Visual Math and Music

This week we studied the many ways math can be represented beyond standard methods. Math is often viewed as a group of rules and formulas one must memorize and master to solve various problems. However, it is also possible to solve a problem by creatively using visual representations to help solve the problem.

The following video explains how one student answered a fraction comparison question.

She drew out the problem in a way that she could understand then worked towards the solution visually. Math does not have to be written as an equation all the time. Flexibility in our approach to problem solving allows students to use their critical thinking skills to solve the problem in a way that makes sense to them.

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This method of solving a problem visually rather than through standard equations reminded me of a piece I sang in my women’s choir during my undergrad. The song was completely lacking standard notation. I remember the first time I looked at it, I was confused. How was I supposed to sing a bunch of squiggles, lines of various lengths and thicknesses, and a circle with random l’s written inside? Where was my standard notes, rhythms, and keys that I was accustomed to? Have a look and listen to the R. Murray Schafer piece that did not give us proper notes:

The piece was written using graphic notation. Graphic Notation is a method of writing music through visual representations of sound rather than the standard mathematical written method music is normally recorded on. Graphic notation is an interesting way to present music. Once we began to sing the piece, we eventually adapted and stretched our brain muscles to interpret the music in a new way. It ended up sounding just as beautiful as if we had read it on a conventional score, and the way I looked at how music can be represented and produced has changed forever.

Having experience now teaching students to sing who do not read music, I have had practical insight on how to use visual presentations to allow them to understand what I needed them to know. I might have them draw out the contours of the melody they are singing, observing breath marks and rests, accents on certain “notes” or sections in their line, dynamics (louds and softs), and even desired qualities of sound through colours and images. The notes are still foreign to that student, but they now have a better understanding of how to understand what the score is asking them to do through visual representations. When they do begin to read the score in the traditional way, the will have a deeper understanding of what the certain marking might really mean.

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For example, forte means loud. On a score, this is just a fancy lower case f. What does that really mean? Well, if a student understand that when something is sung forte, the line might feel/look bigger, and they might draw it with a thicker line. Now the student who see f will know, oh this part will be sung bigger. By drawing the contours of the melody, students learn the order of pitches and the relationships between them based on their placement on the staff. An E on the lower line will be lower than the B on the middle line, but an E on the top space is higher than the B on the middle line. By visually seeing the contours, they will have a better understanding of the placement on the staff correlates if a pitch is high or low, and how it relates the pitches before and after it. 

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The above image used by the Toronto Symphony Orchestra shows how a piece of music is organized visually, rather than presenting the full score. This allows listeners to identify various parts of the piece without significant musical background knowledge. 

Only people who can read musical notation would find the excerpts in the [written] traditional [listening] guides useful, so I wanted a graphic way to represent what is being heard so anyone could understand.
- Hannah Chan-Hartley, Toronto Symphony Orchestra

The Schafer graphic notation score allowed for anyone to sing from it, rather than only someone who knows the secrets of reading musical notation. This is like understanding a math problem through visual representations. Visual representations might allow a student who doesn’t quite understand the math formulas and rules still attempt to interpret and solve the problem successfully.

So we talked about math can be solved using visual representations. But what about how we teach math? Could we teach math using visual representations without words?

This gentleman raises the question of how we can teach math without using words. He suggests using methods such as game based play to provide “visual feedback” while exploring mathematical concepts. While we cannot always use games in our instruction, it would be an interesting path to research how to provide more visual feedback for students to understand how to do math and help them solve their problems. He suggests that this method also allows students who struggle with language such as English Language Learners or students with learning disabilities. Providing questions visually, as well visual feedback, also allows us to differentiate for all our learners.

As mentioned in class, last year in placement I used a free online tool to help my student visually play with fraction concepts in a visual way that provided visual (as written) feedback for students who were struggling to grasp some of the concepts we had moved past. Being able to walk through the concepts at their own pace, with visual manipulatives, allowed them to finally understand the missing piece that was holding them back from success.

