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Math education: It’s not about numbers, it’s about learning how to think

• taneq 9y ago
  As my old boss once said, "never confuse mathematics with mere arithmetic."
  • debbiedowner 9y ago
    Exactly. That's why there are three kinds of mathematicians, those who can count, and those who can't.
  • joejoebob 9y ago
    A professor of mine who was quite bad at arithmetic used to say, "I'm a mathematician, not an arithmetician."
  • netcan 9y ago
    I don't understand why this point seems so important to people. Seems like a semantic point, if we're talking about math education in schools. I mean, there is theory & practice in every field, technique and discovery. A medical scientist studying the development of aberrant cells in a petry dish has a pretty different job to a pediatrician or radiologist. They generally seem to coexist happily under the "medicine" label. For first year undergraduates, they would probably need to take most of the same classes.
    • sp332 9y ago
      Arithmetic is theory too. Graph theory has lots of practical applications. I don't think it's that kind of divide. Arithmetic is just a very small part of mathematics.
  • Koshkin 9y ago
    Yet, - arithmetics (a.k.a. Number Theory) is a vast branch of modern pure mathematics; - numeric calculations, i.e. how to do them efficiently, is a very important part of the (applied) mathematics.
    • sanderjd 9y ago
      The parent commenter's old boss certainly did not mean Number Theory by "arithmetic".
  • dahart 9y ago
    I've always liked that line, but it's clearly being cheeky and there are times it's applicable and times it's not. All arithmetic is math, but not all math is arithmetic. Except that if you dig deep enough, all math is arithmetic under the hood. Algebra is doing the arithmetic on variables before they're numbers. Calculus is just useful shorthand for compound arithmetic; the derivative is a divide and the integral is a sum. Exponentiation is repeated multiplication, which is repeated addition. Linear algebra is arithmetic on blocks of numbers. So we abstract over the individual arithmetic operations for our own convenience, there are too many of them. But it's all still fancy arithmetic. Maybe we should call math "fancy arithmetic" to help all the people who are intimidated by math...
    • bjl 9y ago
      Actually, the opposite is true. If you dig deep enough, it turns out that arithmetic is far more complicated than most other areas.
      • dahart 9y ago
        Then maybe instead of calling math fancy arithmetic, we should call arithmetic "hard math"? :P Seems like you agree more than disagree; math and arithmetic aren't easily separable, if you're being serious instead of cheeky?
        • bjl 9y ago
          Well, I was trying to make two points. 1) Arithmetic (number theory) is far harder than most people realise, to the point that its considered one of the most complicated mathematical disciplines. 2) Most subject areas in math bear little to no resemblance to arithmetic, so it doesn't make sense to call all math 'Fancy Arithmetic'.
          • dahart 9y ago
            Sure, understood and I agree. I was also trying to make one point: all math is built on arithmetic operations if you dig down to first principles, so it does make some sense to call all math 'fancy arithmetic'. There is no math at all without the concept of addition sitting somewhere under it. But I'm not actually proposing that, just sharing a cheeky point of view to contrast @taneq's boss, so don't take me too seriously. ;)
  • jack_h 9y ago
    As someone who went to university for mathematics (and computer science) I can tell you this is the most common misconception. People think that I spent years studying how to become a calculator. I love talking about both subjects I went to university for actually and both can be difficult to get people interested in. There are massive misconceptions about both.
    • lordnacho 9y ago
      Yes, this is one of the most annoying things about people who don't know anything about math. - They see a shopping bill or a phone number and ask you if you can add them. - They see random equations, often in the background, and ask you if you understand what it means.
      • taneq 9y ago
        It's like when you study mechanical engineering and suddenly everyone wants you to fix their car.
• lordnacho 9y ago
  I think a major issue with math problems in school is that they're obvious. By that I don't mean it's easy. But when you're grappling with some problem, whatever it is, eg find some angle or integrate some function, if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?" And you won't have any excuses for why you didn't think of it. Because math is a bunch of little logical steps. If you'd followed them, you'd have gotten everything right. Which is a good reason to feel stupid. But don't worry. There are things that mathematicians, real ones with PhDs, will discover in the future. By taking a number of little logical steps that haven't been taken yet. They could have gone that way towards the next big theorem, but they haven't done it yet for whatever reason (eg there's a LOT of connections to be made).
  • throwawayjava 9y ago
    Reminds me of this quote: "The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles." Alan Turing
    • CarolineW 9y ago
      You might want to edit that to fix the name. It's "Turing", not "Turning".
      • throwawayjava 9y ago
        Thanks. That's what I get for posting on my phone :)
  • hasenj 9y ago
    > if you don't find the answer, someone will show you, and you'll think "OMG why didn't I think of that?" Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time. The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules. It's as if when learning programming, you would learn fragments of various languages here and there but never connect the various pieces into anything meaningful. Lesson 1: boolean variables in Java Lesson 2: If statements in python Lesson 3: Three-part for-loop in C Mid-term exam: Q1: In the C language, how do you separate the components of a for loop? (correct answer: semicolons) Q2: What can you put in the right hand side of the '=' symbol in a boolean variable declaration in Java? (correct answer: an expression that evaluates to either true or false). Q3: You are supplied a function that can take any number and determine if it's a Crastomarian number. We have 20 foot ball players each wearing a T-shirt with its own number. The numbers go from 5 to 24. We want to find how many players have Crastomarian numbers. The program to find the correct answer will use a for loop. Write the header part of the for loop. (correct answer: `for(i = 5; i < 25; i++)`) ... And this is supposed to be how you learn programming. Also note: there's almost no way to know the answers without rote memorization. Of course you will never absorb these concept by osmosis because your day to day activity will not involve writing programs that do things. Instead, you're just memorizing the rules given to you and doing exercises that help you recall which rule to use for which situation. If you haven't memorized the for syntax and written 25 different variations of it, you will not be able to know it. And when I say that you write 25 different variations, I mean you write them as part of pointless exercises of course because these are not in the context of a program. It's more like just repeating these lines until the pattern somehow hopefully sinks into your mind. for(x = y; x < z; x++) for(y = 10; y < j; j--, y++) .. etc
• listentojohan 9y ago
  The true eye-opener for me was reading Number - The Language of Science by Tobias Dantzig. The philosophy part of math as an abstraction layer for what is observed or deducted was a nice touch.
