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How to Teach Math?

115 points by LittlePeter 5y ago

quacked 5y ago
I don't know how to teach math, but I know how not to teach math. 1. Don't teach that math is a "ladder". Trigonometry is not "harder" than geometry, although many concepts in trigonometry are expressed as geometry. Calculus is not "harder" than algebra, although many concepts in calculus are indeed expressed as algebra. Teaching that math is a ladder allows people to believe that once they've covered the "basics" of a field of math (as if you could ever learn all of "algebra"), they shouldn't have to think about the basics again. Try telling a pianist that once they've learned a set of scales they don't need to study them any more, or try telling Richard Feynman that once he understood the standard atomic model there was no need to continue examining it in greater depth. 2. Don't teach that math was "discovered" in its current form. What most students think of when they think "math" is in fact western notation for patterns observed in reality, handed down by a bored teacher as universal law. A sheet full of squiggles isn't "math", it's just a set of the most-predictable and best-notated patterns made up by past mathematicians. "Math" is the process of discovering and refining those squiggles. The relationship between the length of the two perpendicular sides of a right triangle and the hypotenuse exists outside of human knowledge, but the Pythagorean Theorem is a only a certain method for expressing that relationship. The numbers 1 through 9 are one possible code for counting, but the quantities I through IIIIIIIIII and onward exist outside of the notation of Arabic numerals. 3. Don't immediately move on from concepts once a student has mastered them. Imagine if every time you figured out how to do your job correctly, your boss moved you to a completely new task requiring a completely new set of skills, giving you no time to enjoy applying your prior mastery. If you've just learned how to factor quadratic equations, why move on? Why not explore programs that factor equations? Why not dig up old exams and show how factoring would have solved earlier problems faster? This is always met with cries of "but they're supposed to use what they've learned to go on to even harder problems!" Sure, I agree, but how can you possibly enjoy any task if the only thing you can really expect is that even if you master your task, you'll be struggling again within the week? Would you join a rec basketball team if every time you started hitting three-pointers consistently, they moved the rim higher and farther away without giving you any time to feel what it's like to be good at basketball? 4. Don't teach that you need to know math because your grades have to be high. For nearly every profession available, grades are immediately forgotten as soon as you start receiving wages. Teach that you need to know math because the world is constructed and controlled by mathematicians and those acting on the advice of mathematicians. If you don't know math you'll be taken advantage of by those who do, whether it's in advertising, gambling, banking, medicine, insurance, politics, entertainment, engineering, programming, or any of the many other fields driven entirely by math. Of course, not teaching those four lessons becomes difficult, because those lessons work in opposition to the central tenets of mandatory state education, which necessarily operates as a giant brainwashing factory working to justify its own existence. Modern ediucators spend most of their time trying as hard as they can to teach people that knowledge is best bestowed by authority and then proven through certificates, that various fields of inquiry exist separately from one another (for instance, that physics and biology can or should be studied independently of history, mathematics, grammar, semantics, or art), and that failure to live up to expectations must necessarily lead to shame and corrective action. Don't send your kids to school if you can avoid it.
- this-pony 5y ago
  I think maybe you mean "don't send your kids to an _American_ school if you can avoid it". Although I have no personal experience with the American school system, I would say that there are many schools in the world that adhere, to some degree, to your 'how not to teach mathematics'. I went to a public school in Europe and have had plenty of enthusiastic teachers that also tried to deliver the beauty of mathematics.
  - rsj_hn 5y ago
    Russia has an amazing math education tradition. I always recommend Russian math books and Russian math programs. They are excellent.
    - quacked 5y ago
      Every Russian and ex-Soviet I've ever met was extremely proud of their school system, which arose from the Soviet system.
      - rsj_hn 5y ago
        They really did a great job at math. The humanities were terrible, it was hopelessly politicized. But if you didn't want to deal with constant lectures about the struggles of the working classes, or constant coursework on marxism-leninism, you studied math or physics, or engineering. It helped, of course, that the government had as a national policy the promotion of STEM. But they did an amazing job at it, and produced a fantastic crop of world-changing talent.
        geomark 5y ago
        What a great approach. The easy social studies courses are made so horribly boring that it drives students to study the hard sciences.
    - geomark 5y ago
      Any titles you can recommend? I'm homeschooling my kid and have been using sample problems from both Russian and Singapore math. But I only have a single source for Russian math problems, which is a pdf of a paper by a Russian educator who taught in the US and then wrote a paper comparing the two educational systems. It's interesting because he talks about how algebra is a big scary subject that is introduced late in the American system, while in Russia they start doing algebraic-type problems early in primary but they are solved with pictures. So when algebraic notation is introduced later it's just a next step to the types of problems that students have already been solving.
tediousdemise 5y ago
The human brain is like a self-compiling compiler. You can teach people math by teaching them how to read, and then suggesting a good math book.
- royaltjames 5y ago
  Can you suggest me a good math book to read?
  - hansvm 5y ago
    Tons, but it really depends on where you're starting. Is there anything in particular you'd like to learn? Every time I've recommended Calculus Made Easy [0] it's been a huge success. The writing is lively and full of motivating examples, and it's an enjoyable read. [0] https://news.ycombinator.com/item?id=17185577 https://news.ycombinator.com/item?id=17185577
  - tediousdemise 5y ago
    While there are many amazing math books out there that specialize in a particular branch, I tend to prefer generalist books that cover a wide range of topics. It you want something that starts with the absolute fundamentals, What Is Mathematics? is a perennial classic. I think that one starts with addition and takes you all the way through advanced calculus. If you want a comprehensive survey of nearly all major branches in mathematics written by some of the most prominent Soviet mathematicians, you’ll want Mathematics: Its Content, Methods and Meaning. If you want more of a reference-style encyclopedia, The Princeton Companion to Mathematics is a good one. If you want to build a solid foundation for math tailored to computer science, look no further than Concrete Mathematics, coauthored by Donald Knuth himself.
- rahimnathwani 5y ago
  I'm trying that with my son (who is not yet 5). He has no trouble reading anything in his math book[0], but 'suggesting' that he work through the book isn't enough. He'd rather play with lego, read a story book, or do pretty much anything else. My solution: short sessions of a few pages at a time, with me sitting by his side, ready to intervene when he gets sidetracked. [0] https://shop.singaporemath.com/index.php/product/dimensions-math-textbook-kb/ https://shop.singaporemath.com/index.php/product/dimensions-...
atoav 5y ago
One important thing is: create realistic problems that need certain math. I had to calculate integrals for a year before our teacher for a brief moment glossed over what problem it might be needed for. In fact I only understood why integrals were useful, when I started learning for the final exam. This stuff can be really cool, but if even your teacher does as if it is the most boring and useless thing in the world, if even your teacher is unable to explain beforehand why you want to know this, how would you?
- jrib 5y ago
  One of my best memories from high school math was our teacher taking us outside to measure the height of our building after teaching us trigonometry. We took a triangular ruler outside, leveled the base and lined it up by eye with the top of the building. Then we measured the distance to the point where we were standing and the distance from the ground to the ruler. I really wish education reduced the amount of rote learning involved and focused more on letting students explore and make mistakes with the educator as a guide.
  - function_seven 5y ago
    My high school took this concept one step further. We were tasked with designing a solar hot dog cooker. The physics class specified the reflective area needed to gather enough sunlight to cook the hot dog. They tossed that over the wall to the algebra class, who determined the parabola shape and focal length from the dog. From there it was sent to drafting class, where the plans were drawn up. Those went to the kids in wood shop, to build the thing from the plans. Then one day we all went outside with our solar cookers and hot dogs and had a cookout. IIRC, there was a lot of variation among the cookers. Different reflectors were used (piece of aluminum sheet, bent and held by end brackets. Mylar film laminated to a wooden parabola "bed". One kid got adventurous and designed his with a vertical dog holder, and a circular parabola that was actually several different profiles stacked to "smear" the focal point along the vertical axis. I don't remember if it worked well or not) I think we were meant to learn some sort of process engineering in addition to our respective subjects? Or just "teamwork", I guess.
dfdz 5y ago
I am really confused by this article. The section "First An Issue" is a great example of how NOT to teach math. First, the class of sequences under consideration is not clearly defined. (I am not going to invest time solving a problem, if I am not sure about the definitions involved, constant coefficient/variable coefficient etc). Second, a question is asked "Is a product of two such sequences also of the same form?" and the meaning of product is also not clearly defined. Third, the author links to an article whose PDF on the publishers website is unreadable, and does not seem to answer the question from what I can tell. The only purpose linking to this article serves is intimidation. There is a happy ending to my rant. I googled the topic alluded to by the author and found a clear post explaining the solution. In this stackexchange post the question and solution are both clearly stated. https://math.stackexchange.com/questions/1348838/sum-and-product-of-linear-recurrences https://math.stackexchange.com/questions/1348838/sum-and-pro...
- this-pony 5y ago
  I think the point the author tries to make in the section "First An Issue", as you say, is exactly to give an example of bad teaching.
