## What Makes Math Interesting Anyway?

What does it mean to be “creative” in math?
What makes math interesting anyway?

Questions I think we all need to dive into!

Many teachers are comfortable allowing their students to read for pleasure at school and encourage reading at home for pleasure too. Writing is often seen as a creative activity. Our society appreciates Literacy as having both creative and purposeful aspects. Yet mathematics as a source of enjoyment or creativity is often not considered by many.

I want you to reflect on your own thinking here. How important do you see creativity in mathematics? What does creativity in mathematics even mean to you?

Marian Small might explain the notion of creativity in mathematics best. Take a look:

## Type 1 and Type 2 Questions

Several years ago, Marian Small tried to help us as math teachers see what it means to think and be creative in mathematics by sharing 2 different ways for our students to experience the same content. She called them “type 1” and “type 2” questions.

Type 1 problems typically ask students to give us the answer.  There might be several different strategies used… There might be many steps or parts to the problem.  Pretty much every Textbook problem would fit under Type 1.  Every standardized test question would fit here.  Many “problem solving” type questions might fit here too.

Type 2 problems are a little tricky to define here. They aren’t necessarily more difficult, they don’t need a context, nor do they need to have more steps.  A Type 2 problem asks students to get to relationships about the concepts involved.  Essentially, Type 2 problems are about asking something where students could have plenty of possible answers (open ended). Again, here is Marian Small describing some examples:

## Examples of Type 1 & 2 Questions

Notice that a type 2 problem is more than just open, it encourages you to keep thinking and try other possibilities!  The constraints are part of what makes this a “type 2” problem! The creativity and interest comes from trying to reach your goal!

## Where do you look for “Type 2” Problems?

If you haven’t seen it before, the website called OpenMiddle.com is a great source of Type 2 problems.  Each involve students being creative to solve a potential problem AND start to notice mathematical relationships.

Remember, mathematically interesting problems (Type 2 problems) are interesting because of the mathematical connections, the relationships involved, the deepening of learning that occurs, not just a fancy context.

## Questions to Reflect on:

• When do you include creativity in your math class? All the time? Daily? Toward the beginning of a unit? The end? What does this say about your program? (See A Few Simple Beliefs)
• If you find it difficult to create these types of questions, where do you look? Marian Small is a great start, but there are many places!
• How might “Type 2” problems like these offer your students practice for the skills they have been learning? (See purposeful practice)
• What is the current balance of q]Type 1 and Type 2 problems in your class? Are your students spending more time calculating, or deciding on which calculations are important? What balance would you like?
• How might problems like these help you meet the varied needs within a mixed ability classroom?
• If students start to understand how to solve type 2 problems, would you consider asking your students to make up their own problems? (Ideas for making your own problems here).
• Would you want students to work alone, in pairs, in groups? Why?
• If you have struggled with developing rich discussions in your class, how might these types of problems help you bring a need for discussions? How might this change class conversations afterward?
• How will you consolidate the learning afterward? (See Never Skip the Closing of the Lesson)
• As the teacher, what will you be doing when students are being creative? How might listening to student thinking help you learn more about your students? (See: Noticing and Wondering: A powerful tool for assessment)

## “Number Boxes”

A few weeks ago I was introduced to Jenna Laib‘s game “Number Boxes” and was very interested in using it as a dynamic game to help students learn a variety of new content — Jenna’s blog explaining the game can be found here: “One of My Favorite Games: Number Boxes“.

Basically the game involves students rolling dice (or spinning a spinner / drawing a card) to generate a random number and placing that number in one of their empty number boxes one-at-a-time. The game can progress in a variety of ways:

As you can see, the game is quite adaptive to the sizes of numbers and concepts your students are comfortable with. As students roll/spin/draw a number, they have to place it on the board. What makes this tricky is not knowing what future numbers will be. In the board above, you can see that there is also a “Throwaway” box that students can use if they do not like one of the numbers rolled/spun/drawn. This game is an excellent example of a “Dynamic Game” or “Dynamic Practice” as students are following the ideals on the right side of the chart below:

Blow is a gallery of some possible adaptations of this game or linked here is a slideshow

## Metric Conversions

I however, wanted to use Jenna’s game to help students practice a concept they often have difficulty with – Metric Conversions. Once students have had many opportunities to estimate and measure various distances, capacities, and masses, they should be able to start making connections between all of the units. I suggest a good balance between using problems that help students make sense of the relationships between the units, and opportunities to practice conversions on their own. However, instead of randomly generated worksheets or other rote practice, I think Jenna’s game could work perfectly. Take a look at some examples:

## Reflection

It is important to offer tasks that allow students to make choices and decisions like the ones offered in this game. Learning needs to be more than handing out assignments, and collecting work… Learning takes time! Students need more time to explore, see what works, have peers challenge each others’ thinking, make important connections… Hopefully you can see these opportunities in this task.

