Dave’s experiment in Maths education:

What if kids chose to do maths?

David Coulsondtcoulson@gmail.com

Conducted at Unlimited Paenga Tawhiti, Christchurch, NZ, 2014

What if maths was an optional class instead of a necessary class?

Obviously if I said this regarding all maths in all schools in all situations, therewould be reasons to say this was a bad idea, just as there would be reasons forsaying it is a good idea.

But consider a high school setting, above the basics of 2+2=4, times table andother elementary arithmetic processes. I’m speaking of the realm in which mathsspreads out and becomes algebra, geometry, trigonometry, probability andstatistics and so on.

What would 13, 14 and 15 year-old kids choose to do with maths if they wereallowed to choose anything they wanted?

This is an age range for which most kids have acquired competency at the basicskills of arithmetic (which I regard as the essentials of mathematics), but who hadnot yet met the higher levels of mathematics based on algebra (which I feel oughtto be optional).

Just as importantly, this is an age range for which exams have not yet become amotivating or demotivating factor. Kids at this age in our school system havegenerally been assessed in mathematics but have not yet started point-hunting inthe formal examinations that start in many schools across the world at age 15.

Whereas I could have chosen to experiment with kids of any age, younger orolder than this target group, I especially chose this age group because of thosefactors. I wanted kids that already had a bit of skill in the subject but had not yetbecome resentful of the subject by the need to sit maths exams.

It would have been nice to have had carte blanche to do what I want on such anexperiment, but we live in a world in which examinations are important as agateway to tertiary study and employment. Therefore when proposing to do thisproject at the school, I could not morally say to the kids “Forget the old way ofdoing things, just study what you want!”, so I imposed a few practical limitationson my enthusiasm and on the project. Please note that it was my suggestion toimpose these limits, not the school’s.

(ONE) The students in my project should also be attending traditional classes inmaths concurrently. My project therefore was a supplement for the training theywere getting elsewhere, not a replacement for it. Such a limitation ensures thatkids do not gamble with their preparation for exams in the years to come. It alsokeeps me onside with teachers who may resent me encroaching on their territoryand the comparisons that could be made.

Having said that much, it would be very easy to view my experiment as nothingmore than an extension class for the gifted. However this is definitely NOT whatmy experiment was about. While I did handpick the kids who would join thisproject, I did not choose kids on prior test results. Furthermore I was not going toaccelerate their passage through the school system by teaching them next year’ssyllabus. The kids themselves would decide what they would learn, whether thatwas on the national education agenda or not.

(TWO) The students should have a demonstrable curiosity about mathematicsand a demonstrable maturity to direct their own education. This is hard tomeasure, in both cases. However, my concern was that some kids would see thisas an opportunity to avoid learning and kick back and let everybody else do thework. I therefore imposed a simple pre-requisite for kids choosing to participatein my experiment: they would have to tell me before joining the group why theywanted to join and what they wanted to get out of it.

This wasn’t as easy as it sounds. A lot of kids did not pass this simple test. Thesewere kids for whom maths had become simply an hour of each day when theyturned up and the teacher did the rest. This attitude towards maths was notgoing to prosper in the environment I was creating. Not only would these kids notlearn anything, they would make it difficult for their neighbours in the class tolearn anything. That is what I see in normal classes where everybody is requiredto participate, motivated or not, and I very much wanted to avoid that.

On the selection day for my project, kids were told “This is a project in which youget to choose to study what you want, week-by-week. No exams, no homework,no curriculum. As a group, you all get to negotiate with each other and choosewhat you want to do.” This of course was met by universal approval. But then kidswere asked individually, “What do YOU want to study?” Excuse me? What? Manykids could not answer that question. I then pointed out, “You know that you willbe asked this question each week while you’re in the project, so it seems to methat if you don’t have an answer now, you will have difficulty coming up with ananswer each week. So, again, what do YOU want to study in mathematics?” Somekids simply could not answer that question, and therefore either decided not tojoin the group after all or were turned away by me. I was quite stubborn on thispoint.

This of course is an imperfect and crude way of judging motivation and maturity.However it was all I could use at the time. And it turned out to be surprisinglyeffective. The kids who I let stay in the group were effective communicators,effective collaborators, good at turn-taking, patient, and keen to try new things.

There were other factors that came into play once the project got started. Inorder to justify to the school that the kids were actually learning something, I hadto provide some evidence of learning at the end of each 5-week block. So how doyou assess learning in a project which has no set currciulum? I threw the questionover to my team of students and they suggested that maybe they could assessthemselves by writing down what they remembered learning in the project overthe preceding 5 weeks. Pretty obviously, if a kid could not remember what (s)hehad learned then (s)he had learned nothing. If, without effort (s)he could reel offa string of comments about what (s)he had done, then some learning hadoccurred. The school administration was comfortable with that. As it turned out,many of the self-reports were quite dazzling in their content, though to be fair, amajority of kids could only write a few sentences.

