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Comment by falcolas | Hacker News Reader

I'm 36, and math in my youth was always about getting the right answer, and proving how I got that answer, even up through college calculus and discrete mathematics. Heck, that's one of the things I really enjoyed about math, when compared to, say, English.

So, if math isn't about getting the right answer, it's been broken a long time.

Even today, math as I've explored it (which admittedly hasn't been much) has right and wrong answers. Sure, there's a bit of fuzz in the answers now, but a probabilistic model which can't determine ham from spam some majority of the time isn't kinda right, or even on the spectrum of right. It's just wrong, and needs to be fixed.

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When I was a kid I used to love solving the same problem in multiple ways. I remember during a written test in high school, I solved some pre-calculus problems in 4 or 5 ways within the allotted time. The coolest part for me was seeing that, no matter the approach, they all led to the same answer. I didn't need to wait for my grades back from the teacher to know if my calculations were right. I got confirmation by simply comparing the answers that each method I used provided. I admit that at times periods in Trigonometry provided a two second scare when the results were equivalent but appeared different on paper. Once I had to get my teacher to change my grade precisely for this reason.

Anyway, getting the answer right and seeing which different angles I could take to get to those results were entirely part of the reason why I loved math. The answer wasn't per se the number at the end. The answer was, for each method I employed, the right sequence of logical and mathematically sound steps to that final value.

I would have never really been so fascinated by mathematics and related disciplines if it weren't for that sense of absolute, meritocratic, objective wrong or right.

Weird, I'm 26 and in college math was all about 'how you got there'. Our tests were like essays, a single problem in either paragraph form or formula form, and several pages to write down how you achieved your goal. Following proper procedure was rewarded, even if the answer is wrong.

In high school, however, my school implemented a 'pass the test first' policy. We learned what the Regents (spelling on this) exam would have and learned only that. Still, most of my teachers always graded the procedure and not the answers (in fact, my teachers suggested we look at the answers afterwards to compare if our answer is right).

Even if you have to show your work, a lot of students will just blindly try a few canned approaches until they get something that looks about right. Actually understanding what you're doing is definitely more efficient, but a lot of people muddle through without really understanding the reasoning.

One of the worst examples of that mentality I saw in a first year university course was a multi-part question where one part had the student calculate something and the next part had the student do something to sanity check the answer. One student got the first part wrong and then fudged the working in the second part to get it to say that the first answer was correct. Of course, this completely misses the point of checking the answer. Worse, they came up to me after class and argued that they deserved full points because the second error was "a consequence of" the first one. They seemed totally perplexed that I didn't agree.

> So, if math isn't about getting the right answer, it's been broken a long time.

There is a problem where children just learn algorithms. But they don't really understand numbers or how this algorithm works. And the bang numbers in and hope for the best.

Show them "speed = distance / time", and give them distances and times and they're fine. Now given them some speeds and times and ask them to find distance and they're stumped.

Math should be teaching children how to think, not just how to put numbers into a calculator.

Math has right answers, but there isn't anyone to tell you what they are. Most of the time you're doing math for real, when you get an answer the next step can't be to check the authoritative answer. The next step has to be something like checking with additional [mathematical] reasoning, running an experiment, (if you're doing math about the world), or consulting with your peers.

"...and proving how I got that answer..."

That's the part that counts, and is largely missing today.

I am not so sure you are talking about about the same thing as the article. Usually, when "proving your answer," I found there was still an expected procedure for you to follow, and you would lose points for deviating from that procedure even if what you did was mathematically correct. What the article talks about is learning how to reason about a problem even if your reasoning is questionable and even if you don't arrive at an answer.

That is unbelievably terrible. You would lose points for a mathematically sound justification?!?!

For example, when solving quadratic equations, I often used trial substitutions and the fact quadratic equations have at most two solutions to justify answers, instead of using quadratic formula. Why wouldn't this get the full mark? (I would actually argue this is a better procedure if you are looking for integer-only solutions, for example.)

What do you mean by "trial solutions"? Were you just substituting numbers for x and seeing if they worked out? If that is what you mean I can't see why it would be a better procedure than using the quadratic formula.

Yes, I meant substituting numbers for x and seeing if they worked out. To get integer solutions, it's enough to try divisors of the constant term. Try it yourself. It's faster.

Yes, it is faster for small integer values. Most numbers aren't small integer values. Test questions are usually designed to give nice clean answers but the real world isn't like that. They were trying to test your ability to solve quadratic equations in the general case. Your approach doesn't demonstrate that ability.

I'm not at all convinced that has changed.

I've read the Common Core (and the Australian equivalent) and they certainly require proving how answers were obtained.

When I was a kid, each subject had its own attractions. The notion of a "right" answer was one of the things that I enjoyed about math. It didn't make me any less inquisitive. I often tried to come up with unorthodox ways of arriving at those answers.

Naturally, there's a side of math involving getting an approximate answer that's good enough to be useful. One thing I've noticed as my kids progress through school is that most science being taught is qualitative or descriptive.

