The underpinnings of proportional reasoning are born when students learn how to count and unitizing. Explore visual representations of this Big Idea.
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]]>Proportional Reasoning is a big idea that is connected in some way to all mathematical strands and stretches across many grades in the Ontario mathematics curriculum. While many focus on the importance of proportional reasoning for students in junior and intermediate grades, I believe that we can see the early development of proportional reasoning in kindergarten and primary grades when students begin unitizing, grouping, and fair sharing. In this post, I’d like to briefly explore some of the progression of proportional reasoning across the Ontario mathematics curriculum.
The Ontario Ministry of Education has done a great job releasing documents in the Paying Attention to Mathematics series and the Paying Attention to Proportional Reasoning document is no exception.
This document does a great job of providing a great overview of proportional reasoning from a research based perspective without getting too heavy. I will share a few take-aways below and build on some of the ideas from the document with my own spin.
When students begin approaching problems multiplicatively instead of additively, such as thinking that 10 is two groups of five or five groups of two rather than one more than nine, students are said to be reasoning proportionally. This multiplicative thinking allows for students to begin comparing two quantities in relative terms rather than absolute terms.
The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.
Paying Attention to Proportional Reasoning Document
Ontario Ministry of Education
Scenario #1: You invest $100 and it grows to $400.
Scenario #2: You invest $1,000 and it grows to $1,500.
Your opinion will change based on whether you make your comparison of the quantities in each scenario in relative or absolute terms.
If you view this comparison in absolute terms, you might believe that Scenario #2 is the best investment since you have earned more money.
However, if you view this comparison in relative terms, you might believe that Scenario #1 is the best investment since you have earned more money relative to the initial investment amount.
As are most big ideas in mathematics, proportional reasoning is an idea that connects to many key ideas including:
Although the ideas of multiplicative thinking begin developing in grade 1 and 2 when students are encouraged to group quantities into units and then formally as multiplication in grade 3, let’s take a look at a couple expectations in the Number Sense and Numeration strand in Grade 4 of the Ontario Grade 1 to 8 Mathematics Curriculum and some sample problems for each:
Question:
There are 4 fish bowls with 5 fish in each. How many fish in total are there?
Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously.
Question:
You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need?
Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously.
Near the end of the Grade 4 Number Sense and Numeration strand, we run into the first reference to proportional reasoning in the Ontario elementary math curriculum. Under this overall expectation, we see the following:
Notice the multiplicative thinking and comparison of quantity in relative terms. We also encounter the formalization of unitizing through whole number unit rates. It is at this stage where we see the ground work being laid for ratios and rates. While I don’t believe it would be developmentally appropriate to make the leap to ratios or rates at this junction, I do think it is important for the teacher to be aware that the tasks students are asked to solve here are ultimately very simple ratio and rate problems in disguise.
For example, each of our previous unitizing problems for multiplication and division involving fish and fish bowls can be represented as a rate of 5 fish to 1 bowl and 3 fish to 1 bowl, respectively.
In Grade 6, we formally introduce the comparison of two quantities with the same unit as a ratio:
It is fairly common to see a rush to the algorithm via an equation of equivalent fractions in order to meet this expectation while minimizing or avoiding the use of concrete materials, drawings and other representations.
Let’s take a look at a sample problem with some possible concrete and/or visual representations in order to promote the idea of concreteness fading as we build our conceptual understanding:
For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies?
While it might be tempting to jump straight to representing this problem in standard fractional notation (i.e.: 2/3 = x/12), let’s consider some concrete and visual approaches to promote a deep conceptual understanding prior to jumping to the symbolic.
The “double array model” is a representation that I feel strongly about as it builds on the idea that proportional reasoning is multiplicative and a relative comparison of two quantities. I am uncertain if I came across this representation somewhere along my learning journey or if it came about organically. In either case, I have yet to find this representation online or in common Ontario resources such as The Guide to Effective Instruction in Mathematics.
Here’s what a double-array model might look like in this case:
While I especially like the explicit connection that we can make from multiplication and division as inverse operations to proportional reasoning that a double array can provide, double number lines (or “clotheslines“) and ratio tables are also common representations that can make proportional relationships easier to conceptualize.
If you’ve read some of my other posts or attended a workshop, you’d probably know that my vision for teaching math involves making tasks contextual, visual and concrete. So, let’s flip back to a 3 act math task I created a while back called Doritos Roulette.
Check out the act 1 video if you aren’t familiar with the task.
After much mathematical discourse amongst students, I always aim to settle on:
How many “hot” chips should we expect in a bag?
