Sunday, May 29, 2011

Merit to Mathematics Labs

There is widespread turmoil among teachers and students when it comes to the practicality of mathematics. School mathematics, at the middle and high school levels, has moved out of the elementary niche of rudimentary skills, but has yet to make it into the realm of complexity necessary to apply it back into the world. Our happy compromise, as teachers, is to go with a two-pronged attack:

1. Tell the students that the practicality comes later
2. Create word problems about trains leaving stations or people tossing balls off cliffs

Every teacher of mathematics (from the wide-eyed rookie to the well-weathered veteran) has encountered "the question" numerous times. We hear it so often, that I would imagine many teachers have a well-rehearsed response to the query. I know I do. Even though we are prepared for this onslaught, the thought of having to employ our answer triggers chills down our spines and sends us retreating into the staff room for cover:

"When are we going to use this?"

There are sects of mathematics that present obvious answers to "the question". Measurement has cooking, geometry has the trades, and probability has gambling; each to their own segment of society. I have already begun to envy the science teachers who bottle themselves up in their back room only to emerge with a concrete and engaging example of when people actually use their discipline. This takes the form of a specimen jar, an interesting chemical reaction, or a proposed perpetual motion machine. I still remember anxiously awaiting "Explosion Fridays" in my Chemistry 30 class in high school. It seems as though these subjects have endless applications, and they harness their practicality in a laboratory.

A lab is where the students get a hands-on, experimental crash-course in their discipline. It is my belief, born partially out of my aforementioned jealousy, that there are situations in the mathematics class where students need the opportunity to play around with mathematics in a laboratory setting.

Before we get too much further, I must give a word of caution. You may have noticed that I always use the term "mathematics lab" over "math lab". Although the latter may be easier to say, speaking too quickly may give your students the impression that they will be creating methamphetamines. Awkward explanation for the administration; I digress.

There have been 3 deliberate labs in my math class this year. My hope is that by briefly detailing the setting and task, you too will begin to see the value in a practical, exploratory, and hypothesizing environment.

1. Monty Hall Problem Mathematics Lab

Students were presented the wildly popular problem verbatim from the game show. (I have found that the newer rendition found in the movie "21" works better as an introduction.) We spent a few minutes hypothesizing and gathering into our lab groups of 3-4 students. The majority of the class was spent re-enacting the situation and creating data. Not only did this build skills in the scientific method, it is, in my view, the only way to fully understand probability. Students must understand that the theoretical calculations we create are actually mimicked in their experiments. The second day, the class reported their data, and it was compiled. Most of the class was spent discussing what the data was telling us, which hypotheses were correct, and how we could alter the experiment to get differing results. (Finding the problem is a simple as googling "Monty Hall Problem". I would encourage you not to research an answer until you have tried the very same lab.)

2. Route to the Commons Mathematics Lab

I gave the lab groups a miniature map of the school that is usually given to substitute teachers. I asked them to resolve the problem of hallway congestion by measuring out the fastest way to the commons and back to class. The problem was left intentionally vague, and the students were given a meter stick, a pencil, a calculator, and a centimeter ruler. Students began suggesting the route they took and locating it on the map. A scale factor was devised and tested on various lengths throughout the school. As students became sure of their results, I challenged them to create a map for their most efficient school day. They began charting their path from class to class. Students were encouraged to test this theoretical shortest distance during lunch in the hallway traffic; soon more crowded hallways needed to be weighted differently. Although I never got this far, a general "congestion constant" could have been developed for each corridor. In a main hall, one meter may be equal to 2 meters on an abandoned one.

3. Fermi Estimation Mathematics Lab

I have used this format a number of times this year to build basic numeracy skills. Students are asked to calculate large or small quantities in a general sense. It is entirely about numeracy-based estimation. How many hairs are on your head? How many hairs are on the floor of the school? Students enjoy creating outrageous questions and tackling them empirically. How many hairs does the average student get caught in their mouth in a given school day? The problems expand to include basic arithmetic, probability, surface area, etc. For ideas of Fermi estimations, see "Guesstimation" by Lawrence Weinstein and John A. Adam.

There are a lot of practical topics in mathematics. Creating disenchanted questions about choosing coloured tiles from a bag does not give math its full due. Setting up mathematics labs creates room for curiosity in math; students take direction of the content. Not to mention that giving students a practical platform to do mathematics will reduce the odds of having to field "the question"


1 comment:

  1. Great ideas. Have used Monty Hall in class to good effect several times but will now look for clip in "21". Excellent exercise to get people to plan routes around school and this could bring in the anomaly that building a dual carriageway to relieve traffic can make matters worse when everyone tries to use it thereby making a bottleneck.

    However, the fundamental problem is that pupils have very little practical experience of problems now. They do not take their toys apart when they break. Gears on bicycles just "work" no thought is given to how. Even adults merely replace items when they break because most are controlled by electronics which are difficult to fix therefore very hard to understand because you cannot "see" anything. Open the bonnet/ hood of a car and what do you see? The "mechanic" in the garage plugs the car into a diagnostic tester.

    People have less and less practical experience so cannot develop their mathematical skills to solve problems that they know nothing / little about.