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Comment by analog31 | Hacker News Reader

This seems interesting because it addresses the issue of "show your work." Many years ago, I spent a semester teaching the freshman algebra course at the nearby Big 10 university. This is the course that you take if you don't get into calculus. My students were bright kids -- they were all admitted to the state flagship school -- but not mathematicians.

There was huge variation in the preparation that kids brought with them from high school. In particular, very few of them understood what "show your work" means. They were told "show your work," but nobody told them what it really entails. Is it just to provide evidence that you did some work, to deter cheating, or is it something else? Many of my students were taught "test taking skills" such as the guess-and-try method. So on one exam, a question was:

x^3 = 27

One student's work:

1^3 = 1

2^3 = 8

3^3 = 27

Answer = 3

I asked the professors to tell me what "show your work" means. None of them had a good answer! These were the top mathematicians in the world. I wanted to talk with my students about it, but I'm not even sure that my own answer was very good.

But if we did well in math, then we just know what it means. It's not just evidence that you did the work. It doesn't mean "turn in all of your chicken scratch along with the answers." It means something along the lines of supplying a step-by-step argument, identifying the premises and connecting them with the conclusion, in a language that is "accepted," i.e., that mimics the language of the textbook / teacher. In fact, the reason to read the textbook and attend lectures, is to learn that language. (It's not so different in the humanities courses).

At least, that's my take on it, as just one teacher with one semester's worth of experience.

In my view, a problem solving tool that actually addresses the process of building the argument and not just determining the answer, would be beneficial to students.

View on HN

Show your work means that the process for doing the problem works in a sufficiently generic situation. The student's methodology would not work for solving the equation:

x^3 = 7

Hence I would mark the student's answer wrong. At least at the level of a college algebra course. Guess and try is fine for getting an intuition for a problem but it is not fine as a solution.

EDIT: I don't care about down votes but I'm interested why there is disagreement with what I wrote. I've been teaching mathematics at the college level for 20 years and I believe my answer above fits with what most college level teachers think about the topic.

So, why do you disagree with what I wrote?

Note that I didn't downvote you. I actually consider the answer wrong too, and the professors agreed too. One of them said: "Obviously the student doesn't understand the problem."

I thought to myself: "You understand the problem because you know the answer." But I held my tongue.

But at the present state of the art, the student has no idea why the answer is wrong. That's where I think their high school math background failed them.

The danger of always using nice numbers! I've taught topics where I thought I had the best lecture, the best examples and that it would be impossible for a student to misinterpret the method. Some still did come up with flawed thinking. It's hard.

> The student's methodology would not work for solving the equation: x^3 = 7

It would if the results they reported were the initial steps of a binary search which got very lucky.

I HATED "show your work". You want a proof, ask me for a proof. It's not my fault I can solve your problem in my head. Especially horrifying is if you want me to solve the problem a particular way. There is no better way of turning the fun and freedom of math into drudgework. You are licensed to ask me to arrive at the correct answer, or to prove a theorem from premises; telling me how to do it is mathematical abuse.

The answer is not important. What is important is that one knows a process for solving the general situation. Knowing how to get the answer in a problem with easy numbers is not the point. Demonstrating how to find it no matter what numbers I used is the point.

There is also the fact that I'm not just teaching the student so they can pass this class. I'm teaching them so that they can pass the next class too. In beginning algebra, our lowest level course, we start with basic problems.

For instance, you drive for 3 hours and travel 150 miles. What's your average speed. Almost every person gets this right. But many can't do this problem. You drive a car for 4/3 an hour and travel 100/6 miles. What is your average speed? Now many can't do this problem. They ask, which way does the division occur?

Our goal is that they know and understand a general process. We are setting them up to solve more advanced problems. Problems that can't be done in their head. If you can't write the steps out in the simple case you'll never understand the harder problems. Problems that involve quadratic functions or trig functions.

Mathematics is a human activity and communication is part of it. Knowing how to communicate what is in your brain to another person in a way that they can understand is very valuable. I give the students nice numbers and in exchange they are expected to tell me a general way of solving that type of problem. Instead of feeding them for a day I want to teach them how to fish.

I strongly disagree with your belief that this mathematical abuse.

Interestingly, I think many of your points would seem obvious if couched in computer programming terms. For instance, how do you write a program that's correct, but that another human can understand and adapt? What happens when a program gets too big to understand all at once in your head? How do you know that you've found all of the possible outputs?

> Knowing how to get the answer in a problem with easy numbers is not the point.

So, don't use easy numbers?

> I give the students nice numbers and in exchange they are expected to tell me a general way of solving that type of problem.

If this quid-pro-quo is explicit, that seems fair. But I still don't see why "nice numbers" are necessary.

In this case nice numbers are not necessary. In other types of problems they are. One does not want to burden students with needlessly complicated calculations when the main point is a simple one.

I should point out that I'm not licensed by the state or anyone else. At the college level we don't have licenses. My contract states that instructors are solely responsible for grading and assigning grades. In the system that I teach in it is my prerogative, more or less, to grade as I see fit. This is where experience and professional judgment come into play.

It is your fault if you can't explain how you solved the problem; anybody can look over someone's shoulder and copy the final answer. Producing the steps allows the teacher to verify that most likely you did the problem yourself.

In first or second grade, I believe I was asked to "show my work" for "2 x 3 = ?". I believe the teacher didn't accept "But I don't have any work to show" as an answer. I don't think most adults would have any work to show either, nor should they—if they actually repeatedly add 2 to itself, or do anything other than a memory lookup, then there's probably something wrong with their prior math education.

In general, if the step that I took to solve a problem was "I looked up a memorized fact in my brain", then that is the truth, and writing down anything else to "show my work" is a lie.

As other people have said, a proper way to ask that question is "Show a derivation of the answer to [problem]." Bonus points if you specify what assumptions they're allowed (e.g., in the above example, "addition").

I don't necessarily disagree with you, but imo it definitely matters how the problem is worded and presented, or if it is otherwise made clear what type of work will be accepted. Silly examples: suppose someone disproves the Riemann hypothesis through a computer search which finds a zero-crossing counter-example. Their counterexample is not a 'wrong' answer, even though they have essentially just done a more complicated version of 1^3 = 1, 2^3 = 8, ... and merely providing whatever computer code they used would probably be accepted by anyone. Example in the other direction might be the original four-color-theorem proof, which was partially by exhaustive computer aided search.