## Final Thoughts:

• If you play one of these games, or your own version, will you first offer a simplified version so your students get familiar with the game, or will you dive into the content you want to teach?
• Would you prefer your students to play this game as a class or with a group, a partner, or independently?
• How will you build in conversations with students so they discuss which numbers they think should be the highest / lowest numbers? How will you offer time for these strategic discussions?
• Should we adapt these to continually offer more challenge and deeper learning, or offer more opportunities to play the same game board? How will we know when to adapt and change?
• What does “practice” look like in your classroom?  Does it involve thinking or decisions?  Would it be more engaging for your students to make practice involve more thinking?
• How does this game relate to the topic of “engagement”?  Is engagement about making tasks more fun or about making tasks require more thought?  Which view of engagement do you and your students subscribe to?
• How have your students experienced measurement concepts like these? Are they learning procedural rules or are they thinking about the actual sizes of numbers / sizes of the units involved?
• As the teacher, what will you be doing when students are playing? How might listening to student thinking help you learn more about your students? (See: Noticing and Wondering: A powerful tool for assessment)

## Math Games – building a foundation for mathematical reasoning

In 2001, the National Research Council, in their report Adding it up: Helping children learn mathematics, sought to address a concern expressed by many Americans: that too few students in our schools are successfully acquiring the mathematical knowledge, skill, and confidence they need to use the mathematics they have learned.

Developing Mathematical Proficiency
The potential of different types of tasks for student learning, 2017

As we start a new school year, I expect many teachers, schools and districts to begin conversations surrounding assessment and wondering how to start learning given students who might be “behind”. I’ve shared my thoughts about how we should NOT start a school year, but I wanted to offer some alternatives in this post surrounding a piece often overlooked — our students’ confidence (including student agency, ownership and identity). If we are truly interested in starting a year off successfully, then we need to spend time allowing our students to see themselves in the math they are doing… and to see their strengths, not their deficits.

[The] goal is to support all students — especially those who have not been academically successful in the past — to develop a sense of agency and ownership over their own learning. We want students to come to see themselves as intellectually capable and competent — not by giving them easy successes, but by engaging them as sense-makers, problem solvers, and creators of meaningful and important ideas.

MathShell – TRUMath, 2016

When we hear ideals like the above quote, what many of us see is as missing are specific examples. How DO we help our students gain confidence becomes a question most of us are left with. Adding It Up suggests that mathematical proficiency includes an intertwined mix of procedural fluency, conceptual understanding, strategic competence, adaptive reasoning and productive disposition. Which again sounds nice in theory, but in reality, these 5 pieces are not balanced in classroom materials nor in our assessment data. Not even close!

So, again we are left with a specific need for us to build confidence in our students. There is a growing body of evidence to support the use of strategy games in math class as a purposeful way to build confidence (including student agency, authority and identity).

To be helpful, I’d like to share some examples of possible strategy games that are appropriate for all ages. Each game is a traditional game from various places around the world.

*The above files are open to view / print. If you experience difficulties accessing, you might need to use a non-educational account as your school board might be restricting your access.

#### How to Play:

Each link above includes a full set of rules, but you might also be interested in watching a preview of these games (thanks to WhatDoWeDoAllDay.com)

### A few things to reflect on:

• Some students have missed a lot of school / learning. Our students might be entering a new grade worried about the difficulty level of the content. Beyond content, what other aspects of learning math might be a struggle for our students? How might introducing games periodically help with these struggles?
• How do you see equity playing a role in all of this? Pinpointing and focusing on student gaps often leads to inequities in experiences and outcomes. So, how can the ideas above help reduce these inequities?
• One of the best ways to tackle equity issues is to expand WHAT we consider mathematics and expand WHO is considered a math person. How might you see using games periodically as a way for us to improve in these two areas?
• If you are distance learning, how might games be an integral part of your program? How do you see including games that are not related to content helpful for our students that might struggle to learn mathematics? (building confidence, social-emotional learning skills, community, students’ identities…)
• If you are learning in person this year, but can not have students working together, how might you adapt some of these strategy games?
• What might you notice as students are playing games that you might not be able to notice otherwise?
• How might we see a link between gaining confidence through playing strategy games and improvement in mathematical reasoning?
• Why do you think I choose the games above (I searched through many)? Hopefully you can see a benefit from seeing mathematics learning from various cultures.

If interested in more games and puzzles? Take a look at some of the following posts:

## Skyscraper Puzzles – printable package

An area of mathematics I wish more students had opportunities to explore is spatial/visualization. There are many studies that show just how important spatial/visual reasoning is for mathematical success (I discuss in more depth here), but often, we as teachers aren’t sure where to turn to help our students develop spatial reasoning, or now to make the mathematics our students are learning more spatial.

One such activity I’ve suggested before is Skyscraper Puzzles. I’ve shared these puzzles before (Skyscraper Puzzles and Skyscraper Templates – for relational rods). With the help of my own children, I decided to make new templates. The package includes a page dedicated to explain how to solve the puzzles, as well as instructions on each page.

## New Puzzles can be accessed here

*The above files are open to view / print. If you experience difficulties accessing, you might need to use a non-educational account as your school board might be restricting your access.

You’ll notice in the package above that some of the puzzles are missing information like the puzzle below:

Puzzles like these might include information within the puzzle. In the puzzle above, the 1 in the middle of the block refers to the height of that tower (a tower with a height of 1 goes where the 1 is placed).

