## “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)

## One-Hole Punch Puzzle Templates

Recently Lowrie et al. published an article in the Journal of Experimental Education where they looked at the effects of a spatial intervention program for grade 8 students. This study followed the 876 grade 8 students across 9 schools as they received 20 hours of spatial interventions (as well as other grade 8 students in schools that conducted “business as usual” mathematics classes). Their findings were quite interesting. They found students who received spatial intervention programs achieved:

• significantly better on spatial tasks (13% higher)
• significantly better on Geometry – Measurement problems that included material not discussed in either class
• significantly better on Number – Algebra problems
• equally well on Statistic – Probability problems

While this study supports many studies showing a link between spatial abilities and mathematics performance (Mix and Cheng, 2012), and others showing how students’ spatial abilities are malleable at any age (can be learned with the right experiences) (Uttal et al., 2013), there seems to be a definite need for teachers to have at their fingertips good examples of experiences that will help our students develop spatial reasoning. At the end of this post are possible examples for us to try.

A recent meta-analysis of 217 studies, representing more than two decades of research on spatial training, found that a variety of activities improve spatial reasoning across all age groups. Not only did the authors find that spatial training led to improvements on spatial tasks closely related to the training task, but improvements were also seen on other types of tasks that were not part of the training.

Taking Shape, 2016

A focus on spatial reasoning, from my experience, has helped the students in my schools make sense of connections between concepts, it has been the underpinning of new learning, and has been the vehicle for so much of OUR learning as educators.

## One-Hole Punch puzzles

You might be familiar with various cognitive tests that ask students to think through mental rotations, 2D/3D visualization, paper folding or other tasks where students’ abilities to visualize are measured. Spatial tasks like these are very predictive of a students’ math success, however, few resources are aimed to specifically help our students develop their spatial reasoning (see Taking Shape as an ideal K-2 example).

Below is an example of one of these cognitive test questions. As you can see, students here are asked to mentally fold a piece of paper and then punch 1 hole through the folds, then imagine what the paper would look like once opened.

## Templates

Instead of trying to measure our students’ abilities by giving sample problems like this, it is far more productive to offer experiences where our students can learn to think spatially. This is why I have created a few sample experiences. Below are sample templates that you can use with your students. They will need squares of paper (linked below), as well as the puzzles they would be aiming to solve. Take a look:

## How to Solve a One-Hole Punch puzzle:

Directions for solving the puzzles are only written on the first page:

“These are One-Hole Punch puzzles.  To complete a puzzle, take a square of paper, fold it using as many folds as needed so that if you punch one hole and unfold it, you will match one of the puzzles.  Complete the following puzzles in any order.  Be prepared to discuss your strategies with others.”

Students will need several blank squares of paper and a single hole puncher. Student will need to visualize how to fold their paper and where to punch their one hole to match the puzzle they are working on. These puzzles are tricky because only 1 hole can be punched to create several holes.

Pro Tip: Make sure you have a lot of squares of paper for every student.

## A few thoughts about using these:

• How will you introduce these puzzles to your students?  How much information about strategies and tips will you provide?  Will this allow for productive struggle, or will you attempt to remove as much of the struggle as possible?
• Would you use these as an activity you give all students, or something you provide to just some.  Why? (Something for students who finish early or something for everyone to try?)
• How would giving a page of puzzles to a pair of students be different than if you gave it to individuals?  Which were you assuming to do here?  What if you tried the other option?
• How will you orchestrate a conversation for your class to help consolidate the learning here? How might this be helpful before you give the next set of puzzles?
• What will you do if students give up quickly?  What questions / prompts will you provide?
• Who is successful at solving these puzzles? Sometimes, those who are often left out of mathematical discussions can become more involved when given spatial tasks. How might you capitalize on increasing your students’ interest in mathematics? Do you see tasks like this helping improve your students’ agency, identity and authority in mathematics?
• How might you see intentional decisions to focus on spatial reasoning as helping your students with mathematics in other areas? This post might be helpful.

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!