• spodek 9y ago
  > it's about learning how to think It's about learning a set of thinking skills, not how to think. Many people who know no math can think and function very well in their domains and many people who know lots of math function and think poorly outside of math.
  • randcraw 9y ago
    Math skills (esp proofs) shape the way you approach problems, giving the problem structure and providing you the tools to reduce it into components, relations, and dependencies. To make mysteries unmysterious, such tools are indispensable. I'd go so far to disagree with you that people do well without learning math skill (those tools of thinking). If you can deconstruct a problem, then you certainly learned that skill somewhere, just not in a math class. Abstraction and reductionism are unusual capabilities to acquire on your own. As I recall, they're almost nonexistent in pre-linguistic societies like hunter-gatherers. If you have no words for abstract concepts, your thinking will be strictly concrete.
    • FLUX-YOU 9y ago
      I feel like there's at least three ways of thinking about problems that are practical: - systematic (math, rule-based) - social (history, emotions, people-based) - creative (art, invention) These all overlap and feed into each other because we're a complex species, but we really only ever hear about the necessity of math to "learn to think".
    • erezsh 9y ago
      Math is a subset of logical thinking. There are deconstructions of thought that math cannot handle, for example in domains where the axioms are too complex, or perhaps even unknown.
• 0xFFC 9y ago
  Exactly, as ordinary hacker i was always afraid of math. But after taking mathematical Analysis I realized how wonderful math is. These day i am in love with pure mathematics. It literally corrected my brain pipeline in so many ways and it continues to do it further and further. I have thought about changing my major to pure mathematics too.
• Koshkin 9y ago
  Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically. The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day that learning them takes a lot of training, years of intensive experience dealing with mathematical structures of one kind or another, so it should come as no surprise that people have difficulty learning math.
  • jdmichal 9y ago
    Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limits is the first time I remember learning something that could not be generalized to a physical form, because it specifically dealt with what happened when a function "broke".
    • hasenj 9y ago
      I find this sentiment somewhat surprising. Limits is incredibly easy to link to real physical things: movement, speed, acceleration. If speed is distance over time, what's the speed of a moving object at a very specific instant?
      • Koshkin 9y ago
        "Ma'am, you were driving at ninety miles an hour." "But, sir, I have only traveled twenty miles!"
      • jdmichal 9y ago
        Are you thinking of derivatives? I agree that those are relatively easy -- it's the rate of change for a function. Or, in geometric terms, it's the slope of the tangent line of the function's curve. Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value would have been had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do in this particular case -- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.
• deleted 9y ago
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• keymone 9y ago
  i always found munging numbers and memorizing formulas discouraging. i think physics classes teach kids more math than math classes and in more interesting ways (or at least have potential to).
  • throwawayjava 9y ago
    Interestingly, my experience was that your two sentences are totally contradictory. Physics and engineering was all about using memorized formulas to solve word problems, and mathematics was where you learned how those equations systematically fall out of some obvious assumptions. I think the real take-away is that both physics and mathematics can be taught poorly, and both can be taught well.
    • keymone 9y ago
      i get a feeling you're not talking about elementary school. but i agree - both can be taught good or bad. bad is dumb memorization, good is understanding real world phenomena.
• d3ckard 9y ago
  Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday struggle for some people. I honestly believe that foundations are the key. Either you're taught to think critically, see patterns and focus on the train of thought, or you focus on numbers and memorization. The latter obviously fails at some point, in many cases sufficiently late to make it really hard to go back and relearn everything. Math is extremely hierarchical and I believe schools do not do enough to make sure students are on the same page. If we want to fix teaching math, I would start there, instead of working on motivation and general attitude. Those are consequences, not the reasons.
  • sixo 9y ago
    Not sure it's exactly the early years. But I think there's zero point to moving past a topic until you have mastery & can think WITH it (not just, about it, or hack through it on paper). And since the early years are often where that fails, they need to be rethought.
    • jamesrcole 9y ago
      I remember listening to an episode of The Science Show podcast where the guest was arguing (based on their experience) that chemistry and physics concepts should be taught to young children as it's actually easier to teach it to them. With the stated reason being that if they're taught it at a later age, as they are now, they've already built a view of the world where those concepts are somewhat unintuitive. If they learn the concepts earlier they are more easily absorbed. I found their point fairly compelling, and I wonder if it applies to mathematics as well. The episode was http://www.abc.net.au/radionational/programs/scienceshow/young-kids-lap-up-molecular-structure-and-the-periodic-table/8405686 http://www.abc.net.au/radionational/programs/scienceshow/you...
      • 9y ago
• JoshTriplett 9y ago
  One of the most critical skills I see differentiating people around me (co-workers and otherwise) who succeed and those who don't is an analytical, pattern-recognizing and pattern-applying mindset. Math itself is quite useful, but I really like the way this particular article highlights the mental blocks and misconceptions that seem to particularly crop up around mathematics; those same blocks and misconceptions tend to get applied to other topics as well, just less overtly.
