Mathochism: Show your work
One woman’s attempt to revisit the math that plagued her in school. But can determination make up for 25 years of math neglect?
It’s been little over a week since I dropped the calculus class. I find myself trying to unpack what went so wrong in this last outing, and what I need to do to prevent myself from going down the wrong path again.
A commenter wisely reminded me I do not necessarily need to be enrolled in a class, with an instructor, homework or tests to learn, and that’s true. Auditing the summer professor’s pre-calc class was a great experience, because I could just sit and absorb it all without stressing about deadlines. Doing this also made the next pre-calc experience less terrifying.
But I also know myself enough these days — a benefit of middle age — and I need the structure of a class, and the deadlines, and the accountability. Blame more than a decade in the newsroom — nothing fuels inspiration quite like desperation.
That said, I plan to use my ample leisure time (ha!) to try and get ahead on the material, and I’m hoping any readers who stick with me, particularly the mathematicians among you, may be willing to let me crowdsource when I come across a baffling problem or technique.
I promise to show all my work. In fact, I came across a lovely little program the other day that graphs equations. There are some limitations — it doesn’t do cube roots or odd roots or anything other than a square root. But it’s still a great tool. I was able to graph the sandwich theorem problem I was agonizing over a month ago, and that helped a lot.
And speaking of showing my work — I have a major peeve. Why is it that instructors and math book authors think it’s okay to skip steps in worked examples? Not all instructors, of course. And not all books. But these days, as a seasoned (albeit overly peppery) math student, I tend to judge the quality of the teacher, and the book, by how willing they are to NOT skip steps.
And when they do, they get the “bzzzzzzzt” of disdain from me.
Yeah, yeah, I get all that about “saving a line.” But I have mad respect for that teacher, or that author, who is willing to cover several blackboards, or several pages, in order to make sure the student doesn’t get lost in some leap from step 14 to step 16. Every time I’ve gotten lost, it’s because there’s been a leap of some sort. And I don’t care how obvious that leap may be to the teacher or author, who unlike me has a PhD. It’s still a leap, and implying I’m stupid because I fell in the gap just makes that individual crappy at their job.
Besides, isn’t math supposed to be precise? Isn’t skipping a step imprecise? Those dreaded geometry proofs never allowed me to just say “for heaven’s sake, it’s an obtuse triangle!” No, I had to go through 10 steps discussing angles and sides, citing theorems and corollaries and using terms like “identity” and “substitution.”
It’s particularly galling when an instructor skips steps on examples in lectures (and so does the book in its ridiculously expensive solutions manual that promises to solve all the odd problems yet leaves out a third of them), then expects me to “show my work,” all my work, and even takes points off for paraphrasing a little on a definition or for not writing f(x) on every step.
To borrow from a famous Seinfeld character, “No leaps for you!”
Isn’t that sort of behavior assy as hell?
AHEM. Disclaimer: The above situation is purely theoretical for the sake of a more amusing blog. It’s not like it just happened on a regular basis, or anything. Or that it also happened frequently in a certain Intermediate Algebra class. Really.
All text copyrighted by A.K. Whitney, and cannot be used without permission.
Speaking of omitting steps, you might want to take a look at: http://lithops.as.arizona.edu/~jill/humor.text and search down for “Definitions of Terms Commonly Used in Higher Math”.
Oof. That sounds painfully familiar! Also, I get the distinct feeling that, had I stuck with STEM, I would now be an engineer.
As a non-mathematician, the whole skipping step thing sounds like an annoying double standard. After all…you’re learning math…seems like you’d need for those steps to be crystal clear.
Surely you’ve seen a variation on this article by now? 36 Methods of Mathematical Proof: http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Challen/proof/proof.html
Ha! Love it!
If you have an iPad, you should get the wolfram alpha app. You can enter problems into it and it will not only tell you the answer but it will show you the steps as well. I just finished a calculus course at Athabasca university and it helped quite a bit.
It’s because the math student is being pushed to fill in those steps themselves, as to make the necessary reasoning into “second nature.” It encourages active reading so that you solve problems as you read, rather than being able to just sit back–and since every problem you get under your belt is another trick you’re more likely to be able to recall later, the practice is profoundly helpful for those of us who are training to become professional mathematicians.
In fact, it is standard practice to write the statement of the theorem or example down, close the textbook, and then force ourselves to solve the problem–then look back in the book to try and pick up a more elegant solution, or find any holes and references we missed in our formulations. If we absolutely can’t figure something out, we might ask a friend, but it’s generally considered rude to trouble a mathematician (or anyone much more skilled than you are) with something elementary, and counterproductive for all involved. They waste time transmitting information, and you waste a problem that could have taught you something. It’s seen as your duty to figure things out on your own.