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

## An Example of Teaching THROUGH problem solving?

Many math resources attempt to share the difference between teaching FOR problem solving and teaching THROUGH problem solving. Cathy Seeley refers to teaching THROUGH problem solving as “Upside-Down Teaching” which is the opposite of a “gradual release of responsibility” model:

And instead calls for us to flip how our students learn to a more active model:

So, instead of starting a unit on Geometry with naming shapes or developing definitions together, we decided to start with a little problem:

Create as many polygons as possible using exactly 2 pattern block pieces. Sort your polygons by how many sides they have.

As students started placing pattern block pieces together, all kinds of questions started emerging (questions we took note of to bring to the whole group in a few minutes):

By the end of a period, students had worked through the definitions of what a polygon is (and isn’t), the difference between concave and convex polygons, defined the term “regular polygon” (which was not what they had been calling “regular” before), and were able to name and create triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, decagons and undecagons. Recognizing a variety of possible ways a shape can look was very helpful for our students who might have experienced shapes more traditionally in the past.

One group compiled their polygons together (with one minor error):

Instead of starting with experiences where students accumulate knowledge (writing out definitions, taking a note, direct instruction), an upside down approach aims to start with students’ ideas. This way we would know which conversations to have with our students, and so our students are actively engaged in the process of learning.

## I want to leave you with a few reflective questions:

• Why might it benefit students to start with a problem instead of starting with the teachers’ ideas?
• Why might it benefit teachers to listen to students’ thinking before instruction has occurred?
• What does it mean to effectively monitor students as they are thinking / working? (See This POST for examples)
• Can all mathematics topics begin with tasks that help our students make connections between what they already know, and what they are learning? Can you think of a topic that can not be experienced this way?
• The final stage in the You-We-I model is where the teacher helps make specific learning explicit for their students. How do you find time to consolidate a task like this? How do you know what to share? (See This POST for an example)
• How might this form of teaching relate to how we view assessment? (See This POST)
• How might this form of teaching relate to how we view differentiated instruction? (See This POST)
• How do you find problems that ask students to actively think before any instruction has occurred? (See This POST for examples)

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

## Spatial Puzzles: Cuisenaire Cover-ups

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 then be 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…

Earlier this year I shared a post discussing how we might decompose and recompose numbers to do an operations (subtraction). But, I would like us to consider why some students are more comfortable decomposing and recomposing, and how we might be aiming to help our students early with experiences that might promote the kinds of thinking needed.

Doug Clements and Julie Sarama have looked at the relationship between students’ work with space and shapes with students understanding of numbers.

“The ability to describe, use, and visualize the effects of putting together and taking apart shapes is important because the creating, composing, and decomposing units and higher-order units are fundamental mathematics. Further, there is transfer: Composition of shapes supports children’s ability to compose and decompose numbers”

Contemporary Perspectives on Mathematics in Early Childhood Education p.82, Clements and Sarama

The connection between composing and decomposing shapes and numbers is quite exciting to me. However, I am also very interested in the meeting place between Spatial tasks (composing/decomposing shapes) and Number tasks that involve composing and decomposing.

A few years ago I found a neat little puzzle in a resource called The Super Source called “Cover the Giraffe”. The idea was to cover an image of a giraffe outline using exactly 1 of each size of cuisenaire rods. The task, simple enough, was actually quite difficult for students (and even for us as adults). After using the puzzle with a few different classes, I decided to make a few of my own.

After watching a few classrooms of students complete these puzzles, I noticed an interesting intersection between spatial reasoning, and algebraic reasoning happening…. First, let me share the puzzles with you:

## Objective:

To complete a Cuisenaire Cover-Up puzzle, you need exactly 1 of each colour cuisenaire rod. Use each colour rod once each to completely fill in the image.

Below are the 5 puzzles:

## Assessment Opportunities

Knowing what to look for, helps us know how to interact with our students.

