids don’t know their math facts! While this may be true for some students, what is more important than simply memorizing multiplication facts is building a conceptual understanding of what multiplication means, how to visualize multiplication and eventually, helping students to generate their own algorithms for multiplication BEFORE they encounter any standard algorithms.

In this post, I want to dive into the math wars and try to clear up some confusion that may exist.

Importance of Math Facts

When addressing the math wars, I think it is really important for us to clear up some of the confusion that may exist around the importance of “math facts” or “multiplication tables” and whether students should be memorizing them.

In Canada, the media often paints a picture as though there is one group fighting for students to memorize math facts like multiplication tables – often referred to as “the back to basics” group and the other group – often referred to as the “discovery math group” is promoted as a group that doesn’t believe that math facts have any importance. Sadly, the picture portrayed in the media is incomplete at best.

And unfortunately, after the debate finally quiets down in the media for a while, it is brought right back to the forefront when standardized test scores or international math rankings such as PISA scores are released.

In the United States, the media paints two similar groups on opposite sides of the spectrum; those who are against Common Core or “new math” suggesting that it confuses kids and leaves parents unable to assist their children and those who believe the common core will provide students with a deeper conceptual understanding of mathematics.

While it might seem easy to blame the Common Core for the math wars in the United States, these two extreme stances have been around for quite some time in mathematics education.

Regardless of whether your beliefs align more closely with a “back to basics” or “discovery and inquiry” viewpoint, the fact is that we all want students to know their math facts. The real conflict exists in different interpretations of what that means and how we can help students get there.

So, let’s spend a few minutes to better understand the similarities and differences of the two stances that are often portrayed in the media in order to find some sort of balance between both.

“Regardless of whether your beliefs align more closely with a ‘back to basics’ or ‘discovery and inquiry’ viewpoint, the fact is that we all want students to know their math facts. The real conflict exists in different interpretations of what that means and how we can help students get there.”

Memorization vs. Automaticity

One of the big differences I see between those who tend to side with the “back to basics” group and those who align more closely with the “inquiry and discovery” group is that of memorization versus automaticity.

When many think of the word memorization, they think about rote learning; committing information to memory through repetition, speed and without the need for meaning.

While humans memorize a lot of information by rote, this memorization technique is used optimally for memorizing information that is difficult to connect to other information we already know such as memorizing the address or phone number of a family member, Learning new information that can easily connect to our prior knowledge through rote can be truly limiting that new learning for recall purposes only, rather than for understanding.

By promoting the learning of math facts and mathematics in general through purposeful mathematical experiences, students are not only able to recall their new learning, but they can also do so with understanding. This approach to learning math facts is often referred to as automaticity.

While different people may have slightly different definitions, I really like how Cathy Fosnot describes both memorization and automaticity in her Minilessons series. Fosnot states that memorization “…refers to committing the results of unrelated operations to memory so that thinking through a computation is unnecessary” while automaticity suggests that “answers to facts must be automatic, produced in only a few seconds … thinking about the relationships among the facts is critical”.

Although it might seem as though the differences are quite subtle, they can really impact how an educator might go about helping students learn their math facts. Ultimately, I believe that if we want our students to build automaticity, we will be providing opportunities to build a conceptual understanding by engaging with interesting mathematics in order to learn math facts, not the other way around.

Let’s commit to exploring interesting math in order to build math fact fluency and automaticity rather than memorizing before exposing students to the beauty of mathematics.

Want to Take This Learning To Go?

Download the 5-page printable Memorization vs. Automaticity Cheat Sheet and finish up the learning with or without internet access from your favourite device!

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Get Students On Task And Engaged With Whiteboards https://tapintoteenminds.com/whiteboards/ https://tapintoteenminds.com/whiteboards/#respond Thu, 12 Jul 2018 03:15:02 +0000 https://tapintoteenminds.com/?p=52686 Another post from Tap Into Teen Minds. Enjoy!

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Research Says That Using Vertical Non-Permanent Surfaces (VNPS) Keep Kids On Task Longer

If you are a long time reader of my blog, you’ve probably noticed many in-class pictures involve students writing on desks with erasable markers. Other times, you’ll see students standing at whiteboards working collaboratively on problems in small groups. Despite the fact that using erasable markers and non-permanent surfaces definitely keeps things lively, I didn’t introduce white-boarding just to mix things up. This approach is actually based on research by Peter Liljedahl in his paper Building Thinking Classrooms. His research showed that students who completed math tasks on non-permanent surfaces like chalkboards or white boards and in particular, when students were standing in a vertical position, improvements in 7 of the 8 measures than when compared against students working on more permanent writing surfaces like chart paper or notebooks.

Effect of Vertical Non-Permanent Surfaces (VNPS)

As you can see from the table above, while the time to task and time to notation was comparable regardless of the writing surface, the measures of eagerness, discussion, participation, persistence, non-linearity and mobility were all winners when students used non-permanent surfaces, in particular when standing in a vertical position. Shortly after I began introducing the use of both vertical non-permanent surfaces (i.e.: whiteboards on the wall) and horizontal non-permanent surfaces (i.e.: students writing with erasable markers on their desks), I began observing the power of this approach.

Gas Guzzler - More Student Solutions to Gas Guzzler

A hypothesis suggests that when you write on paper, even if with a non-permanent writing utensil like a pencil, students psychologically feel like what they write is permanent. This can have the effect of making students feel as though they should be able to determine a direct path to the answer without doing any sort of brainstorming or strategizing on the page. Sadly, I’m sure that the many years of me demanding that my students line up their work neatly by lining up the equal-sign in a straight vertical line with a clear and concise therefore statement at the end probably fuelled this pitch for perfect. While I had thought I was helping train students to become better mathematicians, I realize now that I was actually hindering them from developing their mathematical mind by fogging it with anxiety and a false fear of failure.

For those of you who have been with me for over five years, you’ll remember when I used to run a completely paperless math classroom with a class set of iPads. Wow! That was a long time ago. While I thought having students use their finger or a stylus to write on a PDF annotation app like GoodNotes 4 or Notability would promote brainstorming, conjecturing and making mistakes before crafting a final solution, what I found was the same issue that Peter Liljedahl found with permanent surfaces like paper. Students appeared to treat the iPad – which makes it very easy to erase while working in an annotation app – as a permanent surface. My only conclusion is that students were aware that their work completed on the iPad would be saved and uploaded to a shared folder in the cloud, whereas when students are writing on a non-permanent surface, they know that their work will have to be erased at the end of class for the next group of students. Even if they were to take photos of their work for safe keeping, it was as though the non-permanent surface allowed them to work much more freely and a little more messy which allowed them to engage in the problem-solving process more deeply.

Although I never collected data like Peter Liljedahl, my own personal observations and conversations with students showed that students stayed on task much longer when working collaboratively at white boards or even standing at the desk working on horizontal non-permanent surfaces in small groups. I’m not alone on this vertical non-permanent surface (VNPS) journey; be sure to check out some of the many other math teachers who are engaging their students with VNPS including Alex Overwijk, Laura Wheeler, Jon Orr, Nathan Kraft, Graham Fletcher, Mark Stamp and many more.

Now that my job has taken me out of the classroom into more of a facilitator and presenter role, I’m constantly going into different spaces to work with teachers and engage in math learning. The downside is that I usually don’t know exactly what the room setup is going to be like. I often don’t know until I arrive whether there will be chalkboards, white boards or other surfaces for my participants to get up and do some math learning on. While I would typically bring a few pads of chart paper and sticky tack, tape and tacks to provide some “permanent” surfaces to write on, more creative folks like Alex Overwijk and Jimmy Pai bring a stack of laminated chart paper on the road with them to use as a portable non-permanent surface solution. I’ll be honest and say I’m not that creative (and maybe a bit lazy?), but I’ve stumbled across a product that helps folks like me. Toby and the team from Wipebook based in Ottawa, Ontario, Canada have got us covered! Wipebook specializes in non-permanent surface products including non-permanent surface charts that can be reused and are quite easy to take with me in the back of my car.

Recently, Jon Orr and I took Wipebook Chart Packs on the road with us to some conferences and district presentations and they worked great! In a breakout session we held at The OAME Leadership Conference in 2017, we had a room of over 100 teachers, consultants, and coaches up at the walls doing a math problem with us. It was fantastic!

We had so many participants asking for more details on how to get their own Wipebook Chart Packs that they are now offering an exclusive Educator Starter Pack:

Educator Starter Pack

The Wipebook Educator Starter Pack includes:

  • A Wipebook Flipchart;
  • A Wipebook Notebook Plain;
  • A Wipebook Workbook Graph;
  • A single correctable marker; and
  • A single tri-plus marker.

You can grab the starter pack for 25% off the list price.

Interested? Grab the starter pack by clicking here for Canadian addresses and click here for addresses outside of Canada.

The great folks at Wipebook were generous enough to give you this great discount and they have offered to give Jon and I a small percentage of every sale without any additional cost to you. WIN-WIN!

Are you using vertical non-permanent surfaces in your classroom?

If so, what kind are they? A traditional chalkboard? Whiteboards? Or have you ripped up your own?

Have more questions about how you might go about getting your students in the vertical and non-permanent world?

Let us know in the comments below!

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The Complete Guide to Spiralling Your Math Curriculum https://tapintoteenminds.com/spiralling-guide/ https://tapintoteenminds.com/spiralling-guide/#respond Thu, 10 May 2018 02:09:52 +0000 https://tapintoteenminds.com/?p=19887 Another post from Tap Into Teen Minds. Enjoy!

