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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Talent vs Hard Work

Am I to Be or Not to Be a “Math Person”, That is the Question

The last few weeks, we have explored how our views in math can hinder us from succeeding. By adopting a growth mindset, we can overcome our struggles and learn from our mistakes. But what if you don’t struggle? We discussed phrases such as “I’m not a math person”, and “I’m just not good at math” and challenged it saying that “everyone can be a math person with the right attitude”. But what if you are a natural “math person” and math comes easily to you innately? Does talent exist or are “math people” just work harder than the rest of us? I believe that some people are naturally better at math than others. This does not mean that if you are not born with natural talent you cannot be a “math person”. It just means that more work is involved to get to the same level. 

Being someone who has struggled with math, growth mindset and learning from mistakes is an applicable strategy that I can embrace. I want to improve, and I know that I will succeed if I persevere through my mistakes and learn from them. I am growing my brain and learning valuable skills through my effort. But how does talent benefit a “math person”? Is it better to be talented or to work hard? 

Level: Minimal Effort

This idea of talent verses hard work reminds me of my musical experiences growing up. I was always told I have a talent for singing, and my experiences reinforced this idea. I barely practiced, yet I succeeded through each lesson. Each year in competitions, I received high marks and often placed in first and second place. In my teens, I represented my city in provincials five years in a row. And I did all this with minimal practice. I breezed by on talent and little effort. I still learned new techniques to improve, but I didn’t have to work very hard to apply the techniques like others in my studio. Even when I went to university for singing, and was rudely awakened to the concept that talent only gets you so far, I was still able to learn songs the night before a lesson and be praised for the “hard work you clearly put in this week”.

I’m not trying to brag. It was just my reality. And I’m not sure having this talent was a good thing. I struggled a lot through university when my talent didn’t cut it, and I think I hit walls harder than those who might not have had it so “easy”.

Does Talent Only Get You So Far? 

A study conducted by David Hambrick and Elizabeth Meinz explores if talent or hard work ultimately fares better in the real world. Their findings conclude that while talent initially fares better, hard work almost always wins out. This is because people who are talented are often LAZY.

Yes, I’ll admit it, I am! Why should I work hard if I can naturally get away with minimal effort and still succeed?

Except my talent only got me so far. When techniques and repertoire being challenging, I didn’t have the growth mindset skills to persevere and work hard. I just despaired and struggled and resisted. I was too lazy to try to work hard. And it killed me that I wasn’t “good” anymore, and I suffered with mental health issues in my final year. Those in my program who might not have been at the same level I was at when I first arrived at school were now surpassing me. Those rare gems who were talented AND knew how to work hard were superstars. But it was the people who knew how to work hard that really understood the craft. They embraced their struggles, learned from the mistakes, took the time to develop their skills. This allowed them to gain a deeper understanding of how to sing.

Mistakes? What Mistakes? 

In our math classrooms, we need to emphasise the importance of making mistakes to learn. We also need to make sure we challenge our students enough to allow them to make those mistakes. We focus a lot on the students who struggle with math. But we also need to remember that maybe those students are better off than those who don’t struggle with math. They will know how to persevere, how to develop a plan to improve, and the many components of a concept. Those with talent might miss out on all those valuable lessons, and become content with a shallow understanding of math.

Creating Hard Working and Talented Superstars 

How can we differentiate our instruction to allow all students to develop perseverance and hard work, even if they are “talented”?

One solution suggested is the use of open ended math problems. These allow all students to explore a variety of solutions to a problem that might not have a definitive answer. Focusing on the HOW they solved the problem rather than only their final answer will teach students how to break down their problem-solving skills and gain an understanding on a deeper level.

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Mathemathics and Me. Volume 2

I'm Back!

Hello, internet! You might have noticed my hiatus from blogging for the last few months. I promise it was not wasted time. I began my internship in my teaching placement, which was quickly followed by my first teaching block. With the help of my lovely associate teacher, I gained rich experience teaching primarily math and music throughout the many weeks of my block. Who would have thought the girl recovering from a bad relationship with math for most of her adolescent life would end up teaching it every single day in her first block? Talk about break away from your comfort zone!

A Short Update:

St. Catharines Museum Education Kits

After successfully completing my teaching block, I went back to classes for a semester. After that, I got a fantastic summer job working at the St. Catharines Museum and Welland Canals Centre as a Program Assistant. I spent the summer organizing tours for little kids, designing and facilitating activities for events, and most importantly, I created a huge teacher resource! My partner and I redeveloped the museum’s Education Kits. These are rented out by teachers for 12 days, complete with lessons and materials. The kit was less than ideal when we got out hands on it. Activities were loosely connected to the curriculum, if at all, and provided little education background. The materials were old or needed to be expanded to be properly used. By the end of the summer, we created 64 new lesson plans for grades 1-8, covering an 8-day unit plan. We also designed, made, and coded all the materials! The documents look official and I’m very excited to see what it looks like when it comes back from the printers. The process of unit planning for multiple grades, and creating all the corresponding materials is going to be so beneficial when I begin unit planning for this year’s classes and teaching blocks.

