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
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
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:
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 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
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?
Did both groups of teachers rely on the learning they had received as students, or learning they had received as teachers?
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!
A few things to reflect on:
How do you use contexts and visuals to help your students make sense of concepts?
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:
Students (and teachers) think that all strategies are equal.
Students are left thinking that there are an unlimited, vast number of “strategies” to solve a problem.
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 how to lead an effective lesson close, 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 think deeper about which contexts are mathematically important, 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:
Thinking Mathematically 