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Just like the graphic notation allows for anyone to read the music and sing, visual representations in math allows for all learners to interpret and explore math from all entry points, regardless if they understand the rules and formulas. 

How might you use visual representations in your math instruction?

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Stretch your Math Muscles

This week in class, we explored what makes a task a rich learning experience for our students. The criteria included accessibility for all learners, engaging material, various entry points, encourages creativity and collaboration, allows students to make connections, and broadens mathematical skills. All ideal points of interest that teachers should consider when planning any learning activity.

A post about rich mathematical tasks fittingly follows my previous post about adapting for gifted learners. Rich tasks allow for all learners to engage in a problem-solving situation at their own level of readiness without planning multiple lessons. I mentioned using open tasks to engage all learners, especially gifted students, last week. During class, we participated in an open task in which our teacher gave us the answer, but we were to give the question. 

The answer was 32 = ?

As a class, we brainstormed a variety of ways to represent 32, and I’m sure it was only the tip of the iceberg. It was an interesting process for me. Initially, I began with simple equations such as 32 = 30 + 2 or 32 = 16 x 2. But then I had an urge to challenge myself and I began to think a little deeper about what 32 holds. It has 3 tens and 2 ones. So I could represent 32 as 10 + 10 + 10 + 1 + 1. Or even employ some BEDMAS with (3 x 10) + (2 x 1). I could break down 32 a little differently. What about taking the 30 and dividing it in half to get 15? I could do (15 x 2) + 2. I could use exponents 25 or (2 x 2 x 2 x 2 x 2). I could use integers and fractions, I could show it on a graph, I could draw a picture…You get the point.

All this came from one simple idea that the answer was 32.

We discussed this same task in Brittany and Kursten’s webinar about differentiated instruction. As a group, we came to the consensus that this task is rich because it promotes creativity, which in turn engages our students, and it allows for various entry points so all students can contribute. I’d like to add that this task allows students to naturally engage their prior mathematical knowledge and make connections between numbers and various representations. It also encourages students to explore number flexibility.

It's like Yoga for Math

Number flexibility allows a person to look beyond the surface of a number and truly understand the components of the numbers. This flexibility will allow students to make connections between numbers and operations to assist them in problem-solving tasks. For example, I could just look at 32 as only the number stated. 32. But when I really think about what 32 is, 3 tens and 2 ones, or 6 fives and 2 ones, or 32 ones or two sets of 16 or two sets of 10 and 6... I am now able to manipulate what those numbers mean. So if I have 3 tens in 32, I also know that I have 4 tens in 42 or 9 tens in 92. I can connect those numbers just by unpacking 32 slightly deeper beyond “It’s 32.” Number flexibility is discovering the relationships between numbers.

http://brownbagteacher.com/number-talks-how-and-why

Some strategies to encourage number flexibility include using the above brainstorming task of finding the question. This presentation given by Sherry Parrish about how to host Number Talks to encourage students to explore relationships between numbers while focusing on the process rather than the answer. Math strings also invoke the same outcome. Exploring number flexibility can begin with allowing students to demonstrate various representations of numbers or simple equations.

Ready, SET, Go 

Subitizing

Another strategy we explored in one of our first classes also reinforces number flexibility and relationships. Our teacher very briefly showed us an image of random dots, and then we had to state how many dots were on the screen.

But what was interesting was the many different strategies everyone had employed to “connect the dots”. This is called subitizing, the ability to know an amount without counting using sets. Below is a fun video with many examples of sets to subitize with students.

To make subitizing rich, allow students to develop and share their process in determining the number in each set. You can even extend the activity by providing students with the opportunity to create their own series of subitizing examples to which the class must solve!

Not only does this task allow for number flexibility, but also practices number fluency, which is something I believe is important for students to make connections in higher level math. 

How do you encourage number flexibility for your students in a rich task? Let me know below!

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