Year Two: Building Bridges

Classes began last week and boy, are we hitting the ground running this year! And yes, I am once again thrown into the gladiator ring with my old nemesis, Mathematics. As I’ve stated above, last year we began to build the broken bridge brick by brick. Each brick contained a base of Growth Mindset, with additives of New Experiences, Research, Determination, and a dash of Fun. The bridge is basic but solid. I can cross it easily with little to fear. But it isn’t anything grand or expansive. I shouldn’t get ahead of myself though. It will take many years to build a masterpiece with Math. This year, I will just work to make some upgrades to further support this bridge with Math.

I was a little unsure on how to approach these upgrades during the first Mathematics class this semester. I came in with the ingredients that had worked last year, prepped and ready to go. My professor started the class with a card trick and tasked us to figure out why it works, and alternative solutions to continue to make it work. My group struggled to figure out the basic mechanics of the trick, forfeiting and asking for help from another group. Now we had to find the next number that would also allow the trick to work. This is where all my ingredients went stale, and it all started with the deterioration of Growth Mindset.

The professor had stated at the beginning of the trick that it was the “simplest card trick”. Yet, we couldn’t even figure out how to do the trick on our own, never mind how to break down the patterning. Already, I felt like Math was beginning its old tricks again, working me up with an overwhelming challenge that I was just too stupid to understand, yet it was supposed to be “simple”. I barely understood what the next step was supposed to be. "Find the next number that would work." I began to rack my brain for mathematical formulas or solutions to find the number. Would fractions work? No, the deck wasn’t confirmed to be accurately divided during the trick, so it wasn’t a case of fractions. Addition and subtraction would also need to be determined by an equally divided deck. What about multiples of 3? Did it have to do with prime numbers? It became more and more overwhelming as the options flooded my mind with to only hit a wall and crash together in a confusing whirlpool. Once that happened, I just gave up. Someone would tell us the answer in a minute. But we didn’t. The worst part is I still don’t know the answer because we never discussed the problem. I don’t know how to problem solve for next time!

And thus, the Growth Mindset I honed all last year disappeared in a flash. The bridge with Math began to crack and crumble. It’s still intact, but it is not as strong as it was when I walked into the classroom. How could my efforts give way so quickly?

Too Good to be True…

This experience is something I want to remember when teaching my future students. I want to remember the confidence I thought was strongly assembled from the year before yielded so easily at the first sign of a challenge. I want to remember how the idea that a task was “simple” made me feel like I was stupid because I didn’t understand it. I want to remember how I gave up on myself and my determination to keep trying because it was just “too hard”. I was never good at math anyway. Why bother trying again? Clearly, I hadn’t learned as much as I thought I did last year.

It was that easy to lose my growth mindset and confidence. Math and I were fighting again after we had worked so hair to repair our relationship. It was easier for me to fall back into my belief that I was not a “math person”, a myth I have internalized for a very long time. And when we didn’t discuss the solution, I was left with the feeling that everyone else had gotten the answer, so it wasn’t taken up. I felt ostracized by math! I want to remember this will happen to my students too.

The Fine Art of Criticism….

We work with students to build their confidence, giving them opportunities for positive learning experiences and encouraging them to foster a growth mindset. But it’s just like criticism, you need three to five positive comments to outweigh one negative comment. One negative experience with math is going to take at least twice if not three times as many positive experiences to reaffirm confidence and secure a strong growth mindset. Positive experiences don’t need to mean that there is no challenge and the student understands the concept easily. Positive experiences mean the student is given the support they need to find the solution. They need to feel safe enough to explore their options. It will give them an opportunity to learn from their mistakes, and do better in a similar situation.

To provide positive learning experiences for our students, we need to be careful how we phrase our questions, avoiding stating something is “simple”. We need to model the tools students might need to solve similar problems. We need to provide encouragement during mistakes and during each step in the problem-solving process. We need to be transparent in discussing the solution and emphasize the strategy of learning from mistakes.

My goal for this year is to find more positive experiences with math to outweigh the negative experiences. I want to upgrade the bridging relationship I have worked so hard to mend over the last year. My goal is to build a bridge strong enough to support my students who might be struggling to build their own crumbling bridge with math too.

How will you encourage your students to continuously foster a growth mindset when the struggle becomes too real?

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