The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.

June 13, 2012 2:32 PM

My younger brother has just finished tenth grade. His math teacher was a martinet. His math teacher next year might turn out to be one too. How can I help my brother cope with this kind of instructor?

A confession: I hated tenth-grade math ("Algebra II with Trigonometry"). It was a grab bag of topics introduced incoherently and without motivation. Textbook problems made assumptions and sometimes failed to make them clear. By comparison, math in eleventh ("pre-calculus") and especially twelfth grade (intro univariate calculus) was a breath of fresh air. I am several years out of college now and I'm taking math classes for fun. Unambiguous definitions are awesome. Rigorous proofs are awesome. Math is awesome.

Math is not my brother's best subject, but he is eager to improve and often turns to me for help with homework. However, he often comes back with feedback that shows that most of the problems I helped with have been marked incorrect or had points deducted.

After some back and forth it became clear that my brother's math teacher was a pedant. He deducted points for mistakes made in students' scratchwork, even when the reasoning was sound and the ultimate answer was correct. He deducted points for "skipping steps." For example, when rewriting fractions with radical expressions in the denominator, he apparently wanted to see an explicit step with multiplication by sqrt(2)/sqrt(2), or whatever. He was a stickler for arbitrary notation: one I discovered through trial and error is that he only wanted sequences and series to be subindexed with n and no other letter. I habitually subindex with i.

My brother's work was bleeding points left and right, in no small part thanks to my help. This, in combination with his otherwise imperfect performance, was making math very frustrating for him.

I help with homework via video chat and do not have easy access to his teachers. How can I best support my brother's learning, especially in math? If you have been in a similar situation (as either the student or the tutor), what strategies worked best?

posted by Nomyte to Education (21 answers total)

He deducted points for mistakes made in students' scratchwork, even when the reasoning was sound and the ultimate answer was correct. He deducted points for "skipping steps." For example, when rewriting fractions with radical expressions in the denominator, he apparently wanted to see an explicit step with multiplication by sqrt(2)/sqrt(2), or whatever. He was a stickler for arbitrary notation: one I discovered through trial and error is that he only wanted sequences and series to be subindexed with n and no other letter. I habitually subindex with i.


Well, some of this your brother needs to make himself aware of; he's the one going to class, so presumably he should be aware of what kind of work the teacher wants (and is modeling). Ideally, you're helping with the ideas but not necessarily the notation (so, if you (and I) index with i, but the teacher wants n, then your brother hopefully would use n when he writes up a clean solution). If the teacher makes it clear that he needs to see all the steps, then your brother should be writing all the steps. Presumably, there can be point dockage on the first few assignments as your brother learns the teacher's expectations, but then once the expectations are clearer, fewer points would be deducted. But the point is that it's your brother who needs to be determining what gets written down on the homework, and that that's part of what he needs to be learning in the course.

Does your brother take notes? Does he take notes other than what the teacher writes on the board? For example, sometimes teachers make comments about what they want to see in terms of good solutions, but don't write it on the board---but that's a good thing to make a note of.

Why is the teacher seeing scratch work, anyway? A good idea---in any math class---is to make a final clean copy to turn in, which can also serve to check your answers (do they make sense when you write things up again?). (On the other hand, if you said scratch work but meant the actual work in the problem---the steps---it's reasonable and typical that points would be deducted, even if magically the right answer appeared at the end.)

I do sympathise, though, with how frustrating it is to have teachers that seem/are unreasonable, and I'm sad to hear that your brother had one. I have taught a number of students who were doing secondary certification in math who didn't seem to actually like math, and I'm sure it's very hard to learn from someone with that attitude.
posted by leahwrenn at 2:48 PM on June 13, 2012 [4 favorites]

There's something of a difference between the examples you gave. For example, although to a good student it is clear that you can multiply top and bottom by the same quantity, it's not entirely unreasonable to require students to show that they did that. Although I think it's a bit dumb to deduct marks for it. (I have a lot of trouble making myself write out arguments fully formally. One of the great joys of doing maths at undergrad was no longer having to show stupid steps, because the markers had faith that I was actually moderately competent.)

On the other hand, deducting marks for the "wrong" dummy variable is just insane and stupid. As is marking the scratch work, which is scratch precisely because you don't want it marked.

Do the grades matter? In this case I would seriously tell your brother that he should just ignore the stupid marks (but not the ones that are actually for important things).
posted by katrielalex at 2:50 PM on June 13, 2012

It is not uncommon for a math teacher to deduct points for errors in scratch work, but I've never had a teacher deduct for skipping steps. It's not just the answer a teacher is looking for, but also the process at which the student arrived at the answer that's important (so the teacher can see if the student is understanding the entire process).