Do your students struggle to retain their learning from math class? Are you finding yourself having to reteach because students don't have the prior knowledge for today's lesson? Then spiralling and interleaving is perfect for you! Take some time and explore the complete guide to spiralling math class!

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3-Part Spiralling Math Class Guide

This post is a summary of the 3-Part Spiralling Math Class Video Course.
Use the buttons below to jump to the different parts of the guide.

Part 1

What Is Spiralling & Why Should I Do It?

Part 2

6 Steps to Spiralling Your Math Class

Part 3

9 Starter Strategies to Get Spiralling

Part 1: What Is Spiralling Math Class and Why Should I Do It?

Spiralling Your Math Curriculum - What Is Spiralling Math Class and Why We Should Do It

Have you been curious about spiralling your math curriculum? Heard colleagues talking about it, but not quite sure what it is, what the benefits are and how you can get started in your own classroom? Then this post is for you!

For the last 5 years or so, I have been experimenting with different ways to spiral my math courses and a recent discussion on Twitter sparked by Jon Orr had me search back to a draft blog post I had been working on, but had left incomplete.

Here’s what Jon had asked the Twitterverse:

With over 20 people throwing out great things to consider, I thought it would be worthwhile to share some strategies to help you get started spiralling in your math classroom.

Before we begin, we should probably clarify what a spiralled math curriculum actually is.

What Is Spiralling the Math Curriculum?

When we spiral curriculum in math class, we are organizing topics that might traditionally be taught in blocks, chapters, or units of study over a short period of time and we are introducing topics in smaller chunks and spreading them out over a longer period of time. While you can do this in many different ways, it is common to come back to the topic multiple times over the duration of the grade or course and going deeper each time. Spiralling is commonly referred to as “interleaving”, “distributing”, “spacing” or “mixing” the topics from the math curriculum, while teaching a concept in one unit or chapter like you see in many textbooks is commonly referred to as “blocked” or “massed” approaches.

Spiralling Your Math Curriculum - Downward Spiral Surface to Deep Learning

Generally when I think of spiralling math curriculum, I picture spiralling through all of the big ideas through tasks early on in the course at a surface level. At the end of this first spiral, we go back through these same ideas to build on our surface knowledge and dig deeper. We continue to spiral through these concepts introducing more complex and rigorous tasks as we help students build their conceptual understanding and develop procedural fluency. I like to see this thinking as very similar to that of John Hattie when he speaks about surface learning and deep learning. By spiralling the curriculum and using well planned, thoughtful guided inquiries and investigations, you can help students develop much needed surface learning and deep learning.

When you read about this spiralling, the big question many may have is “why can’t I do this by organizing my curriculum in units like I always have?” Well, the research suggests that loading up all of the learning for a concept over a continuous block of time just doesn’t have the same effect as mixing it up and spreading it out.

Why Interleaving Is Better Than Blocking

When we interleave math concepts throughout the duration of a course rather than approaching that concept in a continuous block over a shorter period of time, research from over the past 100 years suggests that students learn concepts more deeply and they retain that information for a longer period of time versus blocking.

Hermann Ebbinghaus & The Learning Curve

Experiments by Herman Ebbinghaus which were conducted on himself were the first to investigate properties of human memory. In his experiments, Ebbinghaus would create a lists of about 20 three letter words. These nonsensical words were created starting with a constant, followed by a vowel, and ending with a constant.

Spiralling Your Math Curriculum - Herman Ebbinghaus List of Nonsensical Words

To test the process of committing new learning to memory, he would read and say each item on the list, before moving onto the next. When he was finished the entire list, he would return to the beginning of the list and repeat the process. As you would expect, as the repetitions increased, so did his ability to recall the items in the list. This work by Ebbinghaus was responsible for the creation of the world’s first learning curve.

Spiralling Your Math Curriculum - Learning Curve Graph

The Forgetting Curve

While these experiments were exciting, what Ebbinghaus is most well known for is the forgetting curve. Using the same types of 3 letter, nonsensical lists of syllables, he then began focusing his experiments on how long he could retain these items in his memory over time. His research showed that once he had “learned” a list, his retention would decrease with each passing day that he did not attempt retrieving the items from his memory. However, when he retrieved a list from his memory after short intervals of time that gradually increased, the forgetting curve would become less steep.

In the graph below, we can see an example retrieving information from memory after 1, 3 and 6 days after initial learning:

Spiralling Your Math Curriculum - Forgetting Curve Animated Gif

Ebbinghaus believed that the speed of forgetting depends on a number of factors such as the difficulty of the learned material (in other words, how meaningful it is to the individual), its representation (such as what connections to prior learning is made with the new learning) and and physiological factors such as stress, sleep or even how open to learning the individual is.

Other Research Supporting Spacing Over Massing

In The Educational Psychology Review, Son and Simon state:

“On the whole, both in the laboratory and the classroom, both in adults and in children, and in the cognitive and motor learning domains, spacing leads to better performance than massing.”

Son, L. K., & Simon, D. A. Distributed learning: Data, metacognition, and educational implications. Educational Psychology Review (2012): 1-21.

Want to save this for later?

Download the Printable Cheat Sheet E-Books!

Surprisingly, much of what we believe to be true about learning is actually false as explained in Benedict Carey’s book How We Learn: The Surprising Truth About When, Where, and Why it Happens:

Let go of what you feel you should be doing, all that repetitive, over-scheduled, driven, focused ritual. Let go, and watch how the presumed enemies of learning – ignorance, distraction, interruption, restlessness, even quitting – can work in your favor.

Carey, Benedict. 2014. How we learn: the surprising truth about when, where, and why it happens (222)

Seems pretty counterintuitive, but interesting none the less.

Wonder and Curiosity Drives Learning

While I won’t be suggesting that we promote distractions, interruptions, restlessness and quitting in our math classes, some of the key ideas from the book have interesting implications for math class and school in general. First and foremost, Carey concludes that learning happens best when it is driven by wonder and curiosity rather than by fear or envy. When you consider the traditional approach to teaching math class is usually blocking or massing concepts in a short period of time followed by a one shot test, it would seem that the learning is more likely to be driven by fear (i.e.: failing) or envy (i.e.: wanting the highest grade) rather than by

In order to promote learning driven by wonder and curiosity, Carey argues that we should help students become curious thinkers – not as a means to do individual tasks like completing a section in a textbook, but for cultivating a love of learning in general. As a teacher who used to teach in units or blocks, I find it much easier to spark curiosity in my students when I spiral math curriculum using 3 act math tasks to teach concepts because solution strategies are much less predictable, students are not expected to use a specific formula or algorithm explicitly taught moments before during a teacher directed lesson and each of these contextual tasks creates an intellectual need for the learning.

I know that if I can get students curious about a problem and get them to put some skin in the game by sharing what they notice and wonder as well as making predictions before all of the required information is shared, students are much more likely to learn and retain this new knowledge.

Effect of Spiralling on Retention

Although the decades of research has clearly indicated that interleaving math concepts and spacing practice is much more effective than teaching in blocks and massing practice, we are still seeing the majority of textbooks and math classes organized in units or blocks.

Why?

One possible reason is because of the illusion of understanding often experienced when we teach or learn using blocked instruction and massed practice. Because students are focusing their attention on few concepts and practicing them repeatedly over a short period of time, the facts, steps and procedures are fresh in their minds and they appear to “know it”. Unfortunately, this perceived fluency is short lived and often results in a lack of retention over time. Many of us and our students have experienced this sort of memory loss when we “draw a blank” on a written assessment and I’m sure every teacher has had their students claim they don’t remember how to solve a problem related to a concept they learned the previous year.

When we distribute or interleave concepts and space practice over time, this forces our brains to work harder to retrieve the information and ultimately builds our retrieval strength. By waiting to come back to a concept just before it feels like it is fully forgotten, we are giving our brains exercise to retrieve those memories and build a stronger neural pathway to that information. Thus, Carey not only recommends interleaving and spacing practice, but also using tests as an effective studying technique to promote retention rather than just as a measurement tool. The logic here is that each test where a student works on problems independently and without the aide of peers or resources is an opportunity for them to practice retrieving that information that is stored deep in their brain.

Imagine that: using a test to study rather than studying for the test.

I like it!

Some Jurisdictions Explicitly Promote a Spiralled Curriculum

Interestingly enough, my colleague Jana LePage-Kljajic brought to my attention the fact that the Ontario Mathematics Curriculum explicitly states that teachers should be teaching in some sort of spiralled format:

“When developing their mathematics program and units of study from this document, teachers are expected to weave together related expectations from different strands, as well as the relevant process expectations, in order to create an overall program that integrates and balances concept development, skill acquisition, the use of processes, and applications.”

The Ontario Curriculum Grades 1 – 8 Mathematics 2005, page 7

Spiralling Your Math Curriculum - Ontario Curriculum Quote About Spiralling Strands

Recently, the Ontario Ministry of Education released a completely spiralled Grade 1 to Grade 8 Math Resource on the EduGains website called TIPS4Math. Definitely a great place to start if you are teaching elementary mathematics in Ontario.

Ontario EduGains Math Resource - TIPS4Math Spiralled Resource

Although I couldn’t find it stated as explicitly, the Common Core State Standards (CCSS) for Mathematics does mention a spiral-like approach:

… not only stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value and the laws of arithmetic to structure those ideas.

Common Core State Standards (CCSS) for Mathematics

Although I’m less familiar with the University of Chicago’s Everyday Math program, it is also a spiralled curriculum and they share their logic behind the organization here.