Assuming students are developmentally ready to take the leap to a symbolic strategy using a proportion via equivalent fractions, it can be useful to connect our understanding of double arrays, ratio tables and double number lines like you see in this video:
While I do believe that building a conceptual understanding through multiple representations of proportional reasoning is very important, I think it should be explicitly stated that taking this approach will not necessarily speed up the learning process. Realistically, it could take longer due to the depth of knowledge we are striving for. I’m a firm believer that anything worthwhile takes time and effort. Our mathematical understandings are no exception.
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]]>The Solar Panels Problem is a 3 Act Math Activity asking students to predict how many there are, then determine how much greenhouse gasses they offset.
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]]>This is a 3 Act Math Task that focuses on “Green Energy” proportional reasoning questions related to solar panels in order to address proportions both visually using area models and proportions written as equivalent fractions. Special thanks goes to Dave Raney from CPE Inc. for allowing us to use his video and photos as well as Kathleen Quenneville, Energy & Environmental Officer at GECDSB for her feedback to create problems around solar energy.
Show students the act 1 video.
Then ask students to do a rapid write of what they notice and what they wonder.
Students will then share out their noticings and wonderings while I jot their ideas down on the whiteboard.
While we may explore some other wonderings, the first question I intend to address is:
How many solar panels are there?
I’ll flash this image on the screen and then ask students to make a prediction.
I’ll give students some time to think independently, then chat with a neighbour to discuss their predictions while encouraging them to share their thinking. Then, we share out to the whole group and jot down predictions.
When students have shared out their predictions, we will show students the blue prints for the solar panel system with the total number of modules (582).
Now that students have invested some thinking into this context, we are going to extend our thinking to this question which may or may not have come up in the notice/wonder rapid write:
How much greenhouse gas emissions can the solar panels offset?
Show students this image showing the carbon dioxide from 4,848 km of driving a passenger car that is offset for every 5 solar panels that are in service on the roof of the school.
Show students the act 3 animated solution video showing three different representations for solving this problem.
Consider asking students the following extension question:
How many homes can all 582 solar panel modules power?
Show students this image that states 4 solar panels provide 1,576 kWh of power per year and the average home consumes 9,000 kWh of power each year.
Click on the button below to grab all the media files for use in your own classroom:
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]]>Students are shopping for clothes with expensive retail price tags. Will the price be more reasonable after applying significant percentage discounts?
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]]>The following lesson resource material provides Real World Math Problems that were created with the Grade 6 Ontario Mathematics Curriculum in mind, but can be used in all intermediate grades to reenforce an understanding of percentages and discounts. Throughout the lesson, students are asked to estimate the cost of each item of clothing after a discount is applied while encouraging the use of friendly numbers and their understanding of percentage as a part (percentage) of a whole (out of 100). Students will then be able to take their estimate and complete the calculation with a calculator to compare their estimate with the actual result.
After our Real World Percentages Math Lesson, I will be able to:
Show students the act 1 video.
Then, ask the students:
What do you notice? What do you wonder?
Give students some time to jot down some of their noticings and wonderings on a piece of paper or on a whiteboard/their desk with non-permanent marker. I generally give a minute of time for students to do a “rapid write” of these ideas.
Then, I have students share out their noticings and wonderings while I typically list them in point form on the whiteboard or in a note on my computer on display for all to see. Attaching names to these ideas can be a nice way to build in some accountability and encourage sharing.
Since the price is clearly blocked out in the video, I hope someone is curious about the price and or discount price of the suit jacket with some ideas like these:
How much does it cost before the discount?
How much will the discount save you?
What will be the sale price you have to pay?
And many more…
While the questions I’m fishing for don’t always come out, that is O.K. The discussion is key in order to hook in my students and their curiosity can be moulded quite easily after they have shared out so many interesting ideas. We often spend some time trying to answer their other curiosities in order to ensure students know that their voice is valued.
Then, I ask students to make a prediction.
I’ll have students share out these predictions, jot down their names and try to get a bit of friendly competition going on in the classroom to bring about student discourse in our non-threatening classroom environment.
We will then have students watch the act 2, scene 1 video to reveal the original retail price.
The teacher can then ask students to have a discussion in their table groups to determine ways that they can go about estimating the sale price after the discount. Some guiding questions:
After the discussion, students can share out via Apple TV or using chart paper in your classroom to model as many creative solutions as possible. Students can then check their estimates using a calculator and possibly encourage them to try and find a more efficient strategy as they move through the remainder of the tasks.
Students will watch the act 2, scene 2 video.
Students can then use estimation strategies to find the discount and the sale price of the item.
Students will watch the act 2, scene 3 video.
Students can then use estimation strategies to find the discount and the sale price of the item.
Students will watch the act 2, scene 4 video.
Students can then use estimation strategies to find the discount and the sale price of the item.
Now, students are asked to calculate a sub-total and determine the total bill we should expect when purchasing all four items.
