Cling to the main vine, not the
Kei hopu tōu ringa kei te aka tāepa, engari kia
mau te aka matua
Thoughts on Teaching and Learning of Mathematics
Lesson #9 • Revised 5/1/19
- 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 youcubed.org 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
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
Here the lines sloping down to the
right represent 10 and 2 or the 12.
How to Solve a Problem
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.
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
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
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
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,
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!
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.
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.
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!"
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.
Array Model for Multiplication
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.
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