• simias 9y ago
  I completely agree. I think we start all wrong too, the first memories I have of maths at school was learning how to compute an addition, a subtraction and later a multiplication and division. Then we had to memorize by heart the multiplication tables. That can be useful of course (especially back then when we didn't carry computers in our pockets at all times) but I think it sends some pupils on a bad path with regards to mathematics. Maths shouldn't be mainly about memorizing tables and "dumbly" applying algorithms without understanding what they mean. That's how you end up with kids who can answer "what's 36 divided by 4" but not "you have 36 candies that you want to split equally with 3 other people, how many candies do you end up with?" And that goes beyond pure maths too. In physics if you pay attention to the relationship between the various units you probably won't have to memorize many equations, it'll just make sense. You'll also be much more likely to spot errors. "Wait, I want to compute a speed and I'm multiplying amperes and moles, does that really make sense?".
  • kungito 9y ago
    A friend's kid did not know how to do subtraction with just numbers but when you said it like he had some currency in a shop then he understood it. That was peculiar although the kid was very young (5-6 I guess) so it's interesting how kids develop ability to do abstract thinking as they age
    • steverb 9y ago
      That's the way my kids were able to grasp basic math. For fractions we used pizzas. Being able to hang the abstract concept on something that they were used to dealing with made it work for them.
• crb002 9y ago
  Programming needs to be taught alongside Algebra I. Especially in a language like Haskell or Scheme where algebraic refactoring of type signatures looks like normal algebra notation.
  • sanderjd 9y ago
    Learning multiple different highly abstract concepts at the same time does not seem like a solution to the difficulty of learning a single highly abstract concept.
    • Jtsummers 9y ago
      My experience with this comes from college, but I disagree. I added a math major late into my education (I was one course from a minor, then 4 more from a major so why not?). I ended up in several math courses at once that were often taken separately, but specifically abstract algebra and a higher level (4xxx) linear algebra course. I pretty much aced both, though I probably would have gotten an A in them taken separately. The reason I did better, though, was that they were both algebra. They were both expressing the same concepts, but on different types of objects. The moment that that clicked in my head, my Monday lesson in linear became the clarified by my Tuesday lesson in abstract, which was further clarified by my Wednesday linear. If you structure it correctly (particularly, with programming, by selecting appropriate problem domains for programming exercises), I'm quite confident you could create a better math and programming education experience by properly pairing the two.
      • sanderjd 9y ago
        Maybe! I just don't think programming (even in Haskell!) is similar enough to algebra for this to be effective, no matter how much we all want it to be. Programming still requires a mental model of how machines compute, which is a good thing to learn about but not (in my opinion!) a strong complement to learning math. Don't get me wrong though, I learned programming and calculus at the same time, I'm not saying you can't learn both at once, I just don't think one helps much in learning the other, for a novice in both.
• dahart 9y ago
  I wonder if a large part of our math problem is our legacy fixation on Greek letters. Would math be more approachable to English speakers if we just used English? I like to think about math as language, rather than thought or logic or formulas or numbers. The Greek letters are part of that language, and part of why learning math is learning a completely foreign language, even though so many people who say they can't do math practice mathematical concepts without Greek letters. All of the math we do on computers, symbolic and numeric, analytic and approximations, can be done using a Turing machine that starts with only symbols and no built-in concept of a number.
  • redial 9y ago
    Symbols are symbols, are symbols. Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).
    • adamnemecek 9y ago
      Let's add Chinese, Amharic and Arabic characters. Why not, expanding is expanding.
      • d8421l01vv4r 9y ago
        In some branches of engineering (traditional engineering, not software), there are sometimes too few symbols in the Latin and Greek alphabets, so they have to rely heavily on subscripts. Expanding the set of symbols would be useful here, but I guess it's hard to get people to accept change unless they absolutely have to.
        • Jtsummers 9y ago
          A part of the problem is character entry. If it were easy to enter in all these symbols APL may have taken off and we wouldn't have gotten ASCII variants of it like J. So it's somewhat easier to just take our latin alphabet and turn a single symbol into a pair of letters or a letter with a subscript. The symbols in math are great for writing by hand, but can become cumbersome when typing.
• jmml97 9y ago
  I'm studying math right now and I have that problem. We're just being vomited theorems and propositions in class instead of making us think. There's not a single subject dedicated to learning the process of thinking in maths. So I think we're learning the wrong (the hard) way.
  • bikenaga 9y ago
    Many undergrad programs (in the U.S.) now have a course in math proof, which is intended to introduce students to proof techniques. It seems helpful to learn how to write proofs without the pressure of learning the content of a subject (like linear algebra, abstract algebra, and so on) at the same time.
• yellowapple 9y ago
  I wish school curricula would embrace that "learning how to think" bit. With the sole exception of Geometry, every single math class I took in middle and high school was an absolutely miserable time of rote memorization and soul-crushing "do this same problem 100 times" busy work. Geometry, meanwhile, taught me about proofs and theorems v. postulates and actually using logical reasoning. Unsurprisingly, Geometry was the one and only math class I ever actually enjoyed.
• jrells 9y ago
  I often worry that mathematics education is strongly supported on the grounds that it is about "learning how to think", yet the way it is executed rarely prioritizes this goal. What would it look like if math curriculum were redesigned to be super focused on "learning how to think"? Different, for sure.
• BrandiATMuhkuh 9y ago
  Disclaimer: I'm CTO of https://www.amy.ac https://www.amy.ac an online math tutor. From our experience most people struggle with math since they forgot/missed a curtain math skill they might have learned a year or two before. But most teaching methods only tell the students to practise more of the same. When looking at good tutors, we could see that a tutor observes a student and then teaches them the missing skill before they actually go to the problem the student wanted help with. That seems to be a usefull/working approach.