Your brother (and maybe a parent) needs to have a discussion with the teacher so he knows what is expected of him and what the teacher wants to see in the scratch work. It's all about communication.
posted by ATX Peanut at 2:52 PM on June 13, 2012

_{I think we have different definitions of "scratch work". I would use it to mean the scrap you do before writing your argument in full. Naturally, in the final answer you have to include all the intermediate steps -- it's no good saying "the answer is three" without the proof.}
posted by katrielalex at 3:05 PM on June 13, 2012

The teacher might just be trying to verify that your brother really does understand the work he's turning in, and that it is his own. I mean, if you work with him and write your subscripts with i all the time, then if your brother truly understands what those subscripts mean and do it should be no problem for him to switch them to n in the final product, right?

The teacher has a case for saying that if he doesn't follow the basic instructions it is evidence of a lack of understanding. This is not to say that your brother doesn't understand it, but, well, the point of turning in homework is to demonstrate understanding, right? This might be showing up in his tests, since you say his other work is imperfect.

So, what to do? Since you video chat with him, make him attempt the problem before you work on it with him. Then you can see for yourself what he is understanding and what he isn't. If the convention in his class is to index series with n, then his attempt should include a series indexed with n and you can avoid the whole problem of differing notation. Does the instructor hand out a syllabus? Information about conventions might be on there.

I'm with the teachers on this. A lot of teaching is assessment of gaps in understanding, and if work is unclear, inconsistant, or unconventional in notation, it's pretty hard to get a grasp of "mistakes" versus "lack of comprehension." You may have problems with the cirrucula as presented but the teacher probably has no control over that, and needs all the tools they can to teach.
posted by newg at 3:09 PM on June 13, 2012 [1 favorite]

If you have been in a similar situation (as either the student or the tutor), what strategies worked best?

Don't try to get around the teacher. Find out exactly what the teacher wants and then give the teacher exactly what the teacher wants.

Also, steer your brother to supplemental sources such as Khan Academy, but make sure he knows that Mr Martinet's requirements trump any different methods your brother might see elsewhere.
posted by pracowity at 3:18 PM on June 13, 2012 [1 favorite]

Are you doing your homework for him or is he doing it himself? It sounds like you're doing the problems for him. The teacher doesn't have a ton of time to grade papers, and he's not going to go out of his way to figure out notation he didn't tell the students to use.
posted by empath at 3:25 PM on June 13, 2012

Your brother needs to use the notation and workflow that the teacher uses in class. He should be able to explain to you, when you are tutoring him, what the notation and required steps are. If not, I think that signals he may not understand what's going on in class or that he didn't take good enough notes. If he can take better notes on how the teacher works the problem in class, that could give him insight into how he should be doing the problems on his homework.

I realize this may not be fair. But, from the teacher's point of view, it can be difficult to grade so many papers each using different notation. The best teachers are more flexible than this. I always tell students that part of doing math is communication. You need to effectively communicate mathematical ideas to whoever is reading your work, and sometimes it means knowing your audience, and using their (at times pedantic) notation and workflow.
posted by bluefly at 3:44 PM on June 13, 2012 [2 favorites]

Some possible reasoning, as understood by the husband of a high school math teacher:

Skipping steps is a bad habit. It's easy to skip steps early on when everything makes sense, but when you get used to doing math in your head, you can't easily track back where you made mistakes. No one can correct your work if they can't see it.

Using methods that were not taught in class could imply that you're copying from someone, and there might be really strict rules against copying in this class or at the school.

The teacher has to grade ~30 students for ~6 classes. Maybe s/he outlined specifically what s/he wanted to see on homework, or s/he expects students to pick up on those details as the class goes on. Either way, as bluefly said, learn how to communicate with the intended audience.
posted by filthy light thief at 3:48 PM on June 13, 2012 [3 favorites]

I have graded homework involving math before, but only at the college level.

If a student showed their work incorrectly but came up with the correct final answer, I would definitely deduct points. That means they accidentally got the correct answer (or cheated) and that doesn't deserve full credit. I don't know what you mean by "scratch work". If a student gives me scratch work, then he presumably expects me to grade it. If it is incorrect, then he gets points off. If it's illegible, he gets points off.

If a student didn't show enough steps, and I had asked them to, then I would also take points off. The truth is that a lot of students cheat, so you need them to show their work to make sure that they understand what is going on. They need to show enough work to demonstrate that. How much work is enough is arbitrary, and defined by the teacher. Hopefully this is communicated clearly to the class.

