For various reasons, the school my daughter attends this year was unable to organize things so that she could be in the math class that followed from the math class she had last year. In fact, the class she was put in this term is working on math that she did 3 years ago.
Fortunately, the teacher of that class does not make her (re)do that "old" math with the rest of the class. During class, my daughter sits by herself and works on her own.
So what does she do in math class? She does math problems that I assign her, based on the math that I teach her in the evenings at home, using an online textbook. That is the temporary solution we worked out with her school: I will teach her math at home.
Why me? Why not her dad?
Because this is what dad-as-evening-math-tutor would be like, we feared:
In contrast, this is what mom-as-evening-math-tutor is like, in theory:
(though perhaps a bit more alert, most evenings).
So, I am the designated parental math tutor, and here is what I have learned so far:
- The things I hate about grading still apply. Grading doesn't become more fun just because you are teaching your own beloved child. That is, just because I am teaching my daughter, who is the light of my life and a truly wonderful human being, doesn't make it any less annoying when she turns in a messy page of homework covered with incomplete erasures and crossed out things and a mystifying sequence of answers in no particular order (and no helpful labels).
- For me, Science is easier to teach than Math. In Science, I know how to explain things. In math, some things can be explained by examples -- perhaps many examples of different sorts -- but some things just are. That is showing my limitations as a math teacher, something I also encounter when I teach a quantitative Science course: I explain why I am doing the math in terms of the Science, but I don't typically explain the math itself. I just do it.
- There are a lot more (imaginary) people in (this) Math textbook than in (my) Science textbooks and I don't like some of them. Most chapters of the math textbook we are using describe an impressive array of enterprising teenagers figuring things out involving math. That's nice -- I like the textbook quite a lot, actually -- but I wonder how much the involvement of people -- even imaginary ones -- affects math-learning. That is, are we each influenced by whether we relate to the imaginary people and their imaginary problems? For example, I am not so interested in Josh's questions about the operation of his remote-controlled car or Delores' attempt to figure out which phone plan to get, but I am intrigued by some of the scientific and sociological datasets and the various things we can learn by analyzing them. And, although I do appreciate the real-world examples, sometimes I get tired of all these perky teens and just want to play with the equations.
- When you teach math at home, in the evening, to your child, you can have ice cream during class.
Anyway, despite my shortcomings as a math tutor, we seem to be doing OK with our math-with-mom-at-home arrangement. Even so, once the schedule is fixed so that she can join the right math class at school again, I will happily hand her (and the grading) over to a real math teacher.
Fortunately, the teacher of that class does not make her (re)do that "old" math with the rest of the class. During class, my daughter sits by herself and works on her own.
So what does she do in math class? She does math problems that I assign her, based on the math that I teach her in the evenings at home, using an online textbook. That is the temporary solution we worked out with her school: I will teach her math at home.
Why me? Why not her dad?
Because this is what dad-as-evening-math-tutor would be like, we feared:
In contrast, this is what mom-as-evening-math-tutor is like, in theory:
(though perhaps a bit more alert, most evenings).
So, I am the designated parental math tutor, and here is what I have learned so far:
- The things I hate about grading still apply. Grading doesn't become more fun just because you are teaching your own beloved child. That is, just because I am teaching my daughter, who is the light of my life and a truly wonderful human being, doesn't make it any less annoying when she turns in a messy page of homework covered with incomplete erasures and crossed out things and a mystifying sequence of answers in no particular order (and no helpful labels).
- For me, Science is easier to teach than Math. In Science, I know how to explain things. In math, some things can be explained by examples -- perhaps many examples of different sorts -- but some things just are. That is showing my limitations as a math teacher, something I also encounter when I teach a quantitative Science course: I explain why I am doing the math in terms of the Science, but I don't typically explain the math itself. I just do it.
- There are a lot more (imaginary) people in (this) Math textbook than in (my) Science textbooks and I don't like some of them. Most chapters of the math textbook we are using describe an impressive array of enterprising teenagers figuring things out involving math. That's nice -- I like the textbook quite a lot, actually -- but I wonder how much the involvement of people -- even imaginary ones -- affects math-learning. That is, are we each influenced by whether we relate to the imaginary people and their imaginary problems? For example, I am not so interested in Josh's questions about the operation of his remote-controlled car or Delores' attempt to figure out which phone plan to get, but I am intrigued by some of the scientific and sociological datasets and the various things we can learn by analyzing them. And, although I do appreciate the real-world examples, sometimes I get tired of all these perky teens and just want to play with the equations.
