My First Class Mathematics project at the St. Erik Catholic School
by Ambjörn Naeve
In 1991 my daughter Ylva started in first class at the St. Erik's Catholic School in Stockholm. Due to the kindness and interest of her successive teachers (Anette & Kristina), I was allowed to work with the children and "talk mathematics with them" for about an hour a week (and sometimes two). From 1991 to 1997, this was my ongoing First Class Mathematics project. It gave me the opportunity to test some of my pedagogical fantasies "in real" so to speak, and it has strengthened my conviction that mathematics can be taught and learned in a first class fashion from the start and all the way up through the educational system.
This is not the place to present the details of the mathematical smorgasbord that I have presented to Ylva's classmates over these 6 years. It has included subjects such as
- summing the first 99 integers (1+2+3+ ... +97+98+99)
- figuring out how many different necklaces that could be made from 4 different beads
- figuring out in how many ways the class could line up in a row and appreciating the size of 25!
- playing with the faculty concept (by projecting Mathematica onto the overhead screen)
- building the Platonic solids, and exploring their duality in connection with Euler's formula
- exploring patterns by the help of MacWallpaper
- cutting up Moebius strips in various proportions, trying to figure out in advance what happens
- figuring out when a graph can be drawn without lifting the pen (leading to Euler graphs)
- doing arithmetic in different bases (how would we have counted if we had had 1 finger on each hand?)
- separating between the number and the figure (= its representation in a certain base),
- figuring out how many molecules of air that fit into an empty milk-brick (Avogadro's number = 6.02x1023),
- treating variables as boxes and solving equations as finding out what's in the box (showing how Mathematica does it)
In fact, I have found that many of the subjects that I have worked with are presented in an excellent little book called Matte med Mening (= Meaningful Mathematics) by Kristin Dahl, which I was not aware of at the time. Kristin presents a wealth of interesting material that can be presented to children at an early age in order to provoke and stimulate their interest and curiosity in mathematics. It's a shame that such a great book should be "too expensive to buy in class" as one of Ylva's teachers so adequately put it.
Anyway, working with these kids has been great fun, and it has strengthened my conviction that the subject of mathematics can - and should - be presented to children in a much more constructive and thought-provoking way. I will give a brief discussion of the pattern part of the project.
Working with MacWallpaper
This was the main activity during the first two years of the project. Using two Mac-II and a Powerbook 170, I assisted the children in working out their own patterns. The children were first encouraged to draw anything they wanted in the editor, and then play with the 17 different pattern-possibilities in which their basic image could be turned into a wallpaper. After modifying the drawing and choosing their favourite type of symmetry, the children were given paper printouts of the corresponding pattern and encouraged to expand on it further - by adding the dimensions of colour.
We also used an LCD-type of overhead-screen projector to display the patterns in front of everybody and discuss them together. In this way we analyzed their symmetry, and the children could practice how to discover the more subtle forms of symmetry - such as e.g. glide reflections. But at the same time we also discussed the holistic "gestalt" of the patterns, i.e. the impression that they made on us as images. The idea was to convey an experience of the double-brained activity of mathematics - as described in Chapter (7.1). It was an amazing experience to feel the exitement of the children when they were confronted with these concepts. The discussions just did not seem to want to end.
Some General Observations
Of course, my smorgasbord of mathematics didn't manage to turn the oiltanker of learning by repetition. The reasons have to do with the general attitude towards early mathematics education, as presented in the discussion of Chapter (6.4). In other words:
The fear of experimenting with new concepts in mathematics education is related to the algorithmic fundamentalism that still dominates the pedagogical thinking within this field.
The X-anxiety pattern.
The problem is not the learners but the teachers. To put it bluntly, the children don't know that they don't know mathematics, but the teachers and the parents do. And this attitude is very infectious: As an example, consider the following typical conversation - depicted in Figure (20): The parent: "Calculating with x is hard. I never understood it when I was at school." The teacher (later): "Now we are going to start calculating with that mysterious letter x that you have all heard about." The learner reacts with something like: "Oh, shit, I'm never going to understand this stuff." The learner (later - at the slightest sign of conceptual difficulties): "Yeah, just as I thought, I simply can't understand this stuff."
A Variable as a Box with a Name and (maybe) a Content
Building variable abstractions: The "Box-calculus" approach.
Teaching the children how to "compute with x" was handled in the following way. I used the analogy between a variable and a box as a starting point. Both have name and maybe a content - if we have put something inside the box - or equivalently - given some value to the variable. In this metaphor, an equation is a relationship between the contents of different boxes, and solving equations is equivalent to finding out what's in the box.
This way of thinking about variables is very "concrete" and easy to conceptualize for children. Moreover, it corresponds to how the program Mathematica treats a variable. If you type in the name of the variable, Mathematica responds with its value (= content) - if there has been something "assigned to it" (= put in the box). Otherwise Mathematica responds with the name itself. By demonstrating this behaviour, and explaining it in terms of "computations with boxes", I managed to communicate the idea of "what a variable is" to each one of the 25 kids in my daughter's class - at a time when they were about 10 year old.
Manuscripts and other material
- Can be found here