You might also be interested in watching a few students discussing how to play:

### A few thoughts about how you might use these:

As always, I’d love to hear from you. Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).

## The Types of Questions we Ask: which categories of questions should we focus on?

I think we can all agree that there are many different ways for our students to show what they know or understand, and that some problems ask for deeper understanding than others. In fact, many standardized math assessments, like PISA, aim to ask students questions at varying difficult levels (PISA uses 6 difficulty levels) to assess the same concept/skill. If we can learn one thing from assessments like these hopefully it is how to expect more of our students by going deeper… and in math class, this means asking better questions.

Robert Kaplinsky is a great example of an educator who has helped us better understand how to ask better questions. His work on Depth of Knowledge (DOK) has helped many teachers reflect on the questions they ask and has offered teachers examples of what higher DOK questions/problems look like.

In Ontario though we actually have an achievement chart that is aimed to help us think more about the types of questions/problems we expect our students be able to do. Basically, it is a rubric showing 4 levels of achievement across 4 categories. In Ontario it is expected that every teacher evaluate their students based on each the these categories. Many teachers, however, struggle to see the differences between these categories. Marian Small recently was the keynote speaker at OAME where she helped us think more about the categories by showing us how to delineate between the different categories of questions/problems:

• Knowledge
• Understanding
• Application
• Thinking

## Knowledge vs. Understanding

Below are a few of Marian Small’s examples of questions that are designed to help us see the difference between questions aimed at knowledge and questions aimed at understanding:

As you can see from the above examples, each of the knowledge questions ask students to provide a correct answer. However, each of the understanding questions require students to both get a correct answer AND be able to show that they understand some of the key relationships involved. Marian’s point in showing us these comparisons was to tell us that we need to spend much more time and attention making sure our students understand the math they are learning.

Each of the questions that asks students to show their understanding also help us see what knowledge our students have, but the other way around is not true!

Hopefully you can see the potential benefits of striving for understanding, but I do believe these shifts need to be deliberate. My recommendation to help us aim for understanding is to ask more questions that ask students to:

• Draw a visual representation to show why something works
• Provide an example that fits given criteria
• Explain when examples will or won’t work
• Make choices (i.e., which numbers, visual representations… will be best to show proof)
• show their understanding of key “Big Ideas” and relationships

## Application vs. Thinking

Below are a few examples that can help us delineate the differences between application and thinking:

These examples might be particularly important for us to think about. To begin with, application questions often use some or all of the following:

• use a context
• require students to use things they already should know
• provide a picture(s) or example(s) for students to see
• provide almost all of the information and ask the student to find what is missing

Thinking questions, on the other hand, are the basis for what Stein et. al called “Doing Mathematics“. In Marian’s presentation, she discussed with us that these types of questions are why those who enjoy mathematics like doing mathematics. Thinking and reasoning are at the heart of what mathematics is all about! Thinking questions typically require the student to:

• use non-algorithmic thinking
• make sense of the problem
• use relevant knowledge
• notice important features of the problem
• choose a possible solution path and possibly adjust if needed
• persevere to monitor their own progress

Let’s take a minute to compare questions aimed at application and questions aimed at thinking. Application questions, while quite helpful in learning mathematics concepts (contexts should be used AS students learn), they typically offer less depth than thinking questions. In each of the above application questions, a student could easily ignore the context and fall back on learned procedures. On the other hand, each of the thinking questions might require the student to make and test conjectures, using the same procedures repeatedly to find a possible solution.

Ideally, we need to spend more time where our students are thinking… more time discussing thinking questions… and focus more on the important relationships/connections that will arise through working on these problems.

## Final Thoughts

Somehow we need to find the right balance between using the 4 types of questions above, however, we need to recognize that most textbooks, most teacher-made assessments, and most online resources focus heavily (if not exclusively) on knowledge and occasionally application. The balance is way off!

Focusing on being able to monitor our own types of questions isn’t enough though. We need to recognize that relationships/connections between concepts/representations are at the heart of expecting more from our students. We need to know that thinking and reasoning are HOW our students should be learning. We need to confront practices that stand in the way of us moving toward understanding and thinking, and set aside resources that focus mainly on knowledge or application. If we want to make strides forward, we need to find resources that will help US understand the material deeper and provide us with good examples.

## Questions to Reflect on:

• What did your last quiz or test or exit card look like? What is your current balance of question types?
• What resources do you use? What balance do they have?
• Where do you go to find better Understanding or Thinking questions?
• What was the last problem you did that made you interested in solving it? What was it about that problem that made you interested? Likely it was a Thinking question. What was it about that problem that made it interesting?
• Much of the work related to filling gaps, intervention, assessment driving learning… points teachers toward students’ missing knowledge. How can we focus our attention more toward understanding and thinking given this reality?
• How can we better define “mastery” given the 4 categories above? Mastery must be seen as more than getting a bunch of simple knowledge questions correct!

