Tap Into Teen Minds https://tapintoteenminds.com Sparking Curiosity in Mathematics Through Tasks that Are Contextual, Visual and Concrete Tue, 24 Apr 2018 16:18:38 +0000 en-US hourly 1 https://wordpress.org/?v=4.9.5 35226203 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 http://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.

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 Rate Reasoning

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


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.

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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 http://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.

Another post from Tap Into Teen Minds. Enjoy!

Count With Your Eyes - Subitizing Guide

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:

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

Count With Your Eyes - Subitizing Tools

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”.


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.

Download the Printable Subitizing Guide

Download The Printable Subitizing Guide

Want to grab a subitizing cheat sheet that you can print to keep handy in your classroom or as a resource to provide parents to raise awareness of the importance of subitizing at home?

Grab it here

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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 http://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!

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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
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.

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Knowledgehook Math PLC Planning Tool https://tapintoteenminds.com/knowledgehook-math-plc-planning-tool/ https://tapintoteenminds.com/knowledgehook-math-plc-planning-tool/#respond Sat, 27 Jan 2018 01:52:04 +0000 http://tapintoteenminds.com/?p=19953 Another post from Tap Into Teen Minds. Enjoy!

Are you a school administrator or math leader looking for tools to support planning your professional learning in mathematics with your colleagues? Check out Knowledgehook's new PLC Planning Tool!

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Another post from Tap Into Teen Minds. Enjoy!

Knowledgehook Professional Learning Tools.001 Featured Image

Have you been using Knowledgehook’s Free Gameshow Tool or the Premium Mission Feature? I was pleased to find out today that Travis, Lambo, James, Arthur and the rest of the team at Knowledgehook have released version 1 of their Professional Learning Communities (PLC) Tool to assist district math leaders, administrators and school math leads to easily and more effectively plan their professional learning sessions.

Here’s a quick overview with some screenshots to give you a heads up on what you can expect from this really valuable resource.

Upon logging into Knowledgehook with your school and/or district login, you’ll arrive at the Administrator Dashboard where you will see a graph showing usage for Gameshows (teacher paced / whole group activities) and Missions (student paced / independent activities).

Knowledgehook Professional Learning Tools.002 District Insights

Underneath, we see the number of student gaps triggered, custom questions created by teachers and Mathalon medals earned thus far as well as the option to view Trending Gaps across the district:

Knowledgehook Professional Learning Tools.003 Student Gaps and Stats

My suggestion to the Knowledgehook team is to also show how many students have remediated the gaps in their learning after reattempting problems in Missions. I have also suggested that the dashboard “auto-magically” suggests which gaps should be on our “short list” to focus on for informing our professional development planning with some sort of confidence scale indicating how confident the algorithm is in their recommendation.

From the dashboard, we can also see the usage from each school as well as gaps triggered in those specific schools.

Knowledgehook Professional Learning Tools.004 School Stats

When we explore Top Student Gaps in the district, we can explore – grade by grade – the top 3 gaps:

Knowledgehook Professional Learning Tools.005 Top Student Gaps

Upon clicking on “View Teacher Support”, we can access instructional guidance including “Math Background” and “Remediation” documents which are downloadable PDF files:

Knowledgehook Professional Learning Tools.006 Teacher Professional Learning Supports

When I clicked on “Math Background”, I got the following document which has a great background around Representing Whole Numbers to 10 000:

Knowledgehook Professional Learning Tools.007 Math Background Guide

Knowledgehook Professional Learning Tools.008 Math Background Guide

The 6-page math background document outlined:

  • Why some schools struggle with representing whole numbers to 10 000

  • A background of the base ten place value system

  • Useful models that can help students develop number sense
  • Strategies for comparing whole numbers (and decimals)

  • Glossary

In the remediation document, you are presented with a 23-page document that highlights:

  • Common misconceptions and what you can do about it

    Knowledgehook Professional Learning Tools.009 Remediation Guide

  • Remediation questions
 and solutions

    Knowledgehook Professional Learning Tools.011 Remediation Questions and Solutions

  • A Teacher Guide

    Knowledgehook Professional Learning Tools.010 Remediation Guide

  • Black Line Masters including blank and scaled number lines, place value charts and more.

Another new option includes the Administrator PLC Tool:

Knowledgehook Professional Learning Tools.012 PLC Planning Tools and Polls

It is listed currently as “PLC Polls” where administrators can choose PLC content with teacher input (hence the poll) and they can optionally use their PLC Guide to organize how the learning will take place.

By Creating a Poll, we are actually creating a PLC Plan.

Knowledgehook Professional Learning Tools.013 Create a PLC Button

Here, we select all the different grades we are working with in the PLC. Let’s say it is a grade 4, grade 5 and grade 6 PLC. After checking those grades and clicking “NEXT”, the administrator can choose as many topics as he/she would like to offer for possible learning during this PLC cycle.

Knowledgehook Professional Learning Tools.014 Select Grades

This school has been focusing primarily on fractions thus far, so the administrator decides to keep only topics related to fractions as options.

Knowledgehook Professional Learning Tools.015 Select Topics

Then, we fill out some details about the PLC including a potential format in order to receive a structured guide for how we might organize the learning, select the date of the first PLC and add a note for teachers.

Knowledgehook Professional Learning Tools.016 Details of PLC

You’ll then get to preview the PLC Poll and CREATE IT!

Knowledgehook Professional Learning Tools.017 Preview PLC Poll

Then, you’re provided with a link you can share with your staff members so they can share their thoughts on topics they would be interested in learning more about.

Knowledgehook Professional Learning Tools.018 Share the PLC Poll

Alternatively, you can opt not to send out the link and just access the PLC content if you are comfortable selecting the topic or if the group has had a verbal conversation and you’ve all agreed on a specific topic.

Then, you can click on “VIEW” to see the poll results as well as access the content for your PLC:

Knowledgehook Professional Learning Tools.019 PLC Planning Dashboard

You’ll be able to see who has voted and finalize which topics you’d like to explore. Note that you don’t need any votes in order to select the topics. Just hit “choose your final topics” to close the poll and select.

If I’d like to focus on comparing fractions visually and numerically, then I would select those topics and hit “Submit your Final Topics (2)”.

Knowledgehook Professional Learning Tools.020 Select Topics for PLC

You are then provided with a sample email with links to content that teachers can access to bring to the PLC or even to read prior to the PLC.

Knowledgehook Professional Learning Tools.021 Share Resources for PLC

On the PLC Polls page, I can click the 3 dots to the far right of the desired Poll and select “PLC Resources” to access the resources for that PLC and the facilitation guide.

Overall, I was really impressed with version 1 of the PLC Tool. It sounds like there will be many more features added as they continue developing it out further.

If you’re interested in trying out the FREE Knowledgehook Gameshow tool, click here.

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Year In Review: 2017 https://tapintoteenminds.com/year-in-review-2017/ https://tapintoteenminds.com/year-in-review-2017/#respond Sun, 31 Dec 2017 18:53:31 +0000 http://tapintoteenminds.com/?p=19785 Another post from Tap Into Teen Minds. Enjoy!

Year in Review 2017: Looking Back and Planning Forward. This past year has been exciting with over 675,000 pageviews! Let's look at the top content accessed

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Looking Back and Planning Forward

Year In Review Featured Image

What an amazing year 2017 has been!

For those of you who have been with me since the beginning, you likely remember this blog as a place to share my learning from a Teacher Learning and Leadership Program (TLLP) project where I went paperless in my math classroom.

Boy, have I learned a ton since 2012.

Over time, this blog has undergone a complete transformation focused on technology based content to specializing in supporting teachers interested in building their K-12 math content proficiency and pedagogical knowledge.

When I look back to the numbers from 2012, I remember being elated to see that I had just over 38,000 pageviews in that first year of blogging. At the end of 2017, I’m shocked to see that the website has had over 675,000 pageviews from educators like yourself all around the world!