Then, you can show what the bill would look like for students to confirm their thinking. Since I live in Ontario, I’ve shown a bill here using the Harmonized Sales Tax (HST) of 13%:
Something else I typically do with my students is have them figure out how much the bill would be in Michigan, since we are so close to the border and many families go across to Detroit and the surrounding area to shop:
Here’s an image with the final bill from Birch Run, Michigan:
Click on the button below to grab all the media files for use in your own classroom:
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]]>This year, I intend to speak less and listen more. I hope you'll join me here as I try to learn this new role - one day at a time.
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]]>Time flies, doesn’t it?
It is hard to believe that this is my first post of the 2016-17 School Year and it is the middle of October. After doing some travelling to present throughout the summer and spending the balance of my time with my family, the beginning of the school year came fast and furious and hasn’t slowed down yet.
There was definitely no “ease in” time for my new role as K-12 Math Consultant for my district. After our board assembled a Math Task Force to make recommendations to address falling EQAO standardized test scores, our Math Task Force Report was released and we are now entering the first year of our plan. The first two weeks of the school year involved providing mathematics professional development to build the capacity of our central office staff in mathematics content knowledge and pedagogy. Less than a week after, we dove into our first Administrator Capacity Training sessions in which all of our Elementary (K-8) principals and vice-principals engaged in similar learning. Shortly thereafter, we jumped into our Math Learning Team sessions where every school math team consisting of administrators, math liaison teachers, and learning support teachers would join us for mathematics learning. Tomorrow is our last session before we begin planning for our next round of professional learning.
While it might seem logical that my absence in the Math Twitter Blogosphere is purely due to a lack of available time, I think there is more to it. Since the early spring, I have made it a personal goal to speak less and listen more – both in face-to-face interactions and online.
For the past few years, I’ve thoroughly enjoyed sharing my thoughts as I work through ideas and beliefs on my blog and on Twitter. Many times, readers are positive and appreciative; other times, not so much. While I have learned so much from these experiences, my biggest take-away is the realization of how little I know.
We’ve all heard the quote from Socrates:
The more I learn, the less I know.
For me, it was always just a quote with no real meaning or connection to my own life. Then, when new learning prompts you to start rethinking what you thought to be true or believe, the quote suddenly makes perfect sense.
Redefining Mathematics Education and what learning math looks like is not easy. Sometimes in the past, when I would get an “ah-ha” moment or epiphany, I would jump to a conclusion about how to fix the problem or make it better. While deep inside I must have known that math is much more complex than that, my lens was probably too narrowly focused on these new ideas rather than widening my perspective to see the complex system as a whole. I think it is important to note that while mathematics education is complex, it is not complicated. One of My goals this year is to try identifying useful components of this complex system as well as the components that unnecessarily complicate the process.
While I do believe this latest revelation is a sign of personal and professional growth, the downside is that I now hesitate to write that next blog post when I stop to consider that there is always more to learn. I’m sure it will be difficult, but I will do my best to speak less and listen more by sharing my thoughts from a learning stance.
I hope you’ll join me here as I try to learn my new role – one day at a time.
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]]>Fast Clapper is a Three Act Math Task by Nathan Kraft involving proportional reasoning and rates of change. Will he beat the record for fastest clapper?
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]]>The following 3 act math task shared by Nathan Kraft (@nathankraft1) involves proportional reasoning with opportunities to address learning goals around rates, proportions, rates of change and creating equations.
Jon Orr and I have used this task in presentations a number of times over the past few months and many have asked us to share our slide deck and other resources with a summary. So, here we go!
Show the following video:
*Note that this is an edited version of Nathan’s video where I have blacked out the number of claps. You can always add. You can’t subtract.
I typically give students a prompt to jot down what they notice or wonder. Giving a minute of rapid writing in point form or otherwise can be a great opportunity for students to get sucked into the problem.
Students might notice:
Students might wonder:
While we will engage in a healthy amount of discourse around these noticings and wonderings, my main focus is to pull out the question:
Will he beat the current record?
I’m then going to ask students to make a prediction. Will he beat the record? How many claps per minute do you think he’s going to get based on what you saw? I might even show the Act 1 Video a few more times to give students an opportunity to use a mathematical strategy to help with their prediction.
We will record student predictions up on the board next to their names for an opportunity to celebrate later.
Show Nathan’s video:
I am very explicit with my words after showing the video. I say something like:
Do you think you can improve your prediction?
The reason I do this is so that every student has an entry point to the problem. I don’t necessarily care about the exact answer the “math” says you should get. I’m happy with a student using friendly numbers to help them get closer. I’m happy with students using some sort of strategy we might have seen in the past (proportions or maybe trying to create an equation from previous years). The best part by doing this is everyone can get closer to the expected actual result.