Part 2: 6 Steps to Spiralling Your Math Class Now

Spiralling Your Math Curriculum - 6 Steps to Spiralling Your Math Class

Since you’ve made it this far in the post, that likely means that you’re interested in exploring this approach further and getting started. Awesome!

While many teachers have very different perspectives when it comes to getting started – from dipping your toes to diving in head first – I’d like to propose some steps that will help guide you along the way.

Spiralling Step #1 – Know What You Need To Teach

In other words, know your curriculum. If you are brand new to teaching a grade or course, trying to spiral for the first time will be challenging. That said, I’m not suggesting you don’t attempt spiralling, but rather be aware that you will have to do some serious digging into the expectations or standards you will be teaching in order to plan and deliver your spiralled curriculum with confidence.

In reality, the work you need to do upfront to learn the curriculum should be done by every teacher regardless of how you intend to structure your long range plan. Unfortunately, teaching math in blocked units as they are in most textbooks can unintentionally make it possible for us to learn new curriculum one unit at a time. When faced with teaching a new grade or new courses, sometimes we can feel so overwhelmed that we tend to plan our year in small blocks or chunks which is not ideal for student learning.

Spiralling Step #2 – Assess By Learning Goal or Standards Based Grading

When I attempted spiralling my MFM1P Grade 9 Applied math course for the first time, it was after I had been assessing students by learning goal and success criteria or as some in the united states might call it: standards based grading (SBG). I personally believe that by assessing in this manner, it forced me to really know and understand the curriculum I was teaching. Not only does this step serve as a way to ensure that you can check step #1 off the list, but this has other benefits for your classroom beyond the spiralling of concepts. Assessing by learning goal is a great way to help identify where students are strong and where they need more support so you can offer specific and timely feedback.

By monitoring how students are doing based on the learning goals you have created for your course, this can also help you determine how frequently to come back to concepts based on student understanding. In contrast, assessing a mark on a unit test alone is much less helpful when it comes to trying to determine what concepts we need to spiral back to and can often lead us to attempting to re-teach the entire unit – wasting valuable class time – rather than focusing on the specific topics where students are struggling.

Spiralling Math Class - Learning Log For Standards Based Grading

LEARN MORE

Spiralling Step #3 – Create Topics, Strands or Big Ideas

I don’t care what you call them. Some might like calling them “topics” or “big ideas”, while others might have their own creative name for them. Regardless of what you want to call them, you definitely want to look at your curriculum, the learning goals you’ve created and start organizing them. The Ontario Curriculum conveniently organizes our expectations into strands, so that is where I began my first time through spiralling grade 9 applied. I also decided to keep the name “strands” although I had subdivided them into smaller groups.

Now that might sound confusing, especially if you’re not from Ontario like I am, so let’s have a close look.

Here’s a glimpse of the Grade 9 Applied Spiralled Course Google Sheet I was using to organize my course, which I will share in the resources under this video. Luckily for me, Jon Orr was also interested in trying to spiral his grade 9 applied course at the same time which made the process so much easier.

While we were collaborating and helping each other think through how we might teach our courses, we would often go in a different order based on our own student learning needs, but we would often times be using many of the same tasks and lessons.

Spiralling Your Math Curriculum - Spiralled Grade 9 Applied Planning Spreadsheet

ACCESS THE GOOGLE SHEET

On the other hand, maybe you prefer planning with physical paper like my colleague Jana LePage-Kljajic who prints out the curriculum and cuts them up into individual expectations. Then, after grouping them, she glues them to colour paper to help her see the big picture.

Spiralling Your Math Curriculum - Jana LePage-Kljajic Planning Spiralling With Paper

Regardless of how you want to do it, I’d recommend coming up with a plan to organize your topics in a way you think would be easiest for your own personal organizational style.

Spiralling Step #4 – Plan Your Cycles, Spirals or Chunks

After you organize your topics, strands or big ideas, you’re going to want to think about how many times you’ll come back to these concepts. Again, naming is not really important. Alex Overwijk, Mary Bourassa and many others like to call them “cycles“, while I often refer to them as “spirals”. Heck, you might decide you want to keep all your colleagues on staff confused for a cheap laugh and call them “units”.

By planning out your spirals including how long each one will be and how many spirals you’d like to have in your course, you will ultimately be planning out a long range plan for your math class. It should be noted that you don’t need each spiral to be the same length of time and you definitely don’t need to commit the same number of days to each topic. This is really up to you based on what you anticipate and, will need to change to suit the needs of your future students.

Want to save this for later?

Download the Printable Cheat Sheet E-Books!

Spiralling Step #5 – Be Comfortable With Changing Your Plan

That last line from step 4 is really important. If you’re planning out your spiralled math course before the class has even begun, you haven’t even met your students yet. The topics you anticipate students breezing through and some of the topics you anticipate students will struggle through might not play out as planned.

Expect to be confronted with making the decision between proceeding as planned or pivoting your plan based on the needs of your students. This means you should prepare yourself for the highly likely chance that you’ll need to modify your spirals to suit the needs of the learners in front of you.

This might sound scary at first, but it really does help you dedicate the limited class time you have with your students to the concepts students need to work on the most.

Spiralling Step #6 – Interleave Your Assessments

We are all assessing students through conversations and observations on a daily basis. However, if you’re spiralling your course, it might seem hard to determine when you will assess students with a written test or quiz and what questions you should be asking.

Since we are spiralling the content of the course, it would make sense that the assessments are spiralled or interleaved as well. For me, I would give students an assessment on “Mastery Day” which was every Tuesday and the assessment was called a “check-in”. For the first half of class, students would work on 4 to 6 questions; approximately half from ideas we worked with the previous week and the other half would be from anything covered previously in the course. The intention here was to give me and the student a true understanding of where their understanding was related to those learning goals at the current time. There was no review day and there was no list of concepts they were to cram the night before. Just questions to give students the opportunity to strengthen their retrieval strength and highlight areas that they should be focusing on over the next week.

Spiralling Your Math Curriculum.051 - Mastery Day - Do You Understand the Content?

After submitting their check-in, the second half of the class would be used to work towards mastering a concept based on the feedback given on previous check-ins. Here’s a post explaining how I shared that feedback, but save that reading for later as it will send you down a whole new rabbit hole for exploring separately.

Spiralling Your Math Curriculum - Check Ins to Build Retrieval Strength

I would try my best to have feedback for students done by the next day not only for the feedback to be timely for them, but also to help me determine whether I should continue the next week of my spiral as planned or whether I need to come back to an idea like Pythagorean Theorem because so many students crashed hard on it. There are also some positives that come from this like realizing that my students really understand linear relations and maybe I can cut a day out of this spiral to commit to something else.

Spiralling Your Math Class - Give Timely Feedback Click to Tweet

Now, although I have interleaving your assessments as the 6th step, I’ll mention that you might also consider doing this step after step 3, if you feel that you aren’t ready to spiral your lessons, but could see yourself interleaving the content on each assessment. This is also a step that may be easier to put into practice during a school year, whereas spiralling the content in your lessons could be thought of as more challenging if you’ve already begun teaching in blocks or units.

Whew! That’s a lot to consider if you want to dive straight into spiralling your math curriculum.

If that seems overwhelming, then maybe you might want to consider just easing into the idea of spiralling, interleaving, spacing and mixing portions of your math class.

Well, you’re in luck! In the next couple of days, I’ll be sharing 9 Spiralling Starter Strategies to Begin NOW!

Part 3: 9 Steps to Spiralling Your Math Class NOW!

Spiralling Your Math Curriculum - 9 Starter Strategies to Begin Spiralling Now

Spiralling is a pretty complex way to organize your course, but with positive implications for student learning. If you’re feeling a bit scared, stressed or anxious, that’s O.K.! I am certain that with all of the ideas I’ll be sharing with you in this video, you will be able to find a method that will help you get started based on your current comfort level.

Although I shared 6 steps to start spiralling in your math class earlier, the reality is that you might not be feeling ready to take the plunge by spiralling the lesson content in your course. Rather than avoiding spiralling altogether, why not leverage the research around spiralling, interleaving, spaced practice and mixing by starting with one of the following easy to begin strategies.

After reading the many ideas shared by the Math Twitter Blogosphere in response to a tweet by Jon Orr, I’ve managed to summarize much of what was shared into these 9 Spiralling Starter Strategies. All of these strategies are really useful ideas that can be used as very non-threatening first steps to spiralling, so let’s dive in!

Spiralling Starter Strategy #1: Find a Buddy

There is nothing more scary than taking on a new idea alone. The best part is, you don’t have to! Reach out to colleagues in your school who might be open to trying something new and work out a plan together.

Maybe you don’t agree on everything, but at least you’ll have a sounding board to share and borrow ideas from.

Mary Bourassa recognizes how big of a jump spiralling is and suggests seeking mentorship from someone who has tried spiralling in some capacity:

Can’t find someone in your building or close by to collaborate with? Mr. Hogg urges you to use online resources that are free and accessible anytime such as reading teacher blogs and connecting with other math educators on Twitter.

Jessica McConnell takes seeking out math teachers online a step further recommending that you not only connect with math teachers who teach the same grade level, but also connect with teachers who teach younger students and with those who teach older students. Then, you can make better decisions of where and when to introduce topics as well as how long to anticipate spending on those concepts.

Spiralling Starter Strategy #2: Spiralling Warm-Up Activities and/or Practice Problems

Whether you call them bell work, bell ringers, warm-ups, or mind busters, many math teachers use some form of warm-up task to begin their lesson.

Early in my career I would make the warm-up problem related directly to the previous lesson, which seemed to work nicely. But why not stretch back and warm up to a problem involving a topic we haven’t seen in a while to build that retrieval strength?