Let’s take a quick look back to some of the most popular 3 act math tasks, blog posts and math visual videos from this past year and I’ll leave you with my reflections from 2017 as well as what you can expect for 2018.

Top 3 Act Math Tasks for 2017

1. Donut Delight

Krispy Kreme Donut Delight 3 Act Math Task - Primary Sequel Act 1 - How Many Donuts in 3 Boxes

Donut Delight was originally crafted to provide an opportunity to spark curiosity around multiplication and fuel sense making around different strategies involving concrete manipulatives, visual representations and finally, standard algorithms.

This task has since expanded in both directions; stretching backwards to helping primary students who are working on early multiplication as well as pushing forwards to providing opportunities for students working on building a conceptual understanding of division and later, proportional reasoning.

Access Task

2. Airplane Problem

Airplane Task - Trip to Toronto - Act 2 Image 1

This task was created to provide an opportunity to implicitly (then explicitly) introduce arrays when working with early multiplication, however many teachers are using it as a way to access students spatial reasoning skills by conceptually subitizing, using strategies such as skip counting and introducing the distributive property by “splitting the array”.

Teachers in later grades can use this task as they introduce order of operations as well as algebraic expressions and equations.

Access Task

3. Gimme a Break

Fraction as Operator - Gimme a Break - What Operation Could It Be?

Gimme a Break is a task that was intended to give students a more contextual situation and visual experience to tackle the idea of operating on fractions. This problem starts with a really low floor by using unit fractions like 1 one fourth and builds to doing some multiplication and even division of fractions.

Be sure to give this one a look and modify to suit the needs of your students.

Access Task

Top 3 Blog Posts for 2017

1. Japanese Multiplication

Why Japanese Multiplication Works - Using Lines to Multiply Is Not a Math Trick

Earlier this year, I had been seeing a Facebook post going around showing how people in Japan supposedly multiply using sticks. The video made it appear as though this was some sort of magic trick. Anyone who reads this blog knows that I don’t like leaving people to believe that math is just a bunch of tricks, so I tried to figure out why it works.

Turns out, the reason it works is fairly obvious when we explore multiplication concretely using base 10 blocks as a starting point.

Read More

2. Counting and Quantity Principles

Counting Principles - Principles of Counting and Quantity Featured Image

Spending so much time working with Kindergarten and primary math teachers has opened my eyes to how important early development of counting and quantity for our young children really is. Having come from the secondary world, I had no idea how complex the ideas behind counting and quantity really are.

This post summarizes 10 principles with visual animations that are really important for students to build their counting skills and their understanding of quantity in general. Give it a read.

Read More

3. Progression of Multiplication

Student Generated Algorithms - Conceptual Algorithm for Multiplication

The third most popular post on the blog is a continuation from where we leave off with unitizing in the counting and quantity post and building into early multiplication, through the standard algorithm as well as how we can leverage these important skills in grade 8 and 9.

Check it out!

Read More

Top 3 Videos & Math Visual Animations for 2017

1. Visualizing the Volume of a Sphere Formula


In this video, we show the summary of a 3 act task called Cones and Spheres where we learn that a sphere can hold twice the volume of a cone with the same radius and height.

From there, we take the formula for volume of a cone and explore what happens when we double it and simplify. Definitely a fun way to go about introducing the Volume of a Sphere instead of just writing down the formula like I did (regretfully) for the majority of my career.

This video had over 74,000 views this year alone and has over 160,000 views total.

Read More

2. OAME 2017 Ignite Speech: The Beauty of Elementary Mathematics


I really had a blast focusing most of my attention on K-8 mathematics over the past 18 months. From all of that learning, I tried to summarize it all in a 5-minute Ignite Speech.

Read More

3. Visualizing the Volume of a Triangular Prism


Finally, another visualization where I try to help conceptually develop the formula for volume of a triangular prism. Nothing super fancy here, but it did receive over 11,000 views in 2017.

Read More

2017 Reflections and Plans for the Year Ahead

Not only did I prepare this Year in Review post to give you a quick summary of some of the big ideas shared over the past 365 days, but also to give me a reason to go back and analyze what content provided the most value for the most people. For those who know me personally, you’d probably agree that I have many ideas and I can sometimes struggle to decide what is most important to focus on.

While I’ve learned more than I can share in a single blog post about myself and my own mathematical journey, I’d like to mention a couple goals for this website for the year ahead.

Goal #1: Be less of a perfectionist.

With two growing children and a very demanding position as K-12 Math Consultant with my district, I have less time to commit to sharing online. Rather than posting less, I want to make it a priority to try avoiding some of the perfectionist qualities that create more stress and anxiety than productive content.

Goal #2: Spend more time fuelling sense making.

As I mentioned in a recent post, I have committed much of the past 5 years blogging focusing on sparking curiosity alone. I now know that a ton of effort can go wasted if we just get students interested in a problem without having a solid plan to help them build their conceptual understanding around the learning goal for that day.

Based on the top tasks, posts and videos from this year, it is clear that there is a thirst for more of this focus.

Goal #3: Build a more structured way to share knowledge.

While my blog has over 250 posts, 52 of my own 3 act math tasks and a ton of Ontario specific course resources, I can see how it would be difficult for anyone to know where to start. I’d like to commit to creating a course that helps summarize all of what I’ve shared thus far and add the piles of content that sits dormant on my hard drive for more to benefit from. I’d love to hear your feedback on how I might be able to accomplish this moving forward to provide the most value possible for my colleagues in the math education community.

With these 3 goals in mind, I am excited to continue learning as we head into 2018 together! 

Here’s wishing that you have a Happy New Year with your Family and Friends.

We’ll be in touch soon!


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Using Tasks to Teach Lessons https://tapintoteenminds.com/tasks-teach-lessons/ https://tapintoteenminds.com/tasks-teach-lessons/#respond Wed, 27 Dec 2017 03:58:44 +0000 http://tapintoteenminds.com/?p=19724 Another post from Tap Into Teen Minds. Enjoy!

Here's how I went from teaching my students lessons so they could do tasks to using tasks that teach students lessons about mathematics.

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Instead of Teaching Lessons to Do Tasks

Using Tasks to Teach Lessons - Featured Image

Over the past 5 years, I’ve been exploring the use of Dan Meyer’s 3 Act Math Task approach in my math classroom and share many of my own tasks when facilitating workshops. After participants experience these tasks in the role of the student, they quickly understand why 3 act math tasks are useful. After their own curiosity is sparked, it would seem reasonable that this type of task protocol would also likely spark curiosity in their students as well. However, what is less obvious to teachers is when they might use a 3 act math task in their classroom.

For some, they want to know when in the unit they should use 3 act math tasks?

  • Do I use a 3 act math task once a unit?
  • Once a week?
  • Everyday?

Then, after deciding when in the unit to use a 3 act math task, the question now becomes when in the lesson should I use the task?

  • Do we start with the task at the start of class?
  • In the middle after I teach the lesson?
  • At the end after I’ve given enough examples?

Because we are all human and humans prefer when decisions in life are black and white, you might be sad to learn that the answer to the above questions is almost always: “it depends”. Since all teachers are unique individuals – just like our students – that means we could (and probably should) have our own thoughts and beliefs around how our own perfect lesson might be delivered.

That said, my intention here is to share some of the ideas that have been developed collaboratively with Jon Orr over the past couple of years around how the structure of our math lessons have changed over time, often involving the use of a 3 act math-style task. Luckily for Jon and I, we were fortunate to have crossed paths at a time when we were both just starting to shift our practice from a largely teacher directed lesson where we would teach at our students to what we might consider more of a teacher facilitated experience where students were at the centre of the learning. We are both very cognizant that this transformation is still a work in progress; however, I’d like to share some of our thinking thus far.

