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.

For details about how to solve a Skyscraper Puzzle, please click here

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:

How would giving a puzzle 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 might using physical blocks be different than paper-and-pencil or electronic versions?

If students are working on these away from school, what resources might you use instead of school manipulatives like snap cubes? Lego bricks? Checkers? Other household items?

Ask any teacher anywhere what some of the most pressing challenges are that they face as a teacher and likely you will hear examples of how difficult it can be to meet the various needs within a classroom. When conversations on the topic arise, there are often discussions from one of two extremes:

One one side you might hear about reasons why a teacher might believe that it is best to make sure that every student be expected to learn the same things. These beliefs often lead to practices where everyone receives the same instruction, followed by individual assistance for students who were not able to follow classroom instructions. Equity here is believed to be related to equal outcomes.

On the other hand, some teachers might believe that it isn’t possible to expect every student to learn the same things at the same time. Their beliefs lead them to focus more attention on determining readiness and offering different groups of students different learning opportunities. Equity here is viewed as meeting each child’s unique need.

While I understand each of these points of view, part of the issue between these two views is the overall view of what mathematics is. If mathematics is seen as a set of linearly learned skills, where each skill is boiled down to a list of subskills, each learned in a specific sequence, it is difficult to imagine anything else. However, when mathematics is seen through the lens of rich connections, we might start to see students’ development of these connections as what can drive our decisions.

One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world.

TIPS4RM: Developing Mathematical Literacy, 2005

The two views mentioned above do not account for a view of mathematics where connections between concepts is a focus. Instead of seeing the issue as simply whether or not we want a classroom of students to be doing the same things or if we should be providing some students with different things, we should also consider what is actually being learned by the students. Below you can see a matrix showing four different examples of how we could tackle the same concept in a classroom:

Same / Different Learning? Same / Different Tasks?

Same Tasks, Same Learning: The teacher offers everyone the same task, expects everyone to be able to follow the same procedures and might offer explicit help to specific students that aren’t following accordingly.

Different Tasks, Same Learning: While everyone is learning the same thing, the teacher offers some groups easier work and other groups more advanced work based on readiness.

Different Tasks, Different Learning: Based on diagnostic assessments, students are placed into groups based on what they need to continue learning. Some groups might be learning different materials within the same class.

Same Tasks, Different Learning: Every student is provided the same task, but there is variance in how and what is being learned.

For the readers here, I encourage you to think about which of the above models might you have experienced as a student, and which you might think would be best for your students.

Taking an Equity Stance

So, what does any of this have to do with equity? In my experience, a lot! Taking an equity stance means that we both believe that every student can achieve, and understand that every student might need different things from us. Keeping equity in mind requires us to analyze who has access to rich mathematical experiences and whose experiences are narrowed or reduced to lower-level skills (Access), whose ideas contribute to the group’s development of mathematical understanding and whose are not heard (Agency and Authority), who identifies with mathematics and who does not (Identity)… Without considering our beliefs and practices, we will never be able to notice which students are being underserviced, which structures promote some groups over others, or see which practices lead to the “Matthew Effect“.

How do we aim for Equity?

When thinking about how we aim for equity in mathematics, there seems to be 2 key tenets that help point us in the right direction:

Expand WHO is considered a math student

Expand WHAT is accepted as mathematics

The question is not whether all students can succeed in mathematics but whether the adults organizing mathematics learning opportunities can alter traditional beliefs and practices to promote success for all.

Principles to Action – NCTM (p.61)

WHO is considered a math person?

Teachers who come to recognize that some students identify with mathematics (and others do not) aim to promote tasks that allow more students to engage in mathematical reasoning via problems/tasks that are easily accessible (low-floor, high-ceiling tasks). If our students are going to see themselves as budding mathematicians, then we need to allow more opportunities for students to share their emerging ideas first!

Dr. Christine Suurtamm does a great job of articulating what this could look like in practice:

Students need to see themselves in the work they are doing. This includes knowing that mathematics is not created for and used by only some people (race/gender…), nor is it only useful for potential futures of some of our students, but is actually used by all of us RIGHT NOW. If we want to make sure our students see themselves as mathematicians, OUR STUDENTS need to be doing more of the thinking, they need to be part of the process of learning, not simply showing that they have mastered procedures.