What do you think? Will he beat the record?
Here’s one question from a Knowledgehook Gameshow that could be used to support this task:
Allow students to check their solutions:
Additional resources can be downloaded from the download link and the Knowledgehook Gameshow can be grabbed here to modify and edit.
How are you using the problem? Please share in the comments and be sure to thank Nathan for sharing on his blog or on Twitter.
Click on the button below to grab all the media files for use in your own classroom:
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]]>Are students suffering in math because we no longer memorize multiplication tables?Understanding multiplication is much more valuable for students to apply...
The post [Updated Post] Does Memorizing Multiplication Tables Hurt More Than Help? appeared first on Tap Into Teen Minds.
]]>Post update: Saturday May 21st, 2016
This post was originally written two years ago and admittedly took an angle that downplayed the importance of automaticity of multiplication skills which was not the original intention. Please read on as I attempt to clarify the original intent of the article.
As you may have heard in the Globe and Mail or in my recent post, Ontario Education Minister, Liz Sandals recently tossed a comment into the media about the need for students to know their math facts:
That’s actually a great homework assignment: Learn your multiplication tables.
Liz Sandals – Ontario Minister of Education
It seems that whenever things aren’t going well in the world of math education – or more poorly than usual – people are quick to claim that it is because students don’t know their basic math facts; namely, memorizing multiplication tables. There is an obvious benefit to knowing your multiplication tables when solving problems, but is it only the memorization of basic math that students lack?
If we recall what memorizing multiplication tables looked like when we were in school, you might picture flash cards, repeating products aloud or writing out your 7’s times tables repeatedly until it was engrained in your mind. There is no doubt in my mind that having multiplication tables memorized allows for making calculations without taxing working memory.
Although the debate between traditional and reformed mathematics has been going on for decades, I think both groups ultimately want the same thing: students to be proficient in mathematics. However when people make statements about memorizing math facts or “going back to the basics”, people likely have very different interpretations as to how we should get there.
There are no silver bullets in math education and the memorization of multiplication tables alone will not solve all of the problems our students face. That said, many suggest that Ontario needs to “go back to the basics”, but did they ever leave? While the memorization of multiplication tables is not explicitly stated in the Ontario math curriculum, there is a huge push to develop a deep understanding of what it really means to multiply and in time, this should promote automaticity. As I mentioned in this post, shortcuts in math are only effective when you know how to take the long way; learning multiplication is no exception.
Check out the wording of an overall expectation in the Grade 2 Number Sense and Numeration Strand of the curriculum:
solve problems involving the addition and subtraction of one- and two-digit whole numbers using a variety of strategies, and investigate multiplication and division.
We could easily change that overall expectation to say:
know multiplication tables up to 5 times 5 without a calculator.
Which one would be more valuable to the student?
The grade 1-8 Ontario math curriculum contains the word multiplication a total of 47 times, which hardly suggests that multiplication is not a focus throughout elementary.
The curriculum doesn’t suggest that students should not know their multiplication tables, but instead has a greater focus on developing multiplicative skills and understanding. Digging deeper into the document, the words “variety of” occurs in the curriculum a total of 179 times and suggests that much more attention should be spent on exposing students to a variety of strategies, using a variety of tools, to show understanding a variety of ways. If done well, I would have to believe students would have a very deep understanding of math concepts including multiplication.
Over time, I’d like to think that an effective implementation of the Ontario Grade 1-8 Math Curriculum would allow for committing multiplication tables to memory over time.
The curriculum document doesn’t explicitly state the memorization of multiplication tables as an expectation, but it does require that concepts be delivered using a variety of strategies and tools/manipulatives. Where the problem may lie is how the curriculum expectations around multiplication are interpreted by the teacher.
What if the teacher isn’t comfortable teaching math? What if they only really understand one way to multiply? Does “a variety” mean three strategies? Four strategies? … Ten?
These are just a few of the questions that pop into my head when I read the curriculum and ponder some of the challenges we continue to experience in mathematics education.
Tom is a teacher, who admits not having a real passion for teaching math. When he teaches multiplication, it might look like this:
Great for patterns and provides a strategy for students to “get there” if they are stuck, but might not be enough for students to build a deep conceptual understanding.
We all know the algorithm and it is likely that teachers will use it moving forward as students begin working with larger numbers. Works like a charm if you do each step correctly. Something very interesting about the algorithm is that it can often be taught as a procedure and nothing more. Knowing how to use the algorithm without a conceptual understanding of how it works can be useful, but possibly only slightly more useful than typing the expression into a calculator. Miss a step or press the wrong button without a deep understanding of how the algorithm works and you might be out of luck.