Teaching Lessons to Do Tasks

For the first 7 years or so of teaching, my lessons looked a whole lot like the way I remember math class from my K-12 experience:

  1. Take Up The Homework: to ensure everybody “got it”
  2. “Teach” the Lesson: to give definitions, rules, formulae, procedures and algorithms
  3. Give Examples: in order to show the tips, tricks and common misconceptions
  4. Assign Practice Problems: to ensure they would be ready for the next day

I would spend hours each evening planning these lessons in order to feel like I was going to give my students the best chance at them succeeding with that concept the next day.

Know what happened?

Some kids still didn’t “get it”.

Don’t get me wrong, I always had a group of students who were well on their way, but I think they would have been fine regardless of how I delivered the lesson.

It was the group of students who “didn’t get it” that I was concerned about.

How could I reach them?

Trying to Use 3 Act Math Tasks in a Teacher Directed Lesson

It wasn’t until I came across Dan Meyer and 3 act math tasks that I began shifting my thinking about how I delivered my math lesson. You’ll notice that in the previous sentence I intentionally avoided using the word “taught”, because I now know that I can’t “teach” my students math, but rather create the conditions where students can construct their understanding of the learning goal I have set out for them each day. This is where I see rich tasks like those using a 3 act math structure can be extremely helpful.

When I first began using 3 act math tasks, I thought that these tasks could only be used after I “taught” students everything they needed to solve the problem. In the first couple of years, this would have been at the end of a unit – maybe on review day – and I thought that I only had “time” to use 1 or 2 per unit of study. Despite the large amount of time and effort I put into seeking out these tasks, planning how I would “fit them in” and figuring out how to best deliver them in class, the response from students wasn’t much better than that of any old task I would typically use from the textbook.

Using Tasks to Teach Lessons - Unit Plan In The Past

They would be intrigued initially by the problem, but when it came to “doing the math”, the classroom vibe reverted back to the lethargic state we would sadly deem as normal. After the curiosity that was sparked during act 1 had fizzled, we were back to students who believed that they couldn’t get started without a significant amount of my guidance and scaffolding.

I know some of the reasons the tasks flopped had to do with poor delivery; I wasn’t very smooth due to my lack of practice and I was also missing some key elements like giving students time to notice and wonder. However, I now realize that the biggest problem I created in my math class was my pre-teaching of all the math throughout the unit and waiting to ask students to do any of thinking until review day. By then, students were lost in a sea of disjointed mathematical ideas, rules, formulae, steps and procedures that they hadn’t yet conceptualized because I hadn’t provided them with the opportunity to construct that understanding.

Doing Tasks to Teach Lessons

Now, I’ve come to realize that I can use rich tasks delivered in 3 acts to spark curiosity in order to fuel sense making around a new mathematical idea. Rather than pre-teaching all of the math, let’s use tasks to create a need for the math.

Using Tasks to Teach Lessons - Unit Plan Now

While I know this might seem really scary to some teachers (especially those who tend to teach in a similar fashion to how most of us learned), but I’m going to argue that we can teach math concepts through the use of a really interesting task that sparks curiosity and opens the door to fuel sense making as we attempt to connect prior knowledge to new learning.

The best part is that most 3 act math tasks can be used to fuel the sense making of many different mathematical ideas.

Consider the act 1 video from the Airplane Problem, for example:

Not only does the act 1 video of this task spark curiosity and generate some great discussion, but it also opens the door for getting at mathematical ideas including (but not limited to):

  • subitizing,
  • early multiplication using arrays,
  • distributive property,
  • order of operations,
  • algebraic expressions,
  • and many more!

By avoiding the urge to pre-teach all of the most efficient strategies we math teachers believe students should know how to do, we allow for students to use their prior knowledge as a way to help us assess where they are and where our teaching of the new learning goal should begin that day while consolidating the task. After the consolidation is when I can then shift into teacher directed mode if necessary to address any misconceptions, specifically target gaps in prior knowledge, and build on student solutions to press them for deeper understanding and more efficient and/or effective strategies.

Using Tasks to Teach Lessons - Tasks Inquiry Connections Consolidate

So When Should I Use a 3 Act Math Task?

So while I think the answer to “when should I use a 3 act math task” is still “it depends”, I truly believe we should at least consider using a 3 act math task approach to introducing new mathematical ideas as often as possible. For some, that might mean once a week, while for others that could be once a day; it really does depend. I might also recommend changing your definition of what a 3 act math task is and what makes them so great. For me, the key is finding ways to spark curiosity with tasks as a means to fuel sense making around a new mathematical idea. These really interesting tasks can definitely be your typical 3 act task with a great act 1 video and act 3 “solution” video, but I’m learning more and more each day that the video itself is not what makes them awesome.

It’s so much more than that.

Check back to the blog for more on this topic as we dive deeper in future posts.

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Why I Ask Students to Notice and Wonder https://tapintoteenminds.com/notice-wonder/ https://tapintoteenminds.com/notice-wonder/#respond Mon, 13 Nov 2017 21:16:02 +0000 http://tapintoteenminds.com/?p=19655 Another post from Tap Into Teen Minds. Enjoy!

Recently, Jon Orr and I received some descriptive feedback from James Francis from Knowledgehook after watching us co-present a workshop titled “Making Math Moments That Matter” at the GECDSB Math Symposium. After sharing some of the pieces he really enjoyed, he also shared some constructive criticism: What I personally didn’t enjoy was the really general […]

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Another post from Tap Into Teen Minds. Enjoy!

Why I Ask Students to Notice and Wonder - Featured Image

Recently, Jon Orr and I received some descriptive feedback from James Francis from Knowledgehook after watching us co-present a workshop titled “Making Math Moments That Matter” at the GECDSB Math Symposium. After sharing some of the pieces he really enjoyed, he also shared some constructive criticism:

What I personally didn’t enjoy was the really general “what did you wonder” questioning that I have experienced in other workshops as well. I feel like if a teacher asked me to notice and wonder, I would be annoyed knowing that it is very likely this task will have nothing to do with what I come up with, so why waste my energy? When people ask for your opinion and they don’t do anything with it, they might become resentful that you would even ask in the first place.

If the idea of “Notice and Wonder” is new to you, check out Annie Fetter from the Math Forum who has done a great job developing this idea and sharing it with the math world.

This isn’t the first time I’ve had workshop participants question the utility of asking students to notice and wonder. Sometimes, I can see a few eyes roll and every now and again I come across some who are reluctant to participate in this portion of the task. However, I feel that this portion of the lesson can often make or break a task. Let’s explore why.

Building Trust and Confidence Through Discourse

If you’ve ever been to workshops led by Jon and I, we make a significant effort to get participants talking as much as possible in non-threatening situations just as we would when working with students. For example, in this past workshop, we asked the group to think about memorable moments in their lives and the math moments they remember from their educational experience as a student in order to share with the group. Taking the time early on in a math lesson for students to talk and share their thoughts where the stakes are low can be helpful to build trust and confidence, while also showing them that we value their voice regardless of their ranking in the invisible – yet very apparent – math class hierarchy. A well led notice and wonder discussion can really go a long way to creating a classroom of discourse that will hopefully over time, develop into mathematical discourse.

Sparking Curiosity

Not only does asking students to notice and wonder give them an opportunity to have a non-threatening discussion with their peers, but it also helps to feed their natural curious mind. I will never forget the first couple of years attempting to use Dan Meyer-style 3 act math tasks in my classroom and how often I felt like the lessons were a flop. What I eventually realized was that I didn’t take enough time to spark the curiosity in my students by developing the storyline of the problem. After taking much time to reflect on what my lessons were missing, I realized that I wasn’t giving my students a reason to get excited about the task or give an opportunity to engage in any thinking until they were ready to actually solve the problem. They knew that I was going to show some sort of video or photo and I would then tell them what to do next. When we ask students to notice and wonder, we are asking them to think, discuss and share their thinking which builds more interest and anticipation for more. And while the teacher should always have a specific direction in mind for where the learning will lead, we can still make each student feel like a contributor to the class discussion and the direction of their learning by writing down their noticings and wonderings for possible extensions and for future lessons.

