## “Making Math Visual”

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:

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?

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

## 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

## Building our Students’ Mathematical Intuition

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

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

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

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

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

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

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

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

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

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

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

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

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

We shared a few estimation strategies:

and a few “Notice and Wonder” ideas:

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

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

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

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

I ended the presentation with a final thought:

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

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

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

A few questions I want to leave you with:

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

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

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