- When you teach math at home, in the evening, to your child, you can have ice cream during class.
Anyway, despite my shortcomings as a math tutor, we seem to be doing OK with our math-with-mom-at-home arrangement. Even so, once the schedule is fixed so that she can join the right math class at school again, I will happily hand her (and the grading) over to a real math teacher.
Explaining things to family can be hard. Somehow I can be the most patient person in the world when teaching Computational Physics to a person who can't do a FOR loop after 10 weeks of the class (true story) despite getting a B in Intro to Programming last year (true story). But when my wife or brother need tech support, I get visibly annoyed.
ReplyDeleteWe ended up home-schooling this year, and the math is handled by the Art of Problem-Solving online courses. Although I could teach calculus myself, it is much easier to let them do it and just provide small amounts of help when my son gets stuck (some of the AoPS problems are tricky!).
ReplyDeleteThe AoPS books and courses don't have the cutesy problems—they are aimed more at kids who love pure math, and are definitely not suitable for all kids (probably not suitable for 80% or more of kids). They've been a great fit for my son, though.
I have been coaching the robotics club and teaching calculus-based physics using the Matters and Interactions book (which I'm learning along with the students).
I did turn over the computer science this year to a different mentor—one of my colleagues at work. I do still end up having conversations with my son about his project, since he needs someone to bounce ideas off of, and I'm the only other person who programs in Python that he talks with more than once every 2 weeks.
My family did something very similar. My father is/was a terrible teacher if you don't understand things the first time, so my mother, who hates math taught me. I am now pretty darn excellent at math, if it gives you anything to look forward to...
ReplyDeleteWe homeschool as the local school system can't handle gifted kids (in fact, they told me they don't even believe in gifted kids). It is both hard and very rewarding to help teach your own child. We hope to keep doing it for a long while but it is hard to juggle it with two full time professor jobs too.
ReplyDelete" In math, some things can be explained by examples -- perhaps many examples of different sorts -- but some things just are."
ReplyDeleteAnd this is where mathematicians and those who teach math (at least those who do it well) shuddered uncontrollably.
The BEST thing about math is that EVERYTHING can be explained. And not just explained with examples. Really, truly explained. As in proved. As in we know we're right and no one can have a different "opinion" or "interpret the data differently" or any of that nonsense.
Proof is the very, very heart of mathematics, and to go without it misses the whole point really. (And I don't know how old your daughter is, but I work with teachers who do elementary versions of proof from 1st grade.)
If that part of the message isn't getting through, your'e not using very good materials. If you post what kind of math you're doing with your daughter, I (and others) might be able to make some suggestions to help you with the 'splaning part.
Note that:
ReplyDelete(1) I didn't say that math can't be explained, just that I find Science easier to explain (ironically, in some cases using math); and
(2) I don't hate math.
Also, we are not "home schooling" her in math because the school school isn't good enough -- in fact, my daughter's (public) school does an excellent job of educating kids of all abilities. There was just a scheduling issue that may be resolved in the nearish future.
ReplyDeleteArt of Problem-Solving
ReplyDeleteLet me recommend this as well, if you're looking for materials. I picked up the Introduction to Algebra book because my sixth grader was getting bored in math, and it's spectacularly clear and non-condescending. If you're irritated with the book you're using, these are really very good.
I LOVED the illustrations, FSP, illustrating a point that cracked me up. More cat pictures and dog cartoons, please.
ReplyDeleteI like the textbook we are using -- this is fortunate because we need to use the one that is being used by the class that my daughter will (eventually) rejoin. I'm not really irritated with it.
ReplyDeleteCome on, FSP -- you totally avoided the point that MathTT is making. In fact, I don't even know why you said what you did at 03:44. You sound like you're trying to defend not explaining math -- because it's not very easy for you? MathTT's point is not that math can be explained, but that doing so is a wonderful thing and is central to math itself, i.e., important. And he/she is even offering to help! I can understand that for practical purposes it wouldn't work out for you to explain as opposed to show by example, but your response was very odd.
ReplyDeleteAnd who accused you of hating math? My my, you sound terribly defensive.