This is where complexity theory comes in; you want a proof that is checkable in polynomial time; listing examples one by one to prove something is not such a proof. If instead the question asked, solve x^3 = 8192; a proof iterating from 0 uptil the cube root of 8192 wouldn't work, because that's exponential in the problem size.

Brute force search to get a counter example to the Reimann hypothesis produces a solution that is checkable in polynomial time; you just evaluate the Reimann-zeta function at the produced point.

I think, in the case of an algebra class, the student could assume something like an integral / rational solution space. Otherwise, the solution is just to put the opposite symbol on the other side of the equation, e.g. (8192)^(1/3), which is maybe(?) what the teacher is asking, however would the teacher actually accept an answer written in the form 27^(1/3)? I would guess that I would not accept that. I understand what you are saying, but in this case, it could be argued that the 8192 case is actually a different question, because the underlying multiplicative group is different.

Why does it matter the complexity of the solution check?

If for some reason the counter example of the Reimann hyp would be checkable in exponential time (whatever that means), and if it would have required 10 years of computer time to check, how would that matter? The solution would still stand (assuming that everybody agrees that no mistake was made).

The question applies to the classroom setting we were talking about; teachers don't have access to supercomputers :P

To expand on my previous response: you want a proof that is checkable in polynomial time; yes - completely agree. I was just playing the devil's advocate case, that maybe integer solutions could be assumed (if it was a low-level class, for example).

Do you expect three solutions, or just the real one? What about for x^2=-1, two or none? Is \sqrt[3]{7} enough, or is a decimal approximation required? If it isn't (and the student doesn't know how to go about it) one could argue that isn't much of an answer, but rather an existence claim, and that the methodology doesn't amount to much.

Personally I feel that you probably wouldn't give a question such as x^3=27 (neither would I), but if you did, marking it as wrong (as in no credit) after seeing the justification 1^3=1, 2^3=8, 3^3=27 would be too harsh. You can't penalize a student for giving out an easy question.

Cube root of 7 is the right answer. It doesn't have a finite decimal or repeating decimal answer so a decimal answer is wrong. But I wouldn't take off points for giving a decimal approximation as the answer.

In college algebra, for most sections, we deal with real number solutions. They haven't reached the point of knowing about non-real solutions. We teach at the level the students are at. Without having had trig finding the roots of unity is hard and not comprehensible to the students so asking them for all three solutions is a bit much.

I would not give a student in college algebra credit for solving x^3 = 27 by guess and check. It demonstrates that they really don't understand what is going on. I give credit for demonstrating understanding. Not demonstrating that they are good guessers.

You have to be very cautious when marking wrong an answer that is correct, because you're at risk of confusing the student. The guess-and-try approach used in this particular example is very brittle, but it is correct. Professional mathematicians use it frequently, especially with the advent of computer-aided mathematics, for instance to find counter-examples to statements.

I see at least three issues with claiming that answer as wrong: first, correctness is essential (in the true sense) in mathematics, and therefore should not be carelessly dismissed in front of the student. Second, students should not be made to believe that guess-and-try is always inappropriate, but rather to understand that it won't always work. Finally, in this particular example the approach chosen is arguably (at least from the student's perspective) simpler than the one expected by the professor. Invalidating a "simpler" approach might give the student the impression that you always need to take the complicated route (ie, "math is hard") when the opposite is true.

My own take on this example would be to give (partial?) marks, with a lengthy comment of the type "fair enough, in this case, but what about if you wanted to solve x^3 =7? Your method wouldn't work, then!". Alternatively, if you don't want to give marks, it should be justified at length by rules clearly explained before the exam, while acknowledging the correctness of the approach.

I think the simplest approach is to use the cube root function. This is the simplest approach because it solves, over the reals, any equation of the form

x^3 = real number

That's the simplest solution. It works in every case. To me the answer is not important. The methodology is important. Giving a counter example is very much a different type of problem. Just about any method is valid in that type of problem.

Passing a class should mean more than I got a lot of answers correct. It should mean an understanding of the material. A college algebra student who solves x^3=27 in the aforementioned manner is lacking a fundamental understanding of the material. Now a third grader who reasons thusly, well that is impressive. The goal is not the right answer. It a demonstration of understanding and abilit appropriate to the level of the course.

You and I think that the cube root function is the simplest approach. The student doesn't necessarily agree. And I disagree with you that the most general approach is necessarily the simplest.

> Just about any method is valid in that type of problem.

Why is that not true for other types of problems?

> Passing a class should mean more than I got a lot of answers correct.

I agree. But you shouldn't penalise the student if the exam question is poorly framed (and we all make such mistakes). Just take a note for later and don't make the mistake again.

Let's take a calc 2 example. I ask students to integrate ln(x). I want to know if they can do integration by parts when one function is 1. Some of them can memorize the answer and just write it down. I don't give them credit for this. I'm giving an easy problem because I just want to know if they know how to do parts with 1 as one of the functions. I don't want to load the test with hard problems so that I can eliminate any possibility of memorization at play.

It's interesting reading all the replies I've gotten. It's nice to see other peoples' perspectives. Including yours.

As you stated I would not give x^3=27 as a problem in college algebra. It's a fine line and I suspect that we mostly agree except on one part.

As a grader I've given full credit for the wrong answer and no credit for the right answer.

So, your proposed solution to integral(ln(x)) involves first noticing that d/dx (1/x) = ln(x), then proceeding to use v=1/x to integrate by parts?

I think I am misunderstanding something, because as far as I can tell, integration by parts where one of the parts is 1 is literally useless.

u = ln(x) and dv = 1

du = 1/x and v = x

integral ln(x) = x ln(x) - integral 1/x times x

integral ln(x) = x ln(x) - x + C

I could give integral arctan(x) but with the advent of computer algebra systems I'm mostly interested in them knowing the basic examples and to not burden them on a test with something more complicated.

EDIT: The derivative of 1/x is not ln(x) as you stated. You got it backwards and my guess is that is the source of your confusion.

> Passing a class should mean more than I got a lot of answers correct. It should mean an understanding of the material.