• J_Sherz 9y ago
  My problem with Math education was always that speed was an enormous factor in testing. You can methodically go through each question aiming for 100% accuracy and not finish the test paper, while other students can comfortably breeze through all the questions and get 80% accuracy but ultimately score higher on the test. This kind of penalizing for a lack of speed can lead to younger kids who are maximizing for grades to move away from Math for the wrong reasons. Source: I'm slow but good at Math and ended up dropping it as soon as I could because it would not get me the grades I needed to enter a top tier university.
  • MengerSponge 9y ago
    You couldn't finish your tests in high school, but you're good at math? Have you ever been evaluated for test anxiety? https://www.adaa.org/living-with-anxiety/children/test-anxiety https://www.adaa.org/living-with-anxiety/children/test-anxie... There's no shame in it (although high schoolers can be assholes), and it can help your school accommodate your needs eg extra time to take your exams.
    • J_Sherz 9y ago
      I've never been evaluated but I don't think test anxiety fits in my case - the speed issue was only ever in math / physics. My accuracy on the questions I got to was very high, I just couldn't go fast enough to complete enough questions. Same deal on SAT type math papers too.
      • alimw 9y ago
        Where I did my undergraduate degree they squared the marks scored for each question before adding them up! Slow but accurate was an advantage.
        • xfer 9y ago
          But, most exams at that level gives enough time to make them accurate.
      • deleted 9y ago
• Nihilartikel 9y ago
  This is something I've been pondering quite a bit recently. It is my firm belief that mathematical skill and general numeracy are actually a small subset of abstract thought. Am I wrong in thinking that school math is the closest to deliberate training in abstract reasoning that one would find in public education? Abstract reasoning, intuition, and creativity, to me, represent the underpinnings of software engineering, and really, most engineering and science, but are taught more by osmosis along side the unintuitive often boring mechanics of subjects. The difference between a good engineer of any sort and one that 'just knows the formulas' is the ability to fluently manipulate and reason with symbols and effects that don't necessarily have any relation or simple metaphor in the tangible world. And taking it further, creativity and intuition beyond dull calculation are the crucial art behind choosing the right hypothesis to investigate. Essentially, learning to 'see' in this non-spacial space of relations. When I'm doing system engineering work, I don't think in terms of X Gb/s throughput and Y FLOPS... (until later at least) but in my mind I have a model of the information and data structures clicking and buzzing, like watching the gears of a clock, and I sort of visualize working with this, playing with changes. It wouldn't surprise me of most knowledge workers arrive have similar mental models of their own. But what I have observed is that people who have trouble with mathematics or coding aren't primed at all to 'see' abstractions in their minds eye. This skill takes years to cultivate, but, it seems that its cultivation is left entirely to chance by orthodox STEM education. I was just thinking that this sort of thing could be approached a lot more deliberately and could yield very broad positive results in STEM teaching.
• yequalsx 9y ago
  I teach math at a community college. I've tried many times to teach my courses in such a way that understanding the concepts and thinking were the goals. Perhaps I'm jaded by the failures I encountered but students do not want to think. They want to see a set of problem types that need to be mimicked. In our lowest level course we teach beginning algebra. Almost everyone has an intuition that 2x + 3x should be 5x. It's very difficult to get them to understand that there is a rule for this that makes sense. And that it is the application of this rule that allows you to conclude that 2x + 3x is 5x. Furthermore, and here is the difficulty, that same rule is why 3x + a x is (3+a)x. I believe that for most people mathematics is just brainwashing via familiarity. Most people end up understanding math by collecting knowledge about problem types, tricks, and becoming situationally aware. Very few people actually discover a problem type on their own. Very few people are willing, or have been trained to be willing, to really contemplate a new problem type or situation. Math education in its practice has nothing to do with learning how to think. At least in my experience and as I understand what it means to learn how to think.
  • jaclaz 9y ago
    I believe that your "failures" are not necessarily linked to "math", in the sense that other teachers probably experienced the same frustration for their own taught subject. Math is definitely "harder" in the common thinking of people, but I suspect that there is often a generic underlying issue. A number of students (I would say a large part) go to school or college not "to learn" but rather to "pass the exam" or to "get the degree", they are not interested in the subject but in what the consequences of having a good grade in that subject are or - sometimes - only do whatever is needed to get a passing grade. I don't blame the students, mind you it is not their fault, it is society in general that somehow has shifted the values, and culture is not anymore valued.
  • joe_the_user 9y ago
    I've had the same experience - teaching math for a short time at a State University. I think the basic situation is that by the time students have reached the university level, the habit of looking for a set algorithm for completing a class is completely ingrained and moreover, the average student has little-to-no free time so they feel a panic concerning anything that will take an unpredictable amount of time.
• gusmd 9y ago
  I studied Mechanical Engineering, and it was my experience that several professors are only interested in having the students learn how to solve problems (which in the end boil down to math and applying equations), instead of actually learning the interesting and important concepts behind them. My wife went to school for Architecture, where she learned "basic" structural mechanics, and some Calculus, but still cannot explain to me in simple words what an integral or a derivative is. Not her fault at all: her Calculus professor had them calculate polynomial derivatives for 3 months, without ever making them understand the concept of "rate or change", or what "infinitesimal" means. For me that's a big failure of our current "science" education system: too much focus on stupid application of equations and formulas, and too little focus on actually comprehending the abstract concepts behind them.