• Which block are students placing first? The largest blocks or the smallest?
• Which students are using spatial cues (placing rods to see which fits) and which students are using numerical cues (counting units on the grid)? How might we help students who are only using one of these cueing systems without over-scaffolding or showing how WE would complete the puzzle?
• How do our students react when confronted with a challenging puzzle?
• Who is able to swap out 1 rod for 2 rods of equivalent length (1 orange rod is the same length as a brown and red rod together)?
• Which of the following strands of proficiency might you be noting as you observe students:

## Questions to Reflect on:

• Why might you use a task like this? What would be your goal?
• How will you interact with students who struggle to get started, or struggle to move passed a specific hurdle?
• How might these puzzles relate to algebraic reasoning? (try to complete one with this question in mind)
• How are you making the connections between spatial reasoning and algebraic reasoning clear for your students to see? How can these puzzles help?
• How might puzzles allow different students to be successful in your class?

If interested in these puzzles, you might be interested in trying:

## “The More Strategies, the Better?

As many teachers implement number talks/math strings and lessons where students are learning through problem solving, the idea that there are many ways to answer a question or problem becomes more important. However, I think we need to unpack the beliefs and practices surrounding what it means for our students to have different “strategies”. A few common beliefs and practices include:

Really, there are benefits and issues with each of these thoughts…. and the right answer is actually really much more complicated than any of these.  To help us consider where our own decisions lie, let’s start by considering an actual example. If students were given a pattern with the first 4 terms like this:

…and asked how many shapes there would be on the 24th design (how many squares and circles in total).  Students could tackle this in many ways:

• Draw out the 24th step by building on and keeping track of each step number
• Build the 24th step by adding on and keeping track of the step number
• Make a T-table and use skip counting to find each new step (5, 9, 13, 17…).
• Find the explicit rule from the first few images’ data placed on a T-table (“I see the pattern is 5, 9, 13, 17. each new image uses 4 new shapes, so the pattern is a multiplied by 4 pattern…. and I think the rule should be ‘number of images = step number x4+1’. Let me double check…”).
• Notice the “constant” and multiplicative aspects of the visual, then find the explicit rule (I see that each image increases by 4 new shapes on the right, so the multiplicative aspect of this pattern is x4, and term 0 might just be 1 circle. So the pattern must be x4+1″).
• Create a graph, then find the explicit rule based on starting point and growth (“When I graph this, my line hits the y-axis at 1, and increases by 4 each time, so the pattern rule must be x4+1”).

While each of these might offer a correct answer, we as the teacher need to assess (figure out what our students are doing/thinking) and then decide on how to react accordingly.  If a student is using an additive strategy (building each step, or creating a t-table with every line recorded using skip counting), their strategy is a very early model of understanding here and we might want to challenge this/these students to find or use other methods that use multiplicative reasoning.  Saying “do it another way” might be helpful here, but it might not be helpful for other students.  If on the other hand, a student DID use multiplicative reasoning, and we suggest “do it another way”, then they fill out a t-table with every line indicated, we might actually be promoting the use of less sophisticated reasoning.

On the other hand, if we tell/show students exactly how to find the multiplicative rule, and everyone is doing it well, then I would worry that students would struggle with future learning.  For example, if everyone is told to make a t-table, and find the recursive pattern (above would be a recursive pattern of+4 for total shapes), then use that as the multiplicative basis for the explicit rule x4 to make x4+1), then students are likely just following steps, and are not internalizing what specifically in the visual pattern here is +4 or x4… or where the constant of +1 is.  I would expect these students to really struggle with figuring out patterns like the following that is non-linear:

Students told to start with a t-table and find the explicit pattern rule are likely not even paying attention to what in the visual is growing, how it is growing or what is constant between each figure. So, potentially, moving students too quickly to the most sophisticated models will likely miss out on the development necessary for them to be successful later.

While multiple strategies are helpful to know, it is important for US to know which strategies are early understandings, and which are more sophisticated.  WE need to know which students to push and when to allow everyone to do it THEIR way, then hold a math congress together to discuss relationships between strategies, and which strategies might be more beneficial in which circumstance. It is the relationships between strategies that is the MOST important thing for us to consider!

## Focusing on OUR Understanding:

In order for us to know which sequence of learning is best for our students, and be able to respond to our students’ current understandings, we need to be aware of how any particular math concepts develops over time. Let’s be clear, understanding and using a progression like this takes time and experience for US to understand and become comfortable with.

