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 25/1/20
- 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 youcubed.org 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
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
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 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
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
from the Huia and Mike Number assessment showing the line.
See link for bigger version, this is to illustrate the RED
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
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.
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?
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
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.
Array Model for Multiplication
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.
Completing the square visual
Hence Lesson #2
Draw and model everything! Having a main vine as a teacher is
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