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.

What I would look to do is have a series of pretty simple tasks that help to ‘sort’ people into different ways of learning the material and then dynamically present or PROMPT for the rest of the material. Some kids might do better with physically manipulating an object and others watching it on the computer. Learning on the computer doesn’t work for everyone.

Animation is going to help a lot of people where static images and words fail.

You gotta hit all the buttons and let kids who ‘get it’ skip through quickly.

No Kid Left Behind should mean we work with everyone to achieve to the best of their ability and we celebrate their strength areas. Not every kid is going to be a math wiz – and the math wiz kids should be allowed to excel in that area. If they are ready for calculus at age 10 they shouldn’t have to keep doing multiplication tables << THAT is a huge problem for kids. In practice it is almost "No Kid Allowed To Excel" — it's SO COMPLETELY BORING for kids who get it already and honestly even worse for the kids who don't. You are just torturing them with the same material year after year after year. Make sure they can balance a checkbook and make change and LET THEM MOVE ON — but challenge & reward kids who want to go further into any given area.

Maybe carpentry is a better way for some kids to "get" slopes? Maybe dress making is?

Let them build and build and build — challenge them to build more complex things so they HAVE to work it out for themselves to get the angles & lengths right.

If we treated education as an investment in our future we might be able to afford it.

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