## 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?

If you are interested in reading more on similar topics, might I suggest:

Or take a look at the whole slide show here

## Making Math Visual

A few days ago I had the privilege of presenting at MAC2 to a group of teachers in Orillia 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 I shared 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.

I asked the group of teachers what mathematics they noticed here… then how they believed parents might have answered the question.  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:

I then asked the group to do a simple task for us to learn from:

After a few minutes of thinking, I shared some research of the different ways we use fractions:

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:

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

Followed by a quick story of when a student told me that the following statement is true (click here for the full story).

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.

We wrapped up with a few suggestions:

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 school be represented visually?
• Why might a visual representation be helpful?
• 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?

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

## The Importance of Contexts and Visuals

My wife Anne-Marie isn’t always impressed when I talk about mathematics, especially when I ask her to try something out for me, but on occasion I can get her to really think mathematically without her realizing how much math she is actually doing.  Here’s a quick story about one of those times, along with some considerations:

A while back Anne-Marie and I were preparing lunch for our three children.  It was a cold wintery day, so they asked for Lipton Chicken Noodle Soup.  If you’ve ever made Lipton Soup before you would know that you add a package of soup mix into 4 cups of water.

Typically, my wife would grab the largest of our nesting measuring cups (the one marked 1 cup), filling it four times to get the total required 4 cups, however, on this particular day, the largest cup available was the 3/4 cup.

Here is how the conversation went:

Anne-Marie:  How many of these (3/4 cups) do I need to make 4 cups?

Me:  I don’t know.  How many do you think?  (attempting to give her time to think)

Anne-Marie:  Well… I know two would make a cup and a half… so… 4 of these would make 3 cups…

Me: OK…

Anne-Marie:  So, 5 would make 3 and 3/4 cups.

Me:  Mmhmm….

Anne-Marie:  So, I’d need a quarter cup more?

Me:  So, how much of that should you fill?  (pointing to the 3/4 cup in her hand)

Anne-Marie:  A quarter of it?  No, wait… I want a quarter of a cup, not a quarter of this…

Me:  Ok…

Anne-Marie:  Should I fill it 1/3 of the way?

Me:  Why do you think 1/3?

Anne-Marie:  Because this is 3/4s, and I only need 1 of the quarters.

The example I shared above illustrates sense making of a difficult concept – division of fractions – a topic that to many is far from our ability of sense making.  My wife, however, quite easily made sense of the situation using her reasoning instead of a formula or an algorithm.  To many students, however, division of fractions is learned first through a set of procedures.

I have wondered for quite some time why so many classrooms start with procedures and algorithms unill I came across Liping Ma’s book Knowing and Teaching Mathematics.  In her book she shares what happened when she asked American and Chinese teachers these 2 problems:

Here were the results:

Now, keep in mind that the sample sizes for each group were relatively small (23 US teachers and 72 Chinese teachers were asked to complete two tasks), however, it does bring bring about a number of important questions:

• How does the training of American and Chinese teachers differ?
• What does it mean to “Understand” division of fractions?  Computing correctly?  Beging able to visually represent what is gonig on when fractions are divided? Being able to know when we are being asked to divide?  Being able to create our own division of fraction problems?
• What experiences do we need as teachers to understand this concept?  What experiences should we be providing our students?

#### Visual Representations

In order to understand division of fractions, I believe we need to understand what is actually going on.  To do this, visuals are a necessity!  A few examples of visual representations could include:

A number line:

A volume model:

An area model:

#### Starting with a Context

Starting with a context is about allowing our students to access a concept using what they already know (it is not about trying to make the math practical or show students when a concept might be used someday).  Starting with a context should be about inviting sense-making and thinking into the conversation before any algorithm or set of procedures are introduced.  I’ve already shared an example of a context (preparing soup) that could be used to launch a discussion about division of fractions, but now it’s your turn:

Design your own problem that others could use to launch a discussion of division of fractions.  Share your problem!

## Strategies vs Models

Earlier this week Pam Harris wrote a thought-provoking article called “Strategies Versus Models: why this is important”. If you haven’t already read it, read it first, then come back to hear some additional thoughts…..