Rather than debating over whether memorizing multiplication tables is necessary, maybe a better question might be:
What is the most effective way to have our students build automaticity with multiplication and other math facts?
Can we promote the automaticity of multiplication facts by building a deep conceptual understanding and spacing the practice meaningfully throughout our math curriculum? Would helping students visualize how multiplication works allow for this memorization to build over time?
Here are just a few visual strategies that might help students build a deeper conceptual understanding than simply learning their times tables through mass practice.
Chunking through the use of an area model is a skill that can be used not only to lower the bar for all learners at every level of readiness to begin multiplying with confidence.
If we promote the use of multiple strategies to complete mental math rather than turning to the calculator, then I believe students will value the skill of multiplication automaticity and feel the need to use it as an efficient strategy. I must be clear in saying that this will not happen unless we promote the use of mental math and multiplication strategies throughout the entirety of our math courses.
If multiplication strategies are used consistently and with purpose, students can build their multiplication automaticity in order to create “friendly chunks”. The model above would suggest that this student may be comfortable multiplying by groups of 10 and thus can save a ton of work by chunking in such a manner until they feel comfortable with multiplying by groups of 12.
While I think there are huge advantages to having multiplication tables memorized, I think we need to be cautious about how we plan to get to that end goal. If we do not take time to clarify the how, we risk promoting the use of repetition through mass practice without promoting a deep conceptual understanding of what multiplication really represents. Even worse, promoting memorization without meaning could push students to dislike math and deter them from building a productive disposition towards the subject area we enjoy so much.
Should We Stop Making Kids Memorize Times Tables? – Jo Boaler – US News
Automaticity and why it’s important to learn your ‘times tables’ – Dr Audrey Tan
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]]>Angle geometry including the Opposite Angle Theorem and patterns involving parallel lines cut by a transversal can be boring. Spice it up with some context!
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]]>From Grade 7 to 9 in the Ontario Math Curriculum, student understanding of angle geometry is extended to include (but not limited to) the following specific expectations:
Grade 7
Grade 8
Grade 9 Applied
The expectations extend nicely through these three courses, but the topics alone can often be a bit of a drag. While I’m still going to approach finding missing angles like a “puzzle” for my students as they seem to enjoy it, I have felt pretty uninspired when trying to introduce the idea of finding what would seem to be completely random missing angles.
When I was asked by Keri K. from Kingsville PS to join her class as they prepared to introduce some angle geometry, I figured I had better put my thinking cap on to attempt making the intro to this unit of study more meaningful for students than I might have in the past. So, I thought we would try to get kids thinking about what they notice and wonder via the 3 act math task approach.
Here we go.
Before we started, I ask students to think about what they notice and what they wonder when they watch the following video.
I played the video a second time. I then asked students to take 60 seconds to create a list of what they noticed and wondered on a piece of paper.
Depending on the task, sometimes students nail the question you’re looking to focus on that day, but other times they won’t. Either way, we can always try to answer some of the questions they have shared and nudge them towards the question we are looking to tackle for the day.
After some good discussion, I move on to the act 2 video.
Or, here’s a screenshot:
Students are now fully aware of the question to ponder:
How big is the angle?
You can frame this as a challenge or possibly play up a story involving the need to reconstruct the railing in your home. I think I prefer posing it as more of a challenge since students have not yet been exposed to any angle theorems involving parallel lines with an intersecting transversal line.
Because this was not my own classroom, I thought I should start with a low floor and ensure students were comfortable with benchmark angles prior to moving on. So, we did a few warm-up questions in Knowledgehook Gameshow prior to attempting to tackle our main question from the video.
Here’s the gameshow I used: [play as student | view/clone as a teacher]
Note that I didn’t do the KH Gameshow prior to introducing the task because that would have got them thinking immediately about angles and the whole notice/wonder piece would probably be a dud. If your students have some knowledge of benchmark angles, then you might consider skipping the warm-up.
Already on the desks of the students were bins including paper, scissors, markers, etc.
I said the following:
Friends:
Take a sheet of paper and fold it twice to create an “X” with the folds.[I held up a piece of paper and folded it over twice to create an “X”]
It doesn’t matter how wide or thin your “X” is and it will probably look different than your neighbour.
Now, take a marker, start at any angle you’d like and number the angles in order, clockwise.
[I modelled this.]
Cut out your four angles and piece them together on the desk.
Take one minute on your own to ‘play’ with the pieces. Jot down anything you notice. Do you notice any relationships?
Now, have a conversation with your neighbour. Ensure both of you have a turn to speak.
After sharing out with the group, students noticed that the sum of the angles is 360 degrees and that the opposite angles were equal. And so, the Opposite Angle Theorem (OAT) was born.