9 Starter Strategies to Begin Spiralling Now - Spiralling Warm Up Activities

Once you’re comfortable interleaving your warm-up problems, you might consider interleaving the independent practice problems you plan to assign. Instead of assigning problems that are all related to the content we focused on today, why not assign some from today and some from the past?

Spiralling Starter Strategy #3: Think Back Thursdays

Consider taking one day of the week to engage in your interleaving investigations. While it can be daunting to think that you are going to spiral all of your well organized and thoughtfully planned units of study, it’s definitely reasonable to think that you’ll commit one day a week to mixing things up. While I’m pretty proud to have come up with “Think Back Thursdays” as an option, the reality is that you can pick any day of the week and call it whatever you’d like!

A variation of this idea was brought to my attention by Norma Gordon when she recommended Monday Make Overs or Friday Fix-Ups where you spiral back to concepts students are struggling on for a re-attempt at learning.

Want to save this for later?

Download the Printable Cheat Sheet E-Books!

Spiralling Starter Strategy #4: Interleave Your Assessments

When I was teaching in units or blocks, I would typically give assessments throughout the unit and at the end of the unit. Consider giving assessments at the same point you would during your current unit of study, but give an assortment of problems related to the current unit and some from previous units. While I think this is a great way to slowly introduce the concepts of spiralling in your math class, I think using this strategy would be most beneficial if introduced by a teacher who is progressive in the area of assessment.

Spiralling Your Math Curriculum - Growing Success Document Assessment is Ongoing Varied in Nature Multiple Opportunities

What I mean by this is that the assessments we are giving our students are actually being used as a way to push the learning further rather than simply labelling a student with a letter. While this strategy is easy to implement on paper, I feel that it would be more of a punishment for students if the teacher using the strategy does not believe in giving students multiple opportunities to learn and demonstrate that learning.

9 Starter Strategies to Begin Spiralling Now - Assessments Should Be Used as a tool to push learning forward not for labelling and sorting ctt

Spiralling Starter Strategy #5: Lag Your Homework

If you are teaching a math course where homework is developmentally appropriate – say Grade 7 or so and up – Henri Picciotto suggests that you “lag” homework. Lagging homework involves having your students practicing concepts from the past – say topics from last week – instead of working on a brand new concept introduced that day. The logic here is that students are given some time to digest the new ideas and are not left all alone at home with an unfamiliar concept to work on.

9 Starter Strategies to Begin Spiralling Now - Lagged Homework Schedule vs Typical Homework Schedule

Spiralling Starter Strategy #6: Look For Multiple Connections in Your Tasks

It is highly likely that you are already using some really rich tasks throughout your course. Both Ve Anusic and Heather Theijsmeijer suggest we can easily start by spending more time on intentionally noticing how the tasks you are already using connect to other topics.

By looking at tasks through the lens of different big ideas or strands in your course, you will have an easier time making multiple topic connections while still feeling as though you’re organizing your curriculum in blocks.

Spiralling Starter Strategy #7: Chop Your Units Into Spirals

If you’ve taught your course in the past using blocks or units, why not take that long range plan and simply chop up the units into smaller chunks. For example, you could take your first 3 days of each unit and make them one spiral, then the next 3 days from each unit for your second spiral, and so on.

The benefit here is that you will still feel like you are organized in units, but your units are just being spread out over longer periods of time.

Spiralling Starter Strategy #8: Spiral a Single Concept

Another great starter strategy was brought to my attention by Sherry Doherty and Deborah Hartmann. They suggest starting with a big idea – like fractions – or a single unit – like say measurement – and sprinkling those lessons, activities and investigations throughout your regular units as a manageable first step.

Ms. Butson also suggests starting with one important concept, but rather than spreading your lessons related to that idea throughout the course, why not focus specifically on how that specific concept connects to other areas of your curriculum and intentionally pulling that concept out as often as possible. Once you get a feel for how to do that with one concept, you move on to try doing the same with another concept. Eventually, you’ll be full blown spiralling and making multiple topic connections daily!

Spiralling Starter Strategy #9: Interleaving Activities and Investigations

Finally, the last of the spiralling starter strategies was mentioned by Dave Lanovaz. He suggests that you get started by keeping your units or blocks in tact, but spiralling activities that you might do throughout a unit with a mix from other units across the course. So while your long range plan will feel like you are maintaining the same order and structure you’ve used in the past, the activities being introduced are mixed from different parts of the course.

Seems pretty doable to me!

So, there you have it; 9 Spiralling Starter Strategies that you can use to get your feet wet with spiralling. If you’ve made it this far, that means that you must be seriously considering implementing spiralling in some capacity.

While you can attempt all 9 of these strategies, I’m going to suggest that you pick one and fully commit to implementing it. Be sure to download and print the spiralling cheat sheets included with each section in this series and come back to it for reflection and planning your next step to integrating spiralling and interleaving in your math class.

Make yourself small, attainable goals and you will find yourself confidently spiralling your classroom in the way that suits YOU and your students best.

Ms. Butson also suggests starting with one important concept, but rather than spreading your lessons related to that idea throughout the course, why not focus specifically on how that specific concept connects to other areas of your curriculum and intentionally pulling that concept out as often as possible. Once you get a feel for how to do that with one concept, you move on to try doing the same with another concept. Eventually, you’ll be full blown spiralling and making multiple topic connections daily!

Spiralling Starter Strategy #9: Interleaving Activities and Investigations

Finally, the last of the spiralling starter strategies was mentioned by Dave Lanovaz. He suggests that you get started by keeping your units or blocks in tact, but spiralling activities that you might do throughout a unit with a mix from other units across the course. So while your long range plan will feel like you are maintaining the same order and structure you’ve used in the past, the activities being introduced are mixed from different parts of the course.

Seems pretty doable to me!

Which Spiralling Starter Strategy Are You Going to Try?

So, there you have it: 9 Spiralling Starter Strategies that you can use to get your feet wet with spiralling. If you’ve made it this far, that means that you must be seriously considering implementing spiralling in some capacity. Awesome!

While you can attempt all 9 of these strategies, I’m going to suggest that you pick one and fully commit to implementing it. Be sure to download and print the spiralling cheat sheets included with each section in this series and come back to it for reflection and planning your next step to integrating spiralling and interleaving in your math class.

Make yourself small, attainable goals and you will find yourself confidently spiralling your classroom in the way that suits YOU and your students best.

I’m extremely exited that you’ve dedicated so much time to learn about spiralling with me. I hope you’ve found this resource useful for your own professional learning and I look forward to continue learning with you in an upcoming blog post, video or course real soon!

Spiralling Resources

Here is an ongoing list of blog posts, books and other resources that are useful if you are interested in Spiralling Mathematics or any other course for that matter. Have something to add? Help the whole community out in the comments!

People From The Math Twitter Blogosphere (#MTBoS) Who Spiral & Their Blog Posts

  • Jon Orr [Twitter | Blog: Teaching With Spiralled 3 Act Tasks]
  • Alex Overwijk [Twitter | Blog: Spiralling Post]
  • Henri Pincciotto [Twitter | Blog: Lagged Homework | Blog: Separating Related Topics]
  • Mary Bourassa [Twitter | Blog: Spiralling Post]
  • Dave Lanovaz [Twitter]
  • Mr. Hogg [Twitter | Blog: Spiralling Post]
  • Jimmy Pai [Twitter | Blog: Spiralling Posts]
  • Jessica McConnell [Twitter]
  • Norma Gordon [Twitter]
  • Ve Anusic [Twitter]
  • Heather Theijsmeijer [Twitter]
  • Dawn Butson [Twitter]
  • Great Books, Articles and Documents Supporting Spiralling and Interleaving

    • How We Learn [AMAZON]
    • Make It Stick [AMAZON]
    • Interleaved Mathematics Practice – Giving Students a Chance to Learn What They Need to Know [ARTICLE]
    • The Key to Interleaving: Jumble It Up! [ARTICLE]
    • The Interleaving Effect: Mixing It Up Boosts Learning [ARTICLE]

    Research Supporting Spiralling and Interleaving

    • Interleaved Practice Improves Mathematics Learning (Rohrer, Dedrick, and Stershic) [PDF]
    • The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems (Rohrer, Dedrick, Burgess) [PDF]
    • Interleaved Practice Enhances Skill Learning and the Functional Connectivity of Fronto-Parietal Networks (Lin, C. H. J., Chiang, M. C., Knowlton, B. J., Iacoboni, M., Udompholkul, P., & Wu, A. D. 2013) [PDF]
    • Why interleaving enhances inductive learning: The roles of discrimination and retrieval (Birnbaum, Kornell, Bjork, Bjork) [PDF]

Spiralling Frequently Asked Questions (FAQ)

Here are some Frequently Asked Questions (FAQ) regarding spiralling, interleaving and mixing math content I’ve received through the release of this 3-part video series.

Q: How might I spiral my curriculum if administrators require common curriculum pacing?

Question 1

One of the objections I face from administrators is that they need all teachers in the same place in the linear pacing guide so that, if needed, students can be moved from one roster to another. Any ideas?

– Dan Hawkins

Answer 1

You bring up a challenge that I am sure many others face when considering spiralling. This might pose a problem if there is only one grade level teacher interested in spiralling while the other teachers are content with moving forward using units. In the case where all teachers are “on-board,” you could co-plan what spiralling might look like for that grade level. That way, you can stay relatively consistent with your long range planning. However, as mentioned in video #3, creating a long range plan without the flexibility to meet the needs of the students in the seats serves our own structural and organizational needs more so than the needs of the students whom we are planning the course for. So while it is convenient that I can move a student from one course to the other and feel as though they haven’t missed a beat, I wonder how many students who do not move from one course to the other are sinking because the course material is being delivered regardless of the readiness of the students in the room.

