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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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?
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 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.
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.
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.
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.
"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.
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.
>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.
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.
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.
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.
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).
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.
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.
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 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.
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.
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 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.
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.
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?
"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.
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.
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.”
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.
> 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.
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.
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).
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.
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.”
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.
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.
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.
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 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.
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.
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").
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.
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).
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?
> 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:
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:
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.
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.
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
> 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.
Has anyone done a study to see if this kind of aided solving actually helps students learn? I'm worried that "Eh, I'll just write this solution down today, I'm sure I'll learn it tomorrow" is what's happens.
In my experience as a teacher, most students don't learn it either way until they have to use the skill to do something else. That's why we joke: calculus is a class where you finally learn algebra, differential equations is a class where you learn to integrate, etc.
There is big difference between understanding something, knowing something and application of that knowledge. It is easy to sit in class and follow along and feel like you get it. Much harder to explain it to the class from scratch.
"If you can't explain it simply, you don't understand it well enough." -Einstein
> The quote "An alleged scientific discovery has no merit unless it can be explained to a barmaid." is popularly attributed to Lord Rutherford of Nelson in as stated in Einstein, the Man and His Achievement By G. J. Whitrow, Dover Press 1973. Einstein is unlikely to have said it since his theory of relativity was very abstract and based on sophisticated mathematics.[1]
Also, see the next answer for a more direct source. The list of misattribution is entertaining, too.[2]
Have millennial children fundamentally learnt to learn differently using computers? I decided to understand the math behind the Kalman filter, and despite having read the Wikipedia page and impemented these, I still had to go back to pencil and paper. (Did you know that the Kalman filter is a least squares estimator?)
I was thinking this morning that if the requisite knowledge needed to make incremental advances continues to increase, we risk a technological platuea as fewer and fewer people will have enough knowledge.
One solution is to teach humans more, and I'm curious if technology has or can facilitate this.
It really seems to boil down to how the software is used. I'm not sure how effective simply presenting the steps is, but some of the most well validated intelligent tutoring systems are built around simple algebra problems like these.
The big difference is that they test students on each step, and try to give useful feedback if they get a piece wrong.
(although, I just glanced at the wikipedia article for a tutoring system and it doesn't seem conclusive, so maybe I need to look again.. https://en.wikipedia.org/wiki/Cognitive_tutor)
I don't know of any studies, I'd love to see some!
But, I'm a little torn on the concept you're getting at, which is whether seeing answers is less helpful than struggling to find answers and arriving at them yourself without having seen the answer first.
We do have a strong and pervasive belief in our society that the struggle itself is important, and that struggling to derive how to get to the answer without someone giving it to you is the only "right" way to learn.
(The same goes for money, btw, but that is a meta topic for another time...)
In many ways, I believe in struggle myself, but I don't have any concrete scientific evidence, I'm just becoming aware that it's a belief and not necessarily a truth. Recently, as a parent, I think I'm seeing some evidence to the contrary. When my kids ask for math help and I force them to struggle through each step and think about how to do it and explain and show their work, it works eventually, but it takes a long time and it is a struggle for all of us. When I show them the answer first, and then we talk about it later, they learn quicker with less struggle. Usually I will make them rewrite anything I show, but I'm starting to feel that learning by example without the forced struggle is a lot more efficient.
I still want them to be curious and interested in researching their own solutions, so of course I'm a little worried that by doing too much handing out of answers, I might do damage to their desire to explore math (or any subject). But so far, I'm not seeing that, I'm seeing increased interest and enjoyment in math, we spend more time talking about subjects beyond homework.
In some ways it makes sense, we learn how to talk and eat and behave by example, some subjects we can only learn by example (like, say, history). Math and physics are weird ones where we pile on extra struggle to derive the rules because we think it's helpful for learning.
Anyway, I'm certain struggling to learn rules is important, I'm just becoming less certain that it's always important. I do believe that learning by example works and is useful and sometimes more effective than learning rules.
They studied using the system in 5 groups (taking the same class) and got a statistically significant delta in the 3 experimental groups' scores vs. the 2 control groups.
Both exp and ctl solved the same set of exercises, both worked with a teacher, but exp groups also used the system.
The results about the learning deltas haven't been published yet, but you can e-mail her at <np at mathdip.org> if you're curious.