Then, I asked students to do the following:
Friends:
Take another sheet of paper and fold it twice to create two parallel lines with the folds.[I held up a piece of paper and folded it over twice.]
It doesn’t matter how far apart your parallel lines are and they may look a bit different than your neighbour.
Now, fold the paper once so a single fold cuts through both of the parallel lines any way you’d like.
Now, take a marker, start at any angle you’d like and number the angles in order, clockwise.
[I modelled this.]
Cut out your four angles and piece them together on the desk.
Take one minute on your own to ‘play’ with the pieces. Jot down anything you notice. Do you notice any relationships?
Now, have a conversation with your neighbour. Ensure both of you have a turn to speak.
Similar discoveries to the previous paper fold activity were discussed. We consolidated as a group and then determined that if we have parallel lines cut by a transversal, we basically have two “groups” of opposite angles.
Then, I showed the act 2 video again. Students were now ready to head out on their way to solve for the missing angle.
After students solved and shared out their work, we could consolidate the learning from this task via the Act 3 Video.
Or the animated gif:
If you try this out, let me know what you did differently in the comments!
Click on the button below to grab all the media files for use in your own classroom:
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]]>When delivering teacher workshops, I almost always include a 3 Act Math Task as a way to model the 4-part math lesson framework and for teachers to experience the power of introducing new concepts by leveraging curiosity. Although my workshops spend a significant amount of time exploring interesting lessons, teachers often want a formal description […]
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]]>When delivering teacher workshops, I almost always include a 3 Act Math Task as a way to model the 4-part math lesson framework and for teachers to experience the power of introducing new concepts by leveraging curiosity. Although my workshops spend a significant amount of time exploring interesting lessons, teachers often want a formal description of what defines a 3 Act Math Task from any other effective task. I thought it was about time that I pull apart what I believe makes an effective 3 act math task and why I have found them so useful in my classroom.
As many are aware, the Three Acts of a Mathematical Story approach shared by Dan Meyer when he began creating media rich math tasks that were structured using the effective storytelling technique of “acts” where:
The hook that introduces the storyline, often leaving you curious with questions you are interested in answering and rising tension.
Tension continues to rise to its highest point as more clues are revealed to help lead you to the climax of the story.
Curiosity is satisfied with answers to your questions and tension is restored to its original state prior to the first act.
Click here for an example of how we can apply the three acts storytelling technique to a math task.
It’s no secret that math isn’t high up on most people’s list of things they enjoy to do in their spare time. Worse yet is trying to teach new concepts to students when more often than not, there are no immediately apparent reasons why we need them. As students move from junior grades into intermediate (or from elementary to middle school) where concepts become more abstract, the purpose of mathematics is less obvious to the learner and interest in the subject often decreases. As student engagement in mathematics falls, teachers are left wondering how they can recapture that abundance of natural curiosity younger children seem to possess.
When discussing the topic of student engagement, I often hear math teachers firmly state: “teachers are paid to teach, not to entertain” – something I’d firmly agree with. However, I do believe that it is our duty as teachers to seek out ways to capture the attention of our students by making math class a compelling environment. Just as storytellers and filmmakers are forced to work hard to hook in their audience, it would make sense that math teachers, especially in the middle to high school grades, would have to work even harder to hook in their respective audience. It is easy to forget that many students never willingly signed up to sit in our classroom – they had to. I think educators need to work just as hard to engage our audience as those in other industries do – some for good, others not so much – if we want to have any sort of influence on our learners.
While there are many beliefs as to why the 3 act math task approach is beneficial, here are just a few of many reasons I’ve compiled over the past couple years as a frequent user of this approach.
While I believe the 3 act approach could be used to introduce any question in any subject area, they are best known as a way to deliver media-rich questions in the math classroom. By showing an image or short video, you can quickly spark curiosity in all learners regardless of their mathematical ability.
Dan Meyer and others using this framework typically encourage students to share their thinking throughout the process. Sharing what they notice or wonder, what questions come to mind, predictions and extraneous variables that could affect outcomes are just some of the common discussion starters that can be used.
Because most 3 act math tasks model math in the world around us, often times applying a standard formula or algorithm may not accurately calculate what will actually happen in the given scenario. Encouraging students to think outside of the box and consider what would happen in the real world vs. in the “fake world” math questions often have us living in.
Those who are lucky enough to have the sense of sight are constantly processing images of the world around them. It seems logical that providing students with a visual to better understand mathematics, especially when the content becomes increasingly abstract over each grade level.
Most tasks provide an opportunity for all students to participate and can be extended to more open ended (or open middle) questioning via “sequels” to the original question.
While there are many different ways to engage students in math class, it is clear to me that using the art of storytelling such as Dan Meyer‘s 3 act math approach is a solid way to raise student interest, curiosity and engagement in my mathematics classroom.