Notice and Wonder Challenges

That said, asking students to notice and wonder isn’t something all students will enjoy at first. Some have said they “feel silly” or that “this is stupid” likely because they aren’t accustomed to being involved in the development of a problem and thus, they aren’t quite sure what they are supposed to do. However, I think that this temporary struggle can be a good thing. One of the reasons I want students to notice and wonder when they think about mathematical situations is so they aren’t so dependent on me telling them everything they are supposed to see or do in math class. Over time, many students learn to enjoy the process however like in other areas of life, some may not. An observation I have made over time is that I often find that my “go-getter” students are the largest group of students who hold out the longest on the notice and wonder – much like workshop participants who dislike the process – because they just want to get to the point. However, if you were to watch a movie or read a book that jumps straight to the conclusion, you’d be pretty let down. We have come to expect that sort of uninspiring and emotionless experience in math class and it shouldn’t surprise me when students push back when I try to push them to get involved in the development of the problem.

Our Unconscious Bias

Something Jon and I have discussed in the past is about how most teachers would likely fit in the “go getter” category since we were likely the students who understood the game of school and specifically, how to succeed in math class. It can be easy for us to believe that all students think and feel the same way we did in the math classroom. However, the reality is that many students do not feel as comfortable or confident as many of their teachers may have when they were in math class. When our experiences learning math are very different than that of many of the students in our classroom, it is easy for us to develop an unconscious bias. This might influence our thinking around whether or not there is a need to create non-threatening opportunities for students to talk and discuss in math class.

Interestingly enough, it is not uncommon for those who oppose the notice and wonder portion of a lesson to also become uncomfortable making predictions when required information is withheld. For example, if I ask a group to make a prediction about how many passenger seats there are in the plane below, some get anxious and a bit scared to throw out a prediction that may be way off despite the fact that they don’t have enough information.

Why I Ask Students to Notice and Wonder - Airplane Task

While I don’t have a definitive answer as to why the high achievers in my class most commonly tried to side-step the notice, wonder and predicting portions of the lesson, my hypothesis is that this process may be perceived as a threat – either consciously or unconsciously – to their position in the math class hierarchy. By no means is it my intent to make any group of students feel uncomfortable, but I do believe that this protocol assists in the levelling of the playing field. By providing more opportunities for all students to participate and feel as though they have something valuable to offer the group, we are taking steps to remove the math class hierarchy and build a learning environment that is equitable for all students.

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Exploding Dots for Global Math Week https://tapintoteenminds.com/exploding-dots/ https://tapintoteenminds.com/exploding-dots/#respond Fri, 06 Oct 2017 02:32:53 +0000 http://tapintoteenminds.com/?p=19601 Another post from Tap Into Teen Minds. Enjoy!

Join our mission to engage 1,000,000 students around the globe in a joyous, uplifting mathematical experience with Exploding Dots!

The post Exploding Dots for Global Math Week appeared first on Tap Into Teen Minds.

Another post from Tap Into Teen Minds. Enjoy!

Let’s Get Over 1 Million Students Engaging in Joyous Math!

Exploding Dots - Global Math Week - October 10th, 2017

If you’ve ever checked out my 3 act math tasks, you will quickly notice that I am all about sparking curiosity in order to fuel sense making for our students. Well, Exploding Dots nails the curiosity AND sense making pieces of the puzzle and with Global Math Week coming up next week, what better time than now to jump right in!

It was only a few months ago when I really started exploring this idea called Exploding Dots, but I was immediately intrigued. Since then, I have fallen in love with this amazing story of mathematics that James Tanton and The Global Math Project are hoping to spread to over 1 Million students around the globe.

Why not join the rest of the world next week for a 15 minute exploration or days of math fun with Exploding Dots. Best of all, there are full technology, low technology and even no technology versions to enjoy! Registration is simple because it only involves a pledge of participation and you can do it in under a minute here.

Enough of me pitching this “joyous and uplifting mathematics experience for all.” Let’s get your feet wet!

DISCLAIMER: READING ON IS LIKE SPOILING THE END TO A MOVIE. I highly recommend just jumping into the first Exploding Dots activity (called an “Island”) instead of reading on. However, if you aren’t going to take the time to give it a shot, I’d rather you read in order to share this great experience with your students.

Start By Sparking Curiosity

Watch this video.

In the “silent solution” style video, you are exposed to the “2-to-1” machine (written 1 <-- 2) and are left hanging at the end when they ask you to figure out how many dots would be required in the rightmost box to have a code of 10011.

Exploding Dots - Global Math Week - How Many Dots for 10011?

So. Give it some thought. Maybe draw it out?

It’s never too late to dive into Island 1 and see if playing with the “1 <– 2” machine can help construct some understanding AND I promise you’ll have fun doing it.

For those of you who still won’t go and just dive in to play with the Exploding Dots app, let’s have a closer look at the first island, Mechania.

Island #1 – Mechania: Exploring The Dot Machines

Global Math Week - Exploding Dots Island Summary - Island 1 Mechania

I’d like to share a quick video at the beginning of the first Island, called “Mechania” to assist you with your exploration.

After watching the video, the Exploding Dots app has you jump into the action asking you questions that press you for understanding. It might seem tricky at first, but remember: this activity isn’t about finding the right answer; it is about sparking curiosity, using strategic competence and building a productive disposition towards mathematics.

Here’s a look at the first two questions:

Exploding Dots - Mechania Island 1 <-- 2 Machine Q1 and Q2 Animation

Pretty simple, right?

The Exploding Dots app starts with a very low floor and has an extremely high ceiling.

Here are the next two questions to explore with the 1 <– 2 machine:

Exploding Dots - Mechania Island 1 <-- 2 Machine Q3 and Q4 Animation

Before you know it, you’ll be tacking problems like this one:

Exploding Dots - Mechania Island 1 <-- 2 Machine Q7 Animation

Wait a second. Didn’t the question posed in the video at the beginning of this post look an awful lot like what we just did in the previous question?

Here it is again:

Exploding Dots - Global Math Week - How Many Dots for 10011?

Do you think you can try it now? Go ahead. Take your time.

Heck, why not try the 1 <-- 2 machine in the Exploding Dots app to help you?

Here’s a strategy someone might use to answer the question:

Exploding Dots - Mechania Island 1 <-- 2 Machine Minds On Animation

Can you find a more efficient way?

What’s The Big Deal About the “1 <-- 2" Machine?

Aside from the fact that curiosity is surging, strategic competence is spewing and productive disposition is oozing at all time levels, you could consider exploring how the 1 <– 2 machine helps computers work. While some may have picked up that the 1 <– 2 machine is the same language that a computer uses, many others may not. The base 2 number system – a series of “on” and “off” digits – is extremely important in computer science. It is the language computers speak!

While you might think the fun is over, it’s just begun as you continue through this activity to explore the 1 <-- 3 machine as well as the 1 <-- 10 machine. Something you may notice rather quickly with the 1 <– 10 machine is that it is the “base 10” number system; the standard number system we use in our everyday lives.

A Quick Look At Each Activity

Once you arrive at the Exploding Dots landing page, you’ll notice that there are quite a few activities (called “Islands”) for you to explore. I believe the best experience would involve students diving in from the beginning, however it might be useful for you to see where students will eventually land as they traverse through each of the Islands.