- yCombLinks 5y ago
  Yes, her exact point is "Here is an example where obvious things are skipped over that make it hard to understand."
teeray 5y ago
I started high school in a program that was more focused on “applied math learning.” (IMP [0]) We’d throw dice and figure out probabilities and stuff. I transferred out of it within two months because it was so focused on finding applications that it completely missed the mechanics of manipulating expressions. It was boring at times, but after three years I had a rock-solid base on how to turn one expression into any other. The applications of those skills came later once I got into Physics and Stats. I don’t know if it worked out better that way, but I think there’s kind of a chicken-and-egg situation. To really understand the applications, you need the math background. To appreciate the math background, you need the applications. [0] https://en.wikipedia.org/wiki/Interactive_Mathematics_Program https://en.wikipedia.org/wiki/Interactive_Mathematics_Progra...
mhh__ 5y ago
The more people I meet the less I sure I am how to teach.
oilostthelast 5y ago
Simpleton here: One, the first thing the professor does is put things in "human terms", the issue is plain that the mode of conveyance used in the textbook is wanting. And this is coming from "Author in Action" videos provided by Pearson, ostensibly from Sullivan himself. I think a portion of this issue is derived from being "too simple to state." Two, calculation takes precendence over everything. I've seldom been offered the opportunity to actually apply any of this myself in real world practice. What's worse is that when professors have included "working" problems they're quickly glossed over, to me it seems like this is absurd - as a non-math major this is precisely why I'm in the class, for application. This branches into arguments about the math taught being entirely abstract and stripped from a model, which makes it infinitely more difficult for someone like me. What compounds it is none of the work asks things like "why do we use exponential functions in the calculation of half-life?" instead, you just translate to TI-84 and plug your answer in. I'd quite like to see more applied concepts, and less calculation. It'd also be nice to self-direct, but I'm sure that's in reality impracticable from an assesment standpoint. I noticed repeatedly that I quickly grasped certain concepts, but was forced to repeatedly compute similar problems, time that would've been better spent working on points where I was lacking.
rsj_hn 5y ago
Here are some things I would recommend: 1. Find a teacher that is knowledgeable and passionate about math. Grad students might fit the bill. Engineering grad students also. Low level math (e.g. calculus, differential equations and below) can be taught by anyone in STEM, it doesn't need to be a trained math teacher. Engineers, physicists, or chemists often do a better job and passionate ones might be easier to find. Passion is important, because a good teacher will be a window onto another world, that you are invited to explore. A bad teacher is a window onto a brick wall. I used to tutor undergrads in math and the worst students were the math education people. I was often told "I hate math" by these students. I would even plead with them to find another subject, that it was unfair that someone who hates math becomes a math teacher, but they insisted that they love "teaching", they just hate math. Welcome to the US Public school system. 2. Include history in the math education. This depends on the personality of the student, but for me, I loved learning about the lives of the people who made mathematical discoveries, and the circumstances of those discoveries. To this end, I recommend books by George F. Simmons, for example: * https://www.amazon.com/Differential-Applications-Historical-International-Mathematics/dp/0070575401 https://www.amazon.com/Differential-Applications-Historical-... * https://www.amazon.com/Calculus-Gems-Memorable-Moments-Spectrum/dp/147045128X/ref=sr_1_4?dchild=1&keywords=George+F+Simmons&qid=1620430006&s=books&sr=1-4 https://www.amazon.com/Calculus-Gems-Memorable-Moments-Spect... 3. Look at the Russians. The Soviet Union had an amazing pedagogical program in math education, with fantastic books, puzzles, newsletters, etc. Some of that survived the turn to capitalism, and today they still punch above their weight. Some resources: * http://www.ascd.org/ASCD/pdf/journals/ed_lead/el_198102_brandt2.pdf http://www.ascd.org/ASCD/pdf/journals/ed_lead/el_198102_bran... * https://www.amazon.com/Moscow-Puzzles-Mathematical-Recreations-Recreational/dp/0486270785/ref=sr_1_8?dchild=1&keywords=Russian+school+of+Mathematics&qid=1620430372&sr=8-8 https://www.amazon.com/Moscow-Puzzles-Mathematical-Recreatio... * http://docshare02.docshare.tips/files/17782/177829869.pdf http://docshare02.docshare.tips/files/17782/177829869.pdf 4. Interdisciplinary approach. A great way of studying math is to present some problem in physics or engineering and develop the mathematical techniques to solve it. This is how much great math was discovered. 5. Ask questions. Rather than telling students techniques and then watching them use those techniques, you can create a dialogue where you start by asking specific questions and guiding the student to discover the math on their own. This of course requires a smart student and a smart teacher, but it's a very effective way to really learn a topic. This was famously done by the Texas Topology department in the mid 20th Century. 6. Teach concepts. If you cannot get the student to discover concepts, you can still emphasize the teaching of concepts. For example, there is no reason why a 5th grade math class can't begin to cover concepts such as simply connectedness, or Euler characteristic, or hamiltonian circuits. These can be done with simple pictures, yarn, paper and scissors. Then encourage the student to generalize to other shapes. Similarly in geometry classes, working with a ruler and compass creates a more memorable hands on experience for young students and can be a technique to introduce them to early theorem proving, for example bisecting a chord.
deleted 5y ago
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nobody0 5y ago
math is not a spectator sport. Just like you cannot learn to program by reading books, people have to really dive into problems and rediscover certain facts by themselves.
- hervature 5y ago
  I taught myself PHP from an old LAMP stack for dummies book years before any formal education in programming. Are you saying you need to actually program in addition to reading? No one is debating you need to work on math problems in addition to reading the definitions to learn math.
  - username90 5y ago
    Programming is a lot easier to self teach than math since you can test your programs by running them while you can't test your math work. The end result is that basically everyone who learns math creates a lot of errors in their understanding of it, and then those errors gets hammered out after countless exams as they progress through their education. Without the pressure of education most people just continue to accumulate errors in their understanding until they no longer can progress since the backlog of stuff is too long and they don't know where to start.
    - hervature 5y ago
      Can't test your math work? That's just completely false. The process is the test. Does each step make sense. Lower level math books include answers which will give you confidence the procedure was correct but no way ensures it. At a higher level, it changes to proofs where again, it doesn't matter if you know something is true or false, as long as each intermediate step it true. Of course, I'm not going to suggest that learning math solely from a book is optimal (neither is learning programming from a book), but it certainly isn't impossible.
jorgenveisdal 5y ago
How to study mathematics, according to Niels Henrik Abel's high school teacher: 1. Never study more than one book at a time and never abandon a book you have chosen without working all the way through it 2. If you face difficulties, do not give up but instead go back twenty times if that should prove necessary and only then allow yourself to investigate another mathematician's solution 3. Skip over those parts that are of no challenge in order to get at what is new to you 4. Reflect over the reading, in particularly how the writer came to the solution and moreover, what the solution leads to 5. Investigate whether or not another transformation or substitution would have solved it in a better manner 6. Always read with pen in hand so that you can work out all the calculations and practice all the questions you encounter 7. Write up lists of subjects that afford you an opportunity to develop your own theories 8. Geometric reflections can be a suitable way of strengthening and securing one's judgement
- deleted 5y ago
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- cgriswald 5y ago
  Most high schools and high school teachers don’t teach in a way that particularly enables students to take this advice. “These are the rules. This is how the math works. Do the math. Also these are the classes you have to take to be an $X.” A relatively good teacher will engage with a student who is enthusiastic about the material in this way but many won’t: “You’ll learn about that in $future_course” without even an attempt to talk about it in the abstract. Without some guidance on which book to choose, (1) could condemn a reader to mathematical purgatory. In any case, I’d also say that taking multiple math courses in high school helped me to understand that math is most often about relationships and that there were multiple ways to describe the same relationships. This was in spite of no one bothering to really teach that idea. Previously I was mostly focused on axioms and rules (probably because of how I was taught), which, while useful, missed some important stuff. I also prefer to read through multiple texts because it gives me a break, let’s me see new connections, and if I come back to the text after working on something else and still know what’s up I know I really grok the material. (It’s insanely easy for me to memorize and apply rules; that’s not indicative of understanding.) (3) is good reading practice in general, but also, it’s a skill most people haven’t developed. It’s very easy to say, “Pfft, I know all this,” and just skip something without actually really grokking the information; it’s too easy to over-estimate understanding. For most people I’d say at least work through any proofs and the most difficult questions in each section. This is actually something I think we should be learning in middle school at the latest.
SneakyTornado29 5y ago
One common shortcoming I see often with math teaching (at a high level): the difference in difficulty between lecture problems and homework/midterm problems is sometimes absurd.
- Koshkin 5y ago
  Well, lecture problems are usually for illustration, while homework ones are for independent study and for training, and so they are expected to be more challenging.
  - hdctambien 5y ago
    Yes, that is more a misunderstanding of the purpose/role of university homework vs highschool homework.
credit_guy 5y ago
