Stepping into math: Open-sourcing our step-by-step solver
blog.socratic.org
blog.socratic.org
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.
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
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...
(Edit) like these: https://orsjoforskoleklass.files.wordpress.com/2014/11/img_3...
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).
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...
We used this video (and channel) and it helped lots:
This has never caused me problems.
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 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!
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.
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.
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.
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.
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.
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!
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...
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.”
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.
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.
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
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).
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.
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?
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.
It would if the results they reported were the initial steps of a binary search which got very lucky.
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.
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 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").
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.
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).
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.
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.
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.
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.
> 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.
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 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.
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.
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.
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
I'm not sure I get what you mean by "working". Can you expand that a bit?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
...
>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.
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.
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.
```
"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.
```
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 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.
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.
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.
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?
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.
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 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.
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.
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!
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.
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.
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.
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?)
Awesome software though.
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 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)
"If you can't explain it simply, you don't understand it well enough." -Einstein
Also, see the next answer for a more direct source. The list of misattribution is entertaining, too.[2]
[1] - http://skeptics.stackexchange.com/a/22409
[2] - https://en.wikiquote.org/wiki/Albert_Einstein#Misattributed
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.
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.
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.
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.
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.
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.
I just googled it and it still exists: http://www.wolframalpha.com/calculators/integral-calculator/...
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".
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!
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!
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'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.
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 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.
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.
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 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:-
ax^2 + bx + c = 0
x = ( -b +/- sqrt( b^2 - 4ac ) ) / 2a
A -> B iff not(B) -> not(A)
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 :)
In spite of my weak math background, this has been the most enjoyable comments section on HN I've read so far.
http://www.wolframalpha.com/input/?i=2*y+-+x+%3D+(8+*+x+%2B+...
I don't think it will have any real effect.
Source: Paid for it for that feature.
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.
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.
https://en.wikipedia.org/wiki/Common_Core_State_Standards_In...
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 :-)