The mathematics of real life

July 2016,

Ainaro, Timor-Leste

Mathematics can be viewed as a language. Like all languages, it takes some time and energy to learn, but once you’ve got it, it’s tremendously useful in communicating with other people who know it.

Few mathematics teachers in Timor-Leste are fluent in mathematics. We see that even the experienced teachers with decent educations rarely stray from the narrow line of the curriculum, and rarely feel comfortable delving into a new area or concept or way of doing a problem that may pop up as a result of our pratika activities. We see that old primal fear rise in their eyes, same as a fourth grader anywhere in the world whose past three years’ teachers were also afraid of mathematics.  Here it’s heightened still more by embarrassment: they know they should know this stuff. Heck, they’re supposed to be teaching it.

Teachers girding up a supply of paper polyhedra to take back to their schools.

It’s equally true in science, but somehow more striking and poignant in mathematics.  Many of these teachers have been either skipping a lot of key material or teaching it wrong for years, sometimes decades.  This implies one of two things: they are not interested in getting it right or they feel helpless as to how to get it right.  We find it’s usually the later.  Lazy teachers are not unknown here, but a teacher who will turn down the opportunity to learn their subject better is unheard of.

To bring mathematics concepts into the concrete world of daily experience is critical, and often not a hard trick at junior high level.  Volume and surface area are no-brainers, for example.  We take the ubiquitous chalk boxes and condensed milk cans and start measuring.  First we plaster them with graph paper and then proceed to count the squares, after which we measure the sides and whip out the formulas for prisms and cylinders to calculate the surface area, and finally compare the two results.  Then, again using the formulas for the two shapes, we calculate the volumes, and then fill them with water (box now lined with a plastic bag), measure those water volumes directly, and again compare the results.  If the results from formula and direct measurement are not the same, they’ve got to work out why.

This teacher has wrapped a piece of graph paper around his milk can.

We’re convinced that you can’t understand these two concepts without going through this process, yet we routinely see that more than half of the teachers in our trainings are partaking in this authentic experience for the first time.  If they have done it before, we graduate them to cones (traditional house roofs), pyramids (chicken cages), and spheres (soccer balls).

Filling the plastic soccer ball with water from the measuring cup.

We always try to roll out a concept in terms of something students are already familiar with.  In terms of pre-algebra and simple expressions, we’ve got several pratika activities that are essentially translating a market situation into an equation.  For example, if Aunt Mica spent 9 dollars and bought 3 fish, how much was each fish?  $3!  But wait: we weren’t asking for the answer; we want to see the question written as an equation in one variable. In other words, we just said it in your language, now you translate that to mathematics language.

Shopping algebra.  Second column from the right shows the key equation.  Rows show, from the top: bananas, pumpkins, eggs, watermelons, notebooks, and soccer balls.  Groups in our trainings call themselves after areas of mathematics.  This one is named Group Geometry.

We’ve seen that this simple activity can be transformative.  Teachers get the hang of it and then make more and more problems of their own, each time translating it to the proper equation.  We slowly add in complexity – paying for a taxi back and forth requires a new term, for example, and buying two items adds a variable – and the teachers rise to the occasion, writing the equations for new situations, each time proud of their originality.  The reality for many of them is that never before has anyone or any book asked them to write their own problems, and never before has anyone been there, qualified and ready to help them do so.  But here we are, ready to hold their hand and stand with them until they get it right.

Graphing is also something that seems to be missing in institutions here that prepare teachers. We have information that certain government ministers and their high-level advisors don’t really know how to read graphs.  We thus do graphing on every possible topic. We want our teachers to be part of the solution to this hazardous predicament.

Constructing a graph of inverse proportion. If you have $10 to spend, and two things to spend it on, how many you can get of one depends inversely on how many you get of the other. Thus, a downward sloping graph.

A graph shows information in a visual way, sort of a step between abstract and concrete.  Making a graph forces one to convert a set of numbers or an abstract expression into a tangible two-dimensional design.  Only when you graph something do you get the full gist of a number line, or the meaning of slope, curvature, or negative and positive parts of line or plane.  Again, we regularly see teachers making their first graph, and it’s a transformative event. They’ll color their graphs handsomely and rush smiling to stick them up in the front of the classroom.

One teacher messed up on numbering his vertical axis, and ended up with his linear equation correctly plotted on the graph paper, but his line was crooked.  That fortuitous error alone was enough to solidify the idea of linearity and a fixed scale for the whole class.

Our lead mathematics trainer of infinite patience, Mestra Angela, working her magic.

Trudging through these problems with the teachers can test my patience, and I’ve been happy to find that my Timorese trainer colleagues are way ahead of me in the patience and empathy department.  After all, they had the same poor education as the teachers we’re working with, the same ill-prepared instructors, the same meager learning resources.  They’re not one bit surprised by the huge knowledge gaps, and have plenty of experience filling gaps of their own.  Once again, I step aside and my colleagues lead the way.

I find that I’m more needed in the science section, because the questions there can easily spiral to infinity.  Mathematics has a way of being nicely confined to its nomenclature, its set of postulates and rules, and the given application.  Unlike in the science section, our mathematics teachers rarely ask a question that my Timorese trainer colleagues can’t answer.  A lot of what’s needed is just practice through multiple exercises with feedback and confirmation.  I guess it’s how we learn and gain confidence in a whole range of subjects.

Two serious mathematics teachers working out how to do the surface area pratika.  We do all work in groups and encourage teachers who have more understanding to help their peers.

So when we find mathematics teachers that have never had that experience, I may think, ‘I can’t believe you’ve never done problems like that!’ But instead I say ‘Let’s get at it, and by tomorrow, you’ll feel confident with this concept and you’ll know how to help your students get it.’  It feels good to speak a new language, and we try to get their eyes on that prize.  CG

When teachers make polyhedra and draw their plans on paper, it’s hard to miss the concept of surface area.
Our statistics pratika involves a popular kids’ game: shooting rubber bands at a stick poked into the earth. Teachers shoot 10 times each from 4 different distances, noting how many times they can ring the stick, and then tabulate and graph the results. Teachers love it as much as kids.
The science group takes advantage of a break in the rain to get plants for the transpiration and water transport pratika.

2 thoughts on “The mathematics of real life”

  1. Congratulation Curt. From Indonesia, we hope that TimorPraktika will help mathematics teachers in helping their students to learn mathematics:
    a. meaningfully or learn with understanding, in this case students actively building new knowledge based on the previous knowledge known to them.
    b. joyfully, students will learn mathematics in novel manner, in pleasure situation and in unthreatening situation.
    c. learn to think, therefore teaching and learning process might be started with problem or activity.
    d. be independent citizen, therefore during the teaching and learning process the teachers will give opportunity to every students to solve the problem or activity independently.


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