While most educational resources are filled with lessons and assessment opportunities, very few offer ideas for us as teachers about what to look for as students are working, and how to respond to different students based on their current thinking. This is what Deborah Ball calls “Math Knowledge for Teaching”:

If any teacher wants to improve their practice, I believe this is the space that will have the most impact! If schools are interested in improving math instruction, helping teachers know what to look for, and how to respond is likely the best place to tackle. If districts are aiming for ways to improve, helping each teacher learn more about these progressions will likely be what’s going to make the biggest impacts!

## Where to Start?

If you want to deepen our understanding of the math we teach, including better understanding how math develops over time, I would suggest:

• Providing more open questions, and looking at student samples as a team of teachers
• Using math resources that have been specifically designed with progressions in mind (Cathy Fosnot’s Contexts for Learning and minilessons, Cathy Bruce & Ruth Beatty’s From Patterns to Algebra, Alex Lawson’s What to Look For…), and monitoring student strategies over time
• Anticipating possible student strategies, and using a continuum or landscape (Cathy Fosnot’s Landscapes, Lawson’s Continua, Clement’s Trajectories, Van Hiele’s levels of geometric thought…) as a guide to help you see how your students are progressing
• Collaborate with other educators using resources designed for teachers to deepen their understanding and provide examples for us to use with kids (Marian Small’s Understanding the Math we Teach, Van de Walle’s Teaching Student Centered Mathematics, Alex Lawsons’s What to Look For, Doug Clements’ Learning and Teaching Early Math…)
• Have discussions with other math educators about the math you teach and how students develop over time.

## Questions to Reflect on:

• How do you typically respond to your students when you give them opportunities to share their thinking? Which of the 3 beliefs/practices is most common for you? How might this post help you consider other beliefs/practices?
• How can you both honour students’ current understandings, yet still help students progress toward more sophisticated understandings?
• Given that your students’ understandings at the beginning of any new learning differ greatly, how do you both learn about your students’ thoughts and respond to them in ways that are productive? (This is different than testing kids prior knowledge or sorting students by ability. See Daro’s video)
• Who do you turn to to help you think more about the math you teach, or they ways you respond to students? What professional relationships might be helpful for you?

If interested in this topic, you might be interested in reading:

## 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 ).

## Rushing for Interventions

I see students working in groups all the time…  Students working collaboratively in pairs or small groups having rich discussions as they sort shapes by specific properties, students identifying and extending their partner’s visual patterns, students playing games aimed at improving their procedural fluency, students cooperating to make sense of a low-floor/high-ceiling problem…..

When we see students actively engaged in rich mathematics activities, working collaboratively, it provides opportunities for teachers to effectively monitor student learning (notice students’ thinking, provide opportunities for rich questioning, and lead to important feedback and next steps…) and prepare the teacher for the lesson close.  Classrooms that engage in these types of cooperative learning opportunities see students actively engaged in their learning.  And more specifically, we see students who show Agency, Ownership and Identity in their mathematics learning (See TruMath‘s description on page 10).

On the other hand, some classrooms might be pushing for a different vision of what groups can look like in a mathematics classroom.  One where a teachers’ role is to continually diagnose students’ weaknesses, then place students into ability groups based on their deficits, then provide specific learning for each of these groups.  To be honest, I understand the concept of small groups that are formed for this purpose, but I think that many teachers might be rushing for these interventions too quickly.

First, let’s understand that small group interventions have come from the RTI (Response to Intervention) model.  Below is a graphic created by Karen Karp shared in Van de Walle’s Teaching Student Centered Mathematics to help explain RTI:

As you can see, given a high quality mathematics program, 80-90% of students can learn successfully given the same learning experiences as everyone.  However, 5-10% of students (which likely are not always the same students) might struggle with a given topic and might need additional small-group interventions.  And an additional 1-5% might need might need even more specialized interventions at the individual level.