It's an n of 1, but when I went back and reviewed calculus and linear algebra last year, using symbolab.com to talk through the steps of various problems was a huge help in refreshing my knowledge. In some cases, it even helped me really grasp a concept that I never really had learned well the first time around.
A system that can break a problem out into steps should be able to assess the students understanding of the steps and even give them appropriate practice problems.
the biggest cheating is done by the TA's and profs who don't even grade homework anymore. they just send you to a website where a $100 per-semester, per-student SaaS daemon runs a glorified strcmp() on your answers and marks your problem wrong even if your answer is logically correct but the strings dont match.
My idea was to use planning and A* search to solve any type of math problem, even create probes for things like the quadratic equation https://en.wikipedia.org/wiki/Quadratic_equation . I gave up after learnt the search space was so big for it that it was impossible to solve. If I had to do it today I will explore deep learning as heuristic, but I think it probably wont work.
I always like to see this type of projects, I hope they succeed where I failed.
It was during my first months in my PhD when it wasn't clear what I will do. My idea was to define the theorems as operators of planning algorithms and then to use the planning techniques to create new theorems or probes. But planning is not good for long plans as theorem proving, so we moved to something simpler: linear equations and A* algorithm. Then we found the math equations have different representations for the same thing and the branch factor is quite big. Moreover the heuristic of trying to minimize the equations doesn't give good results some times. That is why the prototype works for the example, but for other examples the a* gets lost in the search because of branch factor and equation representation.
The problem with the equation representation is that if you don't find a good one, then you cannot make searches in hash tables efficiently. You end up with a lot of equations duplicated with different representations. And the representation, I used trees for it, was important for the operators.
Planning was a very nice idea because the algorithms already deal with the heuristics. But algorithms as FF http://www.cs.toronto.edu/~sheila/2542/s14/A1/hoffmannebel-F... wouldn't work due to the branch factor and the relaxation of the problem it performs.
It's a nice program and I can see it being both helpful and harmful. From my perspective, as a teacher of mathematics at a community college, students are unwilling to engage in thought about a problem. If they can't see the solution in a few minutes then they want to look at a complete solution. Mostly they are not willing to struggle through a problem.
I vacillate on whether, with the advent of computer algebra systems, it is necessary for students to master algebraic manipulations. I started to think that conceptual questions are better.
For instance, give me an example of an equation with no solution. Explain how a baseball player can have the highest batting average the first half of a season and in the second half of a season but not have the highest overall average. Draw the graph of a function defined on [0, 1] but has not maximum or minimum.
Students can't do those types of problems either. They are very frustrating problems for students because it requires you to really think about what the words mean and to think of extreme situations. So I've reverted back to the traditional style of teaching math. Manipulation of symbols.
When I was in year 12, 15 years ago, my math teacher introduced our class to a website called "Wolfram's Integrator" which could simplify complicated integrals automatically for you.
At the time I thought it was pretty cool in a passing trivia kind of way but didn't make much use of it.
A year after that I was a first year university student all of our linear algebra tutorials were taught in the labs using a computer program called "Maple". I really struggled to wrap my head around it. I didn't do well in the class until I started writing out the problems myself and solving them on paper.
I found at least for me personally that inputting problems into a computer and having it spit the answer out wasn't teaching me anything (besides which functions to call), in other words I was learning the programming language and not the underlying concepts.
Nowadays I work with FEA and LP solvers and I rely on computer assistance all the time to do my job. I'd like to think having a firm grasp of the underlying math is advantageous and makes me a better engineer but I know there are people around that get by just by "plugging things into Ansys".
people expect to work in school, not think. the sad part is i think they only have that attitude through rigorous conditioning. in earlier education it always seems like you should be able to do any of the work you're given easily, given the proper effort; if you can't, then there's something wrong with you. maybe that makes sense for teaching little kids fundamental age-appropriate skills, but it just seems to stick around.
Worked on something like this as a hobby project a while ago, but to avoid the complexities associated with solving arbitrary exercises, instead I had it set up as an algebra exercise generator: you start with the solution, which you then (algorithmically) obfuscate by splitting terms and recombining things for a couple of rounds. Never got around to finishing it, but the neat thing is that you've already generated one possible way to solve the problem, it's just how you generated the exercise in reverse.
Another thing that's quite easy to do is to check intermediate steps in a solution for equivalence. You don't even really need CAS, just brute force the problem by probing the equations: set all variables to randomly chosen values, n times and if the sets of results are the same for both equations, you're good.