I was at first reluctant to accept the self-report requirement but it did proveuseful later in the year when new kids would come along and want to join in. Insome cases it was only necessary for me to say “A self-report will be required fiveweeks from now” for those kids to lose interest and go elsewhere.

The point was to create an hour in each day when the only kids who wereactually in the class were kids who wanted to be there. That is a very refreshingexperience for teachers and kids alike. Providing these tiny but significant hurdlesmade access to the project feel like a privilege not a necessity.

The structure of our school is that kids can swap to other classes at the end ofeach five-week block if they wish, or stay with the same course if they wish. Istarted with 12 kids in one group and over the course of the year this reduced toa core group of about 5 who stayed for the whole year and typically 3 moreparticipants who would opt out after dabbling with the project for 5 weeks. Istarted a second group in a different schedule to see how the dynamics of thetwo groups differed. The second group started with 4 kids and stayed that way forthe whole year. On some occasions, a newcomer would ask to join the group butafter seeing what my requirements were, chose not to join. Therefore the group-of-four was very stable throughout the year and a LOT of good learning tookplace.

A few words about the kids who opted out midway through the year. You mustremember that my project did not operate in a vacuum. There were other goodcourses going on at the same time as mine and the teachers were very good atwhat they did. Therefore it is not necessarily true to say that kids opted outbecause they didn’t enjoy my project or want to stay in it. So far as I could see, allof them without exception enjoyed being in the project. I could see this in themanner in which they all participated. In many cases it was simply that otherclasses proved to be more attractive or more useful as preparation forexaminable subjects. They certainly weren’t dropouts in the customary sense.

My project was set up to create an environment where enthusiastic kids could beunapologetically enthusiastic about their interest in mathematics. Some of mykids were very poor performers academically but were nevertheless keen to playaround with maths. For some it was as innocent as preschoolers biting into clodsof dirt in the sandpit: they didn’t know what this stuff called maths tasted like butthey were keen to find out.

So what did the kids choose to learn in this unrestrained environment? You wouldbe surprised. Many kids expressed an interest in algrebra. This rather surprisedme too because I thought kids would choose more exotic subjects such as spacetravel and war-based themes (they were mostly teenage boys, after all).Throughout the entire year, boring, stiff and starchy algebra kept coming backtime and again as one of the three most popular themes.

The second noteworthy subject that kept coming back is what my group came tocall ‘Splat problems’. These are problems in which characters fall off cliffs or areshot out of cannons or get thrown out of aeroplanes, all problems easily solvedby the formulas of physics.

Splat problems enabled kids to get used to memorising and handling formulae.Their ability to memorise was attrocious. Across an entire year and perhaps adozen or more revisitations to these formulae, none of the kids could quote themfrom memory. However, the kids could use them like experts. They were able tosubstitute values into the formulae appropriately and rearrange themalgebraically to solve the questions we posed. So while the kids thought theywere learning about things going splat into the ground and into walls, they werereally learning algebra again.

A word here about the whiteboard. I’m a big fan of collaboration. I like it whenkids lean over a table and look down at a common subject and think as a singleunit. Therefore I used a whiteboard that I could lay down flat on the table, notone that hangs on the wall. A table-top whiteboard encourages heads to cometogether, to collaborate. A wall-hanging whiteboard discourages it.

The third topic that kept coming back was what we called ‘number-crunching’ butwhich more accurately would be called approximation. This is not something thatis even taught in most schools, so far as I know. Yet it is generally the mathematicsthat we do in our working lives if we use mathematics at all. It is what we do withnumbers when we know we do not have to be accurate and want to get ananswer effortlessly and don’t have a calculator or a piece of paper. It is to me themaths-iest of all mathematics, and I was very pleased to see the kids develop aninterest in it.

In some instances, approximation became estimation of distances by squeezing afaraway object between the fingers, or walking the length of something andcounting the number of paces. This is still mathematics, and a very practical partof it, that I never see taught in schools.

Beyond the big three choices (algebra, splat problems and number-crunching)kids showed an interest in things that they’d heard of but knew nothing about,tgings they were not likely to encounter in class for many more years to come, ifat all. Such things as complex numbers, Einstein’s relativity, binary numbers, thebending of beams, planetary motion and wind resistance on moving objects,probabilities in card games, the maths of electrical circuits. These topics are quiteadvanced and yet provided plenty of scope for teaching basic mathematics indisguise. For example, the equations from relativity provided opportunity to seehow square roots are calculated and what they mean. A class in probability isreally a class in fractions and decimals. Beam deflection was horrendouslycomplicated mathematically, yet provided an opportunity to see how numbers gointo a formula; algebra again.

One of my favourite examples is when a kid nominated Fibonacci numbers as asubject to look at. Everyone asked him why and he pointed at a poster on the wallabove their heads which had the words Fibonacci sequence as a title. He didn’tknow what the hell it was yet thought it might be interesting. And of course itwas.

