Isn’t the point of math education in grade school to learn how to think consistently and creatively about formal systems? (Something something recursively something, also.)
If they can do that, they’ll pick up (insert field-specific techniques) very easily once they get to college.
I disagree. The two fields aren’t symmetrical in that way. You really can’t start physics without Calc, trig and analytic geometry. Not having incoming students ready to take Calc I would mean physics majors wouldn’t be able to take the first course in their field till sophmore year.
And its not just physics. Most hard sciences require Calc I be taken in the first year. You might be able to get a little further without them in a less mathematical science like Biology, but still not past the first year.
Comp-sci isn’t the same way. It isn’t just a historical artefact that the beginning comp-sci classes are taught with little need for symbolic logic, etc. These things simply aren’t needed to start, or depending in the program, even finish a degree in computer science.
Indeed, looking at the degree requirements for a BA in comp-sci at two universities, both require two calculus classes and a linear algebra class in the first two year of study, before any more computer science specific math classes.
Again, kvetching about what kids aren’t taught in HS is a science professors favorite pass-time. They certainly do imagine it could be different. They just don’t imagine kids knowing more graph theory.
Are we talking about what every kid should know, or what every kid should have available to them?
Yeah, we didn’t learn graph theory at all, except as part of our Jr. level Analysis of Algorithms class, which is where it’s actually used. Same for formal logic- it was nice to know, but was taught as part of a required Jr. level math course that also included combinatorial and discrete math. Set theory and set math was taught as part of our Database concepts class… also a Jr level required course.
We were all more than adequately prepared to go be everday garden-variety programmers after the first 2 years of our degree plan, so I’ll say that there’s absolutely nothing to gain from teaching that sort of math to high school students.
But that brings up the question that I was trying to address. Why do physics, chemistry, and biology degree programs require calculus in the first year? They do so because they can do so, because the math curriculum that we have allows them to do so. They do not do so because it’s carved in stone.
Imagine, for instance, that our high school math curriculum did not require students to take the track focused on algebra, geometry, and trigonometry, but instead required them to take two years of statistics, covering what we now consider to be college level statistics. If so, then freshman-year college courses in biology and chemistry would look extremely different. They would be more focused on those topics in biology and chemistry that can be approached with statistical methods. Meanwhile, topics in biology and chemistry which required calculus would be pushed back towards later years in the program.
I’m not saying that this would necessarily be better. I don’t know if it would be better or not. But it’s certainly something that could be done. We consider it natural that students entering these programs have to take calculus freshman year, because that’s the ways it’s “always” been done. But there really isn’t any reason why calculus-focused science courses need to be top right from the start.
Or look at it this way. Of all the papers that a chemist or biologist is likely to read in his or her career, how many will actually involve calculus? And how does that compare to the number that will actually involve statistics? In my admittedly limited experience, statistics shows up a lot more, and I’ve seen some examples of statistical cluelessness among professional scientists. So it seems to me that we could shift the entire mathematical foundation of our educational system, decreasing emphasis on calculus and increasing emphasis on statistics.
I’ve seen a lot of students who are bad at math shy away from Calc in college in favor of “Math Studies” or Statistics but they find it equally as challenging.
“Math Studies” has a lot of matrices, systems of equations, and just generally a hodgepodge of advanced algebra topics that don’t really fit into any particular discipline. It’s basically remedial Linear Algebra without any of the number theory.
Statistics is doubly as hard because they have completely wrong expectations for it and it’s the worst of all worlds in that it is still math intensive and entirely predicated on word problems: something “weak” math students fail miserably at.
I don’t think the curriculum at a HS/college level is the problem. The problem is in the roots where kids have bad habits, poor arithmetic fundamentals, and a lack of problem-solving intuition.
Algebra is the basis for everything. If you don’t have a good algebraic base, there’s no way you can do any sort of meaningful math on any level in any capacity in any discipline.
My kids got set theory and Boolean Algebra in elementary school. 30 years before I got set theory in 7th grade and no Boolean algebra at all. I don’t remember if I got to do arithmetic in other bases (I predate the New Math) but it was not hard to pick up.