Anyhow, Socratic looks great and a great deal more advanced and useful than what I came up with, so kudos!
Hopefully the kids of tomorrow end up getting less homework because of things like this.
Consider a system that combines practice and assessment. It could individualize both, reducing the need to force students that have mastered a concept to do repetitive practice.
It might be a big challenge to get such a thing to work well, but let's not look back at our schooling as an anchor for what students today must do.
I once designed myself a system for teaching math online. In addition to using an engine like what they're showing here, my plan was to start from more-or-less the beginning, but also track and categorize the errors a given student makes over time. Errors are not generally made randomly, there's a pattern to them.
Then, we could allow the student to "solve" an equation the way they really should, by skipping over two or three steps at a time, but when we see them do something wrong, we can use our table of "errors the student is most likely to make" to explore the space of possible errors between step 4 and step 5, and give focused feedback about what they did wrong. Using that table there's really only a couple hundred possibilities; if that fails we can always ask the student themselves to break it down more tightly. Presumably if someone were making a business out of this, there would be someone on the lookout for errors the computer can't figure out to add what rules they can to the system. (Though there will always be an irreducible residue of incomprehensible error.)
Teaching a student math would then be about reducing each of these errors to zero over time. You'd have the computer custom create problems that hold a constant probability of the student making an error at some point during solving it; say 20% or so. Then as the student demonstrates mastery, you naturally make the problems more complex as you have to put in more steps to make the probability of error go that high.
Instead of a klunky, chunky "ok now we learn this and you blindly practice it, now you learn this and you blindly practice it, and we hope at the end you've learned everything we taught", you would in theory get a naturally progressive, customized difficulty curve that keeps the student continuously engaged with being about 80% correct, but always progressing forward. This approach also naturally ensures that just because we're covering the quadratic equation this week does not mean you get to forget fractions; once you've seen the simple stuff with integers, we're naturally going to fold fractions back in to the problems, for instance.
There's some elaborations on the theme after that, such as pre-examining the generated problems to ensure that the most likely mistakes are all distinguishable by producing different error output.
But I don't have time to do this. I'm at least reasonably confident it would work, though.
As a teacher I would kill for something like this, even if it's entirely paper-based instead of digitized.
As you said, creating something like this is incredibly time intensive. There is a reason lots of free and paid teaching resources are basically worksheets, a textbook, and an answer key for "documentation." Anything more complicated is so much work.
Even if you knew exactly what you wanted to do and how you wanted to implement it, you still have "beta test" it with students, determine the space of possible errors and misconceptions[1], make corrections, etc.
I've thought about creating something like this for science, but I "only" teach the same lesson 4-5 times a day. Anything I learn, any corrections I want to make have to wait until next year when I teach that lesson again. Then if those changes don't work or I realize I need to take different changes, I have to wait another year to make them. Not exactly rapid iteration.
[1]Not quite as easy as you might think. Students usually can't explain their misunderstanding very well or at all and you probably understand the material too well and it's hard to understand how to solve the problem any way but the correct way.
Interesting. I'm literally 15 minutes' walk from them right now. I wish them well; I'd love to have something like this for my own kids in the 3-4 years it'll take for it to be useful for them.
A lot of things in life you get good by doing over and over. Some people are born with innate gifts, but a lot of people have to put in work to get where they are.
Indeed. But if you can do assessment on an ongoing basis, you can eliminate the work for the students that have already mastered a given concept and individually focus the work of other students, so that they aren't wasting their time trying to solve problems that they lack the conceptual foundation to solve (math is often like this, missing 2 abstractions needed to solve a problem undermines the utility of the practice, it will be basically impossible).
I'm not suggesting that it would remove the need for practice. I'm suggesting that it could be used to make the practice more effective, for students that are doing well and for students that have fallen behind.
I remember feeling pretty awesome because my TI-89 could solve most integrals and even some differential equations.
Towards the end of my education, Wolfram Alpha came out and would not only solve those equations, but also show step-by-step solutions. Although the solver could be used for cheating, I think it could also be beneficial. Previously, a student who reached a point where they couldn't understand a problem would have trouble finding the right information to understand the solution (e.g. in Diff EQ there are many "tricks" that are necessary to understand in order to solve an equation and which are not immediately obvious). However, with Wolfram Alpha and the like, the student can work through that problem and understand how to solve it.