  • acjohnson55 9y ago
    Nailed it. I remember "hitting the wall" in Honors Algebra II, when I just couldn't keep up with all the new things I was expected to compute: radicals, synthetic division, logarithms, etc. It was a major dent in my math confidence, which up until that point had be reinforced by being told I was "good at math". I didn't start regaining that confidence until Chemistry, when I learned what pH actually meant. That sparked the epiphany that scale was deeply important in nature, and with that motivation, logarithms seemed much simpler. We so quickly forget that most math wasn't discovered as some abstract exercise. But we still teach it that way, with the concepts before the applications.
    • kaitai 9y ago
      This is compounded in the weird American class called "precalculus," which is just a random grab bag of math crap that might maybe needed before calculus but might not. In most precalc classes I've encountered, not only is there no explanation of why the concepts will be used in calculus, but there is little attempt to relate the concepts covered in the class. Let's go from linear equations to complex numbers to polar coordinates! And let's not mention that complex numbers and polar coordinates are related! To top it all off, a lot of young college students taking precalculus are taking it as a terminal math class -- they'll never take calculus! It's just for a credit! So you've got some poor kid who hit that wall in high school and is repeating precalc and going through some unit on polar coordinates who will never take another math class again and will somewhat justifiably think this whole thing is stupid. In theory, I strongly support "math for liberal arts"-type classes instead in college. When done well they might cover voting systems and how you can prove there isn't a perfect one, gerrymandering and how you can use math to get several solutions, perspective drawing and its relation to projective geometry and Renaissance art, a bit of number theory and basic cryptography....
• g9yuayon 9y ago
  Is this a US thing? Why would people still think that math is about numbers? Math is about patterns, which got drilled into us by our teachers in primary school. I really don't understand how US education system can fuck up so badly on fundamental subject like math.
  • Jtsummers 9y ago
    It is a US thing. And it's because until middle school (age 11-13) your teacher is likely not that familiar with math (they have taken math courses, but likely little formal math beyond geometry and trigonometry, perhaps some courses on math pedagogy). So for them, math is all about the numbers, they barely got to the more abstract concepts. In middle school in the US you start getting teachers specialized in their teaching subject matter, but often tangentially. Any one with enough college math credits can teach middle school math (with some other certifications, usually). High school you may, finally, get a math teacher who is a mathematician. If you're lucky.
    • mamon 9y ago
      What kind of education is required from teachers in US? In Poland it is Bachelor's degree in relevant topic (for elementary and middle school) or Master's degree (for high schools) plus a course in pedagogy. Most teachers however get Master's degree even if not required (they get salary increase for that). It is hard for me to imagine math teacher not good in match themselves.
      • Jtsummers 9y ago
        For all grade levels a BS and certifications is all that's required in most of the US. Sometimes a specialized degree is required or helpful (this is regional) for certain areas. Like a degree in primary education may get you a job as a 1st grade teacher in some state where a degree in English wouldn't (because that degree, ostensibly, conveys the knowledge and ability to specifically help young students like identifying learning disabilities and modalities, or some roles bordering on social services like identifying psychological trauma). At the higher grades (starting at 6th grade or so in some areas, but more often just high school) a degree in the subject matter is preferred or required. But this can often be worked around by demonstrating capability (certification exams and classes) or having enough course work in the subject (like most STEM degree holders have enough math to qualify as a math educator).
• bitwize 9y ago
  Only really a problem in the USA. In civilized countries, there's no particular aversion to math or to disciplined thinking in general.
• jeffdavis 9y ago
  My theory is that math anxiety is really anxiety about a cold assessment. In other subjects you can rationalize to yourself in various ways: the teacher doesn't like me, or I got unlucky and they only asked the history questions I didn't know. But with math, no rationalization is possible. There's no hope the teacher will go easy on you, or be happy that you got the gist of the solution. Failure in math is often (but not always) a sign that education has failed in general. Teachers can be lazy or too nice and give good grades in art or history or reading to any student. But when the standardized math test comes around, there's no hiding from it (teacher or student).
  • mbizzle88 9y ago
    Marking a math test is only objective when an answer is entirely correct. What mark do you give a student who makes a minor arithmetic error on a single step of a multi-step problem?
    • chrisdbaldwin 9y ago
      0 pts - space shuttle exploded.
    • jeffdavis 9y ago
      Objective tests for math ability are pretty good. Arguing that they are only 99% objective doesn't really change my point. People aren't afraid of making tiny mistakes. They are afraid of looking at a problem similar to a dozen they saw in lecture; and having no idea what to do, and then being told under no incertain terms that they did not succeed.
  • findoer 9y ago
    Funnily enough, this goes away to some as you move along. Say I'm taking an exam and don't know how to do a proof. I pray and think that some equation is important, so I write it down with a little context. If I've done a great job up to that point, my professor will be more likely to think that I hit upon the key idea and give me 3/4 credit. If not, I'm probably more likely to just be marked wrong. You can also lose points for stylistic reasons if your professor is opinionated about them (-1 with "it is clear that" underlined and "don't write this ever again" in red ink :) ). Some professors will fail you if you get too technically correct with logical symbols everywhere instead of writing mathematical English. If you show too many steps you're wasting everyone's time; if you don't show enough or you don't seem confident you might get marked down. Etc
• EGreg 9y ago
  There just needs to be faster feedback than once a test. https://opinionator.blogs.nytimes.com/2011/04/21/teaching-math-advanced-discussion/ https://opinionator.blogs.nytimes.com/2011/04/21/teaching-ma...
• dbcurtis 9y ago
  Permit me to make a tangentially related comment of interest to parents reading this thread: This camp for 11-14 y/o kids: http://www.mathpath.org/ http://www.mathpath.org/ is absolutely excellent. My kid loved it so much they attended three years. Great faculty... John Conway, Francis Su, many others. If you have a math-loving kid of middle-school age, I encourage you to check it out.