The RTI model assumes that we, as a group, have had several different learning experiences over several days before Tier 2 (or Tier 3) approaches are used.  This sounds much healthier than a model of instruction where students are tested on day one, and placed into fix-up groups based on their deficits, or a classroom where students are placed into homogeneous groupings that persist for extended periods of time.

Principles to Action (NCTM) suggests that what I’m talking about here is actually an equity issue!

We know that students who are placed into ability groups for extended periods of time come to have their mathematical identity fixed because of how they were placed.  That is, in an attempt to help our students learn, we might be damaging their self perceptions, and therefore, their long-term educational outcomes.

###### Tier 1 Instruction

While I completely agree that we need to be giving attention to students who might be struggling with mathematics, I believe the first thing we need to consider is what Tier 1 instruction looks like that is aimed at making learning accessible to everyone.  Tier 1 instruction can’t simply be direct instruction lessons and whole group learning.  To make learning mathematics more accessible to a wider range of students, we need to include more low-floor/high-ceiling tasks, continue to help our students spatalize the concepts they are learning, as well as have a better understanding of developmental progressions so we are able to effectively monitor student learning so we can both know the experiences our students will need to be successful and how we should be responding to their thinking.  Let’s not underestimate how many of our students suffer from an “experience gap”, not an “achievement gap”!

If you are interested in learning more about what Tier 1 instruction can look like as a way to support a wider range of students, please take a look at one of the following:

###### Tier 2 Instruction

Tier 2 instruction is important.  It allows us to give additional opportunities for students to learn the things they have been learning over the past few days/weeks in a small group.  Learning in a small group with students who are currently struggling with the content they are learning can give us opportunities to better know our students’ thinking.  However, I believe some might be jumping past Tier 1 instruction (in part or completely) in an attempt to make sure that we are intervening. To be honest, this doesn’t make instructional sense to me! If we care about our content, and care about our students’ relationship with mathematics, this might be the wrong first move.

So, let’s make sure that Tier 2 instruction is:

• Provided after several learning experiences for our students
• Flexibly created, and easily changed based on the content being learned at the time
• Focused on student strengths and areas of need, not just weaknesses
• Aimed at honoring students’ agency, ownership and identity as mathematicians
• Temporary!

If you are interested in learning more about what Tier 2 interventions can look like take a look at one of the following:

Instead of seeing mathematics as being learned every day as an approach to intervene, let’s continue to learn more about what Tier 1 instruction can look like!  Or maybe you need to hear it from John Hattie:

Or from Jo Boaler:

###### Final Thoughts

If you are currently in a school that uses small group instruction in mathematics, I would suggest that you reflect on a few things:

• How do your students see themselves as mathematicians?  How might the topics of Agency, Authority and Identity relate to small group instruction?
• What fixed mindset messaging do teachers in your building share “high kids”, “level 2 students”, “she’s one of my low students”….?  What fixed mindset messages might your students be hearing?
• When in a learning cycle do you employ small groups?  Every day?  After several days of learning a concept?
• How flexible are your groups?  Are they based on a wholistic leveling of your students, or based specifically on the concept they are learning this week?
• How much time do these small groups receive?  Is it beyond regular instructional timelines, or do these groups form your Tier 1 instructional time?
• If Karp/Van de Walle suggests that 80-90% of students can be successful in Tier 1, how does this match what you are seeing?  Is there a need to learn more about how Tier 1 approaches can meet the needs of this many students?
• What are the rest of your students doing when you are working with a small group?  Is it as mathematically rich as the few you’re working with in front of you?
• Do you believe that all of your students are capable to learn mathematics and to think mathematically?

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?

## Noticing and Wondering: A powerful tool for assessment

Last week I had the privilege of presenting with Nehlan Binfield at OAME on the topic of assessment in mathematics.  We aimed to position assessment as both a crucial aspect of teaching, yet simplify what it means for us to assess effectively and how we might use our assessments to help our students and class learn.  If interested, here is an abreviated version of our presentation:

We started off by running through a Notice and Wonder with the group.  Given the image above, we noticed colours, sizes, patterns, symmetries (line symmetry and rotational symmetry), some pieces that looked like “trees” and other pieces that looked like “trees without stumps”…

Followed by us wondering about how many this image would be worth if a white was equal to 1, and what the next term in a pattern would look like if this was part of a growing pattern…

We didn’t have time, but if you are interested you can see the whole exchange of how the images were originally created in Daniel Finkel’s quick video.

