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

Lesson #12 Revised 25/1/20

know that enagement is not just quiet
without engagement learning is very haphazard
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!

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.
a. Behavioural
b. Emotional
c. Cognitive
d. Social
Behavioural is about being self managed and making the effort and giving the respect to be part of the group.
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 of Students.

These became
a. Behavioural Measure
b. Emotional Measure
c. Cognitive Measure
d. Social Measure
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 scale.

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 create.

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 result.

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 plank.

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 fib

And so much learning, addition, discourse, debate, collaboration, perseverance, division, patterns and many other things has happened. Better; no one noticed.

Hence Lesson 12

Some questions...
Describe why being engaged is important.
How do you craft enagement?
Adult literacy and numeracy website for rubrics on listening etc.

Teacher TASK
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?