• ouid 9y ago
  When people talk about the failure of mathematics education, we often talk about it in terms of the students inability to "think mathematically". It's impossible to tell if students are capable of thinking mathematically, however, because I have not met a single (non-mathlete) student who could give me the mathematical definition of... anything. How can we evaluate student's mathematical reasoning ability if they have zero mathematical objects about which to reason?
• archeantus 9y ago
  If we want to teach people how to think, I propose that math isn't the best way to do it. I can't tell you how many times I complained about how senseless math was. The real-world application is very limited, for the most part. Contrast that to if I had learned programming instead. Programming definitely teaches you how to think, but it also has immense value and definite real-world application.
• khana 9y ago
  Absolutely. Critical thinking has never been more important.
• cosinetau 9y ago
  As a someone with a degree in applied mathematics, I feel the problem with learning mathematics is more often than not a problem or a fault of the instructor of mathematics. Many instructors approach the subject with a very broad understanding of the subject, and it's very difficult (more difficult than math) to shake that understanding and abstract it to understandable chunks of knowledge or reasoning.
• pklausler 9y ago
  How do you "learn to think" without numbers? Depressing.
• CoolNickname 9y ago
  School is not about learning but learning how to think. The way it is now it's more about showing off than it is about anything actually useful. They don't reward effort, they reward talent.
• calebm 9y ago
  I agree, but have a small caveat: math does typically strongly involve numbers, so in a way, it is about numbers, though it's definitely not about just memorizing things or blindly applying formulas. It just bugs me sometimes when people make hyperbolic statements like that. I remember coworkers saying things like "software consulting isn't about programming". Yes it is! The primary skill involved is programming, even programming is not the ONLY required skill.
• humbleMouse 9y ago
  On a somewhat related tangent, I think about programming the same way. I always tell people programming and syntax are easy - it's learning to think in a systems and design mindset that is the hard part.
• alexandercrohde 9y ago
  Enough "I" statements already. It's ironic how many people seem to think their personal experience is somehow relevant on a post about "critical thinking." The ONLY sane way to answer these questions: - Does math increase critical thinking? - Does critical thinking lead to more career earnings/happiness/etc? - When does math education increase critical thinking most? - What kind of math education increases critical thinking? Is with a large-scale research study that defines an objective way to measure critical thinking and controls for relevant variables. Meaning you don't get an anecdotal opinion on the matter on your study-of-1 no-control-group no-objective-measure personal experience.
• k__ 9y ago
  I always had the feeling I failed to grasp math because I never got good at mid level things. It took me reeeally long to grasp things like linear algebra and calculus and I never was any good at it. It was a struggle to get my CS degree. Funny thing is, I'm really good at the low level elementary school stuff so most people think I'm good at math...
• jtreagan 9y ago
  You say "it's not about numbers, it's about learning how to think," but the truth is it's about both. Without the number skills and the memorization of all those number facts and formulas, a person is handicapped both in learning other subjects and skills and in succeeding and progressing in their work and daily life. The two concepts -- number skills and thinking skills -- go hand in hand. Thinking skills can't grow if the number skills aren't there as a foundation. That's what's wrong with the Common Core and all the other fads that are driving math education these days. They push thinking skills and shove a calculator at you for the number skills -- and you stall, crash and burn. The article brings out a good point about math anxiety. I have had to deal with it a lot in my years of teaching math. Sometimes my classroom has seemed so full of math anxiety that you could cut it with a butter knife. I read one comment that advocated starting our children out even earlier on learning these skills, but the truth is the root of math anxiety in most people lies in being forced to try to learn it at too early an age. Most children's brains are not cognitively developed enough in the early grades to learn the concepts we are pushing at them, so when a child finds failure at being asked to do something he/she is not capable of doing, anxiety results and eventually becomes habit, a part of their basic self-concept and personality. What we should instead do is delay starting school until age 8 or even 9. Some people don't develop cognitively until 12. Sweden recently raised their mandatory school age to 7 because of what the research has been telling us about this.
• brendan_a_b 9y ago
  My mind was blown when I came across this Github repo that demonstrates mathematical notation by showing comparisons with JavaScript code https://github.com/Jam3/math-as-code https://github.com/Jam3/math-as-code I think I often struggled or was intimidated by the syntax of math. I started web development after years of thinking I just wasn't a math person. When looking at this repo, I was surprised at how much more easily and naturally I was able to grasp concepts in code compared to being introduced to them in math classes.
  • sethammons 9y ago
    I found programming (started my first year in college) to be easily accessible, and quite a natural next step due to my success in math at the time. It is interesting how folks develop a mental model of a problem space and scaffold knowledge onto previous models to understand problems at hand. What do you feel made programming accessible when you started?
• quantum_state 9y ago
  Wow ... this blows me away ... in a few short hours, so many people chimed in sharing thoughts ... It is great ... Would like to share mine as well. Fundamentally, math to me is like a language. It's meant to help us to describe things a bit more quantitatively and to reason a bit more abstractly and consistently ... if it can be made mechanical and reduce the burden on one's brain, it would be ideal. Since it's like a language, as long as one knows the basics, such as some basic things of set theory, function, etc., one should be ready to explore the world with it. Math is often perceived as a set of concepts, theorems, rules, etc. But if one gets behind the scene to get to know some of the original stories of the things, it would become very nature. At some point, one would have one's mind liberated and start to use math or create math like we usually do with day to day languages such as English.