Adding some quantitative science to the K-12 curriculum would be a way to teach a more fuzzy approach to math. Naturally, science has its own notion of "right" that could be learned through the formation and testing of quantitative hypotheses.

There were many courses where I could get an "A" by writing a mountain of drivel that anticipated the teacher's social biases. I was thankful for the boost to my GPA, but the "no right answer" aspects of those subjects frustrated me. Oh well, different strokes.

One thing the standardized tests might be doing is to focus too much on memorized algorithms. And it may not be necessary. My son had a teacher in grade school who didn't even teach the standard algorithms for things like two column addition. Instead, the kids spend time trying to come up with their own methods. Yet his students get excellent scores on the state tests.

> math as I've explored it (which admittedly hasn't been much) has right and wrong answers

Try spending some time digging into the foundations of set theory. A good starting point is looking into the Axiom of Choice and the Continuum Hypothesis: are they true or false? Is there a "right" answer to those questions?

Proofs don't have a single correct answer they have a range of acceptable solutions. Any time your told to show your work there tends to be a multitude of acceptable answers.

I think he likes math because it's consistent. E.g. if I prove that some natural number "x" is prime in decimal, then "x" is also prime in binary, ternary, octal, etc. Counting "x" in a different base doesn't change the symmetry. Likewise, arriving at a theorem via different proofs just describe the same mathematical object in different ways.

That's not the point: they have a correct sequence of steps.

Hardly any math has a correct sequence of steps. (Unless you meant a sequence of correct steps?)

But you are aware that tests on math, particularly for school kids, generally do have a correct sequence of steps, right? The point isn't the answer, but rather to demonstrate mastery of a particular technique. In this case, the test should present problems where one particular set of steps is arguably the best or most obvious one. If the student uses a different set of steps but gets the correct answer, then perhaps the question needs to be asked in a different way (e.g. "Use Euler's Method to determine...").

Not really. Consider the following question, for example (taking from an actual eighth grade math test):

"Combine the equation 8x+y/2 = 1/24 with a second equation so that the resulting system of linear equations has (a) exactly one (b) zero solutions. Justify your answer."

That's leaving aside the fact that there are multiple ways to solve a system of linear equations with two variables to begin with (since you don't even need to solve such a system to answer the question).

But if you're teaching a particular technique, then the test would not be whether you can answer the problem but whether you understand how to use the technique.

And that, in a nutshell, is what's wrong with American math education. Too much focus on rote learning rather than actual understanding. What you're describing is an accounting course, not mathematics. Mathematics is about problem solving, not plugging numbers into a black box and blindly following a prescribed sequence of algorithmic steps. You do not want to just test whether students can follow a procedure; you want to test whether they actually know what they're doing when they're using it (even if you specify the procedure, e.g. Gaussian elimination).

This even applies to simple problems. See the comment by Aur Saraf responding to the original article: http://powersfulmath.wordpress.com/2014/04/30/who-or-what-br...

How is a student going to do any math if you never teach techniques? And how are you going to know if the student actually learned it if you don't test them? I never said they should be the focus, but you need to test them sometimes.

Also, algorithmic thinking is one of the more powerful ideas we have right now. You can trust that a Turing machine has certain properties without being a genius yourself, for example.

Take something real simple "Solve X + Y = 2; X + 2Y = 4 using substitution."

At the basic level that has 4 choices, you can substitute X or Y in the top or bottom equation.

WTF? A proof certainly does have a correct sequence of steps. You can't just reorder the steps in a derivation for example.

What kind of proof? Here are 102 proofs of the Pythagorean Theorem. http://www.cut-the-knot.org/pythagoras/index.shtml

Pretty much all of them?

Take the Euclidian proof (#7). The order of the reasoning is important, because each step builds upon what was proved in the previous step.

You can think of proofs as chains of logic. Assuming A, B, and C.

  (A+B) -> X, (A+C)-> Y
  (X+Y) -> Z

Now you need to prove both X and Y before Z. But, you could prove Y before X so It's not a strict question of order but one of dependence.

Going back to the proof 7 the line For the same reason also means the order is not strict and you could swap those lines without changing the logic.

Till my senior year of HS, I thought math was easy, but not very interesting. This was basically for the reason you describe -- there's one obvious right answer, and a set of steps you can mechanically apply to reliably get there. When I finally reached the creative math classes[0], I was surprised to find that math could be so engaging (and difficult relative to all the previous courses). All math may ultimately have a right answer, but there's much more flexibility and even subjectivity as you progress

That's also a big part of why I got into Computer Science. Computer Science[1] courses are basically a flavor of the interesting kind of math. I really enjoyed working through algorithmic and computability problems because it wasn't always obvious how I should approach them or if I'd even be able to figure them out.

I think it's sad the vast majority of students will never even reach a math course where they aren't performing calculations that could trivially programmed into a computer. It seems like there should be a way to give students a taste of that earlier on, but I'm not sure what the best approach would be.

0: This kinda started with integrals in Calc II, but didn't get really interesting till the my first 3000-level University course on proof by induction and the like. 1: Not programming or engineering, although those can be fun in their own ways.

