Math for Future Scientists: Require Statistics, Not Calculus

WBahn

Joined Mar 31, 2012
27,389
Sounds like Neil deGrasse Tyson didn't have a very good calc teacher in high school.

EDIT NOTE: I likely misinterpreted part of his story -- I thought he was given a bunch of equations on day one and then started learning how to use them. But he possibly (probably) was just looking at the inside cover of the book completely on his own, meaning that there is no indication of how his teacher actually presented the material. So the rest is reflective of what I have seen on numerous occasions and not specific to his experience.

It sounds like his teacher approached it from the standpoint of throwing a bunch of equations at them and then teaching them how to work with the equations, very possibly without ever strongly tying them to what those equations mean.

I was fortunate and, I believe, had a very good teacher. The first thing he did, which us students didn't like at first, was he said that we would only have calculus in the second semester (whereas the other high school in our district was going to get a full year -- hence we felt we were getting shortchanged). But he believed that it was better to have a semester of analytic geometry before delving into calculus. Of course, none of us had ever heard of analytic geometry, so we couldn't see how it could be helpful or do anything but leave us a semester behind all the students everywhere that were getting two semesters of calculus going into the AP exams.

But he was the teacher, so we had no choice -- and boy was that a good thing in the end, because we blew the doors off our sister high school on the AP exam because the calculus concepts made perfect sense from the very beginning because we had that analytic geometry framework to establish everything in.

Beyond that, we never started with an equation and then learned how to manipulate it and then, maybe, learn what it meant. Instead, we started with a problem that we wanted to solve, then applied the concepts we already knew to evolve the relationships and equations that would let us solve it more elegantly than we could have done the day before.

I can't even count the number of students that I have seen, not only as an instructor but all the way back to my fellow students when I was a freshman in college, who had gotten straight As in two or even three semesters of calculus (obtained elsewhere), but yet if you gave them an equation for the position of an object as a function of time couldn't tell you what it's speed was as a function of time because they had zero grasp on how position and speed were related conceptually, let alone how the derivative was involved in that relation.

One of my lab partners in Physics I was a transfer student that had two semesters of calculus from another college, with As. He was completely confused by the math from day one and failed the first exam badly. To his credit, he made the decision to drop the course and retake Calc I, barely making the deadline to add the course. The next semester he enrolled in Physics I again and aced every exam. He took Calc III that same semester (skipped Calc II, which he already had transfer credit for) because seeing how the concepts related to the physical world in a good Calc I course was all he needed to fill in the missing gaps.

One thing my high school calc teacher told us that I remember to this day and have validated on numerous occasions was, "You can teach the concepts of calculus to a third grader and they can understand them, but the algebra will kill them." Whenever I get the opportunity to work with a third or fourth grader who is good at math for their level, I see if this is true, and it almost always is. When my daughter was in third grade she had two of her friends over for a sleepover and I sat down with them and made up a game with them (well, it was mostly buying treats by solving math puzzles) in which they could solve the puzzle if they could apply basic calculus concepts to the problem at hand, but without doing any algebra. For instance, I drew a rectangle on a sheet of graph paper. They had just learned how to find the area of a rectangle in school. So I first had them figure that out. Then I showed them how we could break that up into a bunch of vertical strips and find the area for each strip and add them together and get the same result. They grasped that without any problems. Then I drew a right triangle (i.e., a straight line with a positive slope going from some starting point on the x-axis to some other point to the right) and showed them how we could break that up into strips but now the strips were long and narrow with a slopped top (i.e., a trapezoid, but they hadn't been exposed to that yet). I then said that we could get an estimate of the are of this trapezoid by assuming that it had a flat top and treating it like a rectangle whose height was simply the height of the left edge (I didn't want to complicate things with dealing with the midpoint). I then asked them to see if they could get close to the already-known area of the triangle using this knowledge. I then went away and let them hash things out and about a half hour later they had it pretty close. I then gave them a printout (on fine graph paper) of a fourth-order polynomial that was pretty curvy on the top and asked them to find the area of the shaded portion (i.e., between two limits) and they set about using pretty course divisions (since that was what I had intentionally used previously) and they came out with a not-bad estimate, but I told them it wasn't good enough and asked them how they could get a better one. They quickly realized, with no hints from me, that they just needed to make the width of the vertical slices smaller.

So, yep, they had no problems grasping the concepts of integral calculus (and had fun doing it), but had I tried to involve algebra in any way shape or form it would have been a disaster. I did the same thing with them the next day, but using differential calculus instead.
 
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MrSalts

Joined Apr 2, 2020
2,350
Sounds like Neil deGrasse Tyson didn't have a very good calc teacher in high school.
I listened to the first part of that three times and I still don't how you come up with that claim. He said the book was handed to him (at the beginning of the course) and he paged through it and started convincing himself he will never learn it (before the first lecture started). Then, as each week of classes progressed, he realized he was able to learn what the squiggly lines and Greek letters meant.

