Every Student Knows Something About Everything

Are teachers and technology wasting that resource?

Welcome New Subscribers

A hearty Mathworlds welcome to the people who subscribed after my last newsletter about chatbots. I write every Wednesday about math, education, and technology, but only occasionally about chatbots and generative AI, which reflects my interest in writing about ideas that are likely to transform learning and teaching. If it wasn’t clear from the post that brought you here, I’m pretty convinced generative AI will do no more than nibble at the edges of that relationship over the next five years.

If you came here for AI commentary, all is not lost. At the end of each post, I write an “Odds & Ends” section, an analysis of provocative writing from around the internet which often includes the latest comings and goings in generative AI.

Shared experiences transform student learning.

Schwartz and Bransford describe, concisely, why lots of explanations fail to help students learn, including explanations delivered by a chatbot:

It is relatively easy to tell a distinction to someone, if that person shares the same set of experiences. However, with respect to the content of instruction, shared experiences are exactly what novices and experts are missing.

My young kid was banging loudly on a small toy this weekend during hours which I have explicitly reserved for “activities other than banging on small toys.” I asked the kid to be quieter. The kid then banged more quietly and asked, “Is that okay?” Like a fool, I said “yes,” mistakenly assuming we were allies in this exchange rather than antagonists. So the kid then banged just a l i t t l e louder and asked, “Is that okay?”

I then understood where all of this was headed and, after sending my kid to his room, reflected on the fact that, somewhere at the level of marrow, he knows about the intermediate value theorem of calculus.

He doesn’t know it in this form (via Wikipedia):

If f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) in the interval.

But he knows about it. He knows that when he does a loud thing, dad is unhappy, and when he does a quiet thing, dad is happy. Therefore, somewhere between loud and quiet is a decibel level where dad turns from unhappy to happy, and we should find it!

If and when my kid learns the intermediate value theorem in calculus, his teacher will have a much easier time teaching that theorem if the teacher can recognize, “This kid doesn’t know nothing about the intermediate value theorem. This kid has known about the intermediate value theorem from a very early age. This kid and I have had some shared experiences that might be useful here”

And we can design shared experiences.

To give advocates of personalized learning some credit, it is hard for a teacher to know which kinds of experiences they share with anywhere from 20 to 40 students. To give chatbots some credit, they can very neatly repackage ideas like the intermediate value theorem into new contexts on demand. For example, I asked Khanmigo, "Can you give me an example of the intermediate value theorem using baseball as a context?”

Khanmigo’s response:

Sure! Let's think about a baseball game. Imagine a player hits the ball at a height of 1 meter and it reaches a maximum height of 100 meters before it lands back on the ground. Can we say there was a point in time when the ball was at a height of 50 meters?

If I were to describe this interaction in a word, it’d be “neat 👍.” This would be more than neat 👍—it would be wonderful!—if not for the fact that students very often:

• lack the metacognition and executive function to request an example of an idea in this manner.

• prefer to share an experience with other students if one is available.

• struggle to meet the literacy demands of this kind of exchange.

One job of a curriculum author is to design experiences that are accessible to a large group of learners, experiences which they can share together.

For example, we can ask students to predict which of these circles definitely do, definitely don’t, and definitely might hide an intersection with the x-axis. Then students see if they were right.

The last screen of this activity offers students and teachers a shared experience of a key caveat of the intermediate value theorem: that the function must be continuous.

Here is how many people experience math in schools:

• the teacher talks about math through their own experience

Here is how I am excited to see more and more students experience math in schools:

• the teacher and students share an experience

• the teacher talks about math through that shared experience

These ideas, that every student knows something about everything and that we can design experiences for teachers and students to share, are enough to direct several lifetimes of meaningful work.

Odds & Ends

Desmos released a 3D graphing calculator. It’s performant and beautiful and I love the “quests” concept they’re using for onboarding.

Glenda Morgan offers fantastic analysis of education technology (generally in higher ed) and always measures her predictions twice. I enjoyed her recent post, Success Porn in EdTech, and probably identified too much with this line: “Those of us who have spent a long time in EdTech all instinctively know when something is being promoted as a success but when the reality underneath is really different.”

Over at LinkedIn, Kim Machnik asks an interesting question: “What are possible roles for AI in a classroom that centers student thinking, values their voices and experience, and prioritizes high-level cognitive skills?” A great question, though I would reframe it as, “What tools would best help us create that classroom?” rather than “How do we find a job for AI here?” which seems to be the cognitive trap a lot of people are falling into right now.

It isn’t my interest or my job to keep any of these AI companies afloat but if it were, I’d find Benedict Evan’s comments about how thinly many of these companies wrap themselves around language models they don’t control pretty interesting. Interesting in a similar way: an early AI unicorn cutting its valuation drastically and changing its CEO.

I haven’t used MagicSchool.ai since I wrote an analysis of teacher copilot tools last month but I am glad I created an account because the founder, Adeel Khan, sends out a weekly roundup of AI news that’s consistently very interesting! If the funding cliff comes for MagicSchool, I hope Khan continues writing the newsletter!

I thought this 90-second clip of a conversation between two basketball players had a strong connection to teaching and I will leave a description of that connection as an exercise for the reader.