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
Pics and links to do
Visual Mathematics, Lesson #9
- to know that models expose understanding for a learner
- to learn how to include visual models as part of the problem
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.
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
I developed a way to approach all problems in mathematics to help
my Year 11 students solve problems.
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 teh number of times
I have read the problem incorrectly astonishes me. Read and
comprehend. This is a 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 drn 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
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" to wards 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. I think I
even said I would give them a reward if they did that.
Day 1, they renewed their friendships and discussed the holidays.
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 teh standard with A or M. No E
grades but we reflected on that.
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?
1. Intro and Relationships, L#1
This is to help look around my
pages. I have tried to make it consistent in all chapters.
The Planned chapters are only ideas at the moment.
The Main Vine, L#2
Beginning a Year, L#3
4. Number and
Geometry and Measurement L#5
Probability and Statistics L#6
Problem Solving L#7
Visual Mathematics L#9
Assessment and Learning L#10
Team and Problem Based Learning L#11
13. The Classroom L#13
14. Being a Teacher L#14
15. Being a Leader L#15
16. Managing the Principal L#16
17. The Importance of Whanau (Family)L#17
18. The Importance of the Student L#18
19. Math Topic A - Squares
20. Teacher Tools
22. Math Phobia