HW#4 - Week in Review

a. What was the most significant thing you learned in class this week?
Last week's class was devoted to learning Geometer Sktechpad (GSP). I've seen it used before, and it is pretty intuitive to learn to use - although not entirely so.

I was excited during class about the idea of getting images from the Web, or movies from my camera, and fitting curves (e.g. Parabolas) to them.

This ended in disappointment. It turns out you can do this by combining GSP and Fathom, but not in a way that is going to make the task enjoyable to students. It is one thing to fight Excel to the death when one is doing a PhD. It is quite another when you are trying to use it to convey a key concept to students.

On the plus side, just that afternoon I had the opportunity to teach 8th grade parallel line construction with compass and pencil. During class I did the GSP exercise on parallel lines. Later, with Robin's perspective, I was able to come to a point where I so each as supplementing the other. Namely, get the students to use pencil, straight edge, and compass to do the first construction. Then transfer this construction with minimum friction to GSP. In GSP they can explore the properties of parallel lines (or whatever) and ask all the what-if questions that solidifies their understanding. The sheer hassle of pencil and compass construction makes it almost impossible for them to ask the same questions without GSP.

I saw a similar thing today, when too quick a transition to a graphing calculator deprives the student the opportunity to grapple with linear inequalities.


b. What questions do you have and what do you want to learn more about?
After the last class I chatted briefly with Robin about my thoughts on story telling. Story telling goes very very deep in all our cultures. I am guessing that it is deeply tied to the evolution of our brain, and from there to the evolution of language.

We introduce children to their new world through stories, and for almost everyone the life-long fascination with stories continues unabated.

Is it just a lack of imagination that over all these centuries we have not introduced our children to numeracy through stories? Or is there something very fundamental about math that to learn to like it is to learn to like an abstraction that is different from the narrative form?

If not the literary arts, then what about the visual and performing arts? There is a lot of abstract symbolism there, with juxtaposition and contra position used to define relationships and hence meaning. How do we nurture our children to love these arts?


c. What applications do you see to classroom practice based on what you learned?
Over the last year I have been forming my own theory of learning, based around the delta of effort needed to achieve the next delta in skill. In theories of Adolescent Development last week we covered Vygotsky's Zone of Proximal Development (ZPD), which got me very excited because it is so close to what I have been thinking.

This makes me even more convinced that there is a very real need for a practice problem generator that that can be tweaked in real-time by the teacher.
This is no science fiction fantasy that I am asking for here.

Imagine your 7th grade class (in what ever physical shape and location) is using the next 30 minutes to work on simplifying algebraic equations (maybe about 10 of them).

They are working with paper and pencil, and you are circulating watching the individuals or groups work through the material. Now imagine the class is small enough that you know each child really well, and you know the material really well, and you have a stack of problems that you know really well. As each child/group finishes a problem, you slip them another one that is just right. They finish 10 problems, each one that challenges them just right. Before they leave, you give them 20 more problems as homework to reinforce the material.

There is nothing fancy here - sports coaches have been doing this for ages.

Fast-forward to the future. They are working with stylus and slate/table-style portable computers. At your desktop computer you see a panel of little windows, each one containing the child's workspace. You can switch to their web-cam view to see their faces when you want to. The module you are working on can generate fresh problems on the fly. Multiple axes exist, either identifying the concepts being worked on, or collating the most common mistakes seen with these problems. There is a slider along each of these axes that you can tweak as they solve each problem. When they have finished for the day, it is only a few mouse clicks to generate twice as many problems for home work for that child. If they work the portable computer at home, you can actually replay the screen for each homework problem they got wrong, and tweak the sliders yet again before the next class starts.

There is nothing fundamentally different between this case and the earlier one (except possibly the class size, but I not arguing for a large class). Of course the technology has changed between the two cases. But adapting problem difficulty based on current performance is something 'educational' software like Reader Rabbit has been doing for 4 year-olds for at least 10 years now. So it is not as if the technology is new.

What would be really new is using everything that computer game designers have learnt in the past 10 years about keeping games just challenging enough. [But for the most part they seem to have come to the conclusion that matching the player against another player in a massively online game is far preferable to matching the player against the computer. The matching problem, although difficult, seems easier than teaching the computer to play a game that a human finds interesting.]