Many teachers around the world have started blogs about teaching, often to fulfill one or both of the following goals:

• To share ideas/lessons with others that will inspire continued sharing of ideas/lessons; or
• To share their reflections about how students learn and therefore what kinds of experiences we should be providing our students.

The first of these goals serves us well immediately (planning for tomorrow’s lesson or an idea to save for later) while the second goal helps us grow as reflective and knowledgeable educators (ideas that transcend lessons).  Pam’s post (which I really hope you’ve read by now) is obviously aiming for goal number two here.

#### Models vs Strategies

In her article, Pam has accurately described a common issue in math education – conflating models (visual representations) with strategies (methods used to figure out an answer).  Below I’ve included a caption of Cathy Fosnot’s landscape of multiplication/division.  The rectangles represent landmark strategies that students use (starting from the bottom you will find early strategies, to the top where you will find more sophisticated strategies).  Whereas the triangles represent models or representations that are used (notice models correspond to strategies nearest to them).

In her post, Pam discusses 3 problems that arise when we do not fully understand the different roles of models and strategies:

1. Students (and teachers) think that all strategies are equal.
2. Students are left thinking that there are an unlimited, vast number of “strategies” to solve a problem.
3. Students get correct answers and are told to “do it a different way”.

I’d like to discuss how this all fits together…

Liping Ma discussed in her book Knowing and Teaching Elementary Mathematics four pieces that relate to a teacher having a Profound Understanding of Fundamental Mathematics (PUFM).  One of these features she called “Multiple Perspectives“, basically stating that PUFM teachers stress the idea that multiple solutions are possible, yet also stress the advantages and disadvantages of using certain methods in certain situations (hopefully you see the relationship between perspectives and strategies). She claimed that a PUFM teacher’s aim is to use multiple perspectives to help their students gain a flexible understanding of the content.

Many teachers have started down the path of understanding the importance of multiple perspectives.  For example, they provide problems that are open enough so students can answer them in different ways.  However, it is difficult for many teachers to both accept all strategies as valid, while also helping students see that some strategies are more mathematically sophisticated.

As teachers, we need to continue to learn about how to use our students’ thinking so they can learn WITH and FROM each other.  However, this requires that we continue to better understand developmental trajectories (like Fosnot’s landscape shared above) which will help us avoid the issues Pam had discussed in her original post.

If we want to get better at helping our students know which strategies are more appropriate, then we need to learn more about developmental trajectories.

If we want teachers to know when it is appropriate to say, “can you do it a different way?” and when it is counter-productive, then we need to learn more about developmental trajectories.

If we want to know which visual representations we should be using in our lessons, then we need to learn more about developmental trajectories.

If we want to continue to improve as mathematics teachers, then we need to learn more about developmental trajectories!

While I agree that it is essential that we get better at distinguishing between strategies and models, I think the best way to do this is to be immersed into the works of those who can help us learn more about how mathematics develops over time.  May I suggest taking a look at one of the following documents to help us discuss development:

Might I also suggest reading more on similar topics:

## Zukei Puzzles

A little more than a year ago now, Sarah Carter shared a set of Japanese puzzles called Zukei Puzzles (see her original post here or access her puzzles here).  After having students try out the original package of 42 puzzles, and being really engaged in conversations about terms, definitions and properties of each of these shapes, I wanted to try to find more.  Having students ask, “what’s a trapezoid again?” (moving beyond the understanding of the traditional red pattern block to a more robust understanding of a trapezoid) or debate about whether a rectangle is a parallelogram and whether a parallelogram is a rectangle is a great way to experience Geometry.  However, after an exhaustive search on the internet resulting in no new puzzles, I decided to create my own samples.

Take a look at the following 3 links for your own copies of Zukie puzzles:

Copy of Sarah’s puzzles

Extension puzzles #1

Extension puzzles #2

I’d be happy to create more of these, but first I’d like to know what definitions might need more exploring with your students.  Any ideas would be greatly appreciated!

##### How to complete a Zukei puzzle:

Each puzzle is made up of several dots.  Some of these dots will be used as verticies of the shape named above the puzzle.  For example, the image below shows a trapezoid made of 4 of the dots.  The remaining dots are inconsequential to the puzzle, essentially they are used as distractors.

