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

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 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.

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

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 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 drn 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.

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" 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. 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 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.

Teacher TASK
Describe how you convey new ideas to learners.
Are you good at drawing your ideas?

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.
1. Intro and Relationships, L#1
2. The Main Vine, L#2
3. Beginning a Year, L#3
4. Number and Algebra, L#4
5. Geometry and Measurement L#5
6. Probability and Statistics L#6
7. Problem Solving L#7
8. Investigations L#8
9. Visual Mathematics L#9
10. Assessment and Learning L#10
11. Team and Problem Based Learning L#11
12. Engagement L#12
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