I know that it is very difficult to make systemic change, but referencing the overwhelming benefits for all students when we interleave content as opposed to blocking to potentially help a small few who switch rosters should at least get decision makers thinking.

Q: How does spacing practice compare to mastery?

Question 2

In your Spiralling Videos, you relate spacing to massing. How does spacing compare to mastery?

You have to understand I’m playing devil’s advocate here. Looking at the top countries in the PISA results from 2016, the top 7 scoring countries for math use mastery as a instructional method for their programming. I know mastery is not the only reason why these countries are strong, but I would assume it plays a significant role.

– Adam S. from Durham, Ontario

Answer 2

I believe that spacing and massing are inversely related since you can give lots of practice on one idea in a short period of time or spread that practice out.

I don’t think we can relate mastery to spacing (or massing) in the same way. Personally, when I’m teaching, I want students to develop mastery of concepts and in order to achieve that mastery, the teacher can choose to mass practice or space practice by spreading it out over time.

If brain science tells us that students will gain a deeper understanding of new learning by spacing practice, then I’d argue that you have a better chance of reaching mastery by spacing that practice.

When you mention that you are playing devil’s advocate, I immediately begin to wonder if you have interpreted my message around spacing practice as if it means “doing less” practice than when massing. If that is the case, I apologize for not being clear. Students need to have significant exposure in order to learn anything through repetition. As mentioned in the spiralling video series, mass practice routinely tricks us into believing students have a deep understanding (i.e.: illusion of understanding) because it is so fresh and familiar in our short term memory. However, the research shows that retention is very poor and spacing practice – while appearing to be less effective in the short term – comes out on top in the long run.

I hope this helps clear up some of the confusion.

Q: Does spiralling mean incorporating all the units of the course at any given time?

Question 3

It sounds like sprialling incorporates all the units of the course at any given time. Is this a good way to think of spiralling?

Alternatively, could I teach all the units in their own compartments and as these units evolve, review material from other parts of the course? If so, do we integrate previous material into the current units or do we review previous material separately.

– Ravi S. from Toronto, Ontario

Answer 3

There are so many different ways to spiral math curriculum and I’d argue that both of those methods you’ve outlined above could be done effectively depending on your own style and comfort level.

I’d like to first respond to your thinking around introducing concepts from all of the units at any given time. While that is a good general view, I would suggest that you would want to pre-plan what content is coming up and when. As you do more thinking about spiralling in this manner, you’ll begin to see natural connections that may not have been apparent to you previously due to the “siloed” manner in which we traditionally teach units in math class. So, in general, I would promote you thinking about why you are picking one concept to come next over another, however without allowing any perfectionist qualities to set in. When we mix up our content in this manner, you’ll begin to better assess where your students are and teach to those needs rather than teaching what you had written down on your unit plan from months before.

Your second method suggests still teaching in traditional units and then “reviewing” content as you go. This is another great spiralling strategy and one that I’ve used for a number of classes. It is a great way to start for those a bit too anxious to break away from unit structures, but to still provide the much needed spaced practice that our brains need to build deep connections and retain information. This seems like a very easy starting point for anyone who is interested in taking advantage of what brain science tells us is effective for learning in math class and in general.

Another method for consideration would be to look at tasks that spark curiosity and create an opportunity to fuel sense making around a big idea in math and work backwards to the curriculum. While I have a “big idea” in mind for each task, I also look for ways to sneak in other parts of the curriculum to essentially create a “spiralled lesson.” When we do this, we find that it forces us to think about one specific context and find interesting problems that we can ask around that one context rather than constantly ambushing students with 10 different problems that focus on 10 different contexts. An example of this might be a problem like Soup Du Jour where we are focusing on finding the volume of a rectangular prism and then asking a follow up question like “if each person will eat 250 ml of soup, how many containers of soup will you need for 14 people?” So now, the lesson does not just focus on volume, but also on the proportional relationship that exists between total volume of soup in millilitres and number of people.


If you know of a friend or colleague who would benefit from this video series, please share on your favourite social media platform.

I’m sure there are many other creative ways to start your journey to spiralling your math lessons that aren’t listed here. Please share your ideas in the comments!

What are you waiting for? Let’s get spiralling!

The post The Complete Guide to Spiralling Your Math Curriculum appeared first on Tap Into Teen Minds.

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Make Math Matter With Concreteness Fading https://tapintoteenminds.com/concreteness-fading/ https://tapintoteenminds.com/concreteness-fading/#respond Tue, 20 Feb 2018 01:57:04 +0000 https://tapintoteenminds.com/?p=20014 Another post from Tap Into Teen Minds. Enjoy!

For years, I was spinning my wheels trying to teach students how to make sense of mathematics through abstract representations, when the key was making math concrete and visual through the concreteness fading model or concrete, representational, abstract (CRA).

The post Make Math Matter With Concreteness Fading appeared first on Tap Into Teen Minds.

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Another post from Tap Into Teen Minds. Enjoy!

Importance of Context and Concrete Manipulatives From Kindergarten Through Grade 12

Make Math Matter With Concreteness Fading

During the first half of my teaching career, I would spend what seemed to be the first half of a math lesson teaching a new math concept by sharing definitions, formulas, steps and procedures. To make things more challenging for my students, I would simultaneously introduce the symbolic notation used to represent those ideas. Then, I would spend the remainder of the lesson attempting to help my students make sense of these very new and often abstract ideas. By the end of the lesson, I could help many students build an understanding, but there was always a group I felt who I would leave behind.

Like many other teachers, I was just teaching in a very similar way to that how I was taught.

I knew no different.

However, if we consider that new learning requires the linking of new information with information they already know and understand, we should be intentionally planning our lessons with this in mind. A great place to start new learning is through the use of a meaningful context and utilizing concrete manipulatives that students can touch and feel. When we teach in this way, we minimize the level of abstraction so students can focus their working memory on the new idea being introduced in a meaningful way.

Concreteness Fading: Concrete, Representational, Abstract (CRA) Approach

When we intentionally start with concrete manipulatives to learn new math concepts, our goal is to help students better construct an understanding of the mathematics in their mind. The goal is not to burden students with a big bag of manipulatives that they must carry around with them anytime they are required to do any mathematical thinking, but rather to ensure that they can build their spatial reasoning skills physically – through the manipulation of concrete objects – so they can begin to visualize mathematics in their mind. When a student is able to “look up” as if they are peering into their mind to visualize their math thinking, we know students are thinking conceptually rather than simply following a memorized procedure.

While students are working with concrete manipulatives, it is helpful for the teacher to model visual representations of the student work for all to see. By introducing these visual drawings of the concrete representations students are creating, it will be easier for students to shift away from concrete manipulatives and towards visual (drawn) representations when they are ready.

When students have built an understanding both concretely and visually, we can then begin moving to the final stage called abstraction where we use symbolic notation. The goal here is that when students use the symbolic notation, they can visualize what the concrete representation of that mathematical statement represents.

Some know this idea as concreteness fading, while others have called this progression concrete, representational, abstract (CRA). In either case, the big idea is the same. Start with concrete manipulatives, progress to drawing those representations and finally, represent the mathematical thinking abstractly through symbolic notation.

Let’s look at a couple different questions at different grade levels where using context and concrete manipulatives can lower the floor and help us progress towards more abstract representations.

Scenario #1: Elementary / Primary Classroom

If we were to ask students in a grade 2 class, I might have them look at the following image and ask them what they notice and what they wonder:

Concreteness Fading.002 What do you notice and wonder

Then, I might have them predict how many doughnuts they believe fit in that box.

After students talk with a neighbour and share out their predictions, I would say:

This box of doughnuts has 3 rows of 4 doughnuts.

Concreteness Fading.003 How many doughnuts are in a box?

This question may seem quite simplistic after giving this new information, however for students who are just beginning to shape their understanding of number including place value and additive thinking, we are now throwing a very heavy and abstract idea at them.

In a perfect world, we could give them real doughnuts (or bagels, to be health conscious) so they could recreate the situation right in front of them:

How many doughnuts are in a box animated gif

Despite the fact that using square tiles or circular counters to represent doughnuts is more concrete than drawing doughnuts or using symbols (numbers and operations), we must understand that concrete manipulatives are still more abstract than using the actual items in the quantity being measured.

Concreteness Fading.009 real objects to concrete manipulatives is abstract

As students understanding of number increases, so too should their ability to begin using concrete manipulatives instead of real doughnuts to work through this situation.

As students use concrete manipulatives to build their conceptual understanding of a new idea, they will begin to feel burdened by the manipulatives and seek out less cumbersome tools and representations to show their thinking. If the teacher has been drawing visual representations of the concrete representations students share with the group along the way, many will eventually transition to drawing their representations rather than building them concretely. However, for other students who have seemingly mastered the concrete representations but are not shifting to visuals, we may need to help scaffold them along.

Concreteness Fading.010 first two stages of concreteness fading

With conceptual understanding continuing to deepen through the use of drawn visual representations, teachers can continue sharing student thinking through the use of visuals and begin introducing symbolic notation. Since students have had a significant amount of time to inquire, investigate and solve problems using both concrete and visual representations, they will develop the ability to visualize representations in their mind. At this stage, it would seem more efficient to use symbolic notation such as numbers and operations to represent mathematical thinking rather than building concretely or drawing visually.