If you haven’t already, please take a look at Marian Small’s entire presentation where she labels understanding and thinking as the “fundamentals of mathematics”

## Decomposing & Recomposing – How we subtract

Throughout mathematics, the idea that objects and numbers can be decomposed and recomposed can be found almost everywhere. I plan on writing a few articles in the next while to discuss a few of these areas. In this post, I’d like to help us think about how and why we use visual representations and contexts to help our students make sense of the numbers they are using.

#### Decomposing and Recomposing

Foundational to almost every aspect of mathematics is the idea that things can be broken down into pieces or units in a variety of ways, and be then recomposed again. For example, the number 10 can be thought of as 2 groups of 5, or 5 groups of 2, or a 7 and a 3, or two-and-one-half and seven-and-one-half…

Understanding how numbers are decomposed and recomposed can help us make sense of subtraction when we consider 52-19 as being 52-10-9 or 52-20+1 or (40-10)+(12-9) or 49-19+3 (or many other possibilities)… Let’s take a look at how each of these might be used:

The traditional algorithm suggests that we decompose 52-19 based on the value of each column, making sure that each column can be subtracted 1 digit at a time… In this case, the question would be recomposed into (40-10)+(12-9). Take a look:

While this above strategy makes sense when calculating via paper-and-pencil, it might not be helpful for our students to develop number sense, or in this case, maintain magnitude. That is, students might be getting the correct answer, but completely unaware that they have actually decomposed and recomposed the numbers they are using at all.

Other strategies for decomposing and recomposing the same question could look like:

The first problem at the beginning was aimed at helping students see how to “regroup” or decompose/recompose via a standardized method. However, the second and third examples were far more likely used strategies for students/adults to use if using mental math. The last example pictured above, illustrates the notion of “constant difference” which is a key strategy to help students see subtraction as more than just removal (but as the difference). Constant difference could have been thought of as 52-19 = 53-20 or as 52-19 = 50-17, a similar problem that maintains the same difference between the larger and smaller values. Others still, could have shown a counting-on strategy (not shown above) to represent the relationship between addition and subtraction (19+____=53).

#### Why “Decompose” and “Recompose”?

The language we use along with the representations we want from our students matters a lot. Using terms like “borrowing” for subtraction does not share what is actually happening (we aren’t lending things expecting to receive something back later), nor does it help students maintain a sense of the numbers being used. Liping Ma’s research, shared in her book Knowing and Teaching Elementary Mathematics, shows a comparison between US and Chinese teachers in how they teach subtraction. Below you can see that the idea of regrouping, or as I am calling decomposing and recomposing, is not the norm in the US.

#### Visualizing the Math

There seems to be conflicting ideas about how visuals might be helpful for our students. To some, worksheets are handed out where students are expected to draw out base 10 blocks or number lines the way their teacher has required. To others, number talks are used to discuss strategies kids have used to answer the same question, with steps written out by their teachers.

In both of these situations, visuals might not be used effectively. For teachers who are expecting every student to follow a set of procedures to visually represent each question, I think they might be missing an important reason behind using visuals. Visuals are meant to help our students see others’ ideas to learn new strategies! The visuals help us see What is being discussed, Why it works, and How to use the strategy in the future.

Teachers who might be sharing number talks without visuals might also be missing this point. The number talk below is a great example of explaining each of the types of strategies, but it is missing a visual component that would help others see how the numbers are actually being decomposed and recomposed spatially.

If we were to think developmentally for a moment (see Dr. Alex Lawson’s continuum below), we should notice that the specific strategies we are aiming for, might actually be promoted with specific visuals. Those in the “Working with the Numbers” phase, should be spending more time with visuals that help us SEE the strategies listed.

#### Aiming for Fluency

While we all want our students to be fluent when using mathematics, I think it might be helpful to look specifically at what the term “procedural fluency” means here. Below is NCTM’s definition of “procedural fluency” (verbs highlighted by Tracy Zager):

Which of the above verbs might relate to our students being able to “decompose” and “recompose”?

#### Some things to think about:

• How well do your students understand how numbers can be decomposed and recomposed? Can they see that 134 can be thought of as 1 group of 100, 3 groups of 10, and 4 ones AS WELL AS 13 groups of 10, and 4 ones, OR 1 group of 100, 2 groups of 10, and 14 ones…….? To decompose and recompose requires more than an understanding of digit values!!!
• How do the contexts you choose and the visual representations you and your students use help your students make connections? Are they calculating subtraction questions, or are they thinking about which strategy is best based on the numbers given?
• What developmental continuum do you use to help you know what to listen for?
• How much time do your students spend calculating by hand? Mentally figuring out an answer? Using technology (a calculator)? What is your balance?
• How might the ideas of decomposing and recomposing relate to other topics your students have learned and will learn in the future?
• Are you teaching your students how to get an answer, or how to think?

If you are interested in learning more, I would recommend:

## Skyscraper Templates – for Relational Rods

Many math educators have come to realize how important it is for students to play in math class. Whether for finding patterns, building curiosity, experiencing math as a beautiful endeavour, or as a source of meaningful practice… games and puzzles are excellent ways for your students to experience mathematics.