I agree, but testing for understanding (as opposed to Socratically probing for it) is more time consuming, complex, and difficult than just testing for correct answers.

To stay within a given test workload, students would have to take far fewer tests. Obviously not the direction the educational system is trending these days. Which is a shame.

>I agree, but testing for understanding is more time consuming, complex, and difficult than just testing for correct answers.

You get 1 mark for the right answer and 2 marks for working. Problem solved. The question doesn't have to change at all. This is how I remember mark schemes working in the general case

> You get 1 mark for the right answer and 2 marks for working.

I'm not sure I get what you mean by "working". Can you expand that a bit?

As someone who has spent time in the math world, I sympathize with proactive's response. Once you get to a "high enough" level, the lessons you learned earlier have to be unlearned.

Not just true in mathematics but in engineering as well. I was taking an engineering course in my sophomore year where I had a system of equations in 3-4 variables. I spent forever trying to solve it (analytically), and failed. So did most of the class. The next lecture, the professor showed us how to do it. A mixture of plots, etc reduced the solution space and the rest was trivial. He also said "You could just use the solver in your calculator/MATLAB".

I wasn't satisfied with his answer. It felt like cheating. I didn't learn the cool way to do things.

But in the real world, if you can get the solution this way, it's perfectly valid. As long as you can confirm that you found a/the solution (trivial to do).

With the x^3=27 answer, it is the onus of the instructor to specify explicitly that "guessing is not allowed". Why? Because as others have mentioned, it is totally appropriate in mathematical circles to guess a solution. Much work in mathematics is done that way.

In various classes (mathematics/engineering/physics), I've both utilized non-standard ways to solve problems on tests, and have seen it done by students on tests I grade. This is to be encouraged. Especially because this is what mathematicians/physicists love to do in their real work.

If your goal is to ensure they understand cube roots, either make a problem that is hard to guess (e.g. x^3 = 24), or be explicit about it. Even with x^3 = 24, if they use Newton's method, that should be graded correct.

Professional mathematicians don't guess answers to theorems. They sometimes guess counterexamples. But no one guesses answers to a theorem. I've seen, "I notice that A is a solution to this equation does anyone know a method for formally solving it?"

Guessing is not a method of solving. It is a method of finding counterexamples. Two different types of problems.

I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers.

>Professional mathematicians don't guess answers to theorems.

Believe it or not, proving theorems is not the goal of many mathematicians. I'm simplifying a bit, but read Freeman Dyson's essay on Birds and Frogs. Essentially "problem solvers" vs "theory builders". While problem solvers often do end up proving theorems, it is not their main goal. If they can "guess" a solution, they are done. It is publishable.

Go to the field of combinatorics, and you'll find it is full of guesswork.

>I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers.

Sorry, but many mathematicians disagree with you. Solving a problem is finding a solution (provided you have a means to verify correctness). It doesn't matter if you merely guessed it.

My training was in commutative algebra as a pure mathematician. I did dabble in combinatorial commutative algebra and went so far as purchase a book on the topic by Sturmfels and Miller. I doubt very much that you can find a published paper in mathematics in which the author guessed the answer to something and did not prove anything. Every paper I read (commutative algebra) proved things or discussed something heuristically with the goal of finding a method of proving.

I have the book by Stanley on combinatorics. There are not results of the form: I guessed A is the answer and it's right. Let's move on. This is does not happen. When one notices something is a solution the mathematician always wonders why. A mathematician wonders what underlying structure there is. Never is one satisfied by a guess.

If someone found a counterexample to the Riemann Hypothesis the first question would be, why is this number a counterexample? What caused the obstruction? You could problem publish a paper that just said, A is a counterexample to the Riemann Hypothesis. But you could not publish a paper that said, I guessed A is a solution to B and it turns out I was right.

Even in commutative algebra, there is a fair amount of guesswork. If I tell you that 2+3=4 in an abelian group with {2,3,4}, then ask you what is 2+4, you will immediately guess 2+4=2. Its just what the millennials term "stupid obvious". Yeah you can do a lengthy proof on why 2+4=2. Proof: Its 2 because 4 must be the additive identity, and 4 is the additive identity because 2 & 3 are not, and they aren't because if 2 was then 2+3=3, and if 3 was then 2+3=2, but because I've told you 2+3=4, by closure 4 must be identity, which implies 2+4=2. That's the whole story. Literally no mathematician I know will sit down & write that lengthy proof I wrote. They'll just tell you 2+4=2, and if you pester them with "But why?" they'll say "Because" and excuse themselves :)

In an elementary group theory course this is the type of problem that would be given when groups are first introduced. It's a good homework/test question and just writing a*c = a as the answer would not suffice (depending on the level of the course and aim of the problem). The point at that level is for the student to learn how to justify their beliefs.

This is especially so if one were in a basic mathematical logic course. Of course, in an algebraic topology course where this group showed up it would be assumed that everyone knows how to find the answer and why. No justification would be needed.

http://www.openculture.com/2015/04/shortest-known-paper-in-a...

The paper you cited supports what I've been saying. You can publish a paper that says, "Here is a counterexample." You can't publish a paper that says, "Here is a solution I guessed to be correct."

Any method of finding a counterexample is accepted. Guessing a solution is not.

EDIT: The paper linked to was published because it was a counterexample to a famous conjecture.

So you're saying that if they left out the "search on CDC" line the paper would not have been accepted.

No. I'm saying it would not have been published if it wasn't a counterexample.

What is the "answer" to a theorem? If by "answer" you mean the conclusion of the theorem, then mathematicians guess these all the time; such guesses are generally referred to as conjecture.

A conjecture is someone, usually a well known expert in the field, saying, "I think this is true but have not been able to prove it." It's not considered a theorem until someone proves it. A famous example is the Goldbach conjecture. No proof has been found but people have been searching for a proof for a long time but so far no one has proven it.

Because I can't edit my above post any more, I'll add it here: I marked it correct. I didn't feel like penalizing him for something that had never been taught.

Here's an idea for a better problem:

Solve the following:

x^3 = 27

y^3 = 21

z^4 = 85

My rationale is that there will be a huge time advantage for the student who works out the solutions by using roots, and a visual "hint" that there might be a general solution.

