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.
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 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.
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.
1.9^3 = 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.
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.
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.
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.
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.
It would if the results they reported were the initial steps of a binary search which got very lucky.
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.
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.
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!
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?)
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!
This has never caused me problems.
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...
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:
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.
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.
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.
Awesome software though.
"If you can't explain it simply, you don't understand it well enough." -Einstein
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.
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)
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.
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.
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.
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.
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
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 :-)