"There is no royal road to mathematics", Euclid is thought to have said to Alexander the Great. Well, I don't think there's a royal road to teaching mathematics. First ingredient if you want to be a good mathematics teacher: preparation. When you go to class, just make sure you went through all the examples, all the exercises and problems, and you know them in and out. Be prepared. When they train you how to teach, they never tell you that. It is just assumed. They tell you about the attention curve (which supposedly looks like a bathtub: high at the beginning and at the end, low in the middle), about body position, voice volume, and other stuff. But for some reason they forget to tell you that first and foremost you need to prepare. Second ingredient: feedback. Ask a colleague to come and observe you, and, if possible, go with the same colleague and observe another teacher whom you want to learn from. They (the colleague) can point at what you did differently from that teacher, what you did well, what you should work on next. If that colleague is good, they will give you a list of 20 things to work on, but also tell you to focus on the first one or two. Are you stopping to look at the kids and see if they understood? Are you making eye contact with a single kid, rather than cycle through all the classroom? Is your voice loud enough (pro tip: it never is, speak louder). Did you leave the kids enough time to think on their own? Did you give them too many hints? Are you biased in always letting only the best kids in class answer out loud? Are you pointing with your finger when you want to emphasize something (it's natural, but don't do it, it intimidates the kids). But all these things are second order things. First thing: prepare for the math you are going to teach. It's much more important to not make any mistakes, such as a sign mistake in the middle of a calculation, than to make eye contact. Mistakes erode the kids' trust in you. Avoid them. Prepare. Once you got to a point where you consistently deliver a flawless lecture, then you can think of becoming the next 3Blue1Brown. But you need to walk before you run.
- catbird 5y ago
  Thanks for your comment. It sounds like you have learned these lessons through a lot of experience - do you have any suggestions on a book to read to get the very basics of teaching? > When they train you to teach... I started a job as a professor this year and never got any of that basic training. Before this I was a postdoc reading papers and writing code all day. In job interviews my lack of teaching experience wasn't seen as an issue. The faculty would say, "oh teaching isn't so hard, you'll figure it out quick enough." So far I have been muddling along ok, but I have this anxious feeling that there are a lot of aspects to this that I don't know that I don't know, and don't know where to get started on finding out about them.
ianai 5y ago
Make the lectures and homework actually prepare for the test. The test questions should be easier than the homework!!!!!! Actually, make the test be a minority of the overall grade - i.e. 80% of grade from homework.
frompdx 5y ago
The linked articles about "Inquiring Based Learning" (IBL) seem to be making the claim that this type of teaching is somehow more effective than lecture based teaching. I'm not entirely convinced this is true. My doubt is in part fueled by the statements on IBL not really being backed by anything other than intuition on the part of the author of those articles. If it is backed by more than intuition I either misread or they talk about it in an unlinked article. The remainder of my doubt is due to the mismatch with my own personal anecdotes. IBL reminds me a lot of breakout sessions or whatever you want to call mini group projects that take place in the span of instructional block. Some people in those groups will already know the answer, some won't care, and some will count on the person who knows the answer to carry the group because that person isn't so great at teaching it themselves. Why would they ask questions about answers they already know? When I was still in school IBL style was used far more in sciences than something like history. But I loved history. Especially lecture based history classes. Part of the reasons for that many professors I had loved to talk about the subject they taught. That doesn't mean the professors didn't love the subject they taught, but talking about it didn't seem to be something the loved doing. I personally struggled with the IBL format and did well with the lecture based format. I'm not an educator myself. I just bring that up because at least for me I did not find the technique helpful and got less out of it than I did with lecture based classes. Everyone is different. I imagine the IBL format does appeal to some more than others.
insert_coin 5y ago
How to teach anything? You teach first to learn. In my opinion learning is not an obvious thing, it is not obvious why things need to be presented as problems for example, and many students rebel against them as it leads to an empty feeling every time they even attempt to solve a textbook math problem. Curiosity is innate, but learning has to be taught. So the clash occurs because they begin by teaching math to people that do not know how to learn, or why to learn.
bradrn 5y ago
The best maths book I’ve ever read was Measurement by Paul Lockhart (yes, the one of Lockhart’s Lament [0] fame). It’s been a long time since I read it in primary school, and I can’t remember exactly why I loved it so much, but there is one thing I remember clearly: the problems he poses are just really, really interesting. I can’t say I love problem solving, but I remember struggling with some of those problems with hours, just because I wanted to know what the answer is. Highly recommended as a way of teaching interested children (and adults!) about geometry and calculus. [0] https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....
adam_arthur 5y ago
Teaching in general is best done by "peeling the onion". By that I mean, start with the highest level explanation first, form an understanding of that, and then repeat the same lesson but at a lower level of abstraction. You have to be able to walk before you can run. First, explain the concept like I'm 5. Use examples that provide real world context for why the concept is useful. Demonstrate examples of using the concept to solve a problem. Then explain the concept again, with an added layer of complexity. Go through the same examples with this layer of complexity. And so on. The more levels, the easier it will be for people to ease into learning the concept. When confronted with something that's too foreign, many people just shut off their brains, and convince themselves they're "bad at math" etc. The hardest part about learning is really breaking through the wall of initial understanding. Once you know the language, everything becomes a bit easier to learn. I honestly think math education can be hugely improved with this approach. This could even be done in a "high tech" way, by having an e-textbook with a slider. Adjusting the slider changes the level of abstraction and granularity that it teaches the concepts at. Final note. Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration? Or whatever, you get the point. People can't relate at all to math when all the examples use generic terms without context. At that point it just becomes rote repetition. Obviously this commentary applies moreso to earlier math education.
- lixtra 5y ago
  > Teaching in general is best done by "peeling the onion". My experience is that best way depends on the student, the thing you want to teach and probably even on the time of the day. The best teachers are able to adapt to that.
- enriquto 5y ago
  > Why use X and Y when you could carSpeed and carAcceleration? (...) People can't relate at all to math when all the examples use generic terms without context. Couldn't disagree more. The whole point is that you don't care about the meaning of the numbers. They are just numbers. A big part of math is learning to strip numbers of their context and look at them bare. Besides, variables with several letters in them are extremely ugly. They look like products. (They should be banished from programming languages also, but I realize that this is a controversial idea.)
chrisweekly 5y ago
https://betterexplained.com https://betterexplained.com does a better job of answering this question than anything else I've encountered.
throwawaysea 5y ago
For those who know the education space better, I am curious as to what we get wrong in the West with STEM education. There has been such a large academic or research oriented focus on determining the best approaches to teaching math. But other nations that have more basic or “crude” math education seemingly do better. My instinct is that we are attempting to make learning too easy and that true learning simply requires that learners push hard and build their own fundamental understanding. Can someone offer a better formed opinion on this?
- rimliu 5y ago
  Your instinct is absolutely right. I have not seen anyone suggesting that learning to play an instrument is or can be effortless, yet for some reason people believe that STEM should be.
TrackerFF 5y ago
When growing up, I struggled with math. Part because my ADD made it incredibly hard to read books from start to finish, part because I had very little interest in STEM classes, and part because we had a math teacher that would focus all his energy on the top students. If you struggled with something, and he didn't deem you worthy, he'd simply direct you to the book "read the book". It wasn't until undergrad that we had this fantastic calculus lecturer - everything just made sense. He started at the very, very beginning; adding together numbers, and worked his way up to pre-calc, in around 2 weeks. He never skipped any steps, and was very good at explaining the implications of various proofs, and pointing out the typical / common errors and fallacies made by students. Of course, if you had a good grasp of math leading up to calculus 1, this was all boring and old news. No big deal, those students would skip classes, it was after all in the very beginning of the semester. But for me, it was simply life changing. And parallel to that, people started posting vids and websites explaining math. I suddenly had multiple different sources of people trying to explain math their way - which also turned out to be invaluable. Suddenly this guy Salman Khan started posting vids on youtube, and I discovered Pauls Math notes. Well, the rest is history for me - I graduated with good grades in all math courses, and a love for math. If anything, it thought me that there's no single way of teaching math. Again, when I grew up, it was one book and one teacher.
chobytes 5y ago
In my experience you learn math by doing. Its takes personal time and effort and it cannot be spoon fed. A teacher can provide a little guidance, but mostly its up to the student.
rustybolt 5y ago
I wanted to write a comment, but the comment by tchow8 already sums it up nicely: > Okay, I’ll bite. You wrote, “What I do not always get is some totally obvious ideas. Does this make any sense? Here is an example.” How is what you say about recurrences an example of a totally obvious idea, or of not getting a totally obvious idea? I don’t understand.
tkgally 5y ago