Last year I published a number of templates to play a game/puzzle called Skyscrapers (see here for templates) that involved towers of connected cubes. This year, I decided to make an adjustment to this game by changing the manipulative to Relational Rods (Cuisenaire Rods) because I wanted to make sure that more students are becoming more familiar with them.

Skyscraper puzzles are a great way to help our students think about perspective while thinking strategically through each puzzle.  Plus, since they require us to consider a variety of vantage points of a small city block, the puzzles can be used to help our students develop their Spatial Reasoning!

### How to play a 4 by 4 Skyscraper Puzzle:

• Build towers in each of the squares provided sized 1 through 4 tall
• Each row has skyscrapers of different heights (1 through 4), no duplicate sizes
• Each column has skyscrapers of different heights (1 through 4), no duplicate sizes
• The rules on the outside (in grey) tell you how many skyscrapers you can see from that direction
• The rules on the inside tell you which colour rod to use (W=White, R=Red, G=Green, P=Purple, Y=Yellow)
• Taller skyscrapers block your view of shorter ones

Below is an overhead shot of a completed 4 by 4 city block.  To help illustrate the different sizes. As you can see, since each relational rod is coloured based on its size, we can tell the sizes quite easily.  Notice that each row has exactly 1 of each size, and that each column has one of each size as well.

To understand how to complete each puzzle, take a look at each view so we can see how to arrange the rods:

If you are new to completing one of these puzzles, please take a look here for clearer instructions: Skyscraper Puzzles

### Relational Rod Templates

Here are some templates for you to try these puzzles yourself and with your students:

4 x 4 Skyscraper Puzzles – for Relational Rods

5 x 5 Skyscraper Puzzles – for Relational Rods

### A few thoughts about using these:

A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity.  Hopefully, this post has helped you consider your own decision making processes!

I’d love to hear how you and/or your students do!

## Reasoning & Proving

This week I had the pleasure to see Dan Meyer, Cathy Fosnot and Graham Fletcher at OAME’s Leadership conference.

Each of the sessions were inspiring and informative… but halfway through the conference I noticed a common message that the first 2 keynote speakers were suggesting:

Dan Meyer showed us several examples of what mathematical surprise looks like in mathematics class (so students will be interested in making sense of what they are learning, and to get our students really thinking), while Cathy Fosnot shared with us how important it is for students to be puzzled in the process of developing as young mathematicians.  Both messages revolved around what I would consider the most important Process Expectation in the Ontario curriculum – Reasoning and Proving.

###### Reasoning and Proving

While some see Reasoning and Proving as being about how well an answer is constructed for a given problem – how well communicated/justified a solution is – this is not at all how I see it.  Reasoning is about sense-making… it’s about generalizing why things work… it’s about knowing if something will always, sometimes or never be true…it is about the “that’s why it works” kinds of experiences we want our students engaged in.  Reasoning is really what mathematics is all about.  It’s the pursuit of trying to help our students think mathematically (hence the name of my blog site).

###### A Non-Example of Reasoning and Proving

In the Ontario curriculum, students in grade 7 are expected to be able to:

• identify, through investigation, the minimum side and angle information (i.e.,side-side-side; side-angle-side; angle-side-angle) needed to describe a unique triangle

Many textbooks take an expectation like this and remove the need for reasoning.  Take a look:

As you can see, the textbook here shares that there are 3 “conditions for congruence”.  It shares the objective at the top of the page.  Really there is nothing left to figure out, just a few questions to complete.  You might also notice, that the phrase “explain your reasoning” is used here… but isn’t used in the sense-making way suggested earlier… it is used as a synonym for “show your work”.  This isn’t reasoning!  And there is no “identifying through investigation” here at all – as the verbs in our expectation indicate!

A Example of Reasoning and Proving

Instead of starting with a description of which sets of information are possible minimal information for triangle congruence, we started with this prompt:

Given a few minutes, each student created their own triangles, measured the side lengths and angles, then thought of which 3 pieces of information (out of the 6 measurements they measured) they would share.  We noticed that each successful student either shared 2 angles, with a side length in between the angles (ASA), or 2 side lengths with the angle in between the sides (SAS).  We could have let the lesson end there, but we decided to ask if any of the other possible sets of 3 pieces of information could work:

While most textbooks share that there are 3 possible sets of minimal information, 2 of which our students easily figured out, we wondered if any of the other sets listed above will be enough information to create a unique triangle.  Asking the original question didn’t offer puzzlement or surprise because everyone answered the problem without much struggle.  As math teachers we might be sure about ASA, SAS and SSS, but I want you to try the other possible pieces of information yourself:

Create triangle ABC where AB=8cm, BC=6cm, ∠BCA=60°

Create triangle FGH where ∠FGH=45°, ∠GHF=100°, HF=12cm

Create triangle JKL where ∠JKL=30°, ∠KLJ=70°, ∠LJK=80°

If you were given the information above, could you guarantee that everyone would create the exact same triangles?  What if I suggested that if you were to provide ANY 4 pieces of information, you would definitely be able to create a unique triangle… would that be true?  Is it possible to supply only 2 pieces of information and have someone create a unique triangle?  You might be surprised here… but that requires you to do the math yourself:)

