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
I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.
In my 2nd year of teaching I was assigned to teach a grade 6/7 split class with 35 students. Being a new teacher, I taught my students the way I was taught… through direct instruction… and in a very procedural way… with the belief that learning happens by me telling, and students following the rules/steps that I shared.
Now, I could look back at my teaching and cringe with disbelief at my actions, however, I was doing the best I knew how! Instead of looking back and thinking about how poor of a job I was doing, or even how far I have come, I think it is far more valuable to recognize the moments that helped me grow as an educator. Here is one such incident:
So, my second year of teaching was well underway when the topic of adding fractions came along. I knew how to add fractions, and so I taught my students the only way I knew how. At the end of the unit I gave a test and was surprised at how poorly many of my students did.
I decided to offer those that did poorly a second chance, but assumed that first I needed to offer some help. I called a student over to my desk and showed him a few of the errors he had made on his test.
His first incorrect answer looked something like this:
Pretty common error right! In fact, for each of the questions I had given where the denominators were different, this was his strategy.
So, slowly I “re-taught” him the steps of how to add fractions… I asked little mini-questions to walk him through each small step in finding the solution. It looked like he understood until he said:
“I get that you are telling me that 3/5 + 2/7 is 31/35, but look, on my test I got 3 of the 5 right on the front and 2 of the 7 right on the back. Why did you put 5/12 on the top of my test?”
I, of course, wasn’t expecting this, and completely didn’t know what to do… so, I again showed him my method of adding fractions, completely dismissing his question.
While I think the example above illustrates an interesting moment where I recognized that I didn’t know enough about the math to understand how to react, it is equally interesting to point out what I did when confronted with something I didn’t understand. Take a look at Phil Daro’s quote below:
When we don’t know the math deeply, we jump to “answer getting” strategies… we tell the students procedures to remember… provide them with tricks… we focus on notations… we provide closed questions that are easily marked as right or wrong….. and our attempts to help students that struggle includes doing the same things over and over again!
This brings us to where we started:
I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.
If we want to get better at being math teachers, we need to learn more about the concepts our students explore! We need to learn from knowledgeable others about how the concepts develop over time, and the experiences our students need to make sense of the mathematics!
For me, this has included learning from wonderful influential math leaders like CathyFosnot,CathyBruce, Van de Walle, and Marian Small (check out the links).
When the Ontario Ministry published Paying Attention to Fractions K-12 I finally saw the connection and differences between my understanding of 3/5 + 2/7 = 31/35 and my student’s comment showing 3/5 + 2/7 = 5/12. Take a look. Can you see how my student was seeing fractions?
While a blog might not be the easiest place to deepen our content knowledge, it can be a platform to encourage you to consider how we are challenging our understanding of the math your students are learning.
Do you have a knowledgeable other learning with you? Do you read resources that challenge your current understanding of the concepts you teach? How are you making connections between different representations, or between concepts? How are you learning more about what is developmentally appropriate?
I think we owe it to our students to be continually learning! This learning is often referred to as “Math Knowledge for Teaching” or MK4T. Take a look:
In the beginning of this blog I explained a time when I recognized that while I understood the content myself, I did not know what to do when my students struggled with the math. This is where we need to spend our time learning! This is what our focus needs to be as professionals!
Thinking Mathematically 