> The only way to learn mathematics is to do mathematics—Paul Halmos Paul Halmos taught for two years in the 1970s at the University of California, Santa Barbara, where I was an undergraduate majoring in linguistics. I was also interested in math, and I ended up taking three courses from him: an introduction to mathematical logic, with about twenty students; a seminar on set theory and the foundations of arithmetic, with four or five students; and a one-on-one tutorial on point-set topology. For the latter two courses, he taught based on his “do mathematics” principle. He would provide us with definitions of basic concepts and then have us work through a series of not-so-difficult problems that led eventually to proofs of the major theorems. He let us do most of the work ourselves, though if we got stuck he would provide pointers to help us keep moving ahead. His method was inspired by that of Robert Lee Moore [1], though Halmos was not as hard-assed about it as Moore reportedly was [2]. In the seminar, he encouraged the students to work together. More than forty years later, I remember many details of the two problem-based courses, while the first course, in which he lectured on mathematical logic, has mostly slipped from my memory. [1] https://en.wikipedia.org/wiki/Moore_method https://en.wikipedia.org/wiki/Moore_method [2] https://mathshistory.st-andrews.ac.uk/Extras/Halmos_Moore_method/ https://mathshistory.st-andrews.ac.uk/Extras/Halmos_Moore_me...
- PartiallyTyped 5y ago
  As a grad student in CS, for theoretical and algorithmic courses, doing the proofs one by one is the only thing that works for me on how to get a good understanding of what tools the proof provides for me, what new pattern of solving a problem exists and making use of it. I found that it significantly expands my repertoir, or grammar and vocabulary so to speak instead of staring at the lecturer trying to explain their thought process.
edtechdev 5y ago
There's actually decades of research on how to more effectively teach math, including the effectiveness of inquiry based learning approaches, which are more effective. See the MAA's Instructional Practices Guide: https://www.maa.org/programs-and-communities/curriculum%20resources/instructional-practices-guide https://www.maa.org/programs-and-communities/curriculum%20re... Here's info on research on the effectiveness of inquiry learning in math: https://theconversation.com/who-learns-in-maths-classes-depends-on-how-maths-is-taught-21013 https://theconversation.com/who-learns-in-maths-classes-depe... https://www.colorado.edu/eer/research-areas/student-centered-stem-education/inquiry-based-learning-college-mathematics https://www.colorado.edu/eer/research-areas/student-centered...
- jcassee 5y ago
  You seem to be knowledgeable in this area. Is it weird to ask you to summarize these links, especially in the context of the more concrete discussions in this thread?
dusted 5y ago
As someone who is math-stupid, I totally find that my main difficulty in learning a new thing is that the "obvious parts" are not mentioned, or only in such passing as to make clear to anyone that they should already know this.. I learn from the top down, I can't reverse that, but I can't "get all the fundamentals right first" and then work out from there... I need to scratch a lot of surface on the subject, then the fundamentals start to crystallize for me, so refreshing me on those obvious things in the context of what I'm learning, helps a lot.
v8dev123 5y ago
I have different take on this. Teach how Humans found Maths. Context is important. Teach what Abstraction really is. Start from a Lion Picture and draw it and abstract by step by step. Teach use cases. Once you done that, teach Algebraic thinking. Then the student can learn anything.
MarkMc 5y ago
Craig Barton's excellent book [1] makes a compelling argument in favour of the 'explicit instruction' for teaching maths. Here's an excerpt: --------------- I was particularly proud of a guided discovery task I came up with for introducing some of the more complex laws of indices to my Year 11 class two years ago. The worksheet looked like this: https://photos.app.goo.gl/dzWPzw32hWVDLvYG8 https://photos.app.goo.gl/dzWPzw32hWVDLvYG8 Nice, eh? Again, I ask the question: what could possibly go wrong? Well, quite a lot, as it turns out. My takeaway When considering a guided discovery task, the question I should have asked myself is: what is the best that can happen? Take the laws of indices lesson. The best that can happen is that all students discover the laws of indices for themselves, leaving no gaps in their knowledge, nor developing any misconceptions, in a reasonable time frame. We can then proceed with the rest of the lesson, maybe moving on to application questions, or interleaving other topics into the examples (see Chapter 12), such as indices involving surds or fractions. How often does that actually happen? In my experience, literally never. What actually happens is that one or two students discover exactly what I wanted them to discover. They are feeling great about themselves, and rightly so – as we have seen in Chapter 2, success is motivating. A handful of students have some kind of idea what is going on, but with an eclectic mix of gaps in their knowledge and newly formed misconceptions. Some of these students are aware they have gaps and misconceptions, others are blissfully ignorant. And the rest of the students do not have a flipping clue what is going on. They are feeling confused and pretty down about themselves when they see their fellow classmates have figured it out. Any form of decent formative assessment strategy (Chapter 11) quickly reveals this disparity between levels of understanding, and as such I cannot move on with the lesson. So what do I inevitably end up doing? Teaching the laws of indices, of course! Maybe I will set those students who seem to have understood it off on the work I hoped everyone else would be moving on to – mind you, I would really like them to hear my explanation and do the worked examples, but how can I justify doing so when they have demonstrated their understanding? Hmmm… Anyway, back to the rest of the class. By this stage, I am 30 minutes into a 50-minute lesson, rattling through a series of worked examples on the laws of indices far quicker and with much less care than I should. There is zero time for the students to practise their newly acquired skills and hence consolidate their knowledge, nor sufficient time for me to do any kind of application questions which would show them the full breadth of the topic. But it is even worse than that. Even if I could somehow freeze time and spend those lost 30 minutes going through carefully structured and well-chosen worked examples (Chapters 6 and 7), I am not back at square one. I am behind square one, because my students are no longer coming at the topic with fresh eyes. Many of those who failed to ‘discover’ the key relationships have already decided that indices are difficult, and yet another area of maths that they don’t understand. It’s going to take more than my magically retrieved 30 minutes to turn that one around. And so I wave goodbye to a group of confused students trundling out of the door, promising that we will pick this up again tomorrow, assuring them it will all be fine. I am already dreading the lesson, wanting to open up proceedings by saying ‘okay, everyone, forget what happened yesterday’. In the past, I blamed my students for this – if only more of them could have figured it out. Now, I know the blame rests squarely at my feet. I never gave them a chance. [1] "How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes" https://amzn.eu/3BkjAsl https://amzn.eu/3BkjAsl
pjmorris 5y ago
I recently poured over 'How I Wish I'd Taught Maths' [0], where an award winning high school math teacher reevaluates his approach based on neuroscience research and empirical evidence and finds his previous approach wanting. Fascinating, worthwhile book. [0] 'How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes', Barton https://www.amazon.com/gp/product/B079K3HVMJ/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1 https://www.amazon.com/gp/product/B079K3HVMJ/ref=ppx_yo_dt_b...
konaraddi 5y ago
My 6th grade math teacher would often ask questions to students when teaching, the idea being that students would lead themselves to the solution/ideas then it would stick better. I do the same when tutoring students one-on-one in Calc 1 and below math. One drawback is asking too many questions in succession can become exhausting for students so sometimes there needs to be a healthy amount of both instruction and questioning or there needs to be breaks.
analog31 5y ago
I taught college freshman math at a big state university during a time period in between other jobs. My students were certainly bright -- they all had good high school grades and test scores -- but were the ones who didn't get placed into calculus. I learned that my students came to me with a wide variety of interesting beliefs about math, such as: 1. Math is subjective! The teacher solves the problem by knowing the "trick" that gets them the answer they want to see. They can't explain where the trick came from, or why your answer is better than theirs. 2. They were explicitly taught "test taking skills," such as the "guess and try" method. (This method was described in a hand-out for parents, that one of my kids brought home from school). 3. They will never use their college math after college. Even students who are destined to become engineers believe this. They heard it from somewhere, not from their math teachers, meaning that culture plays a role in math education. In addition, I perceived that the curriculum itself reinforced a problem solving method that is unrealistic: 1) Identify the "form" of the problem, which is one of the "forms" studied in the most recent chapter. 2) Extract the parameters from the problem statement. 3) Perform the algorithm associated with that form, to produce an answer. For instance, they only learned one function with an extremum: The parabola. So in the chapter on "maxima and minima," every single problem boiled down to finding the parabolic equation representing the problem statement, and then finding the vertex of the parabola. This bears virtually no relationship to any useful or even theoretical math topic. I decided that with as much ** as had been handed to them, the least I could do was help them get good grades that were needed to get into some of their intended majors such as Business. But not wanting to completely abandon my integrity, I adopted a compromise: Instead of test taking skills, I would teach math learning skills. Imagine you are working on a homework problem, and have not been paying attention in class at all, but you have the textbook. How do you approach the problem? It taught them how to learn from the textbook, which is at least kind of analogous to how most of us do math today -- by looking things up. And I promised them that if they added just one more ingredient -- repetition -- they would get good grades on the exams. I avoided pulling up facts from memory. Instead I would say "I know the answer, but let's find it in the textbook." There were dozens of sections of the course, and one great big exam, that was graded by a team of graders. So I got to see how my kids scores stacked up against the "competition," and they actually did quite well. I also had kids from other sections dropping in to my lectures. So I can't claim to have some foolproof method, but it at least got me through my teaching stint with my conscience intact. As a pet peeve, school math isn't even real. It doesn't represent how anybody does math: Mathematicians, STEMmies, or laypeople. Nobody does anything with math, without a computer in front of them. In my perfect world, I would place a much greater emphasis on computation when teaching math.

