Q2: there is an old video from NCTM that includes footage of young students, with the question, “what does the “=” sign mean?” And many of the answers boil down to “ding! It means find the answer!” So many students come through early math not understanding the basic concept of equivalence. This explains, in part, why algebra is so challenging, and why such science topics as chemistry lead to difficulty for many students.

Q3: I saw this issue in play in a school in a Southern state, where teachers were truly proud of their math program. There was a call-and-response structure in play, and many student were able to name the pieces of a cross-mutliplication problem. But what would happen if they forgot what to do in such a problem– cross multiply or simply multiply across? Without true understanding of what was called for, students who hadn’t completely memorized the algorithm would be stumped.

Q5: It’s easy, and immediately useful, to tell young students that multiplication is repeated addition. Ok, and in easy exercises, that works. But then, how do you explain multiplication by fraction? 8 x 1/4, for example. How can multiplication yield a product that is LESS than the first number? That requires explaining, particularly if students have internalized the idea that addition makes the number bigger, because in this case multiplication make the initial number smaller. The problem, then, is a long-term one, given the challenge of understanding the power of multiplication. The simple definition works initially, but then risks betraying students who can’t see beyond the “rule” they learn in elementary schoo.

Q6: This is a version of the infamous NAEP army bus problem. The challenge here is that students don’t necessarily equate math-class numbers with real-world problems that require not only pure mathematics but also clear, context-related thinking. Given the parameters of the problem, do you need an additional table or not? This is a significant real-world implication of math, rather than one that resides simply in textbook pages.

Q7: This is definitely a question that the common core math standards unfortunately do not address: given basic math structures, when do we use them? When do they work? When don’t they? Certainly, we can easily calculate mean, median, and mode. But when should we use which, given a context that extends beyond the tight structures of conceptual math? How do we know? And how can we make a decision, given our calculated answers? These are truly critical questions in science, medicine, the social sciences, and even the “data” on student performance that educators are constantly asked to analyze..

Q8: Without a rule, how would we solve a problem like 4 x 3 – 2? You could group the numbers in one way, I could group them in another. But who would be right? Unfortunately, there is nothing inherent in math to say that multiplication and division take precedence over addition and subtraction, or that powers of a number supersede both. What to do, then, to ensure clear and consistent procedures? There only option is to impose a human rule, to make sure that everyone solves complicated problems in the same order. Hence, PEMDAS. There is nothing inherent in mathematics to compel this. It is purely a human construction. And yet how many math teachers understand this? And how many help their students understand it?

Q12; Euclid begins his Elements with a definition: “a point is that which has no part.” Ok, some of us understand that, but how powerful a starting point is that for a system that develops all of geometry? The key question here is the extent to which Euclidean geometry is a perfect (divine?) system, versus the extent to which it is a bridge between a pure system and human understanding. If you accept Euclid’s first assumption then, by the end of book 3, you can prove the Pythagorean theorem. If you can’t accept it, where are you?

Q13: “Goofy” was probably not the best qualifier here. The real question is “what can everyone accept and assume,” vs. “what needs to remain unresolved?” The question still stands among serious, high-level mathematicians trying to extend the parameters of pure mathematics.

I know: much of this may be annoying. But I spent many years in deep discussion with Grant on these issues, and I know that they resonated with him until he died. He was utterly committed to separating “absolute truth” (to the extent that there is such a thing) from “systemic truth” (my term, to indicate a necessary rule or law to ensure the consistency of a system) from “human truth,” (the understanding that individual people bring to a specific instance). The challenges contained within the differences here are truly important, and the require close attention.

Denise Wilbur (Grant’s wife and professional partner)

]]>Then right an essay explaining the sequence of thought that lead to your answer highlighting exactly where the error slipped in :-)

]]>As for the last question, here’s my answer, borrowed by Heath, Euclid’s translator: (BTW it assumes that there is a difference between axioms and postulates, a distinction made by Euclid and other Greek mathematicians, a props your parallel postulate point). “We have to postulate certain things as true. At first you may not grasp why we do or why something is a valid postulate, but you will see that this assumption is necessary for things we want to prove (e.g. the Pythagorean Theorem.” This is why, according to many math people, Euclid proved everything he could BEFORE introducing the parallel postulate, and why the objection that the postulate was not as self-evident as others is a misunderstanding. Axioms, not postulates, are self-evident. Hence, the distributive property.

Curious: what did you think of the test after taking it? That to me is the next interesting question. After all, my questions were more suggestive than metrically sound. What did it make you think?

]]>