Until I was the 8th grade I memorized most of the formulas of math and I didn’t understand what they signified in the real life.I had some asshole teachers who couldn’t describe me the relation between the formula and the way you use it in the real world.All they did is solved some problems which involved quadratic equations,cubic equations,geometrical shapes in 2d and in 3d and that’s all.
Is math about how you make a bond between objects or just memorize all the formulas not knowing what to do with them?
It’s sad that so many people associate math with rote memorization.
For the most part (with the exception of conventions about notation and such), math consists of stuff that you could figure out for yourself—if you were smart enough and thought logically enough and spent enough time and happened to think of things the right way.
And even if you didn’t figure it out for yourself, someone who already understood it could explain to you where it comes from and why it is the way it is.
Still, a good memory is an asset. For example, you can figure out that 2 x 3 = 6; but if you don’t remember things like that and you have to keep re-figuring them out over and over, things get slow and tedious.
As for real life applications, in my experience, some of the people who like math like math because of the real life applications, and some of them like it in spite of the real life applications. Some people are struck by the beauty of mathematical reasoning and logic and relationships, while for others, math only really came alive when they discovered what it could be used for in the real world.
Supposedly math is really more of a way of thinking in a structured, logical and ordered way. Formulas and stuff are more in the realm of arithmetic.
And, FYI, you’re not the first person to encounter this issue. Read this essay by a mathematician about the current state of math education, and you’ll notice a lot of common ground there.
I would say a good dose of both. In theory all of the math I learned through my undergraduate (and even some graduate) education is applicable to something, but it was often so abstract that I couldn’t really make the connection to application.
I think “math people” really like the abstraction of it and see it as beautiful because it can be used so many different ways. The problem is that they end up being the teachers to people who need a good solid connection to the real world to make the subject “pop.”
Regarding your first statement, I agree this is probably the most important benefit for a young mind. The ability to put thoughts together from step one to step two, etc. This skill is applicable across all disciplines.
But forgive my personal ignorance, as it was never explained to me: math <> arithematic??
All arithmetic is math, but not all math is arithmetic. Mathematics can also include Algebra, Matrices, Calculus, Geometry, and many other disciplines that may or may not even be related to arithmetic,
We’ve done Singapore Math with my son. It’s the math curriculum they use in Singapore, and it’s very conceptual. It’s been so much fun, and he LOVES math. One thing that stands out to me is that in Singapore math, you start multiplication and division much sooner–counting leads to adding leads to skip counting leads to multiplying–and this has had a huge impact on his enjoyment of math. Once you can multiply, you can think flexibly and abstractly about things (Six sixes is three 12s!) and it’s just more fun. The way I was taught, we didn’t get to multiplication at all until, I think, 3rd grade, and once we got there we had to memorize it immediately because everything that came after–multi-digit multiplication, long division–pretty much required math facts at your fingertips. By starting with multiplication sooner, we’ve had a couple years to play with it and he likes finding patterns and different ways to do it.
Even now, he knows a lot of his times tables, but he enjoys multiplying up in his head because he likes the tricks and the feel of it coming together.
I just want to say, as someone who has taught math, the one thing that my students hated and complained about more than anything else is “word problems” . So its a with a bit of irony that I read people complaining that they weren’t taught enough about applications.
The OP sounds like someone who thinks a master carpenter is just someone who’s memorized the sizes of lumber and screws.
Yeah, you need to know those things but the real skill in making something is quite artistic requiring creative use of basic knowledge plus years of experience.
I had a Differential Equations prof who did a great job setting up example problems in his lectures based on real world examples. It was amazing to watch.
On the other end of the scale are the overwhelming majority of “Math” teachers in grade schools and quite a few in high schools. They don’t like the topic and their lack of enthusiam gets transmitted to the students.
For example, one of our kids had a question about the answer to a homework problem in Algebra. The teacher could only point to the answer in their textbook, they couldn’t explain why it was right (it wasn’t) and show why our kid’s answer was wrong (it wasn’t). And this from one of the best high schools in the state.
There’s a serious problem in the US with an astonishingly high percentage of students learning at an early age a very negative and distorted attitude towards Math, Science, etc.
I sort of found out the distinction when I was in college. I was never good at math courses, but it turns out I’m a fine computer programmer. I can do the logical ordered thinking, but for whatever reason, I’m not that good at picking up on the actual mechanics of the type of thing that’s done in algebra and calculus classes. Interestingly enough, geometry was an absolute breeze for me, while most of the rest were not.
In this context, I note that there are lots of what I consider surprisingly good math-themed videos on YouTube. I think all of these argue that math is more than memorization, or logic, or applications - art comes into it.
I can’t resist mentioning what may be the best: 3Blue1Brown. Who’d have imagined a video that makes Fourier transforms comprehensible and even beautiful.
These are the creation of Grant Sanderson, whose mellifluous voice makes him sounds like your wise and understanding uncle. In fact, he’s a recent Stanford graduate.
The math as taught in schools is some rote memorization, some rote applying the rules they teach you.
All the points - all the tangible credit on the graded tests - depends about 95% on your ability to write down and manipulate various symbols according to rote rules. Maybe 5% depends on a bigger picture understanding, like reading a word problem or recognizing which shape of a curve is x^2 vs x^3 or whatever.
These were from attending arguably one of the highest rated public high schools in the U.S., followed by a mid tier state university where I took math up until differential equations.