###### Final Thoughts

Graham Fletcher in his closing remarks asked us a few important questions:

• Are you the kind or teacher who teaches the content, then offers problems (like the textbook page in the beginning)?  Or are you the kind of teacher who uses a problem to help your students learn?
• How are you using surprise or puzzlement in your classroom?  Where do you look for ideas?
• If you find yourself covering information, instead of helping your students learn to think mathematically, you might want to take a look at resources that aim to help you teach THROUGH problem solving (I got the problem used here in Marian Small’s new Open Questions resource).  Where else might you look?
• What does Day 1 look like when learning a new concept?
• Do you see Reasoning and Proving as a way to have students to show their work (like the textbook might suggest) or do you see Reasoning and Proving as a process of sense-making (as Marian Small shares)?
• Do your students experience moments of cognitive disequilibrium… followed by time for them to struggle independently or with a partner?  Are they regularly engaged in sense-making opportunities, sharing their thinking, debating…?
• The example I shared here isn’t the most flashy example of surprise, but I used it purposefully because I wanted to illustrate that any topic can be turned into an opportunity for students to do the thinking.  I would love to discuss a topic that you feel students can’t reason through… Let’s think together about if it’s possible to create an experience where students can experience mathematical surprise… or puzzlement… or be engaged in sense-making…  Let’s think together about how we can make Reasoning and Proving a focus for you and your students!

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

## Reasoning and Proving

This week I had the pleasure to see Dan Meyer, Cathy Fosnot and Graham Fletcher at OAME’s Leadership conference.

Each of the sessions were inspiring and informative… but halfway through the conference I noticed a common message that the first 2 keynote speakers were suggesting:

Dan Meyer showed us several examples of what mathematical surprise looks like in mathematics class (so students will be interested in making sense of what they are learning), while Cathy Fosnot shared with us how important it is for students to be puzzled in the process of developing as young mathematicians.  Both messages revolved around what I would consider the most important Process Expectation in the Ontario curriculum – Reasoning and Proving.

###### Reasoning and Proving

While some see Reasoning and Proving as being about how well an answer is constructed for a given problem – how well communicated/justified a solution is – this is not at all how I see it.  Reasoning is about sense-making… it’s about generalizing why things work… it’s about knowing if something will always, sometimes or never be true…it is about the “that’s why it works” kinds of experiences we want our students engaged in.  Reasoning is really what mathematics is all about.  It’s the pursuit of trying to help our students think mathematically (hence the name of my blog site).

###### A Non-Example of Reasoning and Proving

In the Ontario curriculum, students in grade 7 are expected to be able to:

• identify, through investigation, the minimum side and angle information (i.e.,side-side-side; side-angle-side; angle-side-angle) needed to describe a unique triangle

Many textbooks take an expectation like this and remove the need for reasoning.  Take a look:

As you can see, the textbook here shares that there are 3 “conditions for congruence”.  It shares the objective at the top of the page.  Really there is nothing left to figure out, just a few questions to complete.  You might also notice, that the phrase “explain your reasoning” is used here… but isn’t used in the sense-making way suggested earlier… it is used as a synonym for “show your work”.  This isn’t reasoning!  And there is no “identifying through investigation” here at all – as the verbs in our expectation indicate!

A Example of Reasoning and Proving

Instead of starting with a description of which sets of information are possible minimal information for triangle congruence, we started with this prompt:

Given a few minutes, each student created their own triangles, measured the side lengths and angles, then thought of which 3 pieces of information (out of the 6 measurements they measured) they would share.  We noticed that each successful student either shared 2 angles, with a side length in between the angles (ASA), or 2 side lengths with the angle in between the sides (SAS).  We could have let the lesson end there, but we decided to ask if any of the other possible sets of 3 pieces of information could work:

While most textbooks share that there are 3 possible sets of minimal information, 2 of which our students easily figured out, we wondered if any of the other sets listed above will be enough information to create a unique triangle.  Asking the original question didn’t offer puzzlement or surprise because everyone answered the problem without much struggle.  As math teachers we might be sure about ASA, SAS and SSS, but I want you to try the other possible pieces of information yourself:

Create triangle ABC where AB=8cm, BC=6cm, ∠BCA=60°

Create triangle FGH where ∠FGH=45°, ∠GHF=100°, HF=12cm

Create triangle JKL where ∠JKL=30°, ∠KLJ=70°, ∠LJK=80°

If you were given the information above, could you guarantee that everyone would create the exact same triangles?  What if I suggested that if you were to provide ANY 4 pieces of information, you would definitely be able to create a unique triangle… would that be true?  Is it possible to supply only 2 pieces of information and have someone create a unique triangle?  You might be surprised here… but that requires you to do the math yourself:)

###### Final Thoughts

Graham Fletcher in his closing remarks asked us a few important questions:

