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 #12 • Revised 25/1/20
• know that enagement is
not just quiet
Engagement and the measure of engagement is a puzzling
problem. We all learn in different ways but mostly as visual or
aural learners. By seeing or hearing. Most people, like myself,
seems to be a complex blend of doing, seeing and hearing. I like
making mistakes and exploring on my own. I am the typical male who
never reads a manual. However when fixing my landrover I have a
well oiled and fingered handbook. Manuals are useful!
• without engagement learning is very haphazard
What is engagement? How does one measure it? Does
engagement lead to student achievement? How do you know?
Engagement appears to be in four parts and it turns out that this
is a global puzzle.
Behavioural is about being self managed and
making the effort and giving the respect to be part of the
Emotional is about being relaxed and mentally resilient,
being supported as a person and not having any great emotional
dilemmas dominating your existence.
Cognitive is about being mentally prepared, interested
and keen to learn more. All with relevant contexts.
Social is about connecting with your peers and being
accepted, participating and contributing.
Measuring all this is even more problematic so I adapted the
rubrics in this reading on Engagement
and I use a unique way to assess them. Open one up and look. I
suggest modifying my measures to have just a few (5?) items and
talking about each a little as they decide where to mark the
I do know as a teacher when my class or "most of" is
engaged. It is buzzing or busy, I am not required, I am noticing
and watching. I may as well walk away. This has happened a few
times in my 30 years of being a maths and physics teacher. It can
be scary and is so when it first happens. It is a surprise when
the presented learning is sufficiently stimulating and rewarding
that no encouragement is needed. It is a very nice atmosphere to
For me, it first happened in a physics class 1981, 3rd year
teaching, when I was trying to create an experiment using a
builder's plank and making a ballistic pendulum to measure the
speed of an airgun pellet. I needed the weight and casually asked
a couple of students to weigh the plank. They came back to me a
few minutes later saying the scales only measured to 5 kg and the
plank must be heavier than that because the scales bottomed out.
I said "Well just weigh one end and double it." I was solving
the problem and did not expect the response.
The class erupted in uncertain laughter and confusion. Then all
the comments started flooding in.
"That won't work. There will be some of the plank
you are not measuring!"
"You will get an incorrect answer"
"Weight does not work like that"
"Hogan, you are all sh..!
So I said "Well, weigh each end and add them up!"
The class started rolling around the floor laughing and was
absolutely convinced I was a crazy man. I started to enjoy this
response. My experiment had been destroyed. Goodbye ballistic
pendulum and welcome cognitive dilemma and engagement. These
students became one of my better classes.
They argued, debated, reasoned, pondered, wrestled, twisted
truth, falsified truth, ignored fact, embraced hysterical crowd
behaviour, fought, resisted and eventually just before the 1 hr
period ended decided that perhaps I was right after testing my
idea on a lighter object. What an astonishing thing to witness.
Learning happening before my eyes. Later I used two bathroom
scales and had students stand with a foot on both and notice the
The students found something they did not agree with and could not
assimilate the suggestion. It did not fit their world view
or understanding. They resisted as all conservative humans do to
change and dilemma. They pondered and created a way to test the
hypothesis. They discovered new truth. They created new knowledge.
They learned that the "up" forces equaled the "down" forces. They
then figured out that for a symmetrical plank weighing one end and
doubling the answer would give the exact weight of the whole
My best lessons are when I get a class enagaged, or even part of a
class, doing an activity and they do not realize they re
learning something. I try to be a bit mystical or magical in
the classroom. I want to surprise then and put them into the
cognitive zone .
A favorite math task is to write 0.618 and 1.618 on a piece of
paper and staple it closed to the wall. I ask the students to
choose 2 numbers, not too big. Then add them and cross out the
one. Repeat adding and crossing out the top one and do this about
15 to 25 times. I then ask them to take the last two numbers they
arrived at and divide them, one way or another. I then unstaple
the piece of paper and show them the answer they got. Either one
of these numbers depending on how you divide.
All students are somewhat stunned by this. They repeat using
different numbers. They may investigate less or more additions.
They all are engaged and we have a nice discussion about why. Why
do you get 1.618 or just 0.618. What crazy numbers have this
property. They are many of them, a whole classful and they always
give the same answer! Is it magic?
The solution is not so hard. It is actually the Fibbonacci Series
1,1,2,3,5,8,13,... where the next is the sum of the last two.
Alittle algebra will show it does not matter what teh first two
numbers are they soon pale into insignificance and we get a stable
1.618 solution. The actual solution is (1 +v5)/√5 or the solution
to x^2 +x-1=0.
It is also the solution to 1+1/(1+1/1+1/1+...))) or more easily
seen visually by
And so much learning, addition, discourse, debate, collaboration,
perseverance, division, patterns and many other things has
happened. Better; no one noticed.
Hence Lesson 12
Describe why being engaged is important.
How do you craft enagement?
literacy and numeracy website for rubrics on listening etc.
Give your class a few questions from one of teh measures above and
try to make sense of what it measured.
Read the reading above.
Is engagement a group or an individual thing?
How do you measure ENGAGEMENT?