Cling to the main vine, not the loose one.
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123Thoughts on Teaching and Learning of Mathematics

Lesson #9 Revised 25/1/20

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. These include place value blocks, multiplication arrays, addition models, trigonometry models, similar triangles, integer modesl and navigation models. Mathematics is full of models and visual representation.

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

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 "dotty talks" and "number talks" is a favorite of Boaler's work. I think it is engaging for students to explain their thinking, and share strategies of solving number problems. Number talk is an eye-opener for other students to witness the way others think and "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 and messy fashion. All the result of poor teaching and learning, attendance issues and lack of a robust professional relationship with a competent mathematics teacher. A complex mess.

I developed much needed knowledge and skills with most of the students at Year 10. 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 and had poor attitude and respect for themselves.

The Steps I used were...
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, answer the task, 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.

STEP 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. The R-Teach model RT3T spends a lot of time helping students understand the problem, explain new words, draw the problem and develop approaches and skills of doing this themselves. Here is what kids said about RT3T.

STEP 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 as adults. Make a model of an even number, an odd number, multiplication! Draw a picture of 1/4 and the only model that turns up is a circle draw into quarters. More creative thinking needed here team!

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! Draw, Draw, Draw. Play pictioary and normalise drawing. Live cartoon on your white board!

STEP 3. DEVELOP A STRATEGY.  Once a picture of the problem is drawn I use that to logically find a path to follow. I draw a dot as the START and a then draw a line down around and through the problem showing the order I am going to follow. At the end I draw an arrowhead to show the way. The details I drew become the headers for the answer and CALCULATION follows naturally. In teh past I think we just went to the CALC and called that a solution. Example from the Huia and Mike Number assessment showing the line.

Small JPG example of Huia and Mike
See link for bigger version, this is to illustrate the RED STRATEGY line.

STEP 4. DO THE CALCULATIONS. As above this is where we all once thought mathematics was located. The line described above supplies the information of order and heading. Students write what they are about to calculate - "Mike earns $890 and saves 35%" Then the actual calculation 890x35% = 311.5. Restated as "Mike saves $311.50 each week." This structure is at MERIT level. The way problem solving is described here encourages "at least MERIT" and much better thinking.

STEP 5. RECORD YOUR SOLUTION, ANSWER THE TASK, AND PONDER. The really important event that has to happen when solving a problem is to answer the problem posed. In teh Huia and Mike task the question asks how long Huia and Mike would have to work to save enough money to complete the trip from NZ to the UK, Spain and home again. Hence "These calculations show that that Huia and Mike would need to work for 36 weeks to save the total of $7560 needed for the trip. I think they should work for 40 weeks or even a bit longer to save some extra money for insurances, unexpected events, extra spending and trips that might happen once on the way." The last part of the statement is from the PONDER and moves a MERIT answer to more of an EXCELLENT answer. Having a big picture understanding of the problem is EXCELLENCE.

Story Time about "That Year 10!".
My Year 10 class were reluctant learners but became Year 11students. 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!", "My brain hurts!".

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. It hurt. I worried. I re-examined many aspects of my teaching. The relationship with them had developed and I was beginning to get trusted. AKO!

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 One 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. I had maths everyday for 1 hr with them.

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 alert you might have noticed I gave you this problem on Monday, Tuesday, Wednesday and Thursday 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 their understanding of the problem. More questions and more focused chat. Progress?

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 One several had 12 credits, had ticked off the Numeracy requirement for NCEA and were well on the way to gaining NCEA L1. All of this the result of being visual and teaching visual mathematics.

My lesson was teach students How to problem Solve! Give them a structure. This is what RT3T does and builds self confidence.

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!
IN the Jan 2020 news from NCTM  I see a nice visual for "Completing the Square". In teh article found by clicking the link you will see models of x^2, x and a constant that I used in a Year 10 class when they were learning how to add like terms. The question of what was "like terms" and what was "not like terms" arose naturally and was pretty much answered by each student for themselves. I like the example here of completing the square. It can be a bit theoretical and esoteric for Year 10/11 students but here it is obvious that and extra 4 squares must be added and to keep the balance subtracted. No need for any obscure (+9 -9 +5) calculation, just add 4 and then subtract 4. It looks straightforward to my mind.

Completingthe square
Completing the square visual

Hence Lesson #2
Draw and model 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.
Visit NCTM!