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]]>End the ongoing debate; inquiry and explicit instruction both serve an important role in the math classroom to build both conceptual and procedural fluency.
The post Is There a Best Way to Learn Mathematics? appeared first on Tap Into Teen Minds.
]]>The ongoing debate over whether math concepts should be taught using a discovery approach versus explicit instruction has been going on for years. These “math wars” (see examples here and here) typically paint effective instruction in mathematics as an either/or situation between the two extremes. Some might interpret inquiry based instruction as necessary to construct conceptual understanding prior to committing algorithms to memory, whereas others might conversely interpret explicit instruction as the requirement to first build procedural fluency necessary to enable a student to better understand why a formula works later. Regardless of where you stand on the continuum, Dr. Daniel Ansari does an excellent job in this video discussing why these math wars between inquiry based learning and explicit instruction should end.
While I have personally stood on both sides of this war – early in my career using heavy doses of explicit instruction and most recently serving up an almost complete inquiry-based approach – only now am I able to see that neither extreme is appropriate. Some mathematical concepts are likely best suited for a direct instruction delivery, while others may seem more natural to introduce from a discovery standpoint. More importantly, I think the decision as to which topics are taught from different teaching styles could and should be different based on the unique characteristics of the educator such as: personality, teaching style, and their own interpretation or understanding of the specific learning goal.
With that said, I will be honest in saying that I almost always prefer when I can find an interesting way to introduce a task from an inquiry approach. Unfortunately, teacher interpretation of what an inquiry math lesson looks like varies drastically. According to E. Lee May, from Salisbury State University, the definition of an inquiry based math lesson is:
…a method of instruction that places the student, the subject, and their interaction at the center of the learning experience. At the same time, it transforms the role of the teacher from that of dispensing knowledge to one of facilitating learning. It repositions him or her, physically, from the front and center of the classroom to someplace in the middle or back of it, as it subtly yet significantly increases his or her involvement in the thought-processes of the students.
As cited on inquirybasedlearning.org
This leaves many different possibilities for lesson ideas to be defined as “inquiry”. For me, I keep a 4-part math lesson in mind when planning.
While I always attempt using the 4-part math lesson framework when possible, it is not always easily achievable, necessary, or appropriate for every concept. Currently, I do not have every lesson planned using this framework, but I am always trying to think of ways to use this approach to improve my lessons, where possible.
So regardless of whether you feel that you sit more on one side than the other, consider exploring ways to use inquiry and explicit instruction where appropriate in your classroom. A good place to start might be trying to structure a 4-part math lesson or just exploring some 3 act math tasks by Dan Meyer and others from around the web. Here are a couple tasks I’ve created that do some of the heavy lifting for the first two parts of the lesson such as Walk-Out, Stacking Paper Tasks, and Tech Weigh In.
Introducing Distance-Time Graphs & Graphing Stories This 3 act math task was created as a simple, yet powerful way to introduce distance-time graphs and other various graphs of linear and non-linear relationships between two variables. In particular, I'm looking to address the following specific expectations from the grade 9 math courses in Ontario: Grade 9 Applied LR4.02 & Grade 9 Acade...
If you have explored using a 4-part math lesson, please feel free to share your reflections in the comments section.
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]]>Exploring fundamentals that help investors maximizing returns in the stock market and how they apply to professional development and teaching.
The post Earning Versus Learning appeared first on Tap Into Teen Minds.
]]>In recent months, I’ve been doing quite a bit of reading about investing in the stock market. I’ve always had an interest in finance and saving money, but I have never really had a deep understanding of how to be a successful investor beyond handing my money over to a financial advisor or picking random mutual funds. As I dug deeper into the investing world, I began to notice that some popular keys to investing seemed to mirror a reasonable approach to implementing effective teaching strategies.
Regardless of whether you feel like you are a pro-investor or a newbie, my guess is that many have at least heard the popular investing saying: “buy low, sell high!”
Seems like a pretty logical concept; in order to make money, you need to buy something at a price and be able to sell it for a higher price.
This concept parallels what an educator is trying to achieve when introducing a new teaching strategy in the classroom:
If I invest time and effort into a new teaching approach, then learning outcomes will improve for my students.
Despite how simple this may sound, things are rarely as easy as they seem. When investing, many are not comfortable making their own investment decisions. A similar lack of comfort is common amongst many educators when conversations about making changes to their teaching practice are explored. My hunch is that while the concept of buying low and selling high is fundamentally straightforward in both cases, determining what to buy when investing or which strategy to use when teaching becomes much more convoluted once you begin to dig in.