115 points by LittlePeter 5y ago

I don't know how to teach math, but I know how not to teach math. 1. Don't teach that math is a "ladder". Trigonometry is not "harder" than geometry, although many concepts in trigonometry are expressed as geometry. Calculus is not "harder" than algebra, although many concepts in calculus are indeed expressed as algebra. Teaching that math is a ladder allows people to believe that once they've covered the "basics" of a field of math (as if you could ever learn all of "algebra"), they shouldn't have to think about the basics again. Try telling a pianist that once they've learned a set of scales they don't need to study them any more, or try telling Richard Feynman that once he understood the standard atomic model there was no need to continue examining it in greater depth. 2. Don't teach that math was "discovered" in its current form. What most students think of when they think "math" is in fact western notation for patterns observed in reality, handed down by a bored teacher as universal law. A sheet full of squiggles isn't "math", it's just a set of the most-predictable and best-notated patterns made up by past mathematicians. "Math" is the process of discovering and refining those squiggles. The relationship between the length of the two perpendicular sides of a right triangle and the hypotenuse exists outside of human knowledge, but the Pythagorean Theorem is a only a certain method for expressing that relationship. The numbers 1 through 9 are one possible code for counting, but the quantities I through IIIIIIIIII and onward exist outside of the notation of Arabic numerals. 3. Don't immediately move on from concepts once a student has mastered them. Imagine if every time you figured out how to do your job correctly, your boss moved you to a completely new task requiring a completely new set of skills, giving you no time to enjoy applying your prior mastery. If you've just learned how to factor quadratic equations, why move on? Why not explore programs that factor equations? Why not dig up old exams and show how factoring would have solved earlier problems faster? This is always met with cries of "but they're supposed to use what they've learned to go on to even harder problems!" Sure, I agree, but how can you possibly enjoy any task if the only thing you can really expect is that even if you master your task, you'll be struggling again within the week? Would you join a rec basketball team if every time you started hitting three-pointers consistently, they moved the rim higher and farther away without giving you any time to feel what it's like to be good at basketball? 4. Don't teach that you need to know math because your grades have to be high. For nearly every profession available, grades are immediately forgotten as soon as you start receiving wages. Teach that you need to know math because the world is constructed and controlled by mathematicians and those acting on the advice of mathematicians. If you don't know math you'll be taken advantage of by those who do, whether it's in advertising, gambling, banking, medicine, insurance, politics, entertainment, engineering, programming, or any of the many other fields driven entirely by math. Of course, not teaching those four lessons becomes difficult, because those lessons work in opposition to the central tenets of mandatory state education, which necessarily operates as a giant brainwashing factory working to justify its own existence. Modern ediucators spend most of their time trying as hard as they can to teach people that knowledge is best bestowed by authority and then proven through certificates, that various fields of inquiry exist separately from one another (for instance, that physics and biology can or should be studied independently of history, mathematics, grammar, semantics, or art), and that failure to live up to expectations must necessarily lead to shame and corrective action. Don't send your kids to school if you can avoid it.

this-pony 5y ago
I think maybe you mean "don't send your kids to an _American_ school if you can avoid it". Although I have no personal experience with the American school system, I would say that there are many schools in the world that adhere, to some degree, to your 'how not to teach mathematics'. I went to a public school in Europe and have had plenty of enthusiastic teachers that also tried to deliver the beauty of mathematics.
- rsj_hn 5y ago
  Russia has an amazing math education tradition. I always recommend Russian math books and Russian math programs. They are excellent.
  - quacked 5y ago
    Every Russian and ex-Soviet I've ever met was extremely proud of their school system, which arose from the Soviet system.
    - rsj_hn 5y ago
      They really did a great job at math. The humanities were terrible, it was hopelessly politicized. But if you didn't want to deal with constant lectures about the struggles of the working classes, or constant coursework on marxism-leninism, you studied math or physics, or engineering. It helped, of course, that the government had as a national policy the promotion of STEM. But they did an amazing job at it, and produced a fantastic crop of world-changing talent.
      - geomark 5y ago
        What a great approach. The easy social studies courses are made so horribly boring that it drives students to study the hard sciences.
  - geomark 5y ago
    Any titles you can recommend? I'm homeschooling my kid and have been using sample problems from both Russian and Singapore math. But I only have a single source for Russian math problems, which is a pdf of a paper by a Russian educator who taught in the US and then wrote a paper comparing the two educational systems. It's interesting because he talks about how algebra is a big scary subject that is introduced late in the American system, while in Russia they start doing algebraic-type problems early in primary but they are solved with pictures. So when algebraic notation is introduced later it's just a next step to the types of problems that students have already been solving.

tediousdemise 5y ago

The human brain is like a self-compiling compiler. You can teach people math by teaching them how to read, and then suggesting a good math book.

royaltjames 5y ago
Can you suggest me a good math book to read?
- hansvm 5y ago
  Tons, but it really depends on where you're starting. Is there anything in particular you'd like to learn? Every time I've recommended Calculus Made Easy [0] it's been a huge success. The writing is lively and full of motivating examples, and it's an enjoyable read. [0] https://news.ycombinator.com/item?id=17185577 https://news.ycombinator.com/item?id=17185577
- tediousdemise 5y ago
  While there are many amazing math books out there that specialize in a particular branch, I tend to prefer generalist books that cover a wide range of topics. It you want something that starts with the absolute fundamentals, What Is Mathematics? is a perennial classic. I think that one starts with addition and takes you all the way through advanced calculus. If you want a comprehensive survey of nearly all major branches in mathematics written by some of the most prominent Soviet mathematicians, you’ll want Mathematics: Its Content, Methods and Meaning. If you want more of a reference-style encyclopedia, The Princeton Companion to Mathematics is a good one. If you want to build a solid foundation for math tailored to computer science, look no further than Concrete Mathematics, coauthored by Donald Knuth himself.
rahimnathwani 5y ago
I'm trying that with my son (who is not yet 5). He has no trouble reading anything in his math book[0], but 'suggesting' that he work through the book isn't enough. He'd rather play with lego, read a story book, or do pretty much anything else. My solution: short sessions of a few pages at a time, with me sitting by his side, ready to intervene when he gets sidetracked. [0] https://shop.singaporemath.com/index.php/product/dimensions-math-textbook-kb/ https://shop.singaporemath.com/index.php/product/dimensions-...

atoav 5y ago

One important thing is: create realistic problems that need certain math. I had to calculate integrals for a year before our teacher for a brief moment glossed over what problem it might be needed for. In fact I only understood why integrals were useful, when I started learning for the final exam. This stuff can be really cool, but if even your teacher does as if it is the most boring and useless thing in the world, if even your teacher is unable to explain beforehand why you want to know this, how would you?

jrib 5y ago
One of my best memories from high school math was our teacher taking us outside to measure the height of our building after teaching us trigonometry. We took a triangular ruler outside, leveled the base and lined it up by eye with the top of the building. Then we measured the distance to the point where we were standing and the distance from the ground to the ruler. I really wish education reduced the amount of rote learning involved and focused more on letting students explore and make mistakes with the educator as a guide.
- function_seven 5y ago
  My high school took this concept one step further. We were tasked with designing a solar hot dog cooker. The physics class specified the reflective area needed to gather enough sunlight to cook the hot dog. They tossed that over the wall to the algebra class, who determined the parabola shape and focal length from the dog. From there it was sent to drafting class, where the plans were drawn up. Those went to the kids in wood shop, to build the thing from the plans. Then one day we all went outside with our solar cookers and hot dogs and had a cookout. IIRC, there was a lot of variation among the cookers. Different reflectors were used (piece of aluminum sheet, bent and held by end brackets. Mylar film laminated to a wooden parabola "bed". One kid got adventurous and designed his with a vertical dog holder, and a circular parabola that was actually several different profiles stacked to "smear" the focal point along the vertical axis. I don't remember if it worked well or not) I think we were meant to learn some sort of process engineering in addition to our respective subjects? Or just "teamwork", I guess.

dfdz 5y ago

I am really confused by this article. The section "First An Issue" is a great example of how NOT to teach math. First, the class of sequences under consideration is not clearly defined. (I am not going to invest time solving a problem, if I am not sure about the definitions involved, constant coefficient/variable coefficient etc). Second, a question is asked "Is a product of two such sequences also of the same form?" and the meaning of product is also not clearly defined. Third, the author links to an article whose PDF on the publishers website is unreadable, and does not seem to answer the question from what I can tell. The only purpose linking to this article serves is intimidation. There is a happy ending to my rant. I googled the topic alluded to by the author and found a clear post explaining the solution. In this stackexchange post the question and solution are both clearly stated. https://math.stackexchange.com/questions/1348838/sum-and-product-of-linear-recurrences https://math.stackexchange.com/questions/1348838/sum-and-pro...

this-pony 5y ago
I think the point the author tries to make in the section "First An Issue", as you say, is exactly to give an example of bad teaching.
yCombLinks 5y ago
Yes, her exact point is "Here is an example where obvious things are skipped over that make it hard to understand."