I learned to program in high school, long before PCs, and I am a computer scientist. Besides dealing with Boolean algebra and numbers in different bases (which is useful even if you never touch a computer) I’m having a hard time thinking of math that would have been useful. Graph theory is important for some problems, but not fundamental to computer science. So is linear algebra.
Engineers need calculus also, not so much computer scientists.
I agree though that statistics should be a bigger part of high school math. That is useful for everyone.
They do so because there’s no way to meaningfully teach mechanics, reaction-rates, etc. without calculus. The order things are taught in isn’t arbitrary. College level science really requires calculus in a way it doesn’t require stats or formal logic. This isn’t because of how its taught, its intrinsic to the science itself.
The common core does require some stats, as it should. But I don’t think you could productively fill two years worth of statistics classes at the High School level.
Sorry but that is a crock of steaming excrement.
Mental math is a skill that is learned with practice. The boring drills and repetition just like doing scales on piano and shooting free throws over and over again. Okay sure musical ability and athletic ability are genetic too, but anyone can practice and become decent.
What you seem to want is to have kids appreciate symphonies but never have to learn scales. Math is boring because it is rarely actually USED in school. Reading is fun because it is not all about learning how to read but the fact that you then use that skill to learn and enjoy. Math until college mostly stays something that you learn to do but do not actually use to learn. Sure let’s spend all this time learning how to use a hammer and a screwdriver and maybe even a drill but never actually build anything. Your answer is to propose teaching how to use a saw instead. It misses the whole point of why kids hate math. The saw is not intrinsically exciting either. Using the tools as a matter of routine (just like reading is used routinely as a tool) to actually accomplish things (before physics, all along, in social studies and early science and all subjects) - that would spark some interest.
Thing is that requires some numerancy in all teachers and many don’t have the chops for it.
What should every kid know?
It’s not a crock of shit. Of course everyone can practice and become decent but the problem is that they don’t. If you want to use the free throw analogy, let’s say FT shooting is a common core class. Everybody lines up to shoot free throws in a designated class every day. There are some kids that can shoot at a 90% clip without even trying. Then you’re the kid who’s shooting in the low 70’s. You could devote massive amounts of your non-class life to practicing but more likely you’ll just call it quits and say “Basketball isn’t my thing” or “Free throw shooting is hard.”
To extend the analogy, let’s say there are no rules about you having to shoot from the FT line. If you wanted to, you can walk up, climb a step ladder, and just drop the ball in. The only difference is that it’ll take longer. The problem is that the coach is pressuring you to shoot from far away because it’s convention, and you don’t want to admit to being the kid in class that needs to use the step ladder when everyone else is shooting from the line.
Yes, technically you’re right in that most people given the motivation, drive, discipline, and effort can level the playing field. However in practice, that simply doesn’t happen very often.
Are you claiming that mathematical ability is primarily innate, that you either have it or you don’t, and if you don’t it takes far more effort than it’s worth to develop it?
Someone in another thread recently (but alas, I can’t remember who or where) linked to this article from The Atlantic that disagrees: The Myth of ‘I’m Bad at Math’.
Beats me, why do you ask?
I’m claiming that there is this one overvalued aspect of mathematical ability - the ability to hold numbers in your head (expands to being able to solve equations with shortcuts, visualize shapes, move from words to numbers, and overall spatial reasoning) - discourages students and unrightfully so.
I am definitely not claiming that it takes fare more effort than it’s worth to develop it. I’m saying that very small changes to the way students are taught math mitigates this discrepancy and overall it’s good for ALL students to do math [basic algebra] the long way rather than the fast way.
Basically what I’m saying is that some people think they’re not a math person because they’re slow with their multiplication tables, or deciding which legs of a right triangle to use for which trig function but slowness does not mean incapability. However the marginal speed with which someone answers a question disproportionately values how “good” someone is at math. As a consequence, bad math students measure themselves to that metric and decide they’re bad and it’s just a snowball effect from that point on. They could rise to the occasion and put in work but most take the path of least resistance and move to something else that comes more easily.
The problem is, they have to take Math whether it’s their “forte” or not and they sleepwalk through it.
