The evolution of mathematical exposition

More rambly and unsubstantiated than usual, apologies —



Theory: Mathematical exposition has evolved just like fiction writing has. However, tradition has held stronger in mathematics (likely due to a need for precision) and it means that clarity in writing is if not actively discouraged at least passively devalued.

Theory: We are not anywhere near the threshold of simplest and clearest explanations in the exposition of mathematical subjects.

Still, what used to be difficult is now considered easier. Various subjects have shifted their supposed level. For instance, not long ago College Algebra was the prestige class at the top of the high school level.

Furthermore, our raw definitions of what each class is has shuffled the actual content of subjects; Algebra I from the 1940s is not the Algebra I of today.

Theory: It would be possible to take a “hard” subject like group theory or transcendental number theory and make it comprehensible at a lower level. However, as there is no requirement to do so there is little motivation to make the subject easier. When a curriculum shift happens to move topics to a lower level, mathematical exposition evolves to catch up.


4 Responses

  1. “Still, what used to be difficult is now considered easier. Various subjects have shifted their supposed level. For instance, not long ago College Algebra was the prestige class at the top of the high school level.”

    You can’t make up a theory and then make up the evidence. The test results say that the vast majority (> 80%) of high school students fail to “get” algebra. My analysis of TIMSS and AP show further that the students doing well in these subjects has declined over the last 20 years. What are you basing your statements on? Only that they attempt to teach algebra to more students now?

  2. You’re opening up a different barrel of monkeys there. I’m just talking about mathematical exposition, not if shifting curriculum down is a good/bad thing and especially not effect on exam scores.

    I’m not sure where you’re getting a decline out of international test scores, though. The FIMS from the 60s had the US near the bottom in all categories (11th in 8th grade, 12th in 12th grade math students, 10th in 12th grade non-math students). There’s never a point where our international test scores have been the envy of the world.

    to pick another data point, SAT math scores have had improvement since the 70s up to where the test was changed in 2005 and has been pretty much steady since (

    Anyway, none of this has anything to do with if algebra-for-all is a good thing. The data I know of is pretty mixed. Using overall test scores as a justification shows no disaggregation so it’s hard to peg blame on one specific policy.

    • Ok, mathematical exposition, I like that, though we need to define it. I am thinking “mathematical exposition” = “explaining math to laypersons”. And that would certainly describe the last 30 years of math education reform.

      I spent a several years studying many exams, including the state algebra exams and the AP calculus exams and what strikes you right away is the number of non-technical “concept” questions on these exams. Questions that you would expect a layperson to answer. Older exams have some of these as well, but not as many. I suppose you can chalk this up to the “conceptualization” of mathematics. The real nail in the coffin though, is the cutoff scores in place today. The number of questions the student must answer correctly in order to pass.

      It is ok for a test to have some conceptual questions (freebies) as long as the cutoff score is 70% to pass. If you get 70% correct then you have made it past the conceptual questions and into the actual subject. You are more than just a layperson. However, if the cutoff score is 30% and there are many conceptual questions, then you haven’t made into the subject much at all. And this is the vast majority of students today. Even in college. They emerge from algebra 1 a layperson. And then they take algebra 2 and emerge from that a layperson, and so on. They might as well skip all the prerequisites and take calculus, and emerge from that, a layperson. You can make all the way through all of these courses, as nothing much more than a layperson.

      If this is “mathematical exposition” then I agree with your theory. In fact, I can prove it.

      For the decline I am speaking since 1995 (when the TIMSS assessment was first given). Not only has the overall percentage of students scoring 90% or better declined, but the number of classes with at least one student scoring 90% has declined. The TIMSS exam is given to whole classes and we can look at such things. I believe the two are connected. When we look at the better performing countries, the class average could be 50%, but there is still at least one student scoring 90% or greater. In fact, the class average in some of these countries must drop to something like 20% for there not to be any student scoring at least 90%. In the US the picture is very different. As the class average falls, the possibility of a student scoring 90% falls with it, quite rapidly. And this has gotten worse since 1995.

      Something is limiting the better students in these classes.

      I’ll have to review the SAT reference you provide, but the SAT had to be scaled down (or up depending on how you look at it) in 1996. Significantly in mathematics. A 420 in math in 1995 would be a 500 today. I can’t see how that is an improvement, but that also includes the effect of having more students take the test.

      It is very hard to quantify what is the state of affairs, but I would defer to teachers if I really wanted an answer. Teachers with years of experience. What do you think a survey of math teachers would reveal? That students are more prepared now, when they enter their class, than before? It is hard for me to fathom that this is the case.

  3. I just feel that if you are going to say “better exposition” then you have to qualify what “better” means. I follow many blogs and I admire several individuals for their ability to state mathematical principals in the most “simplest and clearest” way. And by my reckoning, if we were to gather up all of these cases we would have a pretty complete, simple and clear description of mathematics. For every principle in mathematics, someone, somewhere has invented the simplest and clearest way to state it. I have incorporated this into my reference curriculum.

    However, I have also noticed that most (math) teachers cannot appreciate these simplest and clearest explanations. Which means that most students cannot either. The reason is that they cannot relate to them. Unfortunately, the simplest and clearest explanation in math isn’t like the simplest and clearest way to get to the grocery store from your house. Anyone can recognize that explanation because they can relate to it. They are familiar with their house, the grocery store, and can recognize a better way to get from one to the other. If only they were that familiar with the requisite elements in mathematical explanations.

    That is the issue now. The textbooks have been rewritten, yet the students still do not even understand the simplest and clearest explanations. Even by your account, the test scores haven’t risen by much. The reason for this I think is pretty clear. The students are incompetent when the enter the classroom. They have been pushed past their competency and no amount of rewriting of textbooks will fix this. The only way to fix this is to stop pushing them past their competency. Do not push them into algebra if they haven’t shown competency in arithmetic. Do not push them into calculus if they haven’t shown competency in algebra. If it were your kid I think you would consider those guidelines not just sensible but the right thing to do. For some reason, schools don’t treat kids as if they were their kid. Well, we know the reason. They used to, but due to the politics, litigation, and self preservation, it just isn’t prudent anymore.

    In any event, I don’t think you can talk about “better exposition” without competent students. Those teachers I admire that have over the years developed the best ways to state things, didn’t do that in a vacuum. They did it in classrooms with competent students. Which probably explains why so many teachers have difficulty appreciating the explanations.

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