Reflecting on WHO believes they are a math person might help us reflect on what messages our students have received over the years. If you haven’t already read about the “Matthew Effect“, I recommend that this might be a great place to help you reflect.

WHAT Counts as “Mathematics”?

To some, mathematics is a very narrow subject. Calculating (adding, subtracting, multiplying, dividing), converting (equivalent fractions), and carrying out other procedures accurately by using the requisite steps… Procedures dominate some textbooks and online practice sites and for some, this narrow vision of mathematics is where some students begin to struggle. However, if we are aiming for equity then we need to allow more opportunities for our students to show us what ARE good at.

One way to expand what counts as mathematics is for us to reflect on how much time we spend on each strand of mathematics (Patterning, Number Sense, Geometry, Measurement, Data Management). Analyzing how much time we spend on each of these strands, and specifically when in the year we might teach these concepts might help us reflect on what messages our students hear when they consider what counts as mathematics. For example, schools in my area typically start with several weeks of patterning because it can be experienced physically (manipulatives) and visually (visual patterns, graphing…), followed by several weeks of Geometry. These moves were strategic, because it allows our students more opportunities to talk, more opportunities to solve problems, more opportunities for our students to use visual/ spatial reasoning and more students to start their year with successes!

Expanding what mathematics means is much more than strands or concepts though. A focus on concrete and visual representations (not solely abstract symbolic representations) can be a path to expand what counts as mathematics. Allowing students to show their strategies, and accepting student strategies as part of the learning process means that preformal representations and strategies can be compared and learned from.

Spatial puzzles and games allow students to think mathematically in ways that differ from typical assignments. A story I often tell is of this young student who had never liked mathematics, and often struggled with mathematics. Here you can see her attempting to solve a difficult puzzle that one of her classmates created. Every child deserves to experience what this student experienced – productive struggle and success. Take a look:

Considerations

If we are aiming for equity in our own personal practices, we need to be aware of our own biases, our own patterns. This isn’t easy! It might mean noticing how we talk about race or gender or socio-economic groups and what our expectations are for each. It might mean reflecting on words we use to discuss students who might currently be struggling to learn mathematics or who are identified learners and what our expectations are of these students. Again, learning more about the Matthew Effect is where I would recommend you start. Planning with providing greater access for students to learn mathematics (same tasks/different learning – spatializing mathematics) is likely a first concrete step we can take.

I want to leave you with a few reflective questions:

How do you see the Same/Different Learning – Same/Different Tasks chart relating to equity? Which quadrant would you like provide for your students to be engaged with more frequently? What barriers are standing in the way?

We need to be aware that when schools group students by ability or assign different tasks to different students, those that are relegated to lower groups/tasks often receive lower level of cognitive demand tasks, which often puts them at a further disadvantage than their peers. How do you combat these inequities in your classroom?

Providing students with rich tasks and access to rich problems isn’t enough. We also need to be noticing our students’ thinking so we know how to respond to our students individually and as a group. This isn’t easy! How do you pay attention to their thinking? What structures do you have in place to listen to students and respond accordingly?

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.

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:

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.

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?

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 ).

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 ).

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.

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?

I’d love to continue the conversation about how we can use these puzzles to further our students’ spatial/algebraic reasoning. Leave a comment here or on Twitter @MarkChubb3

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

Many mathematicians are good at searching for patterns in numbers, however, an area that I think we all need to continue to explore is Visualizing.

Instead of just looking for procedural rules, or numeric patterns I encourage you to take one of the following and actually VISUALIZE what is going on.

Pick one of the above that interests you. Answer some of these questions:

What relationships do you notice here?

What are you curious about?

What visual might be helpful to represent this/these relationships?

Will these relationships work in other instances? When will it work/ when won’t it work?

How might a visual help others see the relationships you’ve noticed?

I’d love to hear some answers. You can respond here below, or via Twitter @MarkChubb3

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?

What resources do you consult to help you develop your own understanding?

I’d love to continue the conversation about how we respond to our students’ thinking. Leave a comment here or on Twitter @MarkChubb3

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

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!