A fully introductory textbook will not skip any steps, and that is good for students who need that. However, a more advanced student needs a textbook that encourages curiosity, exploration, and observations about the material. A more advanced student needs a textbook with holes, and one that explains absolutely everything does us a disservice.
I hope that clarifies the situation somewhat.
The reason that your professor is make you write all the steps is because math is a langauge. He is checking that you understand how the language works. Being able to consistently write the equations correctly is the most important part of math.
Disclaimer: I have a degree in math and also abhor skipping steps. The solutions will lose me too if steps are skipped. I believe there is a BIG problem with how math is taught in the US. I firmly believe that no one is bad at math, they just had bad teachers.
Oh, I hear what you are saying, Vector. But I still disagree.
There is no reason an advanced book shouldn’t also be comprehensive. Relating advanced concepts to the simpler ones in a clear, no steps-missed way can make a huge difference to the student. It also reduces frustration and encourages curiosity.
As for troubling a more skilled mathematician with a simpler problem, I’m not speaking of random mathematicians here, but about the more skilled individual that has been hired to teach the subject to you in a classroom. If a student is made to feel it is rude to ask for help, or that they are stupid for needing it, that’s a real problem. Asking questions, even simple ones, is part of the process.
The process you describe makes it sound like everyone is just pretending to know the answer and then struggling needlessly, because not knowing means being made fun of.
“If you want to improve, be content to be thought foolish and stupid.”
I definitely think the student should write down all the steps! I just don’t agree that the teacher or book shouldn’t, because it sets a bad example. And speaking of learning a language, it’s a bit like insisting a student always pronounce some words perfectly, and always use perfect English grammar, but it’s okay for you to mumble and say “ain’t”.
As you can tell from the references Andy and I supplied, missing or incomplete proofs are a running joke among mathematicians and math students. That said, there’s actually a serious point lurking among the wisecracks. The ability to carry through a formal proof is the key to advanced mathematics. Part of why a geometry class insists that you fill in all the details is that this is where formal proofs are traditionally introduced, and they want to make sure that the students really understand how to justify every step. But once they feel they can count on the reader/student to know how to fill in the missing details for the routine steps, teachers and authors tend to take a few short cuts. Spelling out every single step can get pretty tedious in a long and complicated proof, and it’s possible to lose the shape of the forest while studying all the details of the trees. As long as the teacher/author judges correctly where the audience may need assistance, and where both sides understand how to fill in the missing details, this approach can work pretty well and make for a more interesting lecture/book. But when they misjudge the audience, or worse, make mistakes or typos, things can quickly get pretty ugly.
One of my professors in grad school (not formally a mathematician, but a closely allied subject) was notorious for the detail and rigor of his proofs, and had written the textbook to match. While I appreciate the demonstration of what it takes to fill in all the details, for day to day work, I much preferred one of the classic textbooks in the field, which was a little less detailed, but much better at showing the bigger picture and how all the theorems they were proving fit together. With my professor’s text, it was far too easy to get lost in the details and not remember the important stuff.
That’s a fair point. I’m rather detail-oriented, though, and really enjoy getting lost in the details. I had an example recently where the solution went from 1 + or – square root of 5 to 6 square root of 5. No explanation. I finally figured out that the 6 square root of 5 came from plugging in the answer to the original, non-derived polynomial. It would have been nice if the author could have mentioned that.
Another mathematician’s perspective:
An instructor should be careful to actively encourage asking questions. This way, if a step needs to be skipped for the sake of time, as long as the step is something that’s already been learned and students are willing to speak up, any difficulties can be resolved in class. Worst case scenario, the question is deferred to office hours. I usually only do this when the question reflects a severe misunderstanding of the prerequisite material.
But that said, steps *must* sometimes be skipped. There simply isn’t enough time in class/pages in a book to give every single detail at this level, since you’re doing pretty complicated math now. The steps that can be skipped in the book or in class are steps that have been clear for at least a chapter, IMO. Which means those are steps you can skip, too! Or, if you don’t understand the intermediate steps while you’re reading, you should try to work them out. And by all means, if all else fails, you should be encouraged to ask the instructor!
At the advanced undergraduate (senior-level math) or graduate level, it is indeed common practice to leave out steps or some of the proofs entirely. This is absolutely essential to the learning experience. Experiment after experiment in math education has confirmed that especially beyond undergraduate mathematics, students who are required to fill in details and eventually whole proofs are usually able to retain the material, and those who are not have almost no chance of being able to prove problems outside of the book. It’s not really just a matter of opinion, but a matter of well-tested scientific theory.
Here is a link to my favorite online graphing calculator, which does do cube (and other!) roots! Hope it’s useful for you- I love the clear display. https://www.desmos.com/calculator
Awesome! Thank you!!!!