I was under the assumption that this was for a college algebra course and that the rational exponents had been covered. If it hadn't been taught then I would give credit. Indeed, I'd be impressed by such reasoning.

At university, they forced us using logs to solve similar equations with extremely large exponents. Otherwise it's nigh impossible to calculate any answer with the allowed calculator.

I love it. Write numerically unstable problems, so they have to be solved symbolically.

Until the age of 18 we were not allowed to use a calculator in the exams, not even in physics. We had to solve everything simbolically. This rigorous teaching method resulted in a couple of gold and silver medals at the Internationale Mathematik-Olympiade. A couple of my former class mates are now profs on the MIT, Berkeley, God knows where.

That was also the case when I was in school. It wasn't so much that calculators were prohibited, but that they were useless, because the problems were designed to be solved without one. That was still the case when I taught the college math class in 1997. One student asked me if they could use graphing calculators, and my response was: "You may use one, but I've seen the exams, and a calculator will be of no help."

But I'm of two minds about it. I love manipulating expressions by hand. It's a relaxing hobby. But it limits the choice of problems that can be solved, which in turn narrows the range of things that can be taught, and even creates a false sense of what is possible in math. And it doesn't reflect how math is used by most people, i.e., with a computer.

I'd rather incorporate more computers into the math curriculum, and maybe merge math and programming into a single subject.

The biggest advantage for the students would be if you would not split up math into algebra, calculus, etc. The equation

x^3 = 27 (calculating the volume of a cube)

is an application of the equation of

x^y = z (potentiate a number)

which is an application of the equation

f(x) = y (apply a function)

The solution is simply

x = f-inverse(y)

if f is invertible.

As this simple example shows math on high school level can not splitted up into geometry, algebra, calculus, etc. Understanding one of these areas helps to understand the others and vice versa. If you want to master one of them you have to master all of them at the same time, with the same speed, parallel.

If you were teaching a more formal method, I can see marking it wrong. But you're mistaken to think the student can't find the answer that way. If I had to solve x^3 = 7 in my head, I know that 2^3 = 8, and my next guess would be 1.9

1.9^3 = 6.859 (ok, I'd need a pencil for this)

Now I know the answer to two digits, "x is just a bit more than 1.9", and one can keep going if more precision is needed.

The student demonstrated that they can do such problems with positive integer solutions. Possibly even negative integer solutions. But certainly they did not show any sort of reasoning along the lines you wrote.

But let's take your method. Your method is wrong because you aren't finding the solution. You are finding a sequence of numbers that converges to the solution. But this isn't the solution to the equation. The solution to the equation in question is a number and not a sequence.

The answer, over the reals, is 7^(1/3).

>Your method is wrong because you aren't finding the solution. You are finding a sequence of numbers that converges to the solution.

...

>The answer, over the reals, is 7^(1/3).

So if the student did not write 2, but wrote 8^(1/3), it is OK?

As for the method being wrong, no - it isn't. The sequence converges. If you want an exact answer, you should specify that you want an exact answer.

>The solution to the equation in question is a number and not a sequence.

Sorry, but every number is a sequence. You can start with rational numbers, and define every real number as a sequence of rationals. Lots of books actually do this to define what a real number is.

8^(1/3) is 2 (when interpreted over the reals).

Your method is wrong because it does not produce the answer. One does not, in college algebra, say the solution is:

lim a_n as n->infinity

For one thing you did not prove convergence. It is understood that solutions to algebraic equations over the reals are numbers and not approximations. Any student who knows about Dedekind cuts or infinite sequences knows to take the cube root of both sides.

Most mathematicians consider it silly (strange, wrong) to say that 2 is an element of 3 even though it is.

Anyone solving x^3 = 27 in the method specified does not know any of these finer points of mathematics. The method is bad. It's useful for positive integer solutions but not for the general situation.

Minor correction: you probably meant that books often define real numbers as equivalence classes of sequences of rational numbers. I've never seen a definition of real numbers as a single sequences in a textbook, not because it is impossible, but because defining arithmetic operations would get really messy.

>The answer, over the reals, is 7^(1/3).

Math major here.

7^(1/3) is defined as the solution to x^3 = 7. In this case I would accept it as an answer because 7 does not have a rational cube root, and the question is presumably testing if the students knows fractional exponents [0]. However, in the case of x^3 = 8, I would not accept 8^(1/3) because the students has not actually found the answer; they have merely written the question in a different way. I am also curious what method you would propose the student use to compute 8^(1/3), as all the methods I know degrade to guess and check in the single digit case.

You could say that simplifying to x^3 = 8 to x=8^(1/3) is the first step to solving it; to which I would reply that simplifying x=8^(1/3) to x^3 = 8 is the first step to solving it.

[0] I could also imagine another math class where I would mark 7^(1/3) as wrong because the student has not actually found the answer, merely written the question in a different way. Presumably we are not talking about such a situation.

This thread is turning into Feynman vs the abacus, is it not?

http://www.ee.ryerson.ca/~elf/abacus/feynman.html

Yes it is. Thanks for the link.

funny story, quoting:

```

"Tell me," he said, "how were you able to do that cube-root problem so fast?"

I started to explain that it was an approximate method, and had to do with the percentage of error. "Suppose you had given me 28. Now the cube root of 27 is 3 ..."

He picks up his abacus: zzzzzzzzzzzzzzz— "Oh yes," he says.

I realized something: he doesn't know numbers... Furthermore, the whole idea of an approximate method was beyond him, even though a cubic root often cannot be computed exactly by any method.

```

I think the problem lies with the question. Simple question gets simple answer. If you want the solution in the general real case, ask x^3 = 7 since it cannot be solved by the easiest method. If you ask to solve x^3 = 8, I firmly believe it is fair game to solve using a method that only works for small integer.

I disagree. Whether a process is general or not is not important, sometimes problems aren't part of an obvious larger class of problems, and even then using a method that only works for a small set of problems is not technically wrong, and can sometimes even be many times easier than using a fully general method. For example, solving x^3 = 27 by remembering 3^3 = 27 is a perfectly acceptable solution. Trial and error shouldn't so readily dismissed.

In my opinion 'show your work' means that you show why you've arrived at a particular answer, why this is a correct solution, and (if necessary) why this is the only solution.