If you’re thinking it can’t be that difficult to find a good company to invest in or an effective teaching strategy to use in the classroom, then you might want to think again. There are over 2,800 different companies listed on the New York Stock Exchange and over 150 teaching strategies listed on The University of North Carolina at Charlotte’s learning resource page. Don’t forget that there are a large number of stock markets around the globe and a wide range of other influences on student achievement that we may not even be aware of.
In the investing world, one could invest in Apple (APPL) at about $100 a share or Microsoft (MSFT) at $50 per share, but which is the better pick? Does Apple having a value double that of Microsoft make it a better pick or does Microsoft seem like a deal you can’t refuse?
In the learning world, a teacher could invest all of their time and effort into direct instruction or an inquiry-based approach, but may still feel uncertain as to whether or not they are maximizing learning outcomes. Is direct instruction a better teaching strategy due to the highly structured and guided approach or does inquiry-based strategies provide a better opportunity for students to construct their understanding of content?
Making a decision based on speculation to invest in a company or in a strategy without specific and logically sound reasoning increases risk. If we allow our decision making process to be influenced without an appropriate amount of evidence, we are ultimately taking a gamble in hope that probability will be on our side. What constitutes an appropriate amount of evidence could look very different when comparing investing and teaching. For example, I think that while many teaching strategies are research-based, there are many other strategies teachers may explore based on their own creativity and an intuition that the strategy could have positive effects on student achievement. On the other hand, some in the financial world believe that intuition can be useful to identify a good buy, but I don’t believe there are any investors claiming to make decisions without thoroughly conducting the necessary research to confirm that their thinking is more than just a hunch.
With that said, my hypothesis is that investors would be less likely to gamble on an investment based purely on intuition than an educator would on a teaching strategy because the investor has more to lose in a very short amount of time. We as educators are protected because it is almost impossible to go “all in” on a strategy and lose it all. A particular teaching strategy may yield no academic gains, but it would seem highly improbable that there are too many (any?) strategies that would cause students to regress.
Despite the seemingly limited risk educators take on when exploring new teaching strategies, teacher resistance to change is common across many school districts. This seems logical since every teacher believes they are doing the best they can by using effective strategies they feel will be most beneficial to student learning. Conversely, I have never met a colleague who openly admitted to doing less than their best by intentionally using ineffective strategies to limit learning outcomes.
So what do education policy makers, districts, consultants and instructional coaches like myself do? We bombard teachers with a huge list of “research-based teaching strategies” to diversify the lesson delivery in classrooms. Just as an investment advisor would tell you to diversify your investment portfolio to limit risk, it would seem that we promote the same logic in education reform.
If exploring new teaching strategies produces minimal risk, why do we need diversification?
Rather than sharing a variety of teaching strategies that educators can use at their discretion, all too often system frameworks are created with an expectation that teachers will follow specific lesson formats regardless of whether the approach aligns with their current (hopefully, ever-changing) beliefs. While research indicates the benefits of using such teaching strategies, the risks may outweigh the rewards when teachers are forced to use them against their will. To make matters worse, we often exacerbate a group of already jaded educators when we forget to celebrate the many great things they are already doing in their classrooms on a regular basis.
This self-induced need for wide diversification in the classroom also arises on Wall Street. The inherent need for diversification when building an investment portfolio has been rejected by some of the most successful stockholders of all time as a “recipe for mediocrity“.
Wide diversification is only required when investors do not understand what they are doing.
What Buffet and other successful investors like him advocate for is focus. Rather than investing into a large list of companies – some that will go up in value, some that will go down – do the necessary research to understand the difference and invest in only those that will maximize the return on your investment.
This thinking can be applied to enhance our teaching practice. Rather than trying to do too many research based strategies without understanding their expected outcomes or how to maximize those outcomes, it seems more reasonable to focus on building our portfolio of teacher moves gradually and with intent.
While the word “formula” is probably more common to us math teachers than those in other subject areas, there is no denying the human tendency to want a simple list of steps to follow in order to achieve a desired result. Whether it is The Formula for Losing Weight, 10 Steps to Getting Noticed by Your Crush, or Five Steps to Make Your House Hunt a Happy One, people are always looking for a quick-fix to their problems. The world of finance and education are certainly not exceptions to this rule.
Unfortunately, there is no formula to getting rich or to being the best classroom teacher. If there was, we’d all be rich and our problems in education would be solved.
Although there is no exact formula that will lead to successful investing or teaching, we can use the following fundamentals to help guide our work:
Avoid widely diversifying without having specific intent or purpose for each of your selected investments/teaching strategies.
The only way you can maximize reward and minimize risk is to deeply understand the investments/teaching strategies you use.
The formula is knowing that there is no one formula for success.
Whether from the perspective of a classroom teacher or an instructional coach, being mindful of these fundamentals can help to ensure that our professional development models are a continuum rather than a teaching strategy checklist.
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