I felt like EC wouldn’t really fit a SBG approach where learning must be shown. Sinced once you show you have mastered a standard, your grade will reflect that growth and the EC would not be needed.

]]>I feel our job as teachers is not to just teach slope but to make connections to concepts that will help the kids store and build a logical framework. The variable parts perspective and exploration into meaning of the numbers 3, 2, and 3/2 in the slope used above is a step in the right direction. I have my students design roof lines for passive solar housing projects, this can go as deep or surface as you please and it helps connect slope with steepness and practical design issues like snow load and sun angle through the year. And it’s fun and engaging to design/plan something meaningful. That’s what math should be about. If we can meaningfully teach more topics students at different levels may find more inspiration and connections that stimulate their creativity and curiosity then we did in junior high and high school. One practical dilemma is the time it takes to develop more connected meaningful concepts and the time we have to fit the complete content of a class into their brains.

]]>The hard “trick” is to make Category Theory digestible for pupils in compulsory school. (Partly renaming of concepts are helpful in that.) The main observation to do is that when we tell the comparison of two values, we mention a number (and an unit, if present), but moreover we mention an operation (!), that often comes in disguise. For example we may say that 5 m are 2 m more than 3 m! The operation becomes visible when we translate “more” to Latin: it is “plus”. As can be expected “less” is “minus” in Latin. Ronald points out that comparisons, like arrows, have a direction. In a comparison, one of the objects we compare, works as a reference object. This object I name “root”, for reasons that will show. Now to the definition (confined in its form for use with numbers in school):

A comparison is a way to tell the size of a value by making reference to a root value and tell what “to do” (operation+value) with the root to get the value to be described.

All this can readily be depicted in a kind of diagram. I use the example above to outline the diagram:

The value 5 m, that we want to describe is placed to the left, and the root 3 m to the right, with a left pointing arrow in between. Above the arrow we write +2 m.

(You may construct this diagram in your blog.)

The left pointing arrow can be read: “compared to”, and the comparison result is expressed above the arrow.

I don’t know what conclusions you may draw from this. I can only tell it took me years to realise all its implications. I would say it have a major impact on didactics and mathematical theory. For instance: Category theory tells you can compose arrows, which means you get expressions like “+2 -5 -3 +4”, where you compose elements with “bound” operators. The expression tells we have a composition of an increase by 2, a decrease by 5, etcetera. The outcome will be a total decrease by 2. (Incidentally, this way of regarding a sum of termes seemes easier for pupils to grasp. E.g. the interpretation “decrease” is more natural than “negative number”.)

But as this is a blog comment I am writing, not a book, I have better end here.