Cling to the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari kia mau te aka matua

123Thoughts on Teaching and Learning of Mathematics

Lesson #9 Revised 5/1/19

Visual Mathematics

- to know that models and visual mathematics expose understanding for a learner
- to learn how to include visual models as part of the problem solving process

Drawing a problem or a concept is an essential skill. In Physics it is always a benefit to visualise a problem or idea. Engineers sketch ideas and these become plans. Mathematics is no different and there is an abundance of mathematical equipment used by teachers to explain ideas such as place value, multiplication, addition, trigonometry, similar triangles, integration and so on. Mathematics is full of models and visual representation.

Jo Boaler at has a lot to say on thsi subject and I suggest every math teacher look at what she says and use some of her ideas.

One of her ideas, not new, but she does explain it well, is to use a pattern of dots and ask "What do you see".

                            In this pattern of 9 dots circle the groups you see when you count them up.

I see several different patterns.

4 groups of 2 and 1.

  2groups of 4 and 1.

4 and a five, probably 5 and 4 in that order.

a 3x3 criss-cross and 2 groups of 2.

I moved a few dots in my head and saw 3x2 and 3 more, then immediately 3x3.

having "dot talks" and "number talks" is a favorite of Boaler's work. I think it is engageing for students to explain their thinking, and like sharing strategies of solving number problems, (Number talks) is eye-opening for other students to witness teh way others "see" problems. All this is visual mathematics.

 Other Illustrations of Visual Mathematics.
The background of this page shows an old way to represent numbers and their product. Unpacking what it all means is making sense of the model and deepening understanding of place value and multiplication.

Mult model
Here the lines sloping down to the right represent 10 and 2 or the 12.
The lines sloping up to the right represent 13 or 10 and 3.
The left most dot is the 10x10 or 100
The two groups of centre dots are the tens or 30 + 20 = 50
The right most dots are the ones or 6

We read 100 +50 + 6 = 156 = 12x13

This works with bigger numbers and with some place value knowledge is very useful to finalize the answer.

How to Solve a Problem
I developed a way to approach all problems in mathematics to help my Year 11 students solve problems. It emerged when I had a Year 10 class who really had no problem solving competence but had been exposed to some mathematical ideas in a very hap-hazard fashion. All the result of poor teaching and learning, attendance issues and lack of a robust professional relationship with a competent mathematics teacher. A mess.

I developed much needed knowledge and skills with most of the students at Year 10 and in Year 11, the first Year of serious assessment, I told them that I assumed they knew all the Mathematics and Statistics they needed and I would concentrate on making sure they could solve problems presented as assessment tasks for NCEA L1. I had also returned all Year 12 students to Year 11 because they had gained no credits at that level worth speaking of.

The Steps
1. Read the problem
2. Draw a picture of the problem
3. Develop a strategy to use
4. Do the calculations
5. Record your solution and ponder.

I made a .ppt of this called How to Solve a Problem. This model follows Polya's ideas and is elsewhere in these pages. Here I want to explain the power of the visual model.

1. READ THE PROBLEM. Easier said than done and the number of times I have read a problem incorrectly astonishes me. Read and comprehend. This is subject specific literacy for mathematics students. Reading to extract information. Reading to make connections. Reading to understand a situation. Reading to draw a model.

2. DRAW A PICTURE OF THE PROBLEM. We all draw as little kids and we love drawing as little kids. In workshops where I have asked teachers to draw 3/4 and models of 3x4=12 and how 1/2 + 1/3 makes 5/6 I am always astonished at how darn difficult the process is for many people. Make a model of an even number, an odd number, multiplication!

These two steps are connected and one feeds the other. You do not draw a picture unless you have read and understood the problem and by drawing the picture you gain a better understanding of the problem. I can not underestimate or overstate the importance of drawing pictures!

Story Time
My Year 11 class were reluctant learners. I had battled with traditional ways of teaching an idea, showing examples, easy to hard problems and applications as many teachers do but this approach did not work. Student voice is always informing so I asked "What is going on here guys? You are all bright kids but just not engaged or interested!" Answers varied as you might imagine but a few answers started to include "I do not know how to start", "What is the problem?", "There are too many steps", "Just too hard Sir!"

I had read Polya's book and had been a Math Advisor for many years. I was distraught and felt very much a failure as a teacher and responsible for their learning and success. Their lack of success became my failure.

This agony led to my decision to push the "responsibility for learning" towards my students and trial a new approach. Give up some control and let them take the reins.

On Day 1 of the new Term I gave the class of 24 students a sample Math Assessment Problem and assured them I knew they had all the skills so read the question and draw a picture of the problem. Steps 1 and 2 above. They were to bring that drawing to me and I would inspect it to see if they had all the information.

Day 1, they renewed their friendships and discussed the holidays. No pictures.
Day 2, they continued to talk and doodle. No Pictures.
Day 3, ditto
Day 4, ditto
Day 5, Friday, "Sir, you gave me this problem yesterday!" I replied "If you had been awake you might have noticed I gave you this problem on Monday, Tuesday, Wednesday as well!"

The next week I explained that we would be working on this problem until everyone had solved it to a Merit of Excellence level. They could work in groups. They could ask questions. But they had to read the problem and draw a picture and bring it to me to look at. I was starting to see a few attempts at drawing and the pictures I looked at got feedback and questions. By Friday there was a small improvement in the picture drawing ability and there understanding of the problem.

Week 3 saw a breakthrough. Two students drew a beautiful picture showing all the key ideas and even had suggested a path of how to do the problem.
Week 4 saw 75% of the cohort gain the standard with A or M. No E grades but we reflected on that.

The group continued to achieve and by the end of Term 1 several had 12 credits, had ticked off the Numeracy requirement and were well on the way to gaining NCEA L1. All of this the result of being visual and teaching visual mathematics.

Here are three .ppts I have used that show visual maths as well.
The Array Model for Multiplication
Visual Models
Check out my .ppt webpage for other resources.

Mathematics is being Visual!

Hence Lesson #2
Draw everything! Having a main vine as a teacher is vital.

Teacher TASK
Describe how you convey new ideas to learners.
Are you good at drawing your ideas? Give an example.
Visit Jo Boalers webpage and find out what she says.
Develop a visual explanation of a math concept you are planning.