On learning (math, in particular)

Here are some bon mots I collected from an online course on math didactics.

When you had the answer wrong, your brain grew…

… when you got the answer right, nothing happened to your brain. This aims to build a work ethic. Dave Panesku tried different starting messages at Khan Academy Videos. Messages that tried to build work ethic (“the more you work, the more you learn”) made students solve more problems, while encouraging messages (“this is hard, try again if you fail first time”) had no effect compared to the control group.

It is also about appreciating mistakes.

Convince yourself, convince a friend, convince a skeptic

Most math students and their parents have difficulties to name the topic or learning goal that is currently covered in class. Discussing different ways of seeing, different paths and strategies to tackle a problem, however, is what learning (math, in particular) is about.

Uri Treisman and his colleagues showed in their minority studies [1] that students who discussed the problems outperformed those who did not.

Where are you, where do you need to be, how to close the gap

Feedback is important. Regular peer and self assessments outperforms control groups, especially low achievers improve [2].

Grading does not provide useful feedback. Diagnostic feedback encourages the students and outperforms grades – as well as grades together with feedback [3].

Pseudo-context problems need redesign

Math problems can be fun if they are presented in an open style that allows multiple entry points.

It is more interesting to construct two rectangles given a perimeter than to find the perimeter of a given rectangle.

“Doing and undoing”, i.e. being able to reason both forwards and backwards with operations, is the central practice in algebraic thinking. For instance, first discuss several methods to solve a problem, then present expressions for new methods and discuss what the method behind the expression might be [4].

There is an web effort on makeover of dull math problems.

Can you do any number between 1 and 20 by using only 4 4s? For example, 20 = (4/4 + 4)*4.

1. Fullilove, R. E., & Treisman, P. U. (1990). Mathematics achievement among African American undergraduates at the University of California, Berkeley: An evaluation of the Mathematics Workshop Program. Journal of Negro Education, 59 (30), 463-478.
2. White, B., & Frederiksen, J. (1998). Inquiry, modeling and metacognition: making science accessible to all students. Cognition and Instruction, 16(1), 3-118.
3. Butler, R. (1988). Enhancing and Undermining Intrinsic Motivation: The Effects of Task-Involving and Ego-Involving Evaluation on Interest and Performance. British Journal of Educational Psychology, 58, 1-14.
4. Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10. Heinemann, 361 Hanover Street, Portsmouth, NH 03801-3912.

Gillray on knowledge gained from books

Have a look at the figure above. In the image we see an experiment that goes terribly wrong. Two persons want to transfer their knowledge on horses and try to domesticate crocodiles. We see a whip, a bridle, a saddle and an instruction manual “education for crocodiles”. But crocodiles are not horses, they bite back, and turn the scene into a blood bath.

200 years later it is still true

Gillray created this caricature in 1799, and it was a comment on politics. It relates to personal letters from enraged French officers in Bonaparte's Egyptian command [1]. It is the trait of a caricature to exaggerate its subject and should of course be interpreted metaphorically. The crocodiles may represent the Egypt people Bonaparte tried to subordinate as he did in many Europian countries. On the other hand, the exaggeration, which is also an abstraction, allows us to take the image out of its context and apply it to other situations. Here come two examples.

My first encounter with this image was at an exhibition on the “Age of Reason”. The topic of that exhibition was the time of the transition from religion to science that took place at the end of the 18th century: The first anatomy atlas of the inside of the human body appeared, the earthquake of Lissabon shattered faith in god, and it was the time of the French revolution. In this context, Gillray's caricature appears as a critic of the new stream of thought at that time. Maybe the people overused the scientific method in the first enthusiasm, and rather than seeing the object openly and attentively with some kind of wonder, the persons in the picture simply apply the knowledge they read in books. This goes wrong, and maybe this overacting, this trying-too-hard, is why the age of reason entered eventually the age of Romanticism, which prized intuition, emotion and imagination over the scientific rationalism.

Today, some people are very enthusiastic about data. They proclaim the age of dataism: cheap storage and lots of mobile sensors everywhere lead to huge amounts of data and many new insights. However, Gillray's caricature can serve as a warning here. Some problems will not be tamed and utilized with the right dataset and some analysis method, but will bite back.

1. Draper Hill Fashionable Contrasts. Caricatures by James Gillray, Phaidon Press, 1966