• Are you the kind or teacher who teaches the content, then offers problems (like the textbook page in the beginning)?  Or are you the kind of teacher who uses a problem to help your students learn?
• How are you using surprise or puzzlement in your classroom?  Where do you look for ideas?
• If you find yourself covering information, instead of helping your students learn to think mathematically, you might want to take a look at resources that aim to help you teach THROUGH problem solving (I got the problem used here in Marian Small’s new Open Questions resource).  Where else might you look?
• What does Day 1 look like when learning a new concept?
• Do you see Reasoning and Proving as a way to have students to show their work (like the textbook might suggest) or do you see Reasoning and Proving as a process of sense-making (as Marian Small shares)?
• Do your students experience moments of cognitive disequilibrium… followed by time for them to struggle independently or with a partner?  Are they regularly engaged in sense-making opportunities, sharing their thinking, debating…?
• The example I shared here isn’t the most flashy example of surprise, but I used it purposefully because I wanted to illustrate that any topic can be turned into an opportunity for students to do the thinking.  I would love to discuss a topic that you feel students can’t reason through… Let’s think together about if it’s possible to create an experience where students can experience mathematical surprise… or puzzlement… or be engaged in sense-making…  Let’s think together about how we can make Reasoning and Proving a focus for you and your students!

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

## The role of “practice” in mathematics class

A few weeks ago a NYTimes published an article titled, Make Your Daughter Practice Math. She’ll Thank You Later, an opinion piece that, basically, asserts that girls would benefit from “extra required practice”.  I took a few minutes to look through the comments (which there are over 600) and noticed a polarizing set of personal comments related to what has worked or hasn’t worked for each person, or their own children.  Some sharing how practicing was an essential component for making them/their kids successful at mathematics, and others discussing stories related to frustration, humiliation and the need for children to enjoy and be interested in the subject.

Instead of picking apart the article and sharing the various issues I have with it (like the notion of “extra practice” should be given based on gender), or simply stating my own opinions, I think it would be far more productive to consider why practice might be important and specifically consider some key elements of what might make practice beneficial to more students.

To many, the term “practice” brings about childhood memories of completing pages of repeated random questions, or drills sheets where the same algorithm is used over and over again.  Students who successfully completed the first few questions typically had no issues completing each and every question.  For those who were successful, the belief is that the repetition helped.  For those who were less successful, the belief is that repeating an algorithm that didn’t make sense in the first place wasn’t helpful…  even if they can get an answer, they might still not understand (*Defining 2 opposing definitions of “understanding” here).

“Practice” for both of the views above is often thought of as rote tasks that are devoid of thinking, choices or sense making.  Before I share with you an alternative view of practice, I’d like to first consider how we have tackled “practice” for students who are developing as readers.

If we were to consider reading instruction for a moment, everyone would agree that it would be important to practice reading, however, most of us wouldn’t have thoughts of reading pages of random words on a page, we would likely think about picture books.  Books offer many important factors for young readers.  Pictures might help give clues to difficult words, the storyline offers interest and motivation to continue, and the messages within the book might bring about rich discussions related to the purpose of the book.  This kind of practice is both encourages students to continue reading, and helps them continue to get better at the same time.  However, this is very different from what we view as math “practice”.

In Dan Finkel’s Ted Talk (Five Principles of Extraordinary Math Teaching) he has attempted to help teachers and parents see the equivalent kind of practice for mathematics:

Below is a chart explaining the role of practice as it relates to what Dan Finkel calls play:

Take a look at the “Process” row for a moment.  Here you can see the difference between a repetitive drill kind of practice and the “playful experiences” kind of  practice Dan had called for.  Let’s take a quick example of how practice can be playful.

Students learning to add 2-digit numbers were asked to “practice” their understanding of addition by playing a game called “How Close to 100?”.  The rules:

• Roll 2 dice to create a 2-digit number (your choice of 41 or 14)
• Use base-10 materials as appropriate
• Try to get as close to 100 as possible
• 4th role you are allowed eliminating any 1 number IF you want

What choice would you make???  Some students might want to keep all 4 roles and use the 14 to get close to 100, while other students might take the 41 and try to eliminate one of the roles to see if they can get closer.

When practice involves active thinking and reasoning, our students get the practice they need and the motivation to sustain learning!  When practice allows students to gain a deeper understanding (in this case the visual of the base-10 materials) or make connections between concepts, our students are doing more than passive rule following – they are engaging in thinking mathematically!

In the end, we need to take greater care in making sure that the experiences we provide our students are aimed at the 5 strands shown below:

You might also be interested in thinking about how we might practice Geometrical terms/properties, or spatial reasoning, or exponents, or Bisectors

#### So I will leave you with some final thoughts:

• What does “practice” look like in your classroom?  Does it involve thinking or decisions?  Would it be more engaging for your students to make practice involve more thinking?
• How does this topic relate to the topic of “engagement”?  Is engagement about making tasks more fun or about making tasks require more thought?  Which view of engagement do you and your students subscribe to?
• What does practice look like for your students outside of school?  Is there a place for practice at home?
• Which of the 5 strands (shown above) are regularly present in your “practice” activities?  Are there strands you would like to make sure are embedded more regularly?

## How many do you see?(Part 1)

A few days ago I had the opportunity to work with a grade 2 teacher as her class was learning about Geometry.  The students started the class with a rich activity comparing and sorting a variety of standard and non-standard shapes, followed by a great discussion about several properties they had noticed.