teeray 5y ago

I started high school in a program that was more focused on “applied math learning.” (IMP [0]) We’d throw dice and figure out probabilities and stuff. I transferred out of it within two months because it was so focused on finding applications that it completely missed the mechanics of manipulating expressions. It was boring at times, but after three years I had a rock-solid base on how to turn one expression into any other. The applications of those skills came later once I got into Physics and Stats. I don’t know if it worked out better that way, but I think there’s kind of a chicken-and-egg situation. To really understand the applications, you need the math background. To appreciate the math background, you need the applications. [0] https://en.wikipedia.org/wiki/Interactive_Mathematics_Program https://en.wikipedia.org/wiki/Interactive_Mathematics_Progra...

mhh__ 5y ago

The more people I meet the less I sure I am how to teach.

oilostthelast 5y ago

Simpleton here: One, the first thing the professor does is put things in "human terms", the issue is plain that the mode of conveyance used in the textbook is wanting. And this is coming from "Author in Action" videos provided by Pearson, ostensibly from Sullivan himself. I think a portion of this issue is derived from being "too simple to state." Two, calculation takes precendence over everything. I've seldom been offered the opportunity to actually apply any of this myself in real world practice. What's worse is that when professors have included "working" problems they're quickly glossed over, to me it seems like this is absurd - as a non-math major this is precisely why I'm in the class, for application. This branches into arguments about the math taught being entirely abstract and stripped from a model, which makes it infinitely more difficult for someone like me. What compounds it is none of the work asks things like "why do we use exponential functions in the calculation of half-life?" instead, you just translate to TI-84 and plug your answer in. I'd quite like to see more applied concepts, and less calculation. It'd also be nice to self-direct, but I'm sure that's in reality impracticable from an assesment standpoint. I noticed repeatedly that I quickly grasped certain concepts, but was forced to repeatedly compute similar problems, time that would've been better spent working on points where I was lacking.

rsj_hn 5y ago

Here are some things I would recommend: 1. Find a teacher that is knowledgeable and passionate about math. Grad students might fit the bill. Engineering grad students also. Low level math (e.g. calculus, differential equations and below) can be taught by anyone in STEM, it doesn't need to be a trained math teacher. Engineers, physicists, or chemists often do a better job and passionate ones might be easier to find. Passion is important, because a good teacher will be a window onto another world, that you are invited to explore. A bad teacher is a window onto a brick wall. I used to tutor undergrads in math and the worst students were the math education people. I was often told "I hate math" by these students. I would even plead with them to find another subject, that it was unfair that someone who hates math becomes a math teacher, but they insisted that they love "teaching", they just hate math. Welcome to the US Public school system. 2. Include history in the math education. This depends on the personality of the student, but for me, I loved learning about the lives of the people who made mathematical discoveries, and the circumstances of those discoveries. To this end, I recommend books by George F. Simmons, for example: * https://www.amazon.com/Differential-Applications-Historical-International-Mathematics/dp/0070575401 https://www.amazon.com/Differential-Applications-Historical-... * https://www.amazon.com/Calculus-Gems-Memorable-Moments-Spectrum/dp/147045128X/ref=sr_1_4?dchild=1&keywords=George+F+Simmons&qid=1620430006&s=books&sr=1-4 https://www.amazon.com/Calculus-Gems-Memorable-Moments-Spect... 3. Look at the Russians. The Soviet Union had an amazing pedagogical program in math education, with fantastic books, puzzles, newsletters, etc. Some of that survived the turn to capitalism, and today they still punch above their weight. Some resources: * http://www.ascd.org/ASCD/pdf/journals/ed_lead/el_198102_brandt2.pdf http://www.ascd.org/ASCD/pdf/journals/ed_lead/el_198102_bran... * https://www.amazon.com/Moscow-Puzzles-Mathematical-Recreations-Recreational/dp/0486270785/ref=sr_1_8?dchild=1&keywords=Russian+school+of+Mathematics&qid=1620430372&sr=8-8 https://www.amazon.com/Moscow-Puzzles-Mathematical-Recreatio... * http://docshare02.docshare.tips/files/17782/177829869.pdf http://docshare02.docshare.tips/files/17782/177829869.pdf 4. Interdisciplinary approach. A great way of studying math is to present some problem in physics or engineering and develop the mathematical techniques to solve it. This is how much great math was discovered. 5. Ask questions. Rather than telling students techniques and then watching them use those techniques, you can create a dialogue where you start by asking specific questions and guiding the student to discover the math on their own. This of course requires a smart student and a smart teacher, but it's a very effective way to really learn a topic. This was famously done by the Texas Topology department in the mid 20th Century. 6. Teach concepts. If you cannot get the student to discover concepts, you can still emphasize the teaching of concepts. For example, there is no reason why a 5th grade math class can't begin to cover concepts such as simply connectedness, or Euler characteristic, or hamiltonian circuits. These can be done with simple pictures, yarn, paper and scissors. Then encourage the student to generalize to other shapes. Similarly in geometry classes, working with a ruler and compass creates a more memorable hands on experience for young students and can be a technique to introduce them to early theorem proving, for example bisecting a chord.

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nobody0 5y ago

math is not a spectator sport. Just like you cannot learn to program by reading books, people have to really dive into problems and rediscover certain facts by themselves.

hervature 5y ago
I taught myself PHP from an old LAMP stack for dummies book years before any formal education in programming. Are you saying you need to actually program in addition to reading? No one is debating you need to work on math problems in addition to reading the definitions to learn math.
- username90 5y ago
  Programming is a lot easier to self teach than math since you can test your programs by running them while you can't test your math work. The end result is that basically everyone who learns math creates a lot of errors in their understanding of it, and then those errors gets hammered out after countless exams as they progress through their education. Without the pressure of education most people just continue to accumulate errors in their understanding until they no longer can progress since the backlog of stuff is too long and they don't know where to start.
  - hervature 5y ago
    Can't test your math work? That's just completely false. The process is the test. Does each step make sense. Lower level math books include answers which will give you confidence the procedure was correct but no way ensures it. At a higher level, it changes to proofs where again, it doesn't matter if you know something is true or false, as long as each intermediate step it true. Of course, I'm not going to suggest that learning math solely from a book is optimal (neither is learning programming from a book), but it certainly isn't impossible.

jorgenveisdal 5y ago

How to study mathematics, according to Niels Henrik Abel's high school teacher: 1. Never study more than one book at a time and never abandon a book you have chosen without working all the way through it 2. If you face difficulties, do not give up but instead go back twenty times if that should prove necessary and only then allow yourself to investigate another mathematician's solution 3. Skip over those parts that are of no challenge in order to get at what is new to you 4. Reflect over the reading, in particularly how the writer came to the solution and moreover, what the solution leads to 5. Investigate whether or not another transformation or substitution would have solved it in a better manner 6. Always read with pen in hand so that you can work out all the calculations and practice all the questions you encounter 7. Write up lists of subjects that afford you an opportunity to develop your own theories 8. Geometric reflections can be a suitable way of strengthening and securing one's judgement

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cgriswald 5y ago
Most high schools and high school teachers don’t teach in a way that particularly enables students to take this advice. “These are the rules. This is how the math works. Do the math. Also these are the classes you have to take to be an $X.” A relatively good teacher will engage with a student who is enthusiastic about the material in this way but many won’t: “You’ll learn about that in $future_course” without even an attempt to talk about it in the abstract. Without some guidance on which book to choose, (1) could condemn a reader to mathematical purgatory. In any case, I’d also say that taking multiple math courses in high school helped me to understand that math is most often about relationships and that there were multiple ways to describe the same relationships. This was in spite of no one bothering to really teach that idea. Previously I was mostly focused on axioms and rules (probably because of how I was taught), which, while useful, missed some important stuff. I also prefer to read through multiple texts because it gives me a break, let’s me see new connections, and if I come back to the text after working on something else and still know what’s up I know I really grok the material. (It’s insanely easy for me to memorize and apply rules; that’s not indicative of understanding.) (3) is good reading practice in general, but also, it’s a skill most people haven’t developed. It’s very easy to say, “Pfft, I know all this,” and just skip something without actually really grokking the information; it’s too easy to over-estimate understanding. For most people I’d say at least work through any proofs and the most difficult questions in each section. This is actually something I think we should be learning in middle school at the latest.

SneakyTornado29 5y ago

One common shortcoming I see often with math teaching (at a high level): the difference in difficulty between lecture problems and homework/midterm problems is sometimes absurd.

