What is Mathematical Understanding?
Unfortunately, I didn't prepare my ideas at all for AIED. So I mostly improvised on the topic of how logic & argumentation games can be used in science education, and the use of formalized theories.
---
What is Mathematical Understanding?
My dream that you should be able to transfer mathematical knowledge from a formalized mathematics proof (as in Coq, Mizar, etc.) to a student was based on the incorrect assumption that formal proofs contain enough information.
The question that needs to be answered is
"What is mathematical understanding and how can we model/represent it?"
I think mathematical understanding of a problem/domain requires somewhat more than formal proofs. It requires:
* mapping formalisms to intuitions (which may be formalized as models) and witnessing that the homomorphism/analogy holds up: checking (weaker versions of) "soundness" and "completeness". While proving the relevant facts (completeness), the user fails to prove absurdities with the formalism (soundness).
* evaluating counterfactuals (proof critics: "would this proof go through without this assumption?")
* etc.
Btw, while some kinds of intuition can evaluate statements and can be relied on to guide formalizers (e.g. I *know* that squares are special cases of rectangles, and any formalism must respect that),
other kinds of intuitions are just intuitions, which tend to be acquired empirically through experience with examples one has already seen or imagined. (see Lakatos's(?) notion of intuition by experience) Alison Pease - A Multi-agent Approach to Modelling Interaction in Human Mathematical Reasoning (2001) )
from Bundy's "A Proof, Conjecture and Theory Editor":
Most work on automatic reasoning assumes a given theory and conjecture and develops a proof in the context of these. However, in practice, especially in common-sense reasoning, both the conjecture and the theory are adaptable. For instance, the failure to prove a theorem might result in a change to the conjecture or the theory in order to make proof possible.
Also, Paul Thompson - The Nature and Role of Intuition in Mathematical Epistemology
seems interesting.
Also, Mobius Stripper quotes reader "Susan":
It’s possible for a non-expert to determine whether a child is literate by asking him or her to read something unfamiliar (ideally both outloud and silently) and to explain what they’ve read in their own words.
We need to define and expect something similar as far as being numerate.
In my experience, this can be done by:
* engaging the student in argumentation
* make him/her imagine a slightly different situation, ask questions. Ask the student to compare: "why does this procedure work here but not there?"
I'll have to organize these ideas later.
---
What is Mathematical Understanding?
My dream that you should be able to transfer mathematical knowledge from a formalized mathematics proof (as in Coq, Mizar, etc.) to a student was based on the incorrect assumption that formal proofs contain enough information.
The question that needs to be answered is
"What is mathematical understanding and how can we model/represent it?"
I think mathematical understanding of a problem/domain requires somewhat more than formal proofs. It requires:
* mapping formalisms to intuitions (which may be formalized as models) and witnessing that the homomorphism/analogy holds up: checking (weaker versions of) "soundness" and "completeness". While proving the relevant facts (completeness), the user fails to prove absurdities with the formalism (soundness).
* evaluating counterfactuals (proof critics: "would this proof go through without this assumption?")
* etc.
Btw, while some kinds of intuition can evaluate statements and can be relied on to guide formalizers (e.g. I *know* that squares are special cases of rectangles, and any formalism must respect that),
other kinds of intuitions are just intuitions, which tend to be acquired empirically through experience with examples one has already seen or imagined. (see Lakatos's(?) notion of intuition by experience) Alison Pease - A Multi-agent Approach to Modelling Interaction in Human Mathematical Reasoning (2001) )
from Bundy's "A Proof, Conjecture and Theory Editor":
Most work on automatic reasoning assumes a given theory and conjecture and develops a proof in the context of these. However, in practice, especially in common-sense reasoning, both the conjecture and the theory are adaptable. For instance, the failure to prove a theorem might result in a change to the conjecture or the theory in order to make proof possible.
Also, Paul Thompson - The Nature and Role of Intuition in Mathematical Epistemology
seems interesting.
Also, Mobius Stripper quotes reader "Susan":
It’s possible for a non-expert to determine whether a child is literate by asking him or her to read something unfamiliar (ideally both outloud and silently) and to explain what they’ve read in their own words.
We need to define and expect something similar as far as being numerate.
In my experience, this can be done by:
* engaging the student in argumentation
* make him/her imagine a slightly different situation, ask questions. Ask the student to compare: "why does this procedure work here but not there?"
I'll have to organize these ideas later.