Shortly after, students started working on following the page as independent work. Take a look:

Take a minute to try to figure out what you think the answers might be.  Scroll up and pick one of the less obvious shapes and count how many you see.

This isn’t one of those Facebook “can you find all the hidden shapes” tasks, it’s meant to be a straightforward activity for grade 2 students. However, I’m not sure what the actual answers are here.  So, I need some help…  I’d love if you could:

• Pick one shape (or more if you’re adventurous)
• Think about what you believe the teacher’s edition would say
• Count how many you see
• Share the 3 points above as a comment here or on Twitter

I’m hoping in my next post that we can discuss more than just this worksheet and make some generalizations for any grade and any topic.

## Building our Students’ Mathematical Intuition

I’ve been asked to share my OAME 2017 presentation on Mathematical Intuitions by a few of my participants.  Instead of just sharing the slides, I thought I would add a bit of the conversations we had, and the purposes behind a few of my slides.  Here is a brief explanation of the 75 minutes we shared together:

I started with an image of the OAME 2017 official graphic and asked everyone what mathematics they saw in the photo:

I was impressed that many of us noticed various things from numbers, to sizes of fonts, to shapes and other geometric features, to measurement concepts to patterns…

I decided to start with an image so I could listen to everyone’s ideas (the group could have simply noticed the numbers visible on the page, or the triangles, but thankfully the group noticed a lot more!).

I then shared a few stories where students have entered into a problem where they have attempted to do a bunch of procedures or calculations without ever doing any thinking, either before or after, to make sure they are making sense of things.

You can read the full stories on these 2 slides here and here.

The bandana problem above is a really interesting one for me because it shows just how likely our previous learning can actually get in the way of students who are attempting to make sense of things.  Most students who learned about how to convert in previous years in a procedural way have difficulty realizing that 1 meter squared is actually 10,000 cm squared!

In an attempt to explain the kinds of mental actions we actually want our students to use when learning and doing mathematics I showed an image shared by Tracy Zager (from her new book Becoming the Math Teacher You Wish You’d Had).  We discussed just how interrelated Logic and Intuition are.  Students who are using their intuition start by making sense of things.  They start by making choices or estimates, which are often based on their previous experiences, and use logic to continue to refine and think through what makes sense.  This process, while often not even realized by those who are confident with their mathematics, is one I believe we need to foster and bring to the forefront of our discussions.

I then shared the puzzle above with the group and asked them to find the value of the question mark.  Most did exactly what I assumed they would do… but none did what the following student did:

Most teachers aimed to find the value of each image (which isn’t as easy as it looks for many elementary teachers), but the student above didn’t.  They instead realized that all of the shapes if you add them up in any direction would equal 94.  This student had never been given a problem like this, so didn’t have any preconceived notions about how to solve it.  They instead, thought about what makes sense.

So, how DO we help our students use their intuition?  Here are a few ideas I shared:

1. Contemplate then Calculate routine (See David Wees for more about this here or here, or purchase Routines for Reasoning by Grace, Amy and Susan)

The two images above show visual representations (thank you Andrew Gael and Fawn Nguyen for your images) where I asked everyone to attempt to think before they did any calculations.  I used Andrew’s picture of the dominoes and asked “will the two sides balance… don’t do any calculations though”.  For Fawn’s Visual Pattern, I asked the group to explain what the 10th image would LOOK LIKE (before I wanted them to figure out how many of each shape would be there, and then find a rule for the nth term).

We shared a few estimation strategies:

and a few “Notice and Wonder” ideas:

However, while I love each of the strategies discussed here (Contemplate then Calculate, Estimation routines, Notice and Wonder) I’m not sure that doing a routine like these, then going about the actual learning of the day is going to be effective!

Instead, we need to make sure that noticing things, estimating, thinking happen all the time.  These need to be a part of every new piece of learning, not just fun or neat warm-ups!

Building our students’ intuition means that we need to provide opportunities for them them to think and make sense of things, and have plenty of opportunities for them to discuss their thinking!

If our goal is for students to think mathematically, and use their logic and intuition regularly, we need to operate by a few simple beliefs:

I ended the presentation with a final thought:

Here is a copy of the presentation if you are interested:

I’d suggest you scroll down to slide 49 and play the quick video of one of my students doing a spatial reasoning puzzle.  It’s one of my favourites because it illustrates visually the thinking processes used when a student is using both their intuition and logic.

To me, there seems to be so much more I need to learn about how to help my students who seem to struggle in math class use their intuition.  Hopefully this conversation is just the beginning of us learning more about the topic!

A few questions I want to leave you with:

• What routines do you have in place that help your students make sense of things, use their intuitions and develop mathematical reasoning?
• Do your students use their intuition in other situations as well (or just during these routines)?
• How can you start to build in opportunities for your students to use their intuition as a regular part of how your class is structured?
• What does it look like when our students who are struggling attempt to use their intuition?  How can we help all of our students develop and use these process regularly?

Special thanks to Tracy Zager’s new book for the inspiration for the presentation.

As always, I would love to continue the conversation here or on Twitter