If you enjoyed these puzzles, I recommend taking a look at Skyscraper puzzles for you to try as well.

## From Experimental to Theoretical Probability

Probability is an interesting topic.  Really, it’s different from most mathematics our students learn.  It’s the only topic in K-8 mathematics that doesn’t follow patterns.  For example, we might know that flipping a coin 10 times should result in 5 heads and 5 tails, but in reality, it is quite likely that we will get some other result than this.

Because of this, I believe that we need to spend our time playing with tasks and making predictions, a lot, before we ever broach the concept of theoretical probability.  We need to have students play the same games several times, where they can change their predictions based on previous experiments, before we provide opportunities for our students to understand the theoretical.

##### Big Ideas of Probability

When planning a problem in any topic it is always a good idea to consult Marian Small’s or Van de Walle’s “Big Ideas”.  Here are 2 big ideas we looked at as we constructed the problem below:

• An experimental probability is based on past events, and a theoretical probability is based on analyzing what could happen. An experimental probability approaches a theoretical one when enough random samples are used.  – John Van de Walle
• The relative frequency of outcomes (of experiments) can be used as an estimate of the probability of an event. The larger the number of trials, the better the estimate will be.  – Marian Small

We explained to students that we had created a new game and wanted to test it out.  While each group would play the same game, the board for each game was slightly different.  We explained that it was their job to play the game and tell us which game board we should keep.

##### Rules for the Game
• The game is for 3 players
• 2 coins are flipped each turn
• Player 1 wins if 2 Heads are flipped
• Player 2 wins if 2 Tails are flipped
• Player 3 wins if one of each are flipped
• The result is placed on the game board (H for 2 Heads, T for 2 Tails, B for both – one of each)
• The winner of game is determined by the player with the most wins in that game.
• Once a game is finished the entire game is coloured 1 colour (Red for Heads, Blue for Tails, Green for Both).  If a tie exists, 2 colours are used equally.

##### The Game Boards

Each student is given a game board with 100 sections, however, each individual game consists of a different number of trials.

Game board 1 – 1 game of 100 trials

Game board 2 – 2 games of 50 trials each

Game board 3 – 5 games of 20 trials each

Game board 4 – 10 games of 10 trials each

Game board 5 – 20 games of 5 trials each

##### Results

After playing several games using each game board here is what we found:

##### What do you Notice?  What do you Wonder?

Students noticed:

• The player who was chosen as “Both” won far more than the other two
• Game boards with more trials (game of 100 or 50 or 20) were coloured all/mostly green.
• Game boards with less trials (game of 10 or 5) had more red and blue sections than other boards, but still mostly green

Students wondered:

• Why did “Both” keep winning?
• Why is a game board with more trials more likely that green will win?

In the end, several students came to the conclusion that getting a Heads and a Tails must be more probable.  Here is what they came up with:

##### Some things to think about

If a Standard/Expectation tells us that our students need to understand that experimental probability approaches theoretical probability with more trials, we can’t just tell students this.  We need to set up situations where students are actually experimenting (including: making predictions, performing an experiment, adjusting predictions/making conjectures, re-testing the experiment…..).

As with anything in mathematics, our students need ample time and the right experiences to make sense of things before we rush to a summary of the learning.

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

## Minimizing the “Matthew Effect”

For the past 5 years I have been a math coach in the same (mostly) few schools in my district. This has afforded me the opportunity to observe students through the years as they’ve been developing as young mathematicians. Being able to watch students year after year has afforded me opportunities to notice the different paths some kids take over time.  For example, go into a grade 8 classroom and really listen to the students as they are talking about their mathematics, observe each student as they are thinking and working…  What you might notice is a huge discrepancy between who is doing the talking or sharing and who is not.  You’ll see some students eager to participate, actively engaged in sense-making during new learning opportunities, and others who might seem to let others participate and do the majority of the thinking.  These observations have got me reflecting on a few questions:

• Why are there such differences between these students?
• What happens throughout the years that cause these differences?
• How can we help create classrooms where all students are engaged in doing important mathematics?