Global Math Week - Exploding Dots Island Summary - All 6 Islands

Since we explored the first island, Mechania in detail above, let’s take a quick look at each of the remaining 5 islands that are active on the Exploding Dots website to expose you to the other mathematical connections that can be made from Kindergarten through Grade 12.

Island #2 – Insighto: Conceptualizing The Dot Machines

After students explore and discover in Mechania, students will be brought to Insighto to begin unpacking the conceptual pieces that make the different dot machines work.

Global Math Week - Exploding Dots Island Summary - Island 2 Insighto

Island #3 – Arithmos: Addition and Multiplication

After building some of the conceptual understanding from the Insighto Island, students will be brought to Arithmos to begin applying our understanding of the 1 <-- 10 machine to addition and later, multiplication. The logic students can build using the dot machines really helps them grasp an understanding of place value and why standard algorithms work.

Have a look at a sample of addition:

Global Math Week - Exploding Dots Island Summary - Island 3 Arithmos Addition

And later on, multiplication:

Global Math Week - Exploding Dots Island Summary - Island 4 Arithmos Multiplication

As students work through the Island of Arithmos, they will eventually arrive at commonly used or “standard” algorithms for addition and multiplication wrapped up with a nice bow of conceptual understanding.

Island #4 – Antidotia: Integers and Subtraction

What? Integers before subtraction?

YES! When students arrive on the Island of Antidotia, they will immediately be introduced to “dots” and “antidots” which intuitively build an understanding of integers prior to looking at subtraction. This is because the concrete and visual representations for both will look identical.

Global Math Week - Exploding Dots Island Summary - Island 5 Antidotia - Integers and Subtraction

Students will eventually be led to standard algorithms we commonly see in elementary school for subtraction while also building an understanding of integers and the zero principle.

Island #5 – Obelus: Division

Get ready to look at division in a brand new light. Imagine possibly getting to a point where you could pretty accurately divide large numbers by focusing just on digits in each place value column.

Global Math Week - Exploding Dots Island Summary - Island 5 Obelus - Division

While I love the entire exploding dots experience, I think the conceptual understanding this particular activity builds in students around division alone is worth the time and effort!

Island #6 – Eks: Multiplication and Division in Any Base

Ok, secondary math teachers. This is the island you’ve been waiting for. This is the reason why you put in the time and effort with your senior secondary math students with the Exploding Dots experience.

Have you ever imagined multiplying or dividing in any base in a way that was not only procedurally possible, but also conceptually understandable for students?

Well, here’s your opportunity. Students can play with big, long, BEAUTIFUL polynomials and they can quickly discover how to divide, handle remainders including The Remainder Theorem and multiplication of polynomials.

Global Math Week - Exploding Dots Island Summary - Island 6 Eks - Division in Any Base

Ready to explore?

Join Us During Global Math Week!

If this blog post hasn’t inspired you to join us during Global Math Week with Exploding Dots, then clearly I have done you wrong.

Take the time to register your class and spread the word with your colleagues.

Together, we can reach over 1,000,000 students!

Or, 0111011100110101100101000000000 students if we use the 1 <– 2 machine.

Or, 2120200200021010001 students if we use the 1 <– 3 machine.

Or, 323212230220000 students if we use the 1 <– 4 machine.


Ok, I think you get the point.

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The Progression of Division https://tapintoteenminds.com/progression-of-division/ https://tapintoteenminds.com/progression-of-division/#respond Wed, 20 Sep 2017 01:52:52 +0000 http://tapintoteenminds.com/?p=19555 Another post from Tap Into Teen Minds. Enjoy!

Learn the Progression of Division where we will explore fair sharing, arrays, area models, flexible division, the long division algorithm and algebra.

The post The Progression of Division appeared first on Tap Into Teen Minds.

Another post from Tap Into Teen Minds. Enjoy!

Division From Fair Sharing to Long Division and Beyond

Over the past school year, I have had an opportunity to work with a great number of K to 8 teachers in my district with a focus on number sense and numeration. As a secondary math teacher turned K-12 math consultant, I’ve had to spend a significant amount of time tearing apart key number sense topics including the operations. While I often hear teachers concerned about multiplication skills of their students, an operation that doesn’t come up too often in discussion is division. However, what I found interesting this year was how much of a struggle it was for teachers to attempt representing division from a conceptual standpoint instead of simply relying on a procedure. After spending quite some time diving into division independently as well as collaboratively with educators through workshops, I will attempt to share what I believe to be some pretty important pieces along the progression of division.


This is by no means a complete progression and would welcome other pieces in the comments that I could add in to build on this post over time. I have found thinking about these pieces as pivotal in my own understanding of how division is constructed over time, but will likely continue changing as my own understanding deepens.

I would recommend first exploring the progression of multiplication prior to jumping into this post focusing on the opposite operation, division.

Unitizing: Creating Groups

Before we begin diving into division, I feel it is important for students to be very efficient with unitizing which I discuss in a separate post with counting principles.

To summarize, unitizing is:

Understanding that every quantity we measure is relative to another pre-measured group we call a unit. For example, our base ten place value system.

Before students can successfully unitize, they must be able to count via one-to-one correspondence. For example, a student successfully counting a group of items, one at a time.

Counting Principles - 5 - One to One Correspondence

After learning one-to-one correspondence and working on other principles of counting and quantity, teachers can begin encouraging students to skip count by 2s, 3s, and so on. This might be considered the beginnings of having students unitize implicitly. For example, by counting by 2s: 2, 4, 6, 8, and so on, students are counting up by a group larger than 1. Over time, students can begin counting the groups of 2 (or whatever unit they are skip counting by) with their fingers to really bring out unitizing explicitly.

Unitizing - Groups of 2 - 2 to 1 Correspondence

Or, maybe in groups of 3 (i.e.: 3-to-1 correspondance):

Unitizing - Groups of 3 - 3 to 1 Correspondence

This ability to create equal groups and keep track of the count is important for students to really begin their journey towards division.

Once students are able to count groups of 10, they are not only well on their way down the progression of division, but also on their journey to conceptually understanding our base ten place value system.

Unitizing - Groups of 10 - 10 to 1 Correspondence Base Ten Place Value

Creating Equal Groups By Fair Sharing

Something that comes quite natural to young children is the ability to fair share a group of items. For example, sharing a handful of candies between siblings isn’t something that is typically taught explicitly, but rather students develop this sense of fairness through play.

Prior to attempting to formalize division as an operator, students should have extensive experience fair sharing items amongst friends, both concretely (by sharing to real people like their classroom peers) and when ready, visually/pictorially (by sharing to groups organized on their desk, on paper or on a whiteboard).

An example of such fair sharing is given in the Ontario Grade 2 Mathematics Curriculum in the Number Sense and Numeration strand:

represent and explain, through investigation using concrete materials and drawings, division as the sharing of a quantity equally (e.g., “I can share 12 carrot sticks equally among 4 friends by giving each person 3 carrot sticks.”);

Fair Sharing 12 Carrots With 4 Friends One-To-One Correspondence

While most students will likely fair share the carrots through one-to-one correspondence (i.e.: grabbing one carrot at a time and giving it to one of the friends), something to note here is that they are engaging in an early form of repeated subtraction. In other words, the student is repeatedly subtracting one carrot from the total and then will count how many each friend has once they run out of carrots to share.

As students become more comfortable with the idea of fair sharing, they may begin unitizing the amount of carrots that they share fairly. For example, in this same situation, an experienced student may notice the large group of carrots and begin to share two carrots at a time. You can understand why skip counting backwards can be helpful in completing this task.

Fair Sharing 12 Carrots With 4 Friends Two-To-One Correspondence

Formalizing Division As An Operation

Once students are comfortable with fair sharing using repeated subtraction in units of 1 or more, we can begin formalizing this idea as an operation we call division.