Using that criterion, simply writing something like '3^3 = 27', or more properly 'Simple trial and error shows x=3 to be a solution, since 3^3 = 27', would suffice.

If the problem was instead x^3 = 7, then sure this method wouldn't work (although they might be able to figure out it's somewhere between 1 and 2) but then again 'By definition x=3√7.' isn't particularly illuminating either, even though it's a full and correct derivation of the answer.

I'd say 7^(1/3) is illuminating in the sense that

1. Someone understanding this is more likely to understand the concept of inverse functions.

2. Someone understanding this is more likely to understand how to solve x^(3/5) = 7. And then more likely to understand how to solve x^(sqrt(2)) = 7.

In a college algebra level type course the ultimate goal is not knowing how to solve x^3 = 27. The ultimate goal is to understand more complicated ideas. The reasoning displayed is a huge red flag. In college algebra a student presenting the solution given will very likely fail the course. That student needs help.

The task given was to solve an equation and to show how it was solved. The student did exactly that. Their answer is obviously correct and they deserve full marks for it.

If you want them to complete a different task you should ask them to do a different task. Eg. The question could be "show a general method for solving equations of the form "x^3=n", where n is any real number". Punishing people for obeying your instructions is a great way to make them hate you. I'm not surprised so many people hate math when it's taught in this way.

A student solving the equation in the way the OP stated will likely fail the course. They are showing they don't understand the concepts. The student needs help.

They are unlikely to understand inverse functions when that topic is taught. They are unlikely to understand how to solve x^(2/5) = 7. Or x^(sqrt(2)) = 7. Probably they don't understand exponents.

The solution given is a huge red flag.

It's disgustingly arrogant to assume that. They are showing exactly the understanding needed to answer the question. They obeyed the instructions, and they paid good money to have that opportunity, but they are being punished despite doing exactly what was asked of them. Whether the instructor has the legal right to do this or not, it certainly feels like fraud.

If the question setter failed to set a good question, that's the question setter's fault, not the student's fault. Punishing the student for somebody else's mistake is injustice.

I've been teaching college algebra for 20 years. There is nothing arrogant in what I said. The student needs help and it is clear that this is so to anyone with experience teaching such courses. After many years and thousands of students I've developed an intuition on when a student needs help and when I don't need to worry about them.

If you had to bet a million dollars, based solely on the solution presented and knowing the course was college algebra, would you put the money on them passing or failing?

It's not a question of probabilities. You're advocating punishing people for wrongdoing you anticipate them doing in the future. This is the argument behind racial profiling, and it's widely considered immoral. Would you mark a black student down because they're statistically more likely to fail in the future? If you suspect a student lacks understanding then you should actually test that understanding, not just assume they're going to fail and punish them preemptively.

The solution provided was not appropriate for a college algebra level course because it is not a valid solution as it is not a way of doing similar problem with a trivial numerical modification. You may disagree with me and that is fine.

In addition to the solution demonstrating an inappropriate level of understanding of the material it demonstrates that the student needs help. I mentioned my experience and intuition because you accused me of being incredibly arrogant when I stated that such a solution indicates that the student needs help. Now you are injecting race and raising the question of racial bias on my part.

I think if you reflect on what you've written in response to me you'll see that your statements are not supported by the evidence and your accusations have been inflammatory. Your conclusions about me punishing students preemptively, and raising racial bias on my part are false, unjustified, and not called for on this site. It is expected that a higher level of decorum be demonstrated by everyone on this site.

We simply disagree on the topic of grading and that is fine. I don't think further discussion with you will be productive for either of us. I will read any response you have but will not further comment to you. Please refrain from responding to me in the future. Based on this present interaction you are not the sort of person I wish to discuss things with.

I disagree that students should be punished for answering the question asked of them. You think it's acceptable, and you could have even justified it by saying that it teaches trickery and deception. I don't think that's a good way to teach, but I could accept that reasoning.

But that's not the justification you gave. Instead you said it's evidence for likely future wrongdoing. By bringing up race I am in no way suggesting you are racially biased. Indeed, it would have been pointless to bring it up if I thought you were. The point was to give an analogous example of punishment for anticipated future wrongdoing that I assumed you would accept as wrong. If it's wrong in one case it's wrong in the other.

Your intuition about the student's abilities is probably correct, but intuition is no basis for justice. People should not be punished before they've done anything wrong. This is a fundamental ethical concept. The student deserves a fair trial by exam question, not instructor vigilantism.

I have a standard rubric for questions like these, and would have scored the response above 1/5 for an attempt being made with an incorrect or unclear argument. My expectations are very clear.

I disagree with your characterization of grading as punishment. Added value in teaching is about 10% instruction and 90% feedback. If I let this opportunity to fix the student's reasoning pass, I am doing a far greater injustice to them than if I fail them on a quiz.

I appreciate the sentiment but I'd argue that if you're looking for generic methodology, you should ask for a generic solution. The question shouldn't be "solve x^3 = 7", it should be "solve x^3 = y" or "solve x^y = z".

If a student produces a correct answer to a numerical question that's correct by drawing a graph or doing a linear/binary search, that seems fine to me.

I worked in college as a Digital Librarian and my degrees were Theology and Library Science (Not many of us)

PROBLEM: It is a socially acceptable to be bad at math and I am talking 4th or 3rd grade math!

2nd Problem: We teach pure math (Think algebra) to soon and place applied maths like Trigonometry and Calculus where only a handful of student will ever even attempt and God help them if they have a weak math teacher.

You need algebra to do calculus with any sort of rigor...

Your suggestions are the real problem with teaching mathematics; do people learn science only to learn practical stuff? Read literature only to gain literacy skills? No! That is not how classes are taught.

Mathematics is seen only as a tool, but if it was taught as an art, or even a science, people wouldn't hate it!

Why do we have to only get to calculus or pre-calculus concepts so late and why is it so extreme like it is all or nothing? There is plenty in Applied math in calculus that doesn't hing on algebra. Children natural love pattern recognition and we don't teach anything pattern based till we have beaten them into boring show your work work sheets?

See (A Mathematician’s Lament by Paul Lockhart) http://www.maa.org/external_archive/devlin/LockhartsLament.p...

"In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education."

> Rigor

My trigger word!!! Rigor, the most hatred word in all of education philosophy!!!!