I can’t speak to chemistry or biology but as an engineer, I say that you cannot study engineering without understanding physics and you cannot really understand physics unless you know calculus. Students who want to study engineering or physics in college will need to study calculus as well. They may not need to take calculus at the high school level but they need to have a solid math education in order to survive engineering school. If the students go on to study programming they will find things like Boolean expressions to be a piece of cake with a solid high school math education.
Exactly. If you think physics can be effectively taught without calculus, let me tell you it’s twice as hard. I took non-calculus physics in high school. Lots of really contrived ways to do things like find the rate of change at a point along a curve, or for measuring the area under a curve to get total force over time.
Those things are trivial if you know calculus- it’s just a simple derivative and a simple integral.
Stuff like velocity and acceleration are literally defined in terms of calculus- velocity’s the first derivative of the function and acceleration is the second derivative, and something called “jerk” is the third derivative (jerk is basically the rate of change of acceleration). Good luck calculating that without knowing calculus.
Okay, that makes sense. I’m not sure whether I agree with it or not, but it makes sense.
It seems to me that, no matter what method you use, some students are always going to be quicker than others. And you’re right that this can be discouraging to the slower students, and make them think they just don’t have what it takes; but I’m not sure if there’s a way around this, other than providing individualized instruction to each student that allows them to work at their own pace, which has its own problems.
You can avoid praising speed, as if the speed itself were the major accomplishment.
You can provide explicit instruction and opportunity to practice holding numbers in your head.
You can, as was mentioned by pancakes 3, make sure that you, as the teacher, are showing intermediate steps so that kids learn them.
You can, as the teacher, slow down just a bit to let kids “get” each step before advancing.
You can have kids write answers to questions and give everyone 5-10 seconds to respond, instead of letting the kid that sees the answer in 2 seconds blurt it out, immediately stopping the thinking going on around him. We aren’t talking about giving the slowest kid in class a full five extra minutes on every problem: the issue is those kids who literally just need a couple seconds more.
You can take the slower kids aside and point out to them what is going on, and emphasize to them how trivial it is in the long run.
IMHO, the vast majority of people who don’t like math fall into one or more of the following groups, which obviously overlap considerably: (though I’m not part of this group, and haven’t studied them at all, so, really, why should you listen to me?)
*People who simply find it hard to work with logical systems. You know, like any kind of math.
I don’t think, for any of these groups, learning topology or set theory instead of geometry or calculus would make much difference. Maybe for some people it would be productive to back into the rigorous proof aspect of geometry, starting with practical problems and only later getting into how we can be sure that a solution is right, and from there to logical proofs. But that’s a tactical change, not a subject matter change.
I do agree that a well-educated population should certainly have more exposure to statistics than I got up through High School, and I’d prefer to see more statistics in middle-to-high school, but other than that, I don’t think changing subjects (as opposed to tweaking approaches) will help things much.
Because it seems to me that the question of “what every kid should know” is key to any proposal of what every kid should be taught.
Look, I am personally glad that I learned calculus but in actuality the math that has been useful to me after HS was basic arithmetic, probability, and Baye’s Theorum. Very little math beyond that was required for me to get through college in a science heavy curriculum, med school, and residency, and many of my fellow physicians still don’t quite grok Bayes Theorum. Much of algebra, all of trig and calculus, only became useful to me when it was time to help my kids learn it.
Don’t get me wrong. I care about my intellectual life and I have learned more along the way for the kicks.
It seems to me that only a relatively small number of students will actually need to use much of the math that is being taught. Most kids do not become engineers, physicists, computer programers, economists, or math majors. Accountants don’t need calc and trig. Or topology. Sure many other subjects benefit from rigorous mathematical analysis being applied but most of the time the math applied is pretty basic stuff.
So what is the stuff everyone should know and why? And how do we get that taught to a level that the average citizen can actually use it in real life for the rest of their lives?
I’m not trying to be argumentative, just trying to understand your perspective - why? Most adults will actually benefit more from an ability to do basic mental math well. What’s cheaper? What are the odds? Does this literally actually add up? Very adults will end up setting up any algebra problems the rest of their lives.
We have endemic basic innumeracy. Yes we need to have a population prepared to pursue the variety of STEM careers, but addressing the former is perhaps more critical lest math becomes something that only some priestly class engages in.