Fantasy High School Part 2
[See: fantasy-high-school for part 1]
Recently I listened to a podcast interview with Conrad Wolfram: Link. I’ve been aware of some of his thinking especially his popular: Ted Talk. But in the past it seemed to hazy and unfleshed out to get my mind around and decide if I agree. In the intervening 8 years, he’s moved forward and there’s a website: http://computerbasedmath.org/ and the beginnings of some concrete curricular ideas.
So I’m going to parse and explore some of the questions I thought while listening again especially in the context of “Is Wolfram’s vision the right path forward for High School?”
To paraphrase his basic arguments:
Math is all about 4 steps:
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Define the problem
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Can we turn it into a symbolic representation?
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Take the question to an answer
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What does it mean? Is that right?
He points out that Step 3 is where technology can be most useful, yet we spend 80% of our math time in schools on that step by teaching hand calculating. Instead, he says, we should be focused on steps 1, 2, and 4 which are what people really need to be good at today.
What this means in practice is using all the tools of Mathematica to explore more complex problems that don’t have neat Diophantine solutions or simple characterizing equations. “Why not use a cubic?”
An example problem:
“2. “Should I insure my laptop?”
Students begin by playing lotteries and calculating the costs and probabilities of winning. They learn how insurance works and then determine their personal utility curves for their laptop and compare the losses and gains made from insuring or playing a lottery. They role–play a competitive insurance market, with student insurance agents determining policy premiums to offer to their student clients. The teacher is able to see the big picture and determine an overall winning insurance agent.” I think there are some interesting ideas brought up here.
- Are we overemphasizing algebra 1 techniques like factoring?
- Is Mathematica the right tool to use instead vs. Desmos or Python, Sage or …?
- Is coding more fundamental? Do we need to emphasize this first?
- The sample material presented is pretty small and still has a very strong statistics bent. It remains to be seen whether this approach is comprehensive and links together.
For a concrete example: Let’s think about number talks. To channel Wolfram focusing on the various ways we use the distributive law etc. to solve a problem like 23 x 5 mentally is the wrong use of time. In real life its going to be 23.049 x 4.67 and we won’t mentally solve it exactly at all. Instead, what’s useful is knowing how to estimate this is approximately 20 x 5 (or any other good estimation) and realizing how to quickly calculate on a phone/calculator/desmos etc and confirm the answer is reasonable.
I’m going to focus on the thorniest question for me. ”Can we get here from there?” How much conceptual understanding and practice with simpler problems is needed to effectively use a sophisticated tool like Mathematica? I’d argue we don’t really have an examples yet of this since we all still are educated in the old path. I’d really like to see experimental data on real kids to see how this all comes together. [And in Wolfram’s defense there seem to be pilot projects ongoing] Likewise, this is fundamentally intertwined with which topics should be explored during High School. All these tools can be applied in multiple domains. Which ones are fundamental and should be done universally?
At this point I see this more viably as an approach to enrich the curriculum. In other words:
- teach enough coding to use tools like Mathematica
- tackle more complicated applied problems using tools.
- Thread these explorations and visualizations through the regular curriculum.
- This has particular relevance to Calculus which I’m still avoiding but in a nutshell what is the value of learning to integrate if this can be done by tools?
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