I called the project, the Connections Project, in honour of a TV show that I oncesaw many years ago that investigated the history of technology and itsimplications. I was hoping that by giving the project this name, kids would wantto investigate the connections that mathematics has with everything else aroundthem, how maths can be useful outside of a classroom, and how mathematics gotto be the way it is, almost accidentally, by people poking around curiously at theworld.

However any time you say to a bunch of kids, “Do what you want,” you can bepretty confident that they are going to go in a direction different to what youmight have hoped. Although these kids showed a great deal of drive and neverran short of topics to investigate, they were almost totally disinterested in thehistory that I was constantly trying to sneak into the lesson. Perhaps history issomething that old folks like me like, not young kids. Be that as it may, the kids’focus was always on mastering mathematical skill, no more and no less.

Some topics were quite frivolous. One of my kids was famous for his interest infarming, and after dedicating one class to the horsepower of tractors, hisclassmates jokingly said “Maybe we could throw one of these tractors out of anaeroplane.” It was a throwaway remark but it seemed to me to be an ideal way ofplaying with a number of mathematical techniques. First there was gravity andthen there was wind resistance and then there was the question of how big acrater might be excavated by the impact of a tractor from a great height. Itformed a very interesting project.

On another occasion, a kid came along with screen images from a game he likesto play, in which a rocket-propelled grenade does considerable damage to otherplayers. He wanted to know how realistic this thing was, and whether we coulddesign one.

Now I want to be clear that I know nothing about weaponry but I do know somethings about rockets. So we looked up some formulae and applied some verybasic geometry and worked out how heavy this thing would have to be to matchthe size it has on the screen. Turns out that the RPG would be too heavy tohandle if it was as big as it appears on the screen, and wouldn’t travel nearly asfar.

Stuff like this is fun to do, but there is another side to this learning project that Iwant to discuss, and that is the willingness of the kids to go and find data toprocess. Whereas in the early stages it was sufficient for me to lead the kidsthrough a topic that they had nominated, I very quickly wanted to get the kids togo find the questions for themselves and concoct ways to obtain answers bythemselves. THAT is real earning.

The kids were never as fully able to do research on their own as I had wanted, butthere was certainly movement in that direction. By the middle of the year, anytime I needed a number such as the density of air at 10,000 metres, I was able tosay to the kids “Go find this number...” and they would bring out their smartphones and swipe away. I wanted these kids to see maths as something theycould do at any time. It wasn’t a classroomy thing. It was as simple as thinking upa question and getting the data to answer that question.

By the final term of the year, some six months after the project had started, mykids in the group-of-four were asking me to give them mathematical projects towork on, unassisted. They wanted to know if they could trust themselves toanalyse a problem and come up with a way of solving it, unassisted. So I askedthem on one occasion how to estimate the number of grains of rice in a half-filledjar. On another occasion I asked them to estimate how much recyclable metalthere was in the legs of the tables and chairs in the classroom, without using aruler to take measurements.

And then I let go the reigns almost completely to see what these kids couldachieve on their own. I said “I’m curious about how fast the world’s fastestanimals can move.” I offered this as a topic that clearly required a bit of internetresearch, but I had no idea of what they would do with such a vague question. Aquestion like that could be answered in five minutes and forgotten just as quickly.But these boys knew they had an hour to fill and rather than do what might havebeen expected of boys this age, they used the answers to some questions to formnew questions to be answered, so that by the time the hour was over they hadfound various speeds of various animals and birds and insects and comparedthem to speeds of cars and aircraft, which in some cases required translatingmiles per hour into kph, or metres per second or metres per hour in the case ofsnails. They then took an interest in seeing how long it would take a snail tocircumnavigate the world, and had an answer for me to see and wow over by theend of the class. I even photographed it, the work was that good.

And then I let the reigns go completely. On the last day of the project I let the kidsdefine their own question, research it and give me their answer by the end of thehour. By this stage they had learned a little about momentum and wanted to seewhat would happen if their friend was run down by a galloping buffalo at fullspeed. So they estimated the weight of their friend and then looked up theweight of a buffalo and how fast such a creature can run, and then looked up theformula for momentum and worked out how fast their friend would recoil (underideal circumstances) upon colliding with a buffalo. The idea was entirely theirs.

This to me is the proof of what can happen when maths is something you chooseto do: you don’t simply stop when no-one is looking. I saw evidence of this kind ofoutcome in another way, in the self-reports of some of the kids in my largergroup. Two kids on separate occasions wanted to show the school administrationthat they had mastered basic linear algebra while studying the subject with Dave.Both kids were keen enough to create an example problem for the reader to see,and then solved it step-by-step, and explained their steps along the way. I did notask them to do this. They did it on their own. They chose to do it because it wasfun to show off what they could do. And that to me is REAL learning.

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