> Three Dirty Words are Killing Education by Deb Jensen

RIGOR VS. RELEVANCE The second problem term is rigor (also known as “high” standards). The term is associated particularly with college readiness. The term might call up images of learned individuals from the 1800s, but today's rigor is imposed artificially — it requires only more algebra or more credits. While the mind needs information to build beyond the concrete to the abstract, much of the random information is actually screened out.

http://journals.sagepub.com/doi/full/10.1177/003172171009200...

Also lost 20 minutes on this article.

https://betterexplained.com/articles/a-gentle-introduction-t...

A note on rigor (for the math geeks) I can feel the math pedants firing up their keyboards. Just a few words on “rigor”.

Did you know we don’t learn calculus the way Newton and Leibniz discovered it? They used intuitive ideas of “fluxions” and “infinitesimals” which were replaced with limits because “Sure, it works in practice. But does it work in theory?”.

We’ve created complex mechanical constructs to “rigorously” prove calculus, but have lost our intuition in the process.

We’re looking at the sweetness of sugar from the level of brain-chemistry, instead of recognizing it as Nature’s way of saying “This has lots of energy. Eat it.”

By "any rigor at all", I meant how do you express the concepts of differentiation without knowing how to manipulate symbols?

A part of the push for rigor in calculus is being able to prove results instead of appealing to intuition. They are both important aspects of mathematics when working together (often intuition guides a slower, rigorous, methodical proof), but we don't want people relying on intuition where a proof is necessary. This kind of critical thinking and logical reasoning are both important cross-disciplinary skills that almost every field requires.

Maths is not pattern matching. There's a lot of beauty to be had in playing around with numbers; some of the biggest, deepest unsolved theorems of mathematics can be expressed so that fifth and sixth graders can understand them. Combinatorics and probability can be easily taught to kids, and even the basics of abstract algebra can be expressed in an easily understandable way.

These topics are much more fascinating than mere calculus could ever be, but this requires people to stop viewing mathematics solely as a tool, and start viewing it as a way to reason about the world.

Wasn't it Randall Monroe who said that most people only need enough math to correctly divide a check for a dinner party where you are covering the bill for the guest of honor?

That's moderately complex math, and goal oriented.

What ends up happening is the people good at math prey on those who are bad at it. You tell a kid "you learn this so people can't take what's yours" and most of them will pay attention.

> Why do we have to only get to calculus or pre-calculus concepts so late and why is it so extreme like it is all or nothing?

I agree. I did some research on teaching the concept of instantaneous speed to 5th graders. Building on their intuitive understanding—from a young age on we are continuously confronted with dynamic systems such as (loco)motion, weather, computer games, cooking, and so on—and connecting to their understanding of a constant speed, I devised a learning trajectory to explore and deepen their understanding of instantaneous speed more mathematically, including quantifying it.

If anyone is interested, you can read more about it here: https://heerdebeer.org/DR/thesis/ch5.html

The basic concepts of calculus (area under a curve, areas inside shapes in 1, 2, 3 dimensions, relationships between forces, etc) don't need anything more than some basic physics experiments and the ability to perceive the world around you.

Calculus can be explained in terms of algebra, sure, but it can also be explained with visualizations and with experiments (and/or you can learn the algebra at the same time).

Explaining the basic concepts of calculus is very different from teaching someone to actually be able to apply them.

Application of calculus to the real world, in its most basic form, is basic physics. The whole point of people who aren't planning on building on calculus with further mathematics taking calculus is to better understand the world and how it works, not to be able to derive stuff, I think.

> PROBLEM: It is a socially acceptable to be bad at math and I am talking 4th or 3rd grade math!

This is a huge issue and some people even take pride in being "bad at math". That's like being proud of being illiterate.

And these people hardly know what math means. They are bad in arithmetic and stopped there.

Totally agree. And now it's extending to all of science.

I don't really think people in general being bad at math is a problem. The real problem is that it's somewhat socially unacceptable -- particularly among less academic people -- to be good at math.

Here's Feynman's take on learning algebra:

Around that time my cousin, who was three years older, was in high school. He was having considerable difficulty with his algebra, so a tutor would come. I was allowed to sit in a corner while the tutor would try to teach my cousin algebra. I'd hear him talking about x. I said to my cousin, “What are you trying to do?” He says, “I'm trying to find out what x is, like in 2x + 7 = 15,” I say, “you mean 4.” He says, “Yeah, but you did it with arithmetic. You have to do it by algebra.”

I learned algebra, fortunately, not by going to school, but by finding my aunt's old schoolbook in the attic, and understanding the whole idea was to find out what x is – it didn’t make any difference how you do it

For me, there was no such thing as doing it “by arithmetic,” or doing it “by algebra.” “Doing it by algebra” was a set of rules which, if you followed them blindly, could produce the answer: “subtract 7 from both sides; if you have a multiplier, divide both sides by the multiplier,” and so on – a series of steps by which you could get the answer if you didn't understand what you where trying to do. The rules had been invented so that the children who have to study algebra can all pass it. And that’s why my cousin was never able to do algebra.”

(from What Do You Care What Other People Think?)

(Video: https://www.youtube.com/watch?v=VW6LYuli7VU)

This equation can not be a real university exam question.

Edit: I have seen the other post in this thread. My apologies, this seems to be the university math level in the states. God help us!

freshman college algebra is really basic in the us. It's about what I took in the 10th grade. Most people getting math degrees will skip it because they got it in high school.

Indeed, I observed that among those of us qualified to teach the course, absolutely none of us had ever taken it. Here's what I learned about it, which may be unique to the US:

Just before starting freshman classes, the students took a math exam, and were sorted into three levels:

1. Calculus.

2. College Algebra, which like you say is a repeat of 10th grade algebra. Its curriculum is defined by the requirements for the minimum level of math that can be offered for credit at an accredited college.

3. "Remedial" math, which cannot be offered for credit, but is a preparation for college algebra.

I estimate that about half of the students were in calculus, and the other half in the lower two tracks. For all intents and purposes, if you're not ready for calculus as a freshman, you're not going to be a math or "hard" STEM major. It's the sorting hat. The opinion of the professors was that the algebra students didn't even belong in college, so the lower math courses were conducted under a black cloud. The rigidity of the accreditation requirements may have led to the rigidity of the curriculum, which I thought didn't belong in any century that I've lived in. For instance the course made no meaningful use of computers.