• WheelsAtLarge 9y ago
  True, Math is ultimately about how to think but students need to memorize and grasp the basics in addition to making sure that new material is truly understood. That's where things fall apart. We are bombarded with new concepts before we ultimately know how to use what we learned. How many people use imaginary numbers in their daily life? Need I say more? We don't communicate in Math jargon every day so it's ultimate a losing battle. We learn new concepts but we lose them since we don't use them. Additionally a large number of students get lost and frustrated and finally give up. Which ultimately makes math a poor method to teach thinking since only a few students can attain the ultimate benefits. Yes, Math is important, and needs to be taught, but if we want to use it as away to learn how to think there are better methods. Programming is a great way. Students can learn it in one semester and can use it for life and can also expand on what they already know. Also, exploring literature and discussing what the author tries to convey is a great way to learn how to think. All those hours in English class trying to interpret what the author meant was more about exploring your mind and your peer's thoughts than what the author actually meant. The author lost his sphere of influence once the book was publish. It's up to the readers of every generation to interpret the work. So literature is a very strong way to teach students how to think.
• gxs 9y ago
  Late to the party but wanted to share my experience. I was an Applied Math major at Berkely. Why? When I was in 7th grade, I had an old school Russian math teacher. She was tough, not one for niceties, but extremely fair. One day, being the typical smart ass that I was, I said, why the hell do I need to do this, I have 0 interest in Geometry. Her answer completely changed my outlook and eventually was the reason why I took extensive math in HS and majored in math in college. Instead of dismissing me, instead of just telling me to shut up and sit down, she explained things to me very calmly. She said doing math beyond improving your math skills improves your reasoning ability. It's a workout for your brain and helps develop your logical thinking. Studying it now at a young age will help it become part of your intuition so that in the future you can reason about complex topics that require more than a moment's thoughts. She really reached me on that day, took me a while to realize it. Wish I could have said thank you. Wherever you are Ms. Zavesova, thank you. Other beneits: doing hard math really builds up your tolerance for building hard problems. Reasoning through long problems, trying and failing, really requires a certain kind of stamina. My major definitely gave me this. I am a product manager now and while I don't code, I have an extremely easy time working with engineers to get stuff done.
• monic_binomial 9y ago
  I was a math teacher for 10 years. I had to give it up when I came to realize that "how to think" is about 90% biological and strongly correlated to what we measure with IQ tests. This may be grave heresy in the Temple of Tabula Rasa where most education policy is concocted, but nonetheless every teacher I ever knew was ultimately forced to chose between teaching real math class with a ~30% pass rate or a watered-down math Kabuki show with a pass rate just high enough to keep their admins' complaints to a low grumble. In the end we teachers would all go about loudly professing to each other that "It's not about numbers, it's about learning how to think" in a desperate bid to quash our private suspicions that there's actually precious little that can be done to teach "how to think."
  • jacobolus 9y ago
    Unfortunately your realization is empirically unsound, and the problems of “every teacher you know” have as much to do with lack of available expert teacher attention per student and standard expected teaching styles and school structure with substantial friction/latency in feedback and huge amounts of wasted time as with inherent student ability. Start here http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf but then there is a vast literature (thousands of studies and other research papers) exploring the general topic of math pedagogy. Some approaches and some teachers are radically more successful than others, given comparable students. If you hand me one of your 20th percentile students who is struggling but willing to learn, and give us the same amount of time which would otherwise be spent in a classroom for a year of one-on-one face-to-face time, with the student also spending a typical amount of time working independently, I can have them outperforming your 80th percentile students by the end of the year, without issue. The difficulty of course is that direct mentoring by an expert tutor is too expensive for society to be willing to pay for at scale. The main reasons that your teacher friends are stuck is because (a) many if not most of their students are unprepared before they arrive in any particular course, which is largely down to structural social factors and school scheduling inflexibility, and (b) the combination of lectures and independent work with inadequate feedback are for the average student a terribly inefficient and ineffective way to learn. Neither of those has all too much to do with biological predestination or whatever.
• lucidguppy 9y ago
  Why aren't people taught how to think explicitly? The Greeks and the Romans thought it was a good idea.
• Mz 9y ago
  Well, I actually liked math and took kind of a lot of it in K-12. I was in my 30s before I knew there were actual applications for some of the things I memorized my way through without really understanding. When I homeschooled my sons, I knew this approach would not work. My oldest has trouble with numbers, but he got a solid education in the concepts. He has a better grasp of things like GIGO than most folks. We also pursued a stats track (at their choice) rather than an algebra-geometry-trig track. Stats is much more relevant to life for most people most of the time and there are very user-friendly books on the topic, like "How to lie with statistics." If you are struggling with this stuff, I highly recommend pursuing something like that.
• mindcrime 9y ago
  This part really resonates with me as well: "You read all the time, right? We constantly have to read. If you're not someone who picks up a book, you have to read menus, you've got to read traffic signs, you've got to read instructions, you've got to read subtitles -- all sorts of things. But how often do you have to do any sort of complicated problem-solving with mathematics? The average person, not too often." From this, two deductions: • Having trouble remembering the quadratic equation formula doesn't mean you're not a "numbers-person." • To remember your math skills, use them more often. What I remember from high-school and college was this: I'd take a given math class (say, Algebra I) and learn it reasonably well. Then, summer vacation hits. Next term, taking Algebra II, all the Algebra I stuff is forgotten because, well, who uses Algebra I over their summer vacation? Now, Algebra II is harder than it should be because it builds on the previous stuff. Lather, rinse, repeat. This is one reason I love Khan Academy so much. You can just pop over there anytime and spend a few minutes going back over stuff at any level, from basic freaking fractions, up through Calculus and Linear Algebra.
  • sdrothrock 9y ago
    I wanted to quote this passage for another reason -- it seems like you're agreeing with it, but I completely disagree with it, especially this: > But how often do you have to do any sort of complicated problem-solving with mathematics? There are a LOT of chances to do that kind of complicated problem solving, especially if you're shopping or comparison shopping on your own. It's not that people don't have the chances, it's that people avoid the work involved in doing those kinds of comparisons.