If a student used notation that was different than I showed in class and different than the book, I would assume someone outside of the class is doing his homework for him. I would probably not deduct points before asking him about it, but I could see deducting for not adhering to the rules, if I had set them out that way. Besides the suspicion of cheating, there's the problem that it makes grading more difficult. You could argue that it doesn't make grading that much more difficult. By the same token, it's really not that hard for your brother to just follow the rules.

I think the best way for you to help your brother is to help him less. It sounds like you are leaving your math style too strongly on his work. He needs to make it his own, and tailor it to his class.

Good luck.
posted by pizzazz at 4:14 PM on June 13, 2012 [8 favorites]

If a student showed their work incorrectly but came up with the correct final answer, I would definitely deduct points. If a student gives me scratch work, then he presumably expects me to grade it. If it is incorrect, then he gets points off. If it's illegible, he gets points off.

Seconding this. I grade everything that is written on the page. There is no exception. I give the students copious amounts of scratch paper which they can scribble on, but the final result is on the original document, and that's what I look at. If they want me to look at their scratch papers, they turn them in, otherwise they throw them away on their way out of the room.

You do have a leg to stand on concerning the i versus n thing. Also, if a student writes 2/ √2 = √2, I am fine with that, because it is correct, even though they didn't "show me" that the numerator and denominator was multiplied by √2. He might be able to successfully argue that also.

You can always argue when the mathematics is correct. Especially on the index thing, and even more especially if you can find instances in the text where the index is sometimes one letter and sometimes another. If he's actually making some mistakes but still getting The Right Answer, it's not worth your brother's time to argue.

Again, there's also the possibility, that the instructor said "you must always use n as an index in my class, I don't care what the text has", in which case your brother needs to learn to be flexible and use the notation desired by others.

Also, the title of your question is screwed up, unless you're referencing Homer after he finds Henry Kissinger's glasses, which I'm pretty sure you are.
posted by King Bee at 4:27 PM on June 13, 2012

He deducted points for mistakes made in students' scratchwork, even when the reasoning was sound and the ultimate answer was correct. He deducted points for "skipping steps."

Not only is skipping steps a bad work habit (and can mask actual misunderstandings), these days affordable-ish calculators and easily accessible online tools will produce well-formed answers for derivatives and integrals. It's just too easy for a student to write a few mumbly guesses at intermediate steps and then--voila!--a correct answer. And i is, at this level, often used to designate imaginary numbers (i.e. i² = −1) so while it's crummy to deduct points if he demonstrably knew that it was simply a correct answer formed in a way he didn't like, it could also have been a simple misunderstanding.
posted by pullayup at 4:34 PM on June 13, 2012

I'm teaching myself some math before I go into Trig class in the Fall.

I have rediscovered that after all these years, my math teachers were right all along. The more steps I write out, the easier it is for me to solve the problem, and get the correct answer. And when I don't, it's easier for me to trace through the steps and see why. So now, I'm a big fan of writing out ALL the steps. Even if it's something trivial like "1+1=2". It really makes things much easier in the end. Same with using a lot of space for each problem, to avoid confusion that can come from crowding all the problems in.

(And I fear for my future teacher, because this means that he's going to get reams of paper every time I turn in homework!)
posted by spinifex23 at 4:36 PM on June 13, 2012 [1 favorite]

pizzazz: I don't know what you mean by "scratch work". If a student gives me scratch work, then he presumably expects me to grade it. If it is incorrect, then he gets points off. If it's illegible, he gets points off.

From the example of a single math teacher, just showing the answers on homework is generally useless. Great, you got to the end, how did you get there? My wife gives partial credit for trying, and she can mark where the student went awry if/when they show all their steps.

College is different than high school in that students pay to go to college, so if they squander their time, it's their (monetary) loss. If you can get the answer, that's most of it. If you cheat, you're cheating yourself, and you won't pass the class, and you squandered your money. But kids are generally required to attend high school, and are required to pass a certain number of math courses. To pass, you need to show that you know what you're doing, so we get back to showing your work, even if you mess up somewhere along the way. Not all teachers teach this way, but it sounds like Nomyte's brother's teacher finds some value in showing your work and doing it correctly (according to the teacher's standards).
posted by filthy light thief at 4:55 PM on June 13, 2012

Response by poster: Yeah, I'm not sure what I was hoping to get when I wrote this question. Clearly, the actual answer is "your brother should be a better student." Can't really do much about that, though.
posted by Nomyte at 6:58 PM on June 13, 2012

I have had the same experience with my daughter's THIRD GRADE math homework. She has a problem with skipping steps, because she just does it all in her head and sometimes doesn't even see that she has combined steps. Her teacher wants to see all the steps. I have the same step-skipping problem, so we always go through after and make sure we didn't, say, do two calculations in the same line of the solution, rather than have a line for each one. I think it's fairly common for a teacher to expect to see all the steps, for all the reasons others have outlined above.