I just don't understand your claim at all.
 
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WBahn

Joined Mar 31, 2012
27,389
He didn't say that he "paired through it", but rather that he opened to the front cover. But I will readily admit I probably read too much into it because I thought the course was starting with, "All the equations you need are in the front cover of the book," something I have seen in numerous classes (usually it's a sheet of equations they are handed on day one). If it was him, independently, looking at those equations and then coming back to them, that's quite different.
 

Papabravo

Joined Feb 24, 2006
19,262
The title of this thread is both naive and simplistic. All of mathematics is interconnected and the topics are commutative and associative. I never was fond of whining.
 

WBahn

Joined Mar 31, 2012
27,389
The title of this thread is both naive and simplistic. All of mathematics is interconnected and the topics are commutative and associative. I never was fond of whining.
It's a never-ending debate, with both a relevant and a pointless side. The issue of which math courses to require to get a high school diploma or a college degree (and the same for which humanities courses, which science courses, which arts courses, which (you name it) courses) is one of give and take. There are just so many courses that a student can realistically take and so if you want to require something that wasn't required before, something has to be taken out, be it another course in the same area, a required course in another area, or an elective (which are also important). So the question of how to evaluate the value, across the student body, of one course versus another needs to be addressed somehow.

The pointless side comes when arguments rest largely, if not completely, on things like, "only a small fraction of people will ever use this in their daily lives." If that's the standard, then there's no reason to require the overwhelming majority of classes in any area and we could easily go back to a system where students were only required to attend school through 8th grade (there's a reason that 9-12 are called "secondary" education instead of "primary"). Even then you could probably trim out quite a bit more if the bar is going to be to keep only what is really essential for the overwhelming majority of people to function in society. After all, how much art and history and social studies and math and science does someone need to punch the picture of the large fry on a cash register? We require all of those courses not so that people graduating from high school can minimally function in society, but so that they have the knowledge and skills and background to be in a position to find a path to a level of success (whatever that means for them) much greater than those that choose to accept staying at that minimally functional level (and there are a lot of people that, consciously or unconsciously, choose to do just that -- they define success by how much fun they can have away from work).

But that doesn't mean that we currently have the optimal set of requirements or that the requirements don't result in a barrier to some people being able to move up. The question is what to do about it. Eliminating the requirements so that a handful that can't meet them can graduate is a disservice to the fullness of education of all of the students that can meet them but that would not take them unless they were required (which would be a significant fraction). The answer probably lies in looking closely at the core requirements (courses that every student, regardless of program, must pass) and requiring that each course that is in the core have to justify why it should be there by answering the question, "Why should a student be prevented from graduating from this school if they can't pass this class?" My guess is that there would be relatively few courses that can adequately answer that question. This process would likely identify a number of degree areas (for colleges) or tracts (or whatever a particular high school calls them) for which you CAN make the argument that no student should get that degree or complete that tract unless they can pass that class. Fine. Add that those programs and tracts.

This process might also identify courses for which there is general agreement that it is a strong positive if every person passes that course, but which it doesn't quite rise to the level of being truly essential. So find acceptable alternatives. One possible way would be to offer lighter courses on that subject that someone can take to meet the requirement, but only if they have failed the full-up version. That way students are not incentivized to just take the easier course, and also that students that do end up taking the lighter course take a GPA hit for doing so (due to the original F). Another way to accomplish the same thing but potentially minimize the wasted time and cost impact of repeating the course (albeit in a lighter version) would be to offer a lower number of grade points for it (just like AP and honors courses in high school often have heaver grade points). That way people can choose to take the lighter version up front, but an A is only worth 3.5 grade points instead of 4.0.
 

300-3056

Joined Sep 9, 2022
26
My ability to understand math at Calculus and beyond fell to pieces when I was in College.
I was able to keep cheat sheets and repeat the equations but I didn't get it and I don't now.

Its a source of shame and frustration to this day for me.
 

WBahn

Joined Mar 31, 2012
27,389
My ability to understand math at Calculus and beyond fell to pieces when I was in College.
I was able to keep cheat sheets and repeat the equations but I didn't get it and I don't now.

Its a source of shame and frustration to this day for me.
I wouldn't worry too much about it. There are many different fields of mathematics and it is rather common even for mathematicians that are experts in one field to really struggle with the relative basics of some of the other fields. Each field involves its own set of ways of thinking. The same is true in other fields, such as chemistry and physics and in most engineering fields as well.

I have a friend that retired as a quite highly placed position at DEC (essentially one step down from being a chip architect) and sat on IEEE standards committees. He took a number of classes for personal interest, including abstract algebra. He did fine in the first course and really enjoyed it. He tried the second course three times, each time getting completely lost in the concepts and dropping it before admitting defeat.
 

BobTPH

Joined Jun 5, 2013
5,743
I went to a Jesuit high school and entered the advanced math and science program in second year. We had the same teacher for three years and he was excellent. We got 3 semesters of calculus, including differential equations.
 
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