Concreteness Fading.011 all three stages of concreteness fading

It is important to note that while the concreteness fading model or concrete, representational, abstract (CRA) approach is a general progression that we want to keep in mind when teaching new concepts in math class, we don’t want to overthink it either.

For example, in the abstract / symbolic phase, you’ll notice the words:

“3 groups of 4 doughnuts is equal to 12 doughnuts”

By no means am I suggesting that we should wait until the concrete and visual phases are mastered before using those words. I would actually suggest that we are verbally saying those words during the concrete stage and even possibly writing down that sentence during the concrete stage since there are no new symbols or abstract ideas being introduced by doing so. With an idea like single digit multiplication, you might consider having students build the concrete representations and the teacher may draw the visual representation as well as the symbolic representations at the same time.

The key with concreteness fading is that we are aware of these three phases and we use our professional judgement to determine when to introduce each phase as to push student thinking forward without overwhelming them with too much abstraction too quickly.

Scenario #2: Upper Elementary Classroom

We could introduce a similar question for say a grade 4 class by simply increasing the complexity of the question such as:

How many doughnuts are in 3 boxes?

Concreteness Fading.012 how many doughnuts are in 3 boxes

If we are asking students to work with a problem that we could consider is a multi-step multiplication problem, the beginnings of volume or a double digit by single digit multiplication problem, my hope would be that students are now comfortable abstractly using concrete manipulatives (connecting cubes, square tiles, etc.) to represent how many doughnuts are in 3 boxes. If a student is struggling with the abstraction of using a concrete manipulative in place of the actual object – like doughnuts in this case – we might need to reassess the readiness of this particular student and do some more work with more accessible problems.

How many doughnuts are in 3 boxes animated gif

In this particular case, the progression of concreteness fading might look something like the following:

Concreteness Fading.017 How Many Doughnuts in 3 Boxes concreteness fading model

Or students might go about it using their knowledge of arrays and extend the idea to area models before finally developing a student generated algorithm:

How many doughnuts are in 3 boxes animated gif v2

Here’s a summary of the concreteness fading progression that may take place if students have been doing work with arrays and area models:

Concreteness Fading.021 array and area concreteness fading model

Assuming students have had a substantial amount of experience building concrete representations of multiplication, you may see students skipping right over the concrete phase to the visual stage creating drawn diagrams of this situation. This is absolutely fine as a student who is able to draw what the concrete representation should look like suggests that she could indeed build that representation if required. Furthermore, this also suggests that this student is now able to create a more abstract representation of that concrete model, which is what we are hoping to develop.

What I would not advocate is completely skipping over the first two phases and focusing only on the symbolic representation. Despite the fact that some students may have a visual of that concrete model clear in their minds, we don’t want to promote students relying solely on procedural fluency and risk forgetting all of that conceptual understanding we worked so hard to build. By giving students enough practice drawing visual as well as abstract or symbolic representations, they are utilizing their conceptual understanding and procedural fluency in tandem, where they can be used most effectively.

Concreteness Fading is Cyclical

As students become more comfortable with the abstract representation of multiplying 2-digit numbers by 1-digit numbers, we might think it is fine to start with the abstract stage of concreteness fading when we progress to 2-digit by 2-digit multiplication. Although we have progressed through the stages of concreteness fading for one concept, as we add a new level of complexity (i.e.: adding another digit to multiplication) we should be cycling back to the concrete stage to lower the floor for all students to access this new learning.

A great example that works well here is the 3 act math task, Doughnut Delight.

After students notice and wonder, we land on the question:

How many doughnuts are in that giant box?

Krispy Kreme Donut Delight 3 Act Math Task - Make a Prediction

After students make predictions and justify their reasoning, I reveal the dimensions to the students:

There are 32 rows and 25 columns of doughnuts.

They are then set off to find a solution using an effective strategy of their choosing. Assuming this is their first exposure to 2-digit by 2-digit multiplication, starting in the concrete stage using base-10 blocks would be appropriate. For students who may already have made connections to the work they have done previously, they may choose to draw out an array or use another visual model to show their thinking.

Here in Ontario, we explicitly introduce 2-digit by 2-digit multiplication in grade 5. Every time I’ve used this task with students in grade 5, most are rushing to the algorithm and making errors due to the lack of conceptual understanding.

I do my best to try and get students to back up a stage or two in order to truly understand the mathematics we are asking them to grapple with. This can be a struggle, because often times students just want to get “an answer” and move on.

However, if they are shown how easy multiplication can be by having a conceptual understanding in their back pocket, they will eventually jump on the opportunity.

Here’s what concreteness fading could look like for this task:

Concreteness Fading - Doughnut Delight 3 Act Math Task

You can read more about the progression of multiplication and download the complete Doughnut Delight task for multiplication and division below:

Scenario #3: Middle School Classroom

In a middle school classroom (end of junior/intermediate classroom in Ontario), the question might sound more like this:

There are 36 doughnuts in 3 boxes.
How many doughnuts are in 7 boxes?

Concreteness Fading.022 how many doughnuts are in 7 boxes

While this may seem like a lot of doughnuts for students to represent concretely, having linking cubes, square tiles or other tools students can use to organize their thinking is important especially for those who have not yet built a conceptual understanding of what this task is asking of them. In this case, we are exploring a proportional relationship where the number of doughnuts is proportional to the count of how many boxes there are.

Despite the fact that proportional reasoning is introduced explicitly in the Ontario Grade 4 Math Curriculum, many of our grade 9 students continue to struggle with this type of reasoning. Many may not have fully conceptualized the prior knowledge necessary for them to be successful at that particular grade level. Before I understood the power of concrete and visual representations, I can recall trying to help students in my grade 9 (and sometimes grade 10) class by breaking down the symbolic representation with more symbols.

For example, with this particular problem, I might have attempted unpacking the problem with students by creating a proportion and solving for the unknown:

Scenario 3 - How many doughnuts in 7 boxes - proportion

While working with and solving for unknowns in a proportional relationship was an expectation in my curriculum and in the curriculum prior to grade 9, I was stuck in my habit of trying to start with abstract symbols and unpacking them. However, the reality is that when we do this sort of work without building the necessary conceptual understanding at the concrete and visual phases of concreteness fading, students are forced to either memorize the steps and procedures or get left behind. For some students, they are able to make their own connections at the symbolic stage based on their prior knowledge from past experiences in school and at home, while other students are left scrambling to understand with stress and anxiety levels building with each passing class.

How might I have approached this same problem had I known and understood the concreteness fading model?

Well, I would definitely start with concrete manipulatives for all of my students. Just because a student is able to solve familiar problems using all the right steps and procedures does not necessarily mean that they have a conceptual understanding of the mathematics they are employing.

One possible idea could be giving students connecting cubes and having them model out the situation. They might start by grabbing 36 cubes and dividing them to the 3 boxes. Then, they could double the 3 boxes of doughnuts to get 6 boxes and add an additional box.

How many doughnuts are in 7 boxes - connecting cubes animated gif

Some professional noticing you will want to engage in would be determining whether students are using additive thinking, multiplicative thinking or a combination of the two. To build an understanding of proportional reasoning, we must help students to think multiplicatively. So while thinking additively is not bad or wrong, we do want to try to prompt students to think multiplicatively.

For example, you might ask students:

How many times bigger is the quantity in 7 boxes than in 1 box (i.e.: 7 times bigger)?

How many times bigger is the quantity in 7 boxes than the quantity in 3 boxes? (i.e.: 2 and 1 one-third times bigger)

While I try to encourage all students to make a concrete model, some may be moving away from physical manipulatives and pushing towards a visual model which would suggest that they are ready to move one step deeper into abstraction.

Here’s an example of how a double number line could be used to help students visualize the situation and problem solve their way to a solution.

How many doughnuts are in 7 boxes - double number line animated gif

From both the concrete models and the visual models students use in the classroom, I can prompt students to attempt modelling their thinking using symbolic notation such as algebraic expressions and equations.

The more I can help my students link their concrete and visual models to more abstract representations, the stronger their conceptual understanding will be to help support any procedural approaches they wish to use to progress towards more efficient methods.

Had I known more about concreteness fading earlier in my career, the progression might have looked more like this:

Concreteness Fading - cubes to double number line to solving a proportion

While the above concreteness fading progression would have been a huge help to all students in my class to better understand proportional relationships, I would later learn from my colleagues in the AMP group that setting up a proportion of equivalent fractions is not a very powerful method mathematically, since it yields only the numerical answer to a single problem. A more powerful approach is to uncover the proportional relationship in the problem situation, since this allows us to immediately solve any problem based on that situation.

Let’s take a closer look.

Fuel Sense Making: Scaling in Tandem and The Constant of Proportionality

I have been blessed to be a part of an amazing group of mathematicians funded through the Arizona Mathematics Project (AMP) to make sense of proportional relationships and this group of 18 mathematics education influencers have landed on some really useful definitions related to this very commonly encountered type of middle school math problem.

When we look at the animation of the concrete model using connecting cubes, you can see the two methods of attacking proportional relationships that the AMP group refers to as:

  • scaling in tandem; and,
  • using the constant of proportionality.

We can see the use of scaling in tandem when we see the doubling the number of boxes and number of doughnuts from 3 boxes, 36 doughnuts to 6 boxes, 72 doughnuts.

We can see the use of the constant of proportionality when we look at the number of doughnuts (12) in a single box – often referred to as the unit rate.

First, we will head a bit further down the concreteness fading continuum by taking our horizontal double number line and represent it as a vertical number line. This is a nice way to progress towards a table of values without losing the relative magnitude between each quantity on the number line.