In the Ontario curriculum, we begin formalizing division in Grade 3 using tools and strategies up to 49 ÷ 7:

Progresson of Division - Ontario Grade 3 Math Expectations for Division

And extend our work with division in Grade 4 to using tools and mental math strategies:

Progresson of Division - Ontario Grade 4 Math Expectations for Division

Two Types of Division: Partitive and Quotative

Something that is not stated explicitly in the Ontario mathematics curriculum and can easily be overlooked by elementary math teachers is that there are two types of division: partitive and quotative.

Partitive Division

Also Known As Fair Share or Sharing Division

The first type of division we will explore is called partitive division. This type of division is possibly the first type of division students intuitively experience when they are young by sharing a group of items with their friends as we did earlier by sharing 12 carrots amongst 4 friends. In other words, partitive division occurs when a scenario requires a student to divide a set of items into a given number of groups, where the number of items in each group is unknown.

To give another example, we could look to the following question from Alex Lawson’s book, What to Look For, where a student is asked to answer the following question:

You buy 15 goldfish. You are going to put them in 5 jars evenly. How many goldfish will you put in each jar?

Progresson of Division - Alex Lawson What to Look For Question - You Buy 15 Goldfish Partitive

Since the student knows how many jars he must divide the 15 fish into, students might fair share by units of 1 or more at a time until all the fish are shared.

Progression of Division - Partitive Division - Dividing 15 into 5 groups evenly

After viewing the visualization above of the student distributing the goldfish by assigning one fish to each bowl at a time, it becomes more obvious as to why we also call partitive division “fair sharing”.

Quotative Division

Also Known As Subtraction or Measurement Division

The second type of division we will explore is commonly known as quotative division. This type of division occurs when a scenario requires a student to divide a set of items into groups with a given amount in each group, where the number of groups is unknown.

To rearrange the previous question as an example of quotative, we might ask the student this question:

You buy 15 goldfish. You are going to put them in jars, with 3 in each jar. How many jars will you need?

Progresson of Division - Alex Lawson What to Look For Question - You Buy 15 Goldfish

Since a quotative division problem tells the student how many items should be in each group, it would seem reasonable to assume that the student would unitize the 15 goldfish into units of 3 until all of the goldfish have been used and then count the number of groups created. When a student completes this problem, possibly using cubes or square tiles to represent each of the goldfish, they are implicitly using repeated subtraction to take away 3 goldfish at a time from the total of 15.

Progression of Division - Quotative Division - Dividing 15 into groups of 3

The visualization here shows the student measuring groups of 3 fish and then repeatedly subtracting those groups from the set of 15 fish until there are none left. Hence the names “measured” and “subtracting” division.

Summarizing Partitive and Quotative Division

In summary:

Partitive Division

  • Number of groups is known; and,
  • Number in each group is unknown.

Quotative Division

  • Number in each group is known; and,
  • Number of groups is unknown.

Progression of Division - Summarizing Partitive and Quotative Division

While I personally am less concerned about teachers being able to name these two types as quotative and partitive, it is important that teachers are aware that two types exist to ensure that they are exposing their students to contexts that address both of these types.

Examples of Partitive and Quotative Without Context

When using division without context like fair sharing carrots or placing goldfish in jars evenly, the student can decide whether to use a partitive or quotative strategy based on what they are most comfortable with. While convenient, a potential pitfall may arise if students divide procedurally using only one type of division while ignorant to the fact that another type of division exists.

For example, if a student were to model 56 ÷ 8 using square tiles, a student could approach this using partitive division by fair sharing the 56 tiles evenly into 8 groups like this:

Progression of Division - Animation of Partitive Division - Fair Share 56 Divided By 8

Or, the student could choose to model this problem by using quotation division by repeatedly measuring out groups of 8 and subtracting them from the set of 56:

Progression of Division - Animation of Quotative Division - Repeated Subtraction or Measured Division - 56 Divided By 8

If students are given large quantities of straight calculation problems and only few contextual problems like the goldfish problem shared above, they may not build the necessary fluency to approach contextual problems successfully when they do arise.

Division With Arrays: Moving From Equal Group “Piles” to Rows and Columns

You’ll probably notice that regardless of which type of division students are using, they often make circular piles of the item they are working with. I believe this to be an important stage in the progression as it seems fairly intuitive to simply create your groups without ordering or organizing the items in each group.

As the dividends and divisors that students work with get larger, it can be helpful to think about organizing the items to help promote unitizing as well as building conceptual understanding and procedural fluency of division using arrays.

For example, if we look at an extreme case of dividing 120 ÷ 8 using individual square tiles, we can still identify partitive and quotative division:

Progresson of Division - Using an Array to Show Partitive and Quotative Division With Large Dividend and Divisor

While I don’t want students spending too much time trying to divide large dividends using square tiles, it could assist in showing students why base ten blocks are helpful.

Just like with multiplication, I think students should have a significant amount of experience dividing with arrays up to 81 ÷ 9 if you hope to help them conceptually understand division with dividends greater than 100.

Progresson of Division - 120 Divided By 8 in an Array

Not sure what is going on in the above image? Be sure to check out the progression of multiplication before going on where we explicitly address arrays and area models which we will be diving into pretty quickly here.

Division With Large Arrays: Base Ten Blocks

It doesn’t take long for students to become annoyed by trying to use square tiles to represent division with large dividends. Thankfully, as we saw in our Progression of Multiplication post, we can turn to base ten blocks to cut the hassle.

For example, if a student wants to model 120 ÷ 12, they could use 1 hundred flat and 2 ten rods to represent 120 and 1 ten rod and 2 units to represent 12. The representation would be similar to multiplication, except in this case, you have one factor (12) and the result (120) and you must find the unknown factor:

Progression of Division - Dividing With Base Ten Blocks 120 Divided By 12

It is easy to see the benefit of using base ten blocks when dividends get large. Instead of using 120 individual square tiles, we can use 3 base ten blocks to represent the same dividend. I intentionally selected a fairly easy example to begin with in order to allow students the opportunity to become familiar with dividing with concrete materials. While it might not seem obvious at first, dividing becomes increasingly difficult when more challenging examples are attempted.

Let’s try another example like 112 ÷ 8.

Your dividend will consist of 1 hundred flat, 1 ten rod, and 2 units, while your divisor consists of 8 units.

Progresson of Division - Dividing With Base Ten Blocks - 112 Divided By 8

As students are given opportunities to explore and discover, they will make observations and I would encourage them to discuss these observations with their peers to come up with rules. For example, some students come to realize that you must convert hundred flats to ten rods when working with a divisor with a value less than 10. Other students may come to realize that since multiplying two numbers yields a rectangle, that we must continue “trading down” the base ten materials until you can make a rectangle with the dividend with one factor being the divisor.

This experience might look something like this:

Progression of Division - Dividing With Base Ten Blocks 112 Divided By 8 Animated

While I will attempt to be as clear as possible in this post, it is important to note that students will need a significant amount of experience at each stage of the progression. Do not attempt to rush or you will find both you and your students frustrated.

Moving From Concrete to Digital Manipulatives

Due to a shortage of concrete base ten materials or to make life a bit easier, students may begin gravitating to digital base ten block manipulatives. While it might seem easier to just jump straight to digital manipulatives, I don’t recommend rushing to this stage as students should really have the opportunity to physically manipulate the base ten materials prior to moving to a digital alternative.

When students have had significant experience manipulating physical base ten materials, you might consider using the Number Pieces iOS app or the web-based version from Math Learning Center.

Progression of Division - Number Pieces App by Math Learning Center

While using a digital manipulative can be more efficient for a student who is experienced using physical base ten blocks to model division with 3 or more digit dividends, it can be a huge hinderance for students who have not been given an opportunity to build their conceptual understanding in this area.

Division With Base Ten Blocks: Increasing The Difficulty

When I deliver workshops specific to the progression of division, I find teachers quickly jump to the conclusion that dividing with base ten blocks is simply too difficult and unnecessary.