Who do you turn to to help you think more about the questions you ask? What professional relationships might be helpful for you?

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”

I’d love to continue the conversation about the questions we ask of our students. Leave a comment here or on Twitter @MarkChubb3

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:

A few days ago I had the privilege of presenting at OAME in Ottawa on the topic of “Making Math Visual”. If interested, here are some of my talking points for you to reflect on:

To get us started, we discussed an image created by Christopher Danielson and asked the group what they noticed:

We noticed quite a lot in the image… and did a “how many” activity sharing various numbers we saw in the image. After our discussions I explained that I had shared the same picture with a group of parents at a school’s parent night followed by the next picture.

The picture above was more difficult for us as teachers to see the mathematics. While we, as math teachers, saw patterns in the placements of utensils, shapes and angles around the room, quantities of countable items, multiplicative relationships between utensils and place settings, volume of wine glasses, differences in heights of chairs, perimeter around the table….. the group correctly guessed that many parents do not typically notice the mathematics around them.

So, my suggestion for the teachers in the room was to help change this:

While I think it is important that we tackle the idea of seeing the world around us as being mathematical, a focus on making math visual needs to by MUCH more than this. To illustrate the kinds of visuals our students need to be experiencing, we completed a simple task independently:

After a few minutes of thinking, we discussed research of the different ways we use fractions, along with the various visuals that are necessary for our students to explore in order for them to develop as fractional thinkers:

When we looked at the ways we typically use fractions, it’s easy to notice that WE, as teachers, might need to consider how a focus on representations might help us notice if we are providing our students with a robust (let’s call this a “relational“) view of the concepts our students are learning about.

Data taken from 1 school’s teachers:

Above you see the 6 ways of visualizing fractions, but if you zoom in, you will likely notice that much of the “quotient” understanding doesn’t include a visual at all… Really, the vast majority of fractional representations here from this school were “Part – Whole relationships (continuous) models”. If, our goal is to “make math visual” then I believe we really need to spend more time considering WHICH visuals are going to be the most helpful and how those models progress over time!

We continued to talk about Liping Ma’s work where she asked teachers to answer and represent the following problem:

As you can see, being able to share a story or visual model for certain mathematics concepts seems to be a relative need. My suggestion was to really consider how a focus on visual models might be a place we can ALL learn from.

We then followed by a quick story of when a student told me that the following statement is true (click here for the full story) and my learning that came from it!

So, why should we focus on making math visual?

We then explored a statement that Jo Boaler shared in her Norms document:

…and I asked the group to consider if there is something we learn in elementary school that can’t be represented visually?

If you have an idea to the previous question, I’d love to hear it, because none of us could think of a concept that can’t be represented visually.

I then shared a quick problem that grade 7 students in one of my schools had done (see here for the description):

Along with a few different responses that students had completed:

Most of the students in the class had responded much like the image above. Most students in the class had confused linear metric relationships (1 meter = 100 cm) with metric units of area (1 meter squared is NOT the same as 100cm2).

In fact, only two students had figured out the correct answer… which makes sense, since the students in the class didn’t learn about converting units of area through visuals.

If you are wanting to help think about HOW to “make math visual”, below is some of the suggestions we shared:

And finally some advice about what we DON’T mean when talking about making mathematics visual:

You might recognize the image above from Graham Fletcher’s post/video where he removed all of the fractional numbers off each face in an attempt to make sure that the tools were used to help students learn mathematics, instead of just using them to get answers.

I want to leave you with a few reflective questions:

Can all mathematics concepts in elementary be represented visually?

Why might a visual representation be helpful?

If a student can get a correct answer, but can’t represent what is going on, do they really “understand” the concept?

Are some representations more helpful than others?

How important is it that our students notice the mathematics around them?

How might a focus on visual representations help both us and our students deepen our understanding of the mathematics we are teaching/learning?

Where do you turn to help you learn more about or get specific examples of how to effectively use visuals?

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

If you are interested in all of the slides, you can take a look here

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:

How will you introduce these puzzles to your students? Would you use the cube version I shared first? 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?

How would giving a puzzle 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 might using Relational rods be different than paper-and-pencil or electronic versions? How might it be different than the version with cubes?

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!

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?

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 ).