Math is getting the right answer. Statistics and probability is not exactly math. They deal with chances. So she is wording it in wrong way.

Also, I should note that at METU no mathematician major student was able to graduate from statistics department as minor. They all gave up. That's what our professors told us. That is why math and statistics are very different subjects. It is like math and chemistry or math and computer engineering. Statistics do not deals with math, it has its own separate topic.

Practical applications of math are about getting answers, and unless you do math at university, that's probably all you'll ever do. But the problem is that if you don't learn to understand how the math works and why it gets you correct answers, you're unlikely to spot mistakes and likely to accept misleading reasoning and calculations at face value.

For me, practical applications of math are about understanding how a system works. More often than not, it really doesn't matter what the exact number is that comes out the end (the precision is often beyond my ability to act), but rather the understanding that well, if I take this action, these other things are going to happen at about this magnitude, so I need to offset them by doing something else, etc. etc.

From what I can tell, the author of the article wanted her students to engage in philosophy rather than mathematics, but maybe didn't make this clear to them.

Mathematics is indeed all about rigor, proof, and finding "correct answers". Philosophy, on the other hand, takes a more lenient approach, with emphasis on the process of thinking, rather than the final result.

Perhaps the children in this class merely thought that a more mathematics-oriented approach was what the teacher desired. That's reasonable, if that's how their previous classes had been done.

Apparently the students themselves adjusted quickly enough, once the teacher made her expectations clear. As the article states, "Things went so much better the second time around with not one student asking me if their answer was “right”".

So this could very well come down to a simple misunderstanding between the teacher and her students, in this rather isolated case.

I disagree with your claim that what the teacher wanted the students to do was not really math. Instead she was claiming that they (and apparently you) have a narrow conception of what doing math involves.

The "right answer" consisting of a valid proof is very different from the "right answer" to a numerical computation. The thinking and communication process is surely more relevant to the former than the latter. And if anything, as you say, the former is what math is really about.

Yes, "mathematics" is a relatively narrow concept, even if it embodies a great deal of knowledge.

When it comes to mathematics, the exact form of the outcome isn't very important. Maybe it's a valid proof in one case, maybe it's a specific number in another case.

It's the degree of rigor that's important. Rigor is what separates mathematics from other fields of study.

Thinking and reasoning about abstract concepts aren't without value, of course. But if this is done without an extreme degree of rigor, it probably should not be considered to be mathematics.

Something tells me you've never sat in on an algebraic topology class where the teacher draws a donut and a pair of pants on the board, along with a line or two, and calls it a proof. A key characteristic of mathematics is that it can be made rigorous, but we certainly don't work in full rigor day to day. A look at Principia Mathematica should demonstrate simply why that would be infeasible.

You just blew my mind. I now want to learn everything I can about algebraic topology.

Q: How can you tell there is an topologist in the cafeteria?

A: Because they can't tell their coffee cup and donut apart.

I up voted this because it made me laugh, even though I don't understand.

Fundamentally they both have only one hole through them, so under the right sort of deformations you can turn one shape into the other: They have the same topology.

As a practicing mathematician, I disagree completely. Much mathematics is done with the understanding that the ideas can be made rigorous, and the point of educating people in mathematics is to allow them to craft arguments and revise when they find holes. One can and should introduce rigor gradually over time, and get the benefits of learning to reason and doing mathematics.

When it comes to mathematics, the exact form of the outcome is extremely important. Proofs that are aesthetically pleasing are better than proofs that are not. Proofs that glean insight are better than proofs that do not. In fact, most people don't care about whether a theorem is true unless the proof provides sufficient insights.

If you want mathematics to be it's own little walled garden free from ambiguity, that's fine. But don't you think it's kind of important that people be able to translate between problems in the messy non-mathematical universe and mathematics.

I think the actual issue is their approach to learning, rather than their approach to math. The exercise with probability was meant to help them understand what probabilities are, so the process was more important than the result in that case. When learning math, it's often more important to make sure you're taking the right path toward the answer. And there's also a place for exploratory activities like this, where you figure out what your assumptions are before worrying about whether they're right.

You are right.

This is just another Common Core hit piece. More subtle than most I've seen, but the story only has one point: Common Core is bad, because standardized testing of students promotes rote learning in schools. I don't agree with this, but I hope it helps you understand the article.

It is absolutely not a Common Core hit piece! The teacher talks about how providing context (teaching!) allowed Common Core material to provide a very worthwhile lesson.

"If we are to truly make progress in getting our students to understand the concepts presented in the Common Core to the depth intended we must help them learn to stop looking for a right answer and start looking for a right reason."

Talking to a lot of teachers it is often not the provided Core material that is the issue, but not having enough time to actually provide the structure and context to help the kids internalize why any of it is important. It's also not a surprise that is the case, given that at least where I live (California) the school year is measurably shorter and class sizes are measurably bigger than they were when I was in the same grades.

Oops, I missed the paragraph under the last picture. I know a few educators at a private school, and they said the hardest part is proving that the classes they already teach cover the Common Core requirements. Admittedly, they have much smaller classes than public schools.

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