We then continued down the path of noticing and wondering about the image above.  After several minutes, we had come together to really understand the strategy called Notice and Wonder:

As well as taking a quick look at how we can record our students’ thinking:

At this point in our session, we changed our focus from Noticing and Wondering about images of mathematics, to noticing and wondering about our students’ thinking.  To do this, we viewed the following video (click here to view) of a student attempting to find the answer of what eight, nine-cent stamps would be worth:

The group noticed the student in the video counting, pausing before each new decade, using two hands to “track” her thinking…  The group noticed that she used most of a 10-frame to think about counting by ones into groups of 9.

We then asked the group to consider the wonders about this student or her thinking and use these wonders to think about what they would say or do next.

• Would you show her a strategy?
• Would you suggest a tool?
Would you give her a different question?

It seemed to us, that the most common next steps might not be the ones that were effectively using our assessment of what this child was actually doing.

Looking through Fosnot’s landscape we noticed that this student was using a “counting by ones” strategy (at least when confronted with 9s), and that skip-counting and repeated addition were the next strategies on her horizon.

While many teachers might want to jump into helping and showing, we invited teachers to first consider whether or not we were paying attention to what she WAS actually doing, as opposed to what she wasn’t doing.

This led nicely into a conversation about the difference between Assessment and Evaluation.  We noticed that we many talk to us about “assessment”, they actually are thinking about “evaluation”.  Yet, if we are to better understand teaching and learning of mathematics, assessment seems like a far better option!

So, if we want to get better at listening interpretively, then we need to be noticing more:

Yet still… it is far too common for schools to use evaluative comments.  The phrases below do not sit right with me… and together we need to find ways to change the current narrative in our schools!!!

Evaluation practices, ranking kids, benchmarking tests… all seem to be aimed at perpetuating the narrative that some kids can’t do math… and distracts us from understanding our students’ current thinking.

So, we aimed our presentation at seeing other possibilities:

To continue the presentation, we shared a few other videos of student in the processs of thinking (click here to view the video).  We paused the video directly after this student said “30ish” and asked the group again to notice and wonder… followed by thinking about what we would say/do next.

Followed by another quick video (click here to view).  We watched the video up until she says “so it’s like 14…”.  Again, we noticed and wondered about this students’ thinking… and asked the group what they would say or do next.

After watching the whole video, we discussed the kinds of questions we ask students:

If we are truly aimed at “assessment”, which basically is the process of understanding our students’ thinking, then we need to be aware of the kinds of questions we ask, and our purpose for asking those questions!  (For more about this see link).

We finished our presentation off with a framework that is helpful for us to use when thinking about how our assessment data can move our class forward:

We shared a selection of student work and asked the group to think about what they noticed… what they wonderered… then what they would do next.

So, let’s remember what is really meant by “assessing” our students…

…and be aware that this might be challenging for us…

…but in the end, if we continue to listen to our students’ thinking, ask questions that will help us understand their thoughts, continue to press our students’ thinking, and bring the learning together in ways where our students are learning WITH and FROM each other, then we will be taking “a giant step toward becoming a master teacher”!

#### So I’ll leave you with some final thoughts:

• What do comments sound like in your school(s)?  Are they asset based (examples of what your students ARE doing) or deficit based (“they can’t multiply”… “my low kids don’t get it…”)?
• What do you do if you are interested in getting better at improving your assessment practices like we’ve discussed here, but your district is asking for data on spreadsheets that are designed to rank kids evaluatively?
• What do we need to do to change the conversation from “level 2 kids” (evaluative statements that negatively impact our students) to conversations about what our students CAN do and ARE currently doing?
•  What math knowledge is needed for us to be able to notice mathematicially important milestones in our students?  Can trajectories or landscapes or continua help us know what to notice better?