Koshkin 5y ago
Well, lecture problems are usually for illustration, while homework ones are for independent study and for training, and so they are expected to be more challenging.
- hdctambien 5y ago
  Yes, that is more a misunderstanding of the purpose/role of university homework vs highschool homework.

credit_guy 5y ago

"There is no royal road to mathematics", Euclid is thought to have said to Alexander the Great. Well, I don't think there's a royal road to teaching mathematics. First ingredient if you want to be a good mathematics teacher: preparation. When you go to class, just make sure you went through all the examples, all the exercises and problems, and you know them in and out. Be prepared. When they train you how to teach, they never tell you that. It is just assumed. They tell you about the attention curve (which supposedly looks like a bathtub: high at the beginning and at the end, low in the middle), about body position, voice volume, and other stuff. But for some reason they forget to tell you that first and foremost you need to prepare. Second ingredient: feedback. Ask a colleague to come and observe you, and, if possible, go with the same colleague and observe another teacher whom you want to learn from. They (the colleague) can point at what you did differently from that teacher, what you did well, what you should work on next. If that colleague is good, they will give you a list of 20 things to work on, but also tell you to focus on the first one or two. Are you stopping to look at the kids and see if they understood? Are you making eye contact with a single kid, rather than cycle through all the classroom? Is your voice loud enough (pro tip: it never is, speak louder). Did you leave the kids enough time to think on their own? Did you give them too many hints? Are you biased in always letting only the best kids in class answer out loud? Are you pointing with your finger when you want to emphasize something (it's natural, but don't do it, it intimidates the kids). But all these things are second order things. First thing: prepare for the math you are going to teach. It's much more important to not make any mistakes, such as a sign mistake in the middle of a calculation, than to make eye contact. Mistakes erode the kids' trust in you. Avoid them. Prepare. Once you got to a point where you consistently deliver a flawless lecture, then you can think of becoming the next 3Blue1Brown. But you need to walk before you run.

catbird 5y ago
Thanks for your comment. It sounds like you have learned these lessons through a lot of experience - do you have any suggestions on a book to read to get the very basics of teaching? > When they train you to teach... I started a job as a professor this year and never got any of that basic training. Before this I was a postdoc reading papers and writing code all day. In job interviews my lack of teaching experience wasn't seen as an issue. The faculty would say, "oh teaching isn't so hard, you'll figure it out quick enough." So far I have been muddling along ok, but I have this anxious feeling that there are a lot of aspects to this that I don't know that I don't know, and don't know where to get started on finding out about them.

ianai 5y ago

Make the lectures and homework actually prepare for the test. The test questions should be easier than the homework!!!!!! Actually, make the test be a minority of the overall grade - i.e. 80% of grade from homework.

frompdx 5y ago

The linked articles about "Inquiring Based Learning" (IBL) seem to be making the claim that this type of teaching is somehow more effective than lecture based teaching. I'm not entirely convinced this is true. My doubt is in part fueled by the statements on IBL not really being backed by anything other than intuition on the part of the author of those articles. If it is backed by more than intuition I either misread or they talk about it in an unlinked article. The remainder of my doubt is due to the mismatch with my own personal anecdotes. IBL reminds me a lot of breakout sessions or whatever you want to call mini group projects that take place in the span of instructional block. Some people in those groups will already know the answer, some won't care, and some will count on the person who knows the answer to carry the group because that person isn't so great at teaching it themselves. Why would they ask questions about answers they already know? When I was still in school IBL style was used far more in sciences than something like history. But I loved history. Especially lecture based history classes. Part of the reasons for that many professors I had loved to talk about the subject they taught. That doesn't mean the professors didn't love the subject they taught, but talking about it didn't seem to be something the loved doing. I personally struggled with the IBL format and did well with the lecture based format. I'm not an educator myself. I just bring that up because at least for me I did not find the technique helpful and got less out of it than I did with lecture based classes. Everyone is different. I imagine the IBL format does appeal to some more than others.

insert_coin 5y ago

How to teach anything? You teach first to learn. In my opinion learning is not an obvious thing, it is not obvious why things need to be presented as problems for example, and many students rebel against them as it leads to an empty feeling every time they even attempt to solve a textbook math problem. Curiosity is innate, but learning has to be taught. So the clash occurs because they begin by teaching math to people that do not know how to learn, or why to learn.

bradrn 5y ago

The best maths book I’ve ever read was Measurement by Paul Lockhart (yes, the one of Lockhart’s Lament [0] fame). It’s been a long time since I read it in primary school, and I can’t remember exactly why I loved it so much, but there is one thing I remember clearly: the problems he poses are just really, really interesting. I can’t say I love problem solving, but I remember struggling with some of those problems with hours, just because I wanted to know what the answer is. Highly recommended as a way of teaching interested children (and adults!) about geometry and calculus. [0] https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament....

adam_arthur 5y ago

Teaching in general is best done by "peeling the onion". By that I mean, start with the highest level explanation first, form an understanding of that, and then repeat the same lesson but at a lower level of abstraction. You have to be able to walk before you can run. First, explain the concept like I'm 5. Use examples that provide real world context for why the concept is useful. Demonstrate examples of using the concept to solve a problem. Then explain the concept again, with an added layer of complexity. Go through the same examples with this layer of complexity. And so on. The more levels, the easier it will be for people to ease into learning the concept. When confronted with something that's too foreign, many people just shut off their brains, and convince themselves they're "bad at math" etc. The hardest part about learning is really breaking through the wall of initial understanding. Once you know the language, everything becomes a bit easier to learn. I honestly think math education can be hugely improved with this approach. This could even be done in a "high tech" way, by having an e-textbook with a slider. Adjusting the slider changes the level of abstraction and granularity that it teaches the concepts at. Final note. Math should be taught with well named variables and contextual examples. Why use X and Y when you could carSpeed and carAcceleration? Or whatever, you get the point. People can't relate at all to math when all the examples use generic terms without context. At that point it just becomes rote repetition. Obviously this commentary applies moreso to earlier math education.

lixtra 5y ago
> Teaching in general is best done by "peeling the onion". My experience is that best way depends on the student, the thing you want to teach and probably even on the time of the day. The best teachers are able to adapt to that.
enriquto 5y ago
> Why use X and Y when you could carSpeed and carAcceleration? (...) People can't relate at all to math when all the examples use generic terms without context. Couldn't disagree more. The whole point is that you don't care about the meaning of the numbers. They are just numbers. A big part of math is learning to strip numbers of their context and look at them bare. Besides, variables with several letters in them are extremely ugly. They look like products. (They should be banished from programming languages also, but I realize that this is a controversial idea.)

chrisweekly 5y ago

https://betterexplained.com https://betterexplained.com does a better job of answering this question than anything else I've encountered.

throwawaysea 5y ago

For those who know the education space better, I am curious as to what we get wrong in the West with STEM education. There has been such a large academic or research oriented focus on determining the best approaches to teaching math. But other nations that have more basic or “crude” math education seemingly do better. My instinct is that we are attempting to make learning too easy and that true learning simply requires that learners push hard and build their own fundamental understanding. Can someone offer a better formed opinion on this?

rimliu 5y ago
Your instinct is absolutely right. I have not seen anyone suggesting that learning to play an instrument is or can be effortless, yet for some reason people believe that STEM should be.

TrackerFF 5y ago

When growing up, I struggled with math. Part because my ADD made it incredibly hard to read books from start to finish, part because I had very little interest in STEM classes, and part because we had a math teacher that would focus all his energy on the top students. If you struggled with something, and he didn't deem you worthy, he'd simply direct you to the book "read the book". It wasn't until undergrad that we had this fantastic calculus lecturer - everything just made sense. He started at the very, very beginning; adding together numbers, and worked his way up to pre-calc, in around 2 weeks. He never skipped any steps, and was very good at explaining the implications of various proofs, and pointing out the typical / common errors and fallacies made by students. Of course, if you had a good grasp of math leading up to calculus 1, this was all boring and old news. No big deal, those students would skip classes, it was after all in the very beginning of the semester. But for me, it was simply life changing. And parallel to that, people started posting vids and websites explaining math. I suddenly had multiple different sources of people trying to explain math their way - which also turned out to be invaluable. Suddenly this guy Salman Khan started posting vids on youtube, and I discovered Pauls Math notes. Well, the rest is history for me - I graduated with good grades in all math courses, and a love for math. If anything, it thought me that there's no single way of teaching math. Again, when I grew up, it was one book and one teacher.

chobytes 5y ago

In my experience you learn math by doing. Its takes personal time and effort and it cannot be spoon fed. A teacher can provide a little guidance, but mostly its up to the student.

rustybolt 5y ago

I wanted to write a comment, but the comment by tchow8 already sums it up nicely: > Okay, I’ll bite. You wrote, “What I do not always get is some totally obvious ideas. Does this make any sense? Here is an example.” How is what you say about recurrences an example of a totally obvious idea, or of not getting a totally obvious idea? I don’t understand.

tkgally 5y ago

> The only way to learn mathematics is to do mathematics—Paul Halmos Paul Halmos taught for two years in the 1970s at the University of California, Santa Barbara, where I was an undergraduate majoring in linguistics. I was also interested in math, and I ended up taking three courses from him: an introduction to mathematical logic, with about twenty students; a seminar on set theory and the foundations of arithmetic, with four or five students; and a one-on-one tutorial on point-set topology. For the latter two courses, he taught based on his “do mathematics” principle. He would provide us with definitions of basic concepts and then have us work through a series of not-so-difficult problems that led eventually to proofs of the major theorems. He let us do most of the work ourselves, though if we got stuck he would provide pointers to help us keep moving ahead. His method was inspired by that of Robert Lee Moore [1], though Halmos was not as hard-assed about it as Moore reportedly was [2]. In the seminar, he encouraged the students to work together. More than forty years later, I remember many details of the two problem-based courses, while the first course, in which he lectured on mathematical logic, has mostly slipped from my memory. [1] https://en.wikipedia.org/wiki/Moore_method https://en.wikipedia.org/wiki/Moore_method [2] https://mathshistory.st-andrews.ac.uk/Extras/Halmos_Moore_method/ https://mathshistory.st-andrews.ac.uk/Extras/Halmos_Moore_me...