Concreteness Fading - Horizontal Double Number Line to Vertical Double Number Line Proportion

For years, I would teach my students with an end goal of setting up and solving a proportion rather than focusing on helping them “own the problem” as Dick Stanley put it at our recent AMP meeting.

What we are referring to here is the limited usefulness of setting up a single proportion for a “rule of 3” problem.

Proportional Relationship Rule of 3 Problem

When we set up a proportion of equivalent fractions, we have set out to solve a single problem and often times, we unintentionally rush to a procedure by setting up and solving for a single unknown. While this might be efficient for finding a single answer to a closed problem, it does not help us efficiently solve multiple problems nor does it promote a deep conceptual understanding of the proportional relationship that underpins this situation.

While I do not want to advocate that we avoid proportions altogether, I would much prefer giving students the opportunity to explore these problems more deeply and allow for the students to stumble upon some of the procedures we see taught explicitly in middle school classrooms.

Let’s look at where we might start.

Ratio Reasoning: Scaling in Tandem

We can see from the animation below that scaling in tandem is responsible for allowing us to solve a proportion using any of the procedures we see taught in many middle school math classrooms:

Concreteness Fading - Solving a Proportion Using Scaling in Tandem Instead of Procedures

By utilizing ratio reasoning by scaling in tandem, students are explicitly introduced to the power of the proportion, but in a much more powerful way.

If students are encouraged to utilize scaling in tandem with double number lines, tables and equivalent fractions, over time we can help students see that some of this scaling can be done more efficiently:

Concreteness Fading - Scaling in Tandem One Step

Over time, students may progress from a vertical number line to a table of values. When students are ready, they may begin disregarding the magnitude of number allowing them to “skip over” some of the values on the number line.

Concreteness Fading - Vertical Number Line to Table of Values

We can use this scaling in tandem strategy to find any unknown in this proportional relationship, but it will take a bit of work.

For example, we can find the number of doughnuts in 9 boxes of doughnuts by scaling in tandem by multiplying both 36 doughnuts and 3 boxes by 9/3 or 3:

Concreteness Fading - Ratio Reasoning and Scaling in Tandem

As students become more fluent using scaling in tandem as a strategy for proportional relationships, we can then begin making generalizations:

Concreteness Fading - Ratio Reasoning - Scaling in Tandem to Explain Cross Multiplication

What we see through this generalization is the conceptual understanding for why the common procedures we see in math classrooms actually work.

Concreteness Fading - Ratio Reasoning and Rate Reasoning

One of the most over-used, but misunderstood tricks from the middle school math classroom is cross multiplication. When we look at the generalization of scaling in tandem, we can see where ideas like cross multiplication comes from:

q/p = d/c

qc = pd

c = pd/q

or

d = qc/p

While I taught tricks like cross-multiplication, “the magic circle” and “y-thingy-thingy” before I constructed a firm conceptual understanding of proportional relationships, the reality is that they provide a dead end pathway to a single answer rather than an understanding that allows you to own the problem.

So rather than simply teaching a trick using steps and procedures, let’s give students an opportunity to build a conceptual understanding of scaling in tandem and challenge them to come up with their own procedures and algorithms.

Rate Reasoning Using the Constant of Proportionality

While we can use ratio reasoning and scaling in tandem to up the conceptual understanding over solving a “rule of 3” problem using a proportion, we still don’t “own” the problem yet.

Under the hood of every proportional relationship lies a constant that we can use to solve ANY problem related to the proportional situation. This constant of proportionality can be found by taking the quotient of any two covarying values in the relationship.

Many know this constant of proportionality as the unit rate.

Concreteness Fading - Rate Reasoning - Quotient of a Ratio to Find Constant of Proportionality

Unlike the ratio reasoning strategy of scaling in tandem where one must determine a new scale factor to find each unknown quantity in a proportional relationship, the rate reasoning strategy of finding the constant of proportionality allows one to use the constant to find any unknown from the relationship.

In the case demonstrated here where the number of doughnuts is proportional to the number of boxes, we can determine the number of doughnuts in any number of boxes by multiplying the number of boxes by 12 doughnuts per box, while we can determine any number of boxes by multiplying the number of doughnuts by 1/12 boxes per doughnut (or dividing by 12 doughnuts per box).

Concreteness Fading - Rate Reasoning - Generalizing With the Constant of Proportionality

So again, while I see huge value in students understanding how they can use ratio reasoning to scale in tandem to solve problems involving proportional relationships, only when we unlock the conceptual understanding behind rate reasoning and the constant of proportionality do we own every problem related to that proportional relationship.

Concreteness Fading - Rate Reasoning - Generalizing With the Constant of Proportionality, k

So rather than suggesting that the concreteness fading progression should end at the creation of a proportion of equivalent fractions and solving for an unknown, I would much rather see students exploring both ratio reasoning by scaling in tandem and rate reasoning through the constant of proportionality. Therefore, a suitable progression might look something like this:

Concreteness Fading - Concrete to Visual Across Number Line to Constant of Proportionality

Scenario #4: Middle and High School Classroom

In grade 8 classrooms here in Ontario, we start making a serious push towards deeper algebraic reasoning and functional thinking which carries over into the deep exploration of linear and quadratic relationships in grades 9 and 10. While the context for doughnuts may not be my favourite – despite all their yummy-ness – it is possible for us to extend our thinking around this context from multiplicative and proportional reasoning to algebraic and functional thinking.

In this case, we’re going to continue exploring some situations where the goal is to figure out:

How many doughnuts are there?

In the first scenario, we’re looking at a 1 row box or “strip” of doughnuts, however we do not know how many are in each box initially:

Concreteness Fading - Extending to Algebraic Reasoning.001 How Many Doughnuts Are There

We are then given an opportunity to take a guess at how many might be in that box considering the width of the box should be approximately the width of 1 doughnut.

Since a proportional relationship exists between the number of doughnuts and number of boxes, we can use our understanding of the constant of proportionality to help us create an equation for this situation.

If we assume (or we are given) the number of doughnuts in each box, we can then determine how many doughnuts in total.

Constant of Proportionality as an equation y = 8x

Constant of Proportionality as an equation y = 8x animated

We can also use this same proportional relationship with the same situation where the total number of doughnuts in 8 longer boxes is known in order to determine the number of doughnuts in each box:

2nd Scenario y = 8x where y is 80 doughnuts animated

Not super interesting, I know. However, I’d like to extend this thinking further to help us dive deeper into algebraic thinking with concrete and visual representations in mind.

In this next situation, we are given 8 square boxes and we know the total number of doughnuts is 72.

72 doughnuts in 8 square boxes how many are there

In this case, students may use either ratio reasoning by scaling in tandem or rate reasoning by jumping straight to the number of doughnuts in each box by taking the quotient.

If a student uses ratio reasoning, scaling in tandem might look something like this using a concrete and/or visual representation:

8 square boxes equals 72 doughnuts concrete visual animation

Students using rate reasoning might look something like the following. I’ve also shown the visual and symbolic representations side-by-side:

8 square boxes equals 72 doughnuts symbolic and visual animation

Now that we’ve introduced square boxes, we might consider playing in the land of single-row boxes and square boxes for students to do some algebraic thinking and problem solving.

How many doughnuts squares strips and a single doughnut

Here, we can have students make a prediction noting that the images are to scale and thus they can use the size of the single doughnut to help them with that prediction.

Having them share with their partner how they came up with their prediction can be extremely useful to see what they notice about each of the boxes. The goal here would be for students to make the connection that each doughnut “strip” box is the same length as the square box dimensions.

You can then share some more information and have them update their prediction:

Showing how many doughnuts in each box

Giving students a concrete representation of this situation by cutting out these shapes on card stock could be extremely useful for them to tangibly work with these quantities. If students are given the opportunity to manipulate these boxes of doughnuts, they may realize that they can create a complete rectangle (or square in this case) and that can also be helpful for their prediction.

Creating rectangles with these boxes

By creating this rectangle, they can more easily come up with a total number of doughnuts using their knowledge of arrays and area models.

creating rectangles and partitioning doughnuts animation

Then, we can allow students to use their same manipulatives to determine the total number of doughnuts in this situation. Same size boxes, just more of them.

6 squares 5 strips 1 doughnut

Over time, we would want students to start noticing patterns and using those patterns to help them come to a total using multiplicative and algebraic thinking.

Initially, they may do this verbally by describing the 6 square boxes, the 5 single row “sleeves” of doughnuts and the 1 extra doughnut. Since they know that there are 9 doughnuts in each square box and 3 doughnuts in each single row “sleeve”, it might sound like this:

6 boxes of 9 doughnuts plus 5 boxes of 3 doughnuts plus 1 doughnut

6 boxes 5 sleeves 1 doughnut word expression

We can help students to write their expression in words and then eventually, using symbols.

Symbolic representation with visual

After having quite a bit of experience doing this type of problem puzzle, we can have them start using algebraic equations using variables and in this case, even explicitly draw out how the squares are literally the single row “strips” of doughnuts “squared”.

Boxes of Doughnuts Algebraic Equation with Symbols

Now that students have come up with an algebraic equation, we could then play with the dimensions of the boxes and have students utilize the equation to come up with solutions. For example, if we give students larger boxes, but keep the number of boxes the same, they can leverage the same equation.

5 doughnuts in each box

Some students may still require time to play with this idea in words and then numerically, while others may feel comfortable jumping straight to the algebraic equation. However, in either case, the task is very accessible by starting concrete and leveraging visuals throughout regardless of whether they are working with symbols with more abstract representations.

5 doughnuts in each box animation

When most students are showing signs of being ready to make the leap towards more abstract thinking, I’ll change the dimensions again to have students make a prediction.