However, I believe quite the opposite.

I agree that using base ten blocks is definitely not the most efficient method a student should use when trying to divide two numbers, but they are very useful for building a conceptual understanding of division as well as a unique opportunity to build strategic competence by problem solving their way to a solution. Less obvious is the experience students are gaining around conversions when they trade in a hundred flat for 10 ten rods, a ten rod for 10 units as well as implicitly building the foundation for factoring quadratics – a grade 10 concept here in Ontario – all the way down in grade 5.

Let’s check out another example: 189 ÷ 9

Progression of Division - Dividing With Base Ten Blocks - 189 Divided By 9 Animated

Check out the conversions from hundreds, to tens, to units.

We could go ahead and rush to the long division algorithm, but why would we want to rob students of the opportunity to look at mathematics as a puzzle waiting to be solved?

Still want more practice?

Try 221 ÷ 13 using base ten blocks or Number Pieces app.

Once you’re done, see if your result looks something like this.

Connecting Arrays to Area Models: Distributive Property

Like anything we do in mathematics to build conceptual understanding, we don’t want the learning to stop there. Ultimately, we hope that the rich learning experiences we offer our students will begin to solidify by creating procedural fluency and automaticity. The next step on our way to the long division algorithm is the area model. This model is very useful when dividing and helps set us up for a clear connection to the long division algorithm, which is our end goal for this progression.

If we think about the distributive property from our multiplication post, you’ll remember that we could use this property by “splitting the array”. We can do much the same with division and it is convenient to do so by using open area models. In other words, rather than using square tiles or base ten material to represent an array for multiplication or division, we will use a not-to-scale rectangle in its place. A little less precise, but still gives us the same visual that an array can offer leaving our minds to visualize the rest.

Let’s have a look at 195 ÷ 15 using an area model to represent division.

Progresson of Division - Open Area Model for Division 195 Divided By 15

When drawing an area model for division, students are able to essentially unitize their own “chunks” to partitively (divide x items into 15 groups) or quotatively (divide into x groups of 15) approach this division problem.

In this case, I’ll give an example of a student who decides to approach this quotatively by repeatedly subtracting groups of 15 from 195 until running out of items:

Progresson of Division - Open Area Model for Division 195 Divided By 15 Quotative Repeated Subtraction

Wow, that isn’t a super efficient way to go about things. However, over time, your students may begin to notice more efficient approaches like the example below where a student notices that when there is 150 remaining, that is 10 times larger than 15.

Progresson of Division - 195 Divided By 15 More Efficiently

If we’ve taken away 3 groups of 15, then another 10 groups of 15, we know that we’ve taken away 13 groups of 15 total.

Progresson of Division - 195 Divided By 15 - 10 Groups of 15

Let’s try another one.

Have a look at a possible approach to solving 888 ÷ 24.

Progresson of Division - 888 Divided By 24 Using Area Model

Contextual Division Problems With Open Area Models

While I’m a huge advocate for using context in math class, I’ve kept things fairly contextless for the majority of this post. However, I want to explicitly show that whether there is context or not, these strategies to promote students conceptual understanding and procedural fluency with division are very helpful.

Let’s have a look at a problem with some context involving a pool.

A pool has a width of 14 m and an area of 700 metres-squared.

What is the length?

Progresson of Division - 700 Divided By 14 Pool Problem

When using open area models, you’ll find students will quickly jump on the use of friendly numbers like multiples of 10, for example.

Progresson of Division - 700 Divided By 14 Pool Problem Animated

You may notice that when I’m using area models, I’m using a symbolic approach to keep track of the repeated addition. That method is used in a number of different places with different names. In Ontario, the Guides to Effective Instruction call that strategy “Flexible Division”, since it is very similar to the long division algorithm, but puts the power in the hands of the student to select how many groups of the divisor to subtract with each iteration.

Eventually, students can opt to skip drawing the open area model and using what looks to be the long division algorithm or a variation like flexible division in order to solve division problems without a calculator.

Applying Division With Arrays and Area Models to Factoring in Algebra

As students enter grade 9 and 10, they will be offered an opportunity to put the conceptual understanding and strategic competence they have been developing through division with arrays and area models to use.

When common factoring, students will encounter problems like this one:

A pool has a width of 3 metres and an area of 12 times a number.

What is the length?

It could be helpful for students to create themselves an open area model as we did in the previous example:

Progresson of Division - Common Factoring With Open Area Model

When moving from working with multiplication and division to algebra, we rename our concrete materials from base ten blocks to algebra tiles. Instead of units, ten rods and hundred flats, we use very similar tools, but call them units, x-rods and x-squared flats. It should be noted that a hundred flat could also be called a “10-squared” flat.

So, instead of using ten rods like we use with base ten blocks, a student may opt to draw in “x-rods” representing the missing number, x. While there are many ways to approach this problem conceptually, I often see students approaching this additively by adding groups of 3 x-rods until reaching an area of 12 x-rods total. Students might also keep track of their work by using repeated subtraction (or flexible division) to the right of their area model:

Progresson of Division - Common Factoring With Open Area Model Animated

Factoring Quadratics With Algebra Tiles Is The Same As Dividing!

Recall our work using base ten blocks and arrays for division earlier in the progression. In our next example, we will look at a similar context using the dimensions and area of a pool to show how all that conceptual work back from grade 5 and 6 can be utilized in Grade 10 to factor both simple and complex trinomial quadratics.

In this example, we’ll explore the following:

A pool has an area of 3x^2 + 11x + 6.

What are the dimensions?

While many students in grade 10 struggle with the idea of factoring quadratics, they may not experience the same level of struggle if they have any experience multiplying and dividing with base ten blocks.

When factoring quadratics, students can simply grab the number of tiles that represent the quadratic they are factoring and then attempt to create a rectangle. Once they find a complete rectangle, they can quickly identify the factors that yield that area. If you can’t make a rectangle, then you know your quadratic cannot be factored with integer coefficients.

Let’s have a look:

Progresson of Division - Factoring Complex Trinomial Quadratics With Algebra Tiles

Factoring quadratics with algebra tiles is actually much easier than dividing with base ten blocks due to the fact that you are not required to convert from hundred flats to ten rods and ten rods to unit tiles! Who would have thought that grade 5 was tougher than grade 10?

Concreteness Fading: Start Concrete Before Abstract!

As I’ve mentioned in previous posts, there is definitely an argument for starting new ideas in mathematics with concrete manipulatives, slowly moving towards visual (or drawn) representations and finally to abstract representations that use symbols to represent the concrete. Although the name suggests that the concrete stage fades away over time, it is important to note that we should be returning to concrete manipulatives with every new layer of abstraction.

For example, when we first introduce division, we might be working with two digit dividends and single digit divisors. Concreteness fading for this idea might look something like this:

Progresson of Division - Concreteness Fading - Two Digit By One Digit Division

When progressing to three digit by one digit division, the stages of concreteness fading may look something like this:

Progresson of Division - Concreteness Fading - Three Digit By One Digit Division

Years later, when factoring complex trinomials in grade 10 academic math courses, the stages of concreteness fading might look something like this:

Progresson of Division - Concreteness Fading - Factoring Complex Trinomial Quadratics

So while this progression of division may not be “the” progression, I certainly hope it shines some light on how important understanding division conceptually through the use of concreteness fading is for promoting the development of a complete understanding for our students.

Interested in checking out some 3 act math tasks that can be used in conjunction with the progression of multiplication and division? Be sure to check out the tasks below:

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Avoiding Ineffective Uses of Technology With SAMR https://tapintoteenminds.com/ineffective-use-of-technology/ https://tapintoteenminds.com/ineffective-use-of-technology/#respond Sun, 13 Aug 2017 12:30:36 +0000 http://tapintoteenminds.com/?p=14333 Another post from Tap Into Teen Minds. Enjoy!