Since this was the state flagship university, the students represented K-12 math preparation throughout the state. The variation is huge. Some school districts offer two years of calculus. The high school that my kids attend has a deal with the university for kids to take college math classes beyond calculus. Some schools have no calculus. Some of the kids in my college algebra class had gotten A's in high school calculus, but somehow couldn't pass an algebra exam.

As a mathematician I may be the wrong benchmark but back in primary/elementary school (I do not know how you call it in the US) in a completey normal class at the age of 12-13 everyone understood the "two column format" and everyone used it. I guess maybe 10-20% of the class coudn't solve equations like this on the fly writing on the board in front of the class. And they were just slower and needed more time than the rest. This was shortly after the fall of the iron curtain in Western Europe.

I remember at the age of 12 being once in a math camp with a kid who used to live in the US before and he told us that the american kids in his class could not calculate the sum of 5 and 6 in their head. I thought this was just a mean joke. Until now.

Show your work means that when you are tested on your knowledge of a specific pen-and-pencil algorithm, you are expected to show a trace of execution of that algorithm. You're not being tested on solving the problem, but on knowing the algorithm.

Until recently I didn't even know there existed such a pen-and-paper algorithm resembling long division:

https://xlinux.nist.gov/dads/HTML/cubeRoot.html

Perhaps math classes should focus less on remembering algorithms and more on inventing your own algorithms by solving novel problems. In that context showing work is much more meaningful.

I don't mean to be mean, but people who don't know basic algebra are unlikely to be making algorithmic breakthroughs.

The parent comment probably meant that students would learn how to derive the necessary algorithm from the problem description, rather than memorizing a bunch of rote algorithms without being able to disassemble and reassemble them.

Then you have parents complaining about the difficulty of "story problems" and claiming math class shouldn't be reading class.

as per the curry howard isomorphism (programs as proofs), there is no difference, except that a few symbols are arbitrarilly bound but without loss of generality (as could be proven by induction).

without training, our brains confuses numbers easily from my experience. my niece is learning multiplication tables right now, and she confuses 8 * 7 = 56. we tried to teach her these numbers with a story, or a shortcut, e.g. 5 6 7 8 . see the groups 56 and 7*8. but still she rather found this kind of symbol processing painful and preferred to play with other things. another approach we tried to show her a visual representation of the multiplication but not sure what her current progress is.

I made this: http://trondal.com/m Maybe it can help her.

Maybe it's because I'm on mobile but I have no idea what's happening. There a number at the bottom with some sort of countdown (I think) happening, I click a number, screen flashes red, there's a small Sudoku board on the right. Total confusion haha, can you explain?

It's not good on mobile. I tried making it language free, and relatively easy to figure out if you explore the buttons and see what happens. The top middle number is your problem. Your task is to click on the solution. If the solution is wrong, the screen flashes red. The bottom shows statistics of how much time you spend on each problem. The top left and right numbers lets you decide what factors are chosen in the problem. The sudoku board lets you toggle showing you relevant solutions based on chosen factors.

I think you succeeded. I had no idea what was going on for a few minutes and just clicked around, but it became obvious how it worked fairly quickly.

I'll give it to my 8/10 year olds tonight and see what they think. Thanks for making it! Times tables are hard slogging to memorize, and anything to make it less boring is very welcome

I remember being taught with "math manipulatives"...the result was that I intuitively grasp the relationship between numbers. Maybe that would let you physically show your niece what "eight sevens" actually looks like (and the similarity to "seven eights").

edit: something like this, but with each number 1-10 as its own block and a different color. http://www5.esc13.net/thescoop/insight/files/2012/08/MaryMat...

that looks interesting! it engages a lot of senses, trying now to google for something like that on amazon, but maybe i can print those myself with a 3d printer. or, maybe someone has open sourced a design for those cubicles already?

found the ones I used: http://store.mathusee.com/catalog/math-u-see/manipulatives/i...

Those were great! I'm pretty good at doing math in my head and my brain still visualizes numbers and their relationships based on lengths and colors. In 1970s Sweden they were called "tiostavar", "ten-sticks". They were made of high quality painted wood and very tactile and pleasant to work with. The lengths were precise so they would add up properly in any combination, and the widths were equal to one unit of length so you could make areas and volumes.

(Edit) like these: https://orsjoforskoleklass.files.wordpress.com/2014/11/img_3...

My early education used sticks and cubes like that. I don't remember different lengths with different colors, but that makes a lot of sense as a way to teach addition.

The 1000-cube fascinated me. I think the classroom only had one of them, so the teacher kept it, and we never really got to use it (and it was hard to gather 10 100-squares to build your own).

> we tried to show her a visual representation of the multiplication

I find that visual methods, open up many ways to deeper understanding of arithmetic. Even advanced topics like binomial coefficients pop up quite naturally when folling through. Here is my take on it:

http://heinrichhartmann.com/blog/2016/06/12/Box-Counting-Ari...

My kid just learned by memorizing the results in order. Like: 8, 16, 24, 32, etc. If the problem is 5x8 then she knows its the fifth number in the sequence. Did not take her a lot of time to then learn the whole formula. Now that I think about it, this is like calculating results in a hash table for easy retrieval later.

We used this video (and channel) and it helped lots:

https://youtu.be/IjPjks1R6Ng

when you get to 5x + x = 6x, you can't simply memorize a similar sequence. Knowing how the sequence is created is much more valuable than memorizing the sequence. You do loose some speed, but critical thinking is more important than speed, especially with the onset of cheap personal computers and calculators.

I never memorized the multiplication table, but I also never failed a test on it. You only need to learn a fairly small portion of the table in order to infer the rest. When I see 8 * 7, I do not immediately know the answer but I do know 8 * 5 immediately and can quickly add 8 to the result twice.

This has never caused me problems.

Same here. The odd thing is, I'm usually pretty good at rote memorization, but the multiplication tables just didn't interest me or something. So, like you, I use a "multiply then add" strategy to multiply anything I don't have memorized. And it's never seemed to affect my ability accomplish anything.