So much this! I had a really time learning calculus, due to these "tricks". Wolfram Alpha's step-by-steps helped me so much. I could have went and harassed a TA, and I still did that occasionally, but I could largely do it on my own with the computer's help.
Mathematica was my tool of choice. I could plot, Manipulate and visualize things. That was very powerful for learning quickly and getting a "feel" for things.
The ability to check if my work was equivalent to the initial and final equations meant I could catch mistakes when I was stumped and trace through my work. I could see where I messed up. When practicing for exams, I would make mental notes of common errors or properties I forgot, and write a portion of "misc idiot lapses" on part of my crib sheet.
> You could always use a calculator but the whole 'show your own working' catch meant you had to do it all manually. Not any more!
You could often cheat (for equation rearrangement questions) if you knew the answer by simply working backwards towards the question, this is often easier than going from problem to solution but still provides all of the steps along the way.
I remember one maths teacher hinting at this trick, especially to understand the derivation of the quadratic formula:-
There is also "solving by assuming the negative". It's a technique for problems like "Given X, prove Y", where you assume "Y is false" and use it to prove "X is false", but X is true, therefore Y has to be true. Sometimes it's easier to do it in that direction than X->Y.
I know this doesn't answer your question, but at Socratic we use an API that we pay the creator of http://mathpix.com/ for. It would definitely be super cool to see an open source library for this :)
My academic math journey stopped at pre-calc, and I had been a C student for quite a long time. HS Algebra II would never have happened for me if I hadn't discovered XMaxima, an emacs-based CAS. Fortunately I took a Discrete Math course before dropping out of college, and it gave me a new admiration for math.
In spite of my weak math background, this has been the most enjoyable comments section on HN I've read so far.
People here keep saying this will change learning and be good for the students, but the only real difference is it's open-source. You can already get step-by-step solutions for more types of problems from Wolfram Alpha, and you can already get API access if you're a 3rd-party developer who needs it:
Very interesting work, and well-explained in the post.
Like many others here, I suppose that in it's basic form this would mostly be used for cheating on homework; although it would certainly be useful for those (few?) students who are truly motivated to self-learn the material, rather than just pass the tests.
One thing which springs to mind is "Benny's Conception of Rules and Answers in IPI Mathematics" ( https://msu.edu/course/cep/953/readings/erlwanger.pdf ), which shows the problem of only focusing on answers, and on "general purpose" problem sets. Namely that incorrect rules or concepts might be learned, if they're reenforced by occasionally giving the right answer.
I think it would be interesting to have a system capable of some back-and-forth interactivity: the default mode would be the usual, going through some examples, have the student attempt some simple problems, then trickier ones, and so on.
At the same time, the system would be trying to guess what rules/strategies the student is following: looking for patterns, e.g. via something like inductive logic programming. We would treat the student as a "black box", which we can learn about by posing carefully crafted questions.
Each question can be treated as an experiment, where we want to learn the most information about the student's thinking: if strategies A and B could both lead to the answers given by the student, we construct a question which leads to different answers depending on whether A or B were used to solve it; that gives us information about which strategy is more likely to be used by the student, or maybe the answer we get is poorly explained by A and B, and we have to guess some other strategies they might be using.
Rather than viewing marking as a comparison between answer and a key, we can instead infer a model of the domain from those answers and compare that to an accurate model of the domain.
We can also use this approach the other way around, treating the domain as a black box (which it is, from the student's perspective) and choosing examples which give the student most information about it.
Now... the only thing remaining is to translate this to common core :).
I say that in jest, but doing so would make common core much easier for parents AND teachers to grasp. There's an enormous divide between those who get it and those who hate it, and providing parents/teachers with something that would help them understand the benefits of common core concepts would be a gigantic win.
Reminds me of how different the learning experience is now. When we were at school (80s/90s), there was nowhere to turn if you didn't have the answer. My parents had an Encyclopedia Britannica set, so at least there was a paragraph to go on. It's amazing how good you became at fleshing out that paragraph into an essay :-)
Now that this exists I think it's worth creating an opensource version of the TI-Nspire for engineers & mathamaticians. Something based on cheap hardware, runs linux, and can implement this + a theorum prover to basically make the most handy lab calculator.
that feels like a nice application of AI in a way. we often use a computer that can help in making a plan (e.g. a kind of map or "steps" as here). this might be nice to help understand problem solving in general. also, nice to see the project is in javascript, that means quite a few non-professional programmers could learn from it.
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.