I now work as a computer engineer and the amount of this math in this form I have had to do is…zero. Absolutely none.
There is never a reason to simplify a formula by hand.
But on the other side of things, unit conversions, such as they way they teach it in chemistry, where you draw little lines and write the english words for the units above and below? I use that all the time.
I was lucky enough to have a HS algebra teacher whose love for his subject was infectious to the point that some of us took year 2 algebra. I see that common FB meme about suffering thru algebra in HS and never using it again, and I wonder just how often those people do use it without knowing it.
As a math teacher my challenges are these:
Conflicting goals: teaching kids actual math skills, preparing kids for future math classes, preparing kids for summative assessments that are required by school admin/government at any level/colleges
Differences in students: some are motivated to get good grades, some are motivated to understand mathematical concepts, some are really good at memorization, some have a good grasp of previous math topics, some have firmly established misconceptions
Uncertain vs. certain results: You can give a relatively diverse group of students memorization type tasks, observe them doing those, observe their pace in completing them to judge progress, and test the results fairly reliably, even with largish groups. Even if they aren’t self motivated, you can get them to do those tasks through external motivation.
Tasks building understanding are fundamentally different. They require the student to want to think about a problem. For most students they have to be at their level, which is hard to judge beforehand, or the frustration of not knowing what to do will add to their dislike of “maths” and word problems in particular. The results are difficult to judge, since students who are adept at memorization will be able to solve any problem that is too similar to the one you gave, and giving them a novel task brings you back to having to make guesses about what students might be capable of again.
And any one teacher generally deals with just one slice of a child’s math education, you can’t fix every issues created by previous bad teachers for every child, nor can you truly immunize them from future bad teachers. (And for some kids those teachers might work perfectly well.)
Which probably means that you’re actually quite good at math, but never realized it, because geometry is the only “math” course most students ever take that actually involves doing any math. Math is proofs. If you’re proving things, you’re doing math, and if you’re not proving things, then you’re not doing math.
This is hyperbole, but it’s not that far from the truth, at least if by “doing math” you mean “doing what mathematicians do.”
(See also the recent thread Understanding the logic of mathematical proofs.)
There’s a distinction some people draw between mathematical problems and exercises. If you can be taught exactly how to do something—if there’s a specified algorithm to follow—if it’s just a matter of doing all the steps and following all the rules—then what you’re working is exercises, not problems. Something’s only a problem if you have to figure out for yourself how to do it.
And yes, most if not all of what goes on in many math classes is solving exercises, not problems, and is in that sense not really “doing math.”
Although, to be fair, the exercises have their place. They’re (arguably) important for developing the tools and skills to use in actual problem-solving.
All I know is that I never sat down to memorize anything in math. I did eventually memorize the times table, but never sat down to do it. My third grade classroom had a large times table along the left side (see, I still remember where it was and what it looked like over 70 years later) which we used to do problems and, eventually, memorized, but only out of laziness. Since that point, I always learned where the formulas came from and could reproduced them as needed. For example, the quadratic formula is derived by completing the square. Of course, I know it, but I never set down to learn it. And matrix multiplication, as odd as it seems, is fully motivated by the connection with linear transformations, an obvious notion.
The problem with students is that all they want to do is be taught formulas that they can memorize and never have to think. That’s why they hate “word problems” which should just called problems.
The last time I taught calculus, I walked in and said to the class that I was going to start by posing two questions that they could all answer. There was a bit of snickering, but in the end they agreed.
Q1. If you drive at steady velocity for an hour and travel 50 miles, how fast were you driving?
Q2. If you drive at a steady velocity of 50 MPH for an hour, how far do you travel?
I then explained that what elementary calculus was about was studying what happens when you drop the word “steady” from those questions. I don’t know if it helped.
That’s a pretty blurry line, though. I just asked my son what 137 was. He thought at least thirty seconds, and whispered “seventy” and later “21” before coming up with 91. So I know he multiplied 710 and 7*3 and added the products. He may well have had to add up 7+7+7. Which are techniques we’ve shown him, and talked about, but, as far as I know, he’s never done that to those numbers. While it’s still not what I would call “problem solving” in your sense, it’s light years ahead of the algorithmic way I was taught: for a 7-year old, that was something approaching problem solving.
Arithmetic is a branch of math. And which branch a formula is in will depend on what formula we’re talking about.
I suspect that none of your teachers ever claimed that there is no logic in math and the only way to learn it is by rote memorization, or refused to answer any and all questions.
Actually, I never met anybody like that until I moved to the US; did meet quite a bunch of those among my students there. I wouldn’t even know how to say “word problem” in Spanish: whether in math or in the sciences that applied it, they were the default. In math class, each solving mechanism would get a few demonstrations where only the solving mechanism was shown; for applications there wasn’t even any of those, we always had to begin by figuring out how to express the situation mathematically and which mechanism to use for solving it.
I think it’s linked to another thing which was much more common in the US than in Spain: multiple-choice tests. In general the US school system is more geared towards a bullet-points way of thinking than the Spanish one. I didn’t have a single multiple-choice test K-12, nor in college; two of my Spanish Masters’ involved multiple-choice tests but they were both long-distance, the second of them with the tests taken via a webpage that randomized the questions.
Wasn’t the whole New math thing supposed to teach kids how to use math analytically rather than rote memorization? That’s why parents got so confused and frustrated trying to help with their kids’ homework.