PartiallyTyped 5y ago
As a grad student in CS, for theoretical and algorithmic courses, doing the proofs one by one is the only thing that works for me on how to get a good understanding of what tools the proof provides for me, what new pattern of solving a problem exists and making use of it. I found that it significantly expands my repertoir, or grammar and vocabulary so to speak instead of staring at the lecturer trying to explain their thought process.

edtechdev 5y ago

There's actually decades of research on how to more effectively teach math, including the effectiveness of inquiry based learning approaches, which are more effective. See the MAA's Instructional Practices Guide: https://www.maa.org/programs-and-communities/curriculum%20resources/instructional-practices-guide https://www.maa.org/programs-and-communities/curriculum%20re... Here's info on research on the effectiveness of inquiry learning in math: https://theconversation.com/who-learns-in-maths-classes-depends-on-how-maths-is-taught-21013 https://theconversation.com/who-learns-in-maths-classes-depe... https://www.colorado.edu/eer/research-areas/student-centered-stem-education/inquiry-based-learning-college-mathematics https://www.colorado.edu/eer/research-areas/student-centered...

jcassee 5y ago
You seem to be knowledgeable in this area. Is it weird to ask you to summarize these links, especially in the context of the more concrete discussions in this thread?

dusted 5y ago

As someone who is math-stupid, I totally find that my main difficulty in learning a new thing is that the "obvious parts" are not mentioned, or only in such passing as to make clear to anyone that they should already know this.. I learn from the top down, I can't reverse that, but I can't "get all the fundamentals right first" and then work out from there... I need to scratch a lot of surface on the subject, then the fundamentals start to crystallize for me, so refreshing me on those obvious things in the context of what I'm learning, helps a lot.

v8dev123 5y ago

I have different take on this. Teach how Humans found Maths. Context is important. Teach what Abstraction really is. Start from a Lion Picture and draw it and abstract by step by step. Teach use cases. Once you done that, teach Algebraic thinking. Then the student can learn anything.

MarkMc 5y ago

Craig Barton's excellent book [1] makes a compelling argument in favour of the 'explicit instruction' for teaching maths. Here's an excerpt: --------------- I was particularly proud of a guided discovery task I came up with for introducing some of the more complex laws of indices to my Year 11 class two years ago. The worksheet looked like this: https://photos.app.goo.gl/dzWPzw32hWVDLvYG8 https://photos.app.goo.gl/dzWPzw32hWVDLvYG8 Nice, eh? Again, I ask the question: what could possibly go wrong? Well, quite a lot, as it turns out. My takeaway When considering a guided discovery task, the question I should have asked myself is: what is the best that can happen? Take the laws of indices lesson. The best that can happen is that all students discover the laws of indices for themselves, leaving no gaps in their knowledge, nor developing any misconceptions, in a reasonable time frame. We can then proceed with the rest of the lesson, maybe moving on to application questions, or interleaving other topics into the examples (see Chapter 12), such as indices involving surds or fractions. How often does that actually happen? In my experience, literally never. What actually happens is that one or two students discover exactly what I wanted them to discover. They are feeling great about themselves, and rightly so – as we have seen in Chapter 2, success is motivating. A handful of students have some kind of idea what is going on, but with an eclectic mix of gaps in their knowledge and newly formed misconceptions. Some of these students are aware they have gaps and misconceptions, others are blissfully ignorant. And the rest of the students do not have a flipping clue what is going on. They are feeling confused and pretty down about themselves when they see their fellow classmates have figured it out. Any form of decent formative assessment strategy (Chapter 11) quickly reveals this disparity between levels of understanding, and as such I cannot move on with the lesson. So what do I inevitably end up doing? Teaching the laws of indices, of course! Maybe I will set those students who seem to have understood it off on the work I hoped everyone else would be moving on to – mind you, I would really like them to hear my explanation and do the worked examples, but how can I justify doing so when they have demonstrated their understanding? Hmmm… Anyway, back to the rest of the class. By this stage, I am 30 minutes into a 50-minute lesson, rattling through a series of worked examples on the laws of indices far quicker and with much less care than I should. There is zero time for the students to practise their newly acquired skills and hence consolidate their knowledge, nor sufficient time for me to do any kind of application questions which would show them the full breadth of the topic. But it is even worse than that. Even if I could somehow freeze time and spend those lost 30 minutes going through carefully structured and well-chosen worked examples (Chapters 6 and 7), I am not back at square one. I am behind square one, because my students are no longer coming at the topic with fresh eyes. Many of those who failed to ‘discover’ the key relationships have already decided that indices are difficult, and yet another area of maths that they don’t understand. It’s going to take more than my magically retrieved 30 minutes to turn that one around. And so I wave goodbye to a group of confused students trundling out of the door, promising that we will pick this up again tomorrow, assuring them it will all be fine. I am already dreading the lesson, wanting to open up proceedings by saying ‘okay, everyone, forget what happened yesterday’. In the past, I blamed my students for this – if only more of them could have figured it out. Now, I know the blame rests squarely at my feet. I never gave them a chance. [1] "How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes" https://amzn.eu/3BkjAsl https://amzn.eu/3BkjAsl

pjmorris 5y ago

I recently poured over 'How I Wish I'd Taught Maths' [0], where an award winning high school math teacher reevaluates his approach based on neuroscience research and empirical evidence and finds his previous approach wanting. Fascinating, worthwhile book. [0] 'How I Wish I'd Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes', Barton https://www.amazon.com/gp/product/B079K3HVMJ/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1 https://www.amazon.com/gp/product/B079K3HVMJ/ref=ppx_yo_dt_b...

konaraddi 5y ago

My 6th grade math teacher would often ask questions to students when teaching, the idea being that students would lead themselves to the solution/ideas then it would stick better. I do the same when tutoring students one-on-one in Calc 1 and below math. One drawback is asking too many questions in succession can become exhausting for students so sometimes there needs to be a healthy amount of both instruction and questioning or there needs to be breaks.

analog31 5y ago

I taught college freshman math at a big state university during a time period in between other jobs. My students were certainly bright -- they all had good high school grades and test scores -- but were the ones who didn't get placed into calculus. I learned that my students came to me with a wide variety of interesting beliefs about math, such as: 1. Math is subjective! The teacher solves the problem by knowing the "trick" that gets them the answer they want to see. They can't explain where the trick came from, or why your answer is better than theirs. 2. They were explicitly taught "test taking skills," such as the "guess and try" method. (This method was described in a hand-out for parents, that one of my kids brought home from school). 3. They will never use their college math after college. Even students who are destined to become engineers believe this. They heard it from somewhere, not from their math teachers, meaning that culture plays a role in math education. In addition, I perceived that the curriculum itself reinforced a problem solving method that is unrealistic: 1) Identify the "form" of the problem, which is one of the "forms" studied in the most recent chapter. 2) Extract the parameters from the problem statement. 3) Perform the algorithm associated with that form, to produce an answer. For instance, they only learned one function with an extremum: The parabola. So in the chapter on "maxima and minima," every single problem boiled down to finding the parabolic equation representing the problem statement, and then finding the vertex of the parabola. This bears virtually no relationship to any useful or even theoretical math topic. I decided that with as much ** as had been handed to them, the least I could do was help them get good grades that were needed to get into some of their intended majors such as Business. But not wanting to completely abandon my integrity, I adopted a compromise: Instead of test taking skills, I would teach math learning skills. Imagine you are working on a homework problem, and have not been paying attention in class at all, but you have the textbook. How do you approach the problem? It taught them how to learn from the textbook, which is at least kind of analogous to how most of us do math today -- by looking things up. And I promised them that if they added just one more ingredient -- repetition -- they would get good grades on the exams. I avoided pulling up facts from memory. Instead I would say "I know the answer, but let's find it in the textbook." There were dozens of sections of the course, and one great big exam, that was graded by a team of graders. So I got to see how my kids scores stacked up against the "competition," and they actually did quite well. I also had kids from other sections dropping in to my lectures. So I can't claim to have some foolproof method, but it at least got me through my teaching stint with my conscience intact. As a pet peeve, school math isn't even real. It doesn't represent how anybody does math: Mathematicians, STEMmies, or laypeople. Nobody does anything with math, without a computer in front of them. In my perfect world, I would place a much greater emphasis on computation when teaching math.