We can also run into problems when I help her because I do things the "old" way, or, sometimes, my own way that makes sense to me. Even math is not absolute! How I help her now is that I ask her to explain to me, in detail, what the teacher wants her to do. If she can't articulate that to me, then I send her back to the teacher for clarification because the teacher wants what she wants, not what I think would be the best way to do things. If my daughter understands what she is supposed to do and how, but that is stumbling on the execution, then I will help her through it exactly the way she has said the teacher wants it.

Anyhow, that would be my approach: have your brother explain what he understands the teacher is asking for and work from there. If the problem is he doesn't even know what the teacher is asking for, then he really needs to go back to the teacher. If he knows what is required but just can't get there, then you can help him along, making sure you hit all the marks the teacher is expecting him to. He may learn more doing things your way, which is fine, but he won't get the marks at school, which may be more important to him.
posted by looli at 7:30 PM on June 13, 2012 [1 favorite]

Echoing the posters who say requiring the student to show all steps, and show them correctly is a pretty common expectation of math teachers. It demonstrates understanding and helps verify that the student is not copying someone else's answers--if you see errors in the scratch work, but correct answers, it means the correct answer was done by luck or found because the student was copying someone's work and mis-copied the scratch work. Notation differences can also be a red flag for copying.

I understand your frustration--I am retaking a math course and am accustomed to the notation I used the first time around and doing some of the steps in my head (like integration). The prof is requiring more explicit work, so I've had to adjust to that.

Here's the thing though--if your brother was a perfect student, got good grades on quizzes and tests and demonstrated a thorough understanding of the material, then the math teacher might be more lenient about some of these things. But you said he's an imperfect student. So if he's not demonstrating good understanding in his in-class work, then the teacher may be strict with him because the teacher is suspecting your brother of being fed homework answers without understanding how to get them. Are you 100% sure that your brother is understanding the work that he's turning in? Is it possible that maybe the math is easy enough to you that you don't need to show all steps, but it would be more helpful for him?
posted by Anonymous at 7:57 PM on June 13, 2012

Nomyte, that's a harsh read of the answers provided here.

It's not about being a better student, but knowing what is expected of you. This lesson applies to a lot more than a few high school math classes, and can start to prepare your brother for future experiences with seemingly arbitrary rules and standards, clarified and not.

Has your brother talked to his teacher about how his homework was graded? I realize some teachers only teach during class, and afterwords can't be bothered to spend time with the kids who didn't get it in class, or who are asking about how something was graded, so maybe your brother tried that and it didn't work.

Thank you for being a sibling who cares enough to spend time teaching your brother, and for caring enough to find out how to make your time tutoring help him more with his grades. And thank you for sharing a love of math, especially that you didn't always find math enjoyable. You can still help your brother, even if you think the teacher's methods are bunk.
posted by filthy light thief at 8:03 PM on June 13, 2012 [1 favorite]

It is also worth noting that being able to gauge the expectations of one's supervisor and meeting their standards, no matter how arbitrary and stupid, is a skill best developed as soon as possible as he'll be using it the rest of his life.
posted by Anonymous at 8:08 PM on June 13, 2012

Response by poster: I'm sorry for putting it too glibly, filthy light thief. My brother does not make it easy to help him, and what he thinks of as "help" and what I think of as "help" are often very different things. I do think that the only "surefire" way for him to improve his performance is to become a better student. Sadly, I can't control that part of the equation.
posted by Nomyte at 8:11 PM on June 13, 2012

The most valuable skill for your brother's future is the ability to be creative in mathematics. Conforming to the teacher may provide temporary success, but it may not actually aid his long term achievement. In my career, they are desperate for creative, original thinkers, not very strict verbose people who do not question assumptions (these people are difficult to work with). Our supervisors want their fundamental expectations to be met, but are especially pleased if we turn those expectations inside out and make something even better.
If I had not dismissed and ignored those who were arbitrary and stupid I would never have made it this far.

It all depends what school system he is in. Does his 10th grade mark count towards university acceptance? If not, can the math class be subverted and the same knowledge be taught in parallel and at a much higher level of effectiveness? This may serve your brother better in the long run.
posted by niccolo at 11:12 PM on June 13, 2012