    • mindcrime 9y ago
      I mean, in a certain sense I agree with you. There are chances to do that stuff, yes. To a point. Basic algebra could come up in a shopping scenario for example. But even then, people aren't going to be doing that stuff every day, or even close to it. And when you get even slightly more esoteric, like, say, exponents... how often are people (especially kids) really going to be using exponents, or logarithms, or the quadratic formula, etc. in their daily lives? Sure, if you go out of your way to actively look for reasons to find that stuff, you can find them. But my point is that when you're a kid learning math, it's summer vacation and you're busy playing with your friends, you aren't out actively looking for reasons to apply the quadratic formula, etc. At least not for most people, from what I've seen.
• alistproducer2 9y ago
  I can't agree more. Math is about intuition of what the symbols are doing. In the case of functions, intuition about how the symbols are transforming the input. I've always thought I was "bad at math." It wasn't until my late 20's when I took it upon myself to get better at calculus and I used "Calculus Success in 20 Minute a Day[0]" did I finally realize why I was "bad" at it; I never understood what I was doing. That series of book really put intuition at the forefront. I began to realize that the crazy symbols and formulas were stand-in for living, breathing dynamic systems: number transformers. Each formula and symbol represented an action. Once I understood Math as a way to encode useful number transformation, it all clicked. Those rules and functions were encoded after a person came up with something they wanted to do. The formula or function is merely a compact way of describing this dynamic system to other people. The irony was I always thought math was boring. In retrospect it was because it was taught as if it had no purpose other than to provide useless mental exercise. Once I started realizing that derivatives are used all around me to do cool shit, I was inspired to learn how they worked because I wanted to use them to do cool shit too. I went through several years of math courses and none of them even attempted to tell me that math was just a way to represent cool real world things. It took a $10 used book from amazon to do that. Ain't life grand? [0]:https://www.amazon.com/Calculus-Success-20-Minutes-Day/dp/1576858898 https://www.amazon.com/Calculus-Success-20-Minutes-Day/dp/15...
  • andrepd 9y ago
    But math isn't about intuition, much to the contrary, it's about rigorous reasoning. In math "intuition" and "common sense" are worth nothing. Only objective proof matters. It's a valuable lesson too, learning not to take anything on a hunch and only take as true what what you carefully and logically prove.
    • ice109 9y ago
      you use intuition to come up with the argument. then you use a proof to convince others.
      • andrepd 9y ago
• katdev 9y ago
  You know what helps kids (and adults) learn math? The abacus/soroban. Yes, automaticity with math facts/basic math is important but what's really important is being able to represent the base-10 system mentally. The abacus is an amazing tool that's been successful in creating math savants - here's the world champion adding 10 four-digit numbers in 1.7 seconds using mental math https://www.theguardian.com/science/alexs-adventures-in-numberland/2012/oct/29/mathematics https://www.theguardian.com/science/alexs-adventures-in-numb... Students are actually taught how to think of numbers in groups of tens, fives, ones in Common Core math -- however, most are not given the abacus as a tool/manipulative.
• deleted 9y ago
  [deleted]
• andyjohnson0 9y ago
  A couple of years ago I did the Introduction to Mathematical Thinking course on Coursera [1]. Even though I found it hard, I enjoyed it and learned a lot, and I feel I got some insight into mathematical though processes. Recommended. [1] https://www.coursera.org/learn/mathematical-thinking https://www.coursera.org/learn/mathematical-thinking
• tnone 9y ago
  Is there any other subject that is given as much leeway for its abysmal pedagogical failures? "Economics, it's not about learning how money and markets work, it's about learning how to think." "Art, it's not about learning about aesthetics, style, or technique, it's about learning how to think." "French, it's not about learning how to speak another language, it's..." Math has a problem, and it's because the math curriculum is a pile of dull, abstract cart-before-the-horse idiocy posing as discipline.
• GarvielLoken 9y ago
  tl;dr A couple of numbers-nerds are sad and offended that math is not as recognized as reading and literature, where there are great works that speaks of the human condition and illustrates life. Also they have the mandatory "everything is really math! ™". "LeGrand notes that dancing and music are mathematics in motion. So ... dance, play an instrument." Just because i can describe history through the perspective of capitalism or Marx theories, does not make history the same thing as either of those.
• bojo 9y ago
  When I first saw it I thought the sign in the mentioned tweet may have been because the deli was next to a mathematics department and the professors/students would stand around and hold up the line while discussing math. Overactive imagination I guess.
• Tommyixi 9y ago
  For me, math has always been a source of unplugging. I'd sit at my kitchen table, put in some headphones, and just get lost in endless math problems. Interestingly, now as a masters student in a statistics graduate program, I've learned that I don't like "doing" math but get enjoyment from teaching it. I really like it when students challenge me when I'm at the chalkboard and I'll do anything for those "ah-ha!" moments. The best is at the end of the semester hearing students say "I thought this class was going to suck but I worked hard and am proud of the work I did." I'm hoping that on some small scale I'm shaping their views on math. Or at least give them the confidence to say, "I don't get this, but I'm not afraid to learn it."
• djohnston 9y ago
  Anecdotally, I was a pretty average math student growing up and a pretty good math student in university. One of the reasons I studied math in college was to improve what was objectively my weakest area intellectually, but I found that once we were working with much more abstract models and theories, I was more competent.
• dorianm 9y ago
  Maths problems are cool too, like counting apples and oranges :) (Or gold and rubies)
• kchr 9y ago
  Dammit, I glanced at the domain and expected a statement from the gangsta rap group... School is cool, kids!