I went in the opposite direction. I rote-memorized the table, and then discovered different ways to compose the numbers while playing around with them in my head later.

Of course, "multiply then add" is the only easy algorithm to use when the numbers are larger than the size of the table that you know, so most things beyond 10*10 (and some squares above that) are going to default back to that anyhow.

Often it's just a matter of developmental maturity. Childrens' minds develop at different rates and it's counterproductive to push hard on teaching concepts that they're not ready for. Try again in a year.

> None of them had a good answer! These were the top mathematicians in the world.

I can believe that.

Why?

I'm not one of "the top mathematicians in the world", but I do hold a Ph.D. in applied math from a world class research university and have published peer-reviewed research in math.

My view is that (A) the problem, solve for x in x^3 = 27, (B) the request "show your work", and (C) the objective of the work of the OP to "simplify" some algebraic expressions are at best flawed introductions to math as pure/applied mathematicians do it and in our educational system as efforts in, call it, pedagogy.

IMHO the goal of "simplify" an algebraic expression is especially flawed; that began to dawn on me in high school, and I so concluded in college and since.

Why? To "simplify" an algebraic expression is mostly a matter of style often without clear criteria or a unique answer; such simplification can at times be to illustrate something particular to the context but not be general.

Really, in math, we manipulate/leave algebraic expressions in whatever form is useful for what we are doing with the expressions and, IMHO, essentially never much for a goal of mere style or simplification.

E.g., an important manipulation of algebraic expressions was taking the algebra for discrete Fourier transforms and, essentially, manipulating it to illustrate how to do the calculations of the fast Fourier transform (FFT) -- work mostly of J. Tukey, supposedly at a US Presidential Science Advisers meeting to answer a question of R. Garwin. The FFT is darned important, and curiously the main point can be discovered and illustrated just by manipulating the algebraic expression to be in one of several particular forms.

Too much in math as commonly taught in K-12 and early college isn't really close to math as done by people really using math in, say, the STEM fields but is stuffed in there by the teachers as part of pedagogy or having a source of exercises and test questions.

In response, generally it would be good to lower the emphasis on such make work pedagogy, get the students through it (minimize it and have lenient grading of it), and get on to what is important in math and its applications, research, etc.

E.g., currently a big problem and hot topic in applied/research math is over fitting. Well, hush, don't tell anyone, but in some important cases can make some surprisingly good progress on over-fitting, realliy, get rid of the concerns, by essentially rewriting some of the algebra and just looking and observing. How 'bout that! No, I don't offer to fill in the details! Uh, in some cases, this work can also be a great way around some really nasty numerical stability problems.

But, right, simplifying some algebra can be important when have an important objective in mine, and style is not such an objective and, really, is not a good guide to what would be a simplification useful for some important objective.

Or, with the FFT and over-fitting, I've given two cases where there is an important objective for simplifying an algebraic expression -- alas, in both cases, without the important objective in mind, neither simplification would be seen to have better style!

> IMHO the goal of "simplify" an algebraic expression is especially flawed

This, too, has occurred to me. I am curious as to what heuristics tools like Mathematica use when you ask them to simplify an expression.

> Too much in math as commonly taught in K-12 and early college isn't really close to math as done by people really using math in, say, the STEM fields but is stuffed in there by the teachers as part of pedagogy or having a source of exercises and test questions.

> In response, generally it would be good to lower the emphasis on such make work pedagogy, get the students through it (minimize it and have lenient grading of it), and get on to what is important in math and its applications, research, etc.

I wish that I encountered proof-based math much earlier, and not the weird two-column proof thing they teach in geometry in high school. When I started working with proofs, math made a lot more sense to me.

Once again, like "show your work," nobody is told precisely what "simplify" means. It would be preferable to teach about "form," and then about manipulating expressions to convert them from one form to another.

For at least one class, I noticed that my daughter's textbook had replaced "simplify" with "show in standard form," where they had been told what standard form is.

I was lucky to go through a K-12 math curriculum that used proofs. And I agree that the two column format is awkward. I'm reminded of Edward Tufte's critique of PowerPoint, that a restrictive template makes it harder to express ideas. I wrote my proofs and derivations the same way that they were presented in the textbook, and in class: In a conversational style. This had the added benefit of being able to learn that style by example. When we talk about "using" math later in life, it's not just using math to get an answer, but being able to explain and justify that answer to other people.

> It would be preferable to teach about "form," and then about manipulating expressions to convert them from one form to another.

Teaching, even defining, form would not be so easy, either. In some cases, maybe for partial fractions decomposition or completion of the square, but generally, no.

Instead there is an easier approach, plenty effective: Just present the student with two algebraic expressions that are equal and have the student show that the two are equaly. So, the student gets practice in manipulating algebraic expressions; the goal, show that the two expressions are equal, is clear; and there are no issues of style or form.

Of course a lot of the work in a common course in trigonometry is of this form. So, sure, when a student gets to manipulating trig functions in calculus, they have lots of practice manipulating trig expressions, maybe even more than commonly needed! :-) Or, maybe a good trig course would trim back some of the manipulation exercises and, instead, move on to some of the trig applications, especially to signal processing, Fourier transforms (just the finite versions if want to avoid calculus), power spectra, etc. Heck covering just overtones in music, how a violin or organ is tuned, would be both good and fun.

> IMHO the goal of "simplify" an algebraic expression is especially flawed

I teach calculus occasionally at a local university, and always make a point to highlight this fact to my class. In their previous algebra courses, the "objective" was more often than not to factor something into the smallest expression possible.

But in calculus, you generally want to expand an expression in to more terms to take advantage of linearity properties of operations like differentiation, integration, etc.

The concept of "simplify this" isn't very well defined, and I tended to not be a stickler for the final form of most things.

What is the expected way to show the work on this? IIRC (long time ago), the method I learned for computing cube roots involved knowing the cubes of single digit numbers (e.g. 3^3 = 27, being one of them).

Presumably, it consists of verifying that 3 x 3 x 3 = 27. But who the hell knows.

Alternatively, maybe it expects a binary search through the decision space? 2 x 2 x 2 = 8 < 27.

Too bad this breaks down when the problem is trivially adjusted - not in the X^3 = 28 way (As binary search can approximate that), but in the X^3.3 = 27 way.

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