##### The Matthew Effect:

An important piece to this puzzle can be attributed to “the Matthew Effect.” The Matthew Effect was coined to describe the process of cumulative advantage, basically, the rich get richer and the poor get poorer. The idea of the Matthew Effect is that those who start school with a small advantage continue to benefit, while those with a slight disadvantage continue to lose ground. While it might be easy for us as educators to notice the differences between students’ abilities or effort, it is far harder to notice any inequities that our classrooms and schools might be causing. More about this in a minute.

##### A few examples of the Matthew Effect:

Soccer:  A group of children join a soccer team for the first time.  Each time a child kicks the ball, or strips the ball from someone else, or passes to a teammate, or dribbles with the ball they learn.  Those students who start off more comfortable with running and kicking spend more time with the ball in a game.  By the end of the season, some students have kicked the ball hundreds of times more than others.  While everyone is learning to play soccer, the gap between those comfortable and uncomfortable with controlling the ball in a game widens.

Reading:  Students enter into kindergarten with differing abilities to recognize letters or words, and differing interests in books.  Every time a child sounds out a word, or uses a cueing system to read a new or challenging word, or thinks deeply about the messages/story the better they get at reading.  Those who start off more comfortable with reading, read more books each having more words.  By the end of the year, some students have read thousands of words more than those who started off struggling.  While everyone is improving, the gap between those confident with reading, and those who are struggling to learn to read increases.

##### The Matthew Effect in Math

In both of the previous examples, there were two factors that led to inequities:

1. The differences in the starting points of each individual
2. The differences in opportunities for each individual

For mathematics, the issues can be quite complicated. To think about how the Matthew Effect can be problematic in mathematics learning it’s important for us to <a href=”http://Mackenzie Condon “>consider what early skills in mathematics are and which are predictive of later success.

But while it might be important to know early indicators, it is FAR more important to consider to think about how we are helping all of our students be successful. This is where we need to minimize the Matthew Effect!

So, how does the Matthew Effect happen? Imagine students in a class where the teacher asks a question of the group and those whose hands go up first always get to answer. Students who might need more processing time come to realize that others will get an answer first and might not even attempt to answer questions anymore because they won’t have enough time or believe somebody else will get picked anyway. Imagine a classroom where students are given different assignments based on their readiness. Students that continually get more advanced work come to think of themselves as more advanced while those receiving remedial assignments disengage because they realize they aren’t good at math. Imagine a classroom where every student gets the same page of closed math questions. Some students work independently and complete the tasks easily while others are unsure what to do. Over time those who struggle to work independently realize they can only be successful if they get direct help, they start to immediately raise their hand and expect their teacher to walk them through each question.

In each of these situations, some students are accessing the mathematics themselves while others are receiving a watered down version or are expecting others to do the thinking for them. Over time the gap in experiences is huge!

##### Complexities of teaching

If I provide everyone with the same task, some will struggle to independently be successful while others might find it too easy or repetitive. But if I provide different tasks to different students based on perceived readiness then I’ve also created inequities because I’ve limited students’ access to the mathematics.

If we keep pace of discussions based on the first few hands raising, then we likely haven’t engaged several students because they haven’t had enough time to think. But if we feel like we always need to wait for every student then we likely won’t have a flow of conversation that is ideal.

When determining groupings, if we place students who are currently struggling with students who are quite confident, there is a potential for an inequity in who is doing the work and who is learning. But if we place students who are struggling with other students who are struggling, then there are just as many inequities.

##### Minimizing the Matthew Effect

Teaching is complex! Helping students who disengage or who don’t identify with mathematics is not an easy task. However, we need to consider ways that we can help all of the students in our care to come to appreciate mathematics and believe in themselves as mathematicians.

If there is a practice that benefits those who are already being successful with mathematics more than those who are striving to be successful, then there are inequities at play in our classrooms.  Whether the inequities are related to us believing who is capable, or related to who has access to rich learning opportunities, we need to understand and confront our own biases and beliefs for the benefit of all of our students.

As we start to think more about the inequities in our schools/classrooms we will start to see more students who are actively constructing their understanding of important concepts via rich problems and experiences… the interactions among students and between students and teachers will show that every student’s thoughts and ideas are valued… that every student can be successful if given the right experiences and feedback.

On the other hand, if we don’t believe that ALL students can learn to the highest levels, then our students won’t believe it either!

For more on these topics, please take a look at:

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