Write an equation to show your thinking

We want to make sure that we help students explicitly make a connection between the doughnut “strip” boxes and the square boxes so they can see the connection to algebra tiles.

Connecting Doughnut Boxes to Algebra Tiles

In this particular example, not only can we play in the land of substituting values into algebraic equations and simplifying, but we can also connect to quadratic relationships and factoring trinomials.

Here’s a visual of substituting a value of x into a quadratic equation and simplifying:

Simplifying quadratic expressions

By taking that same quadratic relationship and arranging as the familiar rectangle we are always trying to create when multiplying with base 10 blocks, we can easily see the two factors that are equivalent to this trinomial:

Quadratic Relationship Being Factored to Show Dimensions

Here’s what concreteness fading might look like in those grade 8 to grade 10 classrooms focusing on expanding and factoring polynomials:

Concreteness Fading - Progression of Concreteness for Quadratics Expanding Factoring

Whew… We just started this post down in primary grades and worked our way up to high school math all through the use of concreteness fading. While the context in this last example might have been a tad contrived, I hope it helps us see how flexible mathematics really can be and how we might consider lowering the floor through the use of concrete and visual representations.

If Prince Knew About Concreteness Fading, Why Didn’t I?

For years, I was spinning my wheels trying to teach students how to make sense of mathematics through abstract representations. However, even musicians are aware of the importance of marketing through a concreteness fading model.

Take the musician Prince for example.

Every successful musician knows that the best way to build a true fan base is to begin at the concrete phase. If you go to see Prince live in concert for example, you will quickly understand why he and his music are so great.

After seeing the show, you might rush to grab the next best thing to seeing him live in order to bring back the energy and positive feelings you had while watching live. Buying albums, videos, posters and magazines are great examples of how we can listen and see Prince in our minds as if he were there in the flesh.

Concreteness Fading.059 prince in concrete and visual representations

For those who are fans of Prince, it is highly likely that you know that after years of being known as Prince, he would legally change his name to a symbol:

Concreteness Fading.060 prince symbolic representation

This would be a career ending move if he didn’t already have millions of fans who had watched him live in concert (concrete) and enjoyed his music and memorabilia (visual). But yet, in math class, we so often begin with symbols and try to make meaning of them after.

My call to action here is to be more Prince and think about concreteness fading during the planning process of each and every lesson. If we do this, we stand a much better chance of Making Math Moments That Matter for students.

Concreteness Fading.061 prince analogy for concreteness fading

How are you using concreteness fading in your lessons? I’d love to hear from you in the comments.

The post Make Math Matter With Concreteness Fading appeared first on Tap Into Teen Minds.

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Counting With Your Eyes: Subitizing https://tapintoteenminds.com/counting-with-your-eyes/ https://tapintoteenminds.com/counting-with-your-eyes/#respond Sun, 11 Feb 2018 03:15:58 +0000 https://tapintoteenminds.com/?p=20000 Another post from Tap Into Teen Minds. Enjoy!

Have you ever looked at a group of items and just knew how many there were without actually counting? This ability to "see" how many items are in a group without counting is called subitizing. Read to learn more.

The post Counting With Your Eyes: Subitizing appeared first on Tap Into Teen Minds.

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Another post from Tap Into Teen Minds. Enjoy!

Have you ever looked at a group of items and just knew how many there were without actually counting? This ability to “see” how many items are in a group without counting is called subitizing.

The ability to subitize is an important part of developing a strong mathematical foundation and understanding of number (Baroody 1987, 115).

Playing with dice, dominoes, and asking children to find a specific number of items will help them develop subitizing skills and a sense of quantity. Asking to guess how many items you are holding will help develop estimation skills, which is another very important skill that will help children develop their mathematical skills.

Counting With Your Eyes: Look At The Dots

An interesting activity to do with children and adults is to have them look at the image of the dots below for just a few seconds and then look away (or remove the dots from their view).

Conceptual Subitizing - Dot Card - Spatial Reasoning - Jo Boaler

Ask them to make a picture in their mind of what they saw.

Then, describe what you saw in your mind to someone else.

It’s highly likely that they will “see” it differently than the person next to them.

Even though we are looking at the same dots, it is quite possible that the way you visualized these dots in your mind was different than the next person. This is because the number of dots you are visualizing is too difficult to subitize in a single group.

Here is a video of just a few of the many ways people describe how they visualized the dots:

Watch a Visual Dot Card Number Talk In Action

Watch Jo Boaler, Professor from Stanford University and co-founder of youcubed.org lead a group of students through a visual dot card number talk using this same exercise!

Perceptual Subitizing

When the number of items we are counting is small, we perceptually subitize to “see” the count suddenly.

Most can develop the skill to perceptually subitize quantities of 5 items or less.

Conceptual Subitizing

When the number of items we are counting is too large to “see”, we conceptually subitize to “know” the count suddenly.

When quantities are larger (say, 5 or more), our brains decompose the group into smaller “chunks” and then add them together.

Subitizing in School and At Home

Tools to Subitize With

You can help develop your student’s and/or child’s foundational mathematics skills in school and at home by making use of the following games and tools for subitizing:

Make 5 / Make 10

Use fingers, dice, playing cards with the corners cut off, dominos or dot plates to “make 5” or “make 10”.

War

Using dice, playing cards, or dot plates, two players roll a die, flip a card or dot plate and each player says their number. Player with the higher number wins the round.

Cover It Up!

One player shows how many counters they have in total. Then, hide some of the counters under the cup while the opponent closes their eyes. How many are under the cup?

First to 20

Using dice, take turns rolling 1 or 2 dice. Say the number rolled and record using a tally chart. First player to 20 wins.

Find It!

Take turns rolling a die. Find the same number of dots and cover it with your colour counter. Get 3 of your counters in a line and you win!

Download the game board here.

ONLINE TOOLS AND RESOURCES

Online Tools and Websites to Promote Counting and Subitizing at Home

You can help develop your child’s foundational mathematics skills at home by making use of the following games and tools for subitizing:

Counting Principles and Videos

Video and image prompts for visualizing subitizing and other principles of counting and quantity.

SPLAT!

Show your kids some dots, then SPLAT! Now, some are covered up. How many dots are under the splat?

Over 50 “SPLAT!” experiences to engage your children in subitizing!

Multiplication Dot Cards

Subitizing dot cards that you can use to begin promoting equal groups and multiplication!

Grab a variety of printable resources for number sense.

Download and Print FREE Dot Card Sets

Included in the FREE Downloadable Subitizing Guide are 6 unique sets of dot cards you can print, cut out and play games with your children and/or students!

Here are all 6 sets:

Want More Subitizing Activities?

You can learn more about perceptual and conceptual subitizing and access useful activities by downloading the subitizing guide.

The post Counting With Your Eyes: Subitizing appeared first on Tap Into Teen Minds.

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Lower the Floor in Math Class https://tapintoteenminds.com/lower-floor/ https://tapintoteenminds.com/lower-floor/#respond Sun, 04 Feb 2018 14:12:03 +0000 https://tapintoteenminds.com/?p=19986 Another post from Tap Into Teen Minds. Enjoy!

Struggling to find a way to make math more accessible for all students in your classroom? In this post, we'll give examples why using concrete manipulatives and visual representations is a great place to start!

The post Lower the Floor in Math Class appeared first on Tap Into Teen Minds.

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Another post from Tap Into Teen Minds. Enjoy!

Lower The Floor In Math Class

Making Math More Accessible Through Concrete Manipulatives and Visuals

What comes to mind when you think back to learning math in school? It would seem that most people I ask typically respond with a negative or neutral response and very few with something positive. Since many of us were taught primarily using procedures and steps, it is unlikely that too many of us could see math as anything more than rules, steps and symbols despite the fact that mathematics was created to help us better understand the world around us.

If this is so, then why aren’t we learning math first with concrete objects that we can touch and feel in order to allow students to co-construct and develop the rules, steps and symbols that represent those real world situations. By doing so, we are helping students develop the ability to visualize the mathematics they are engaging in and they will have an opportunity to see mathematics very differently to that of our generation.

Have a look at the visual below.

How many seats do you see?

Lower The Floor in Math Class - Manipulatives and Visuals

I bet you see 18, right?

Here’s the fun part.

How many different ways can you write a numerical expression to represent those 18 seats. I’m going to guess that you all can come up with at least these two:

9 + 9
and
2 x 9

An assumption I’ll go with in this post is that “2 x 9” is read “2 groups of 9”. However, there are other interpretations that would match a different visual.

How many others can you come up with?

While this is a fun activity to give students practice writing expressions, the most important element here is the concrete representation (if you were using square tiles) or the visual representation (say images of the seats as we are doing here).

There are just a few of the many representations you could come up with:

Using Concrete Manipulatives and Visuals to Lower The Floor

By using concrete manipulatives like square tiles for this activity and allowing students to progress towards drawing visual representations when they are comfortable and able, we can give students the opportunity to build a conceptual understanding of how mathematical expressions are created and make conjectures as to what generalizations can be made about simplifying them.

While my representations are based on the assumption that a single seat represents the whole, you could also explore other scenarios such as having the entire plane represent the whole for exploring expressions with fractions, decimals and percentages.

If you’ve been trying to find a way to make math more accessible for all students in your classroom, using concrete manipulatives and visual representations is definitely a great starting point.

I plan to come back to this idea on a regular basis, so be sure to stick around for that. In the meantime, you might consider exploring some of my previous posts related to visualizing mathematics.

The post Lower the Floor in Math Class appeared first on Tap Into Teen Minds.

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