It's easy to forget why we integrate technology in our math class. Let's use the SAMR Model to plan our lessons using technology with purpose and intent.

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Another post from Tap Into Teen Minds. Enjoy!

While in Austin, Texas for iPadpalooza back in 2014, Tim Yenca (@mryenca) was kind enough to give me a lift to dinner with a group of presenters. After noticing a parking lot with a sign advertising $5 parking, we found ourselves a spot and made our way to pay the fee.

After looking around the lot for a payment machine, I ended up back at the sign near the entrance.

Have a look:

Public Parking Pay By Phone Sign

After taking a glance at this sign, I couldn’t help but shake my head.

Not only did the sign have over 45 words, but the process to pay for parking was long and tedious. Even the owner of the parking lot knew it was overwhelming enough to write that the “simple instructions” would take only 1 minute.

SAMR Parking Lot - Ineffective Uses of Technology - Follow Simple Instructions

I think we can all agree that cutting back on the amount of text on the sign would be a great idea, but I’m more concerned about the actual payment process for this parking lot. Not only does it require the customer to use their cell phones to send a text message, wait to receive a text message back, click on a link and then manually enter your credit card details on your smartphone, but there is actually a parking lot attendant there while you do all of this work.

SAMR Parking Lot - Ineffective Uses of Technology - Paying a Lot Attendant

Wait. I lied. There isn’t a single human being there who could make this process much easier by swiping your credit card or taking cash – there are TWO!

Public Parking Pay By Phone - Two Attendants

You can imagine how upsetting this could be to someone who is struggling to get through the “simple steps” as indicated on the sign. Luckily, I was travelling with friends who had United States cell phone plans because my Canadian cell phone roaming data charges would have made it cheaper for me to just get towed and pay the fee later.

So while this company has good intentions to use technology to make the payment process easier, they are experiencing the opposite result. Not only does this payment process limit revenue by restricting their customer base to only those with smartphones, but even smartphone users may possibly seek out a parking lot with easier payment options.

Good Tech Intentions; Poor Follow-Through

The parking lot experience reminded me a lot of what we often do in education when it comes to technology use. In Simon Sinek’s book, Start With Why and in summary during his TED Talk, How Great Leaders Inspire Action, he explains the importance of starting with why you are doing something rather than what or how you do it. In education, large budgets (how) are dedicated to the purchase of technology (what) prior to determining the reason (why) it is needed. In the case of the parking lot, the decisions of what and how to utilize technology as a tool for payment was made prior to determining why it was being considered in the first place. In the parking lot business, I’d have to assume most decisions are made with the goals of increasing revenue, decreasing expenses and possibly to increasing customer satisfaction. I think it is pretty clear to most of us that having a customer send a text message, receive a message back, click a link, enter your address and credit card details to finally pay for parking while employing two workers to do nothing but direct customers to instructions on a sign does nothing but hurt your business.

“Tech for Tech’s Sake”

I’m sharing this story as an example of how we can easily lose focus on why technology is so prevalent in the world today. We all see that technology is ever increasingly influencing how we live our lives each day including in education, however it is easy to forget why. When I encountered the long and tedious payment process for this particular parking lot, I immediately thought of how quickly I lost focus of why I had started integrating technology in my math class back in 2012. My initial goal was to use technology to more efficiently teach my math class and make the learning mathematics a richer experience for my students. However, there were many times when I would use technology and inevitably make the teaching and learning of mathematics more complex with little to no added benefit.

Using The SAMR Model

The SAMR Model - Enhancing and Transforming Math Lessons

It was only after a number of years teaching with iPads in my math classroom that I came across research by Ruben Puntadura which he summarized as the SAMR Model. According to his research, he suggests that we can use technology in education to either enhance or transform learning.

Furthermore, he claims that enhancing a lesson consists of either substituting what you are currently doing with technology or using the technology to augment your lesson, while technology use that is transformative consists of significantly modifying or completely redefining your instruction.

Let’s look at the SAMR Model more closely.

SAMR Model: Technology to Enhance Math Lessons

SAMR Model - Enhancement - Functional Improvement

Substitution – No Functional Improvement

When we use technology as a replacement for something we have always done previously, but have not gained any functional improvement, our technology use is considered to be a substitution. An example of this might include using iPads to read static handouts that would typically be given out in a workbook and are not accessible in any cloud based storage for access from other devices via the internet.

Augmentation – Functional Improvement

The most common stage I tend to find technology being used at in the classroom is the augmentation stage. In this stage, technology is being used to provide some functional improvement to how teaching and learning may have taken place without technology present. For example, using iPads or Chromebooks as a tool for students to create digital portfolios to share with their teachers and parents or a teacher re-creating content currently presented in black and white transparencies on an overhead projector to slides with full colour diagram and details. The teacher may also choose to share the content on the internet so students can access the information from anywhere on any device.

SAMR Model: Technology to Transform Math Lessons

SAMR Model - Transforming Math Lessons


The use of technology in math class is considered to be transformational when the tools are used to redesign tasks significantly. To introduce the idea of systems of linear equations for example, the teacher could use their smartphone to record a video of different groups of items being weighed on a scale and have kids use their intuition to figure out the weight of one of the items and then share their thinking to the class using a tool like Knowledgehook Gameshow.


Another stage of transformational technology use in math class occurs when students are able to engage in tasks that were previously inconceivable without the use of the technology. For example, instead of the teacher giving definitions, rules and procedures to explain distance-time graphs, she could create a video of herself walking and ask students to sketch a prediction of what the graph might look like on their device using a tool like Desmos Activity Builder or PearDeck. Then, the students could use their device to record a partner walking and then challenge classmates to draw the matching graph.

When Can Technology Impact Student Learning?

While using technology can be fun and engaging for teachers and students, what we really need to be aware of is when the use of technology can actually impact student learning outcomes.

Although using technology as a substitute or to augment your lessons might improve the workflow of your math class, research suggests that you shouldn’t expect to see any significant improvements to student learning outcomes since the actual teaching and learning remains relatively unchanged.

However, when teachers find ways to transform their lessons with technology by significantly redesigning tasks or engaging in activities that were previously inconceivable without the technology tools, there is a potential (not a guarantee) to impact student learning outcomes.

Technology to Enhance and Transform Your Lessons

Something important to note about the SAMR Model is that you are not stuck to any one stage for any given period of time. One teacher can use technology for one portion of the lesson as a substitute, while completely redefining how another portion of the lesson might have been delivered without technology. I believe the key for educators is to try to find opportunities to transform how they teach mathematics with technology when appropriate, while avoiding falling into the trap of forcing it in situations where we aren’t at least getting some functional improvement.

SAMR Parking Lot

The parking lot owner from the beginning of this post has to make very similar decisions to those that we must make as educators when it comes to using technology. We may have good intentions when we attempt to use technology in our business or in our classrooms, but it is easy to lose track of why we consider using a technology tool in the first place. If I can give any advice based on the lessons I’ve learned over the past five years teaching with 1:1 iPads in my math class, I’d suggest reminding yourself to reflect on the desired student learning outcomes you have set for the lesson/unit/course before you determine which tools – whether digital or not – give you the best chance to achieve those outcomes.

Only then will we be able to make appropriate decisions regarding which technology tools we should use in our classroom and when they are most beneficial to the students we serve.

Interested in reading more about the SAMR Model? Check out these blog posts:

Taking a Dip in the SAMR Swimming Pool – Carl Hooker
SAMR Swimming Lessons – Carl Hooker
SAMR Model Explained for Teachers – Educational Technology and Mobile Learning
Using SAMR to Teach Above The Line – Susan Oxnevad
Resources to Support the SAMR Model – Kathy Schrock’s Guide to Everything

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