I'm not a mathematician, but I do appreciate the fact that math is the only place where anything can be proven.
Have you considered using the Khan Academy online? It is free and has practice problem. I highly suggest it and your daughter could learn in during the day at school with a laptop and head phone.
ReplyDeletehttp://www.khanacademy.org/
".. so my mother, who hates math taught me. " (Doc, 8:30)
ReplyDeleteI think that is the relevant reference about math-hating moms. Seems reasonable to clarify.
In a way, the comment that says 'All things in mathematics can be explained' is missing the point.
ReplyDeleteYes everything in mathematics can be proven, but only from certain axioms. At some point, someone can ask 'Why are all right angles congruent?' (That is Euclid's Third Postulate.) To that, there is no possible answer, especially if you take seriously the idea that Euclid reduces geometry to Aristotelian logic so that one can agree to all the theorems without having any geometric intuition. All you can say at that point is that mathematicians have agreed to certain rules as the starting points of the game.
In contrast, in science, one can always point to empirical experience as a motivation prior to science itself.
I refuse, refuse, refuse to grade my own kids' work. Period. Not gonna happen.
ReplyDeleteWith my older boy, I expect him to take some sort of standardized exams at the end of any course of study. So last year, he studied US History, and then he took the CLEP exams for both semesters. I figure that is sufficiently objective proof that he understood the content.
My younger son is doing his math through Stanford's EPGY program, and I am SO glad they do all the assessment and whatever else. We just hand it over to the school when they send us info on completion and such.
I'm curious why you couldn't just send the homework to the teacher at the school and have hir grade it rather than dealing with the hassle yourself.
And the one thing I have to disagree with: I find it easier to teach math than anything else. But to each their own.
The part about science teaching being different from math teaching makes sense to me. Just because someone is good at one, doesn't mean they are automatically good at the other, even if we use math in our science work. I think it is an interesting point.
ReplyDeleteAt the risk of sounding defensive, I grade my daughter's math homework for the same reason we grade our students' homework -- it's the best way to see if our students are understanding the material, if there are particular problem concepts, and so on. After I look over the homework, we talk about the mistakes, discuss strategies etc. It's necessary, alas.
ReplyDeleteI apparently misunderstood. I check my kids homework all the time to look for errors and help them through. However, I refuse to assign grades for assignments or classes. I don't know that I can be objective, and I doubt anyone would think I was. Therefore, when I've homeschooled my kids, I expect them to take some sort of assessment given by someone else. When you said grading, I understood that to mean 'assigning a letter grade', not simply checking the homework.
ReplyDeleteOh, I see. I misunderstood you as well. I used the term "grading" in a vague way, I see now. I don't actually assign a number or letter grade. Every once in a while, my daughter shows her teacher what we are doing and the teacher says "keep up the good work". Eventually there will be some grade assigned, not by me, but perhaps with some input from me.
ReplyDelete"I didn't say that math can't be explained..."
ReplyDeleteWell, no. But you said *some* things can be explained by examples. I disagree. Explanation and example are completely different things. You can search for examples that are what I might call "general in principle," and then use those to build an explanation. But they are not the same thing, and you can't explain with examples.
(And I did read "some things can be explained.." to imply that "other things cannot be explained; they just have to be taken on faith," which is really how a lot of people feel about math... that it's just a random set of rules that must be memorized. Sorry if that was not your intention, but it seems a reasonable way to read the sentence you wrote.)
The bit about dependence on axioms is a bit of a red herring, really. I'm not talking about teaching from axioms in elementary school. I'm talking about things like: once we know what "even" and "odd" mean, we can explain why even + even = even and odd + odd = even. We can explain why, based on how the operations work, negative * negative = positive. We can explain, based on how multiplication and division are related, why it's "illegal to divide by 0." And so on.
But the idea that *explanation* and *generalization* are central to mathematics should really be part of the game from the very beginning.
Otherwise, it would be like teaching science and never talking about observation, or experimentation, or hypotheses.
Dependence on axioms is not a red herring, because most math (at least up through the first semester of high school algebra) is quite close to axioms (or at least facts that should be taken as axioms at that level).
ReplyDeleteFor example, for the question of why the product of two negative numbers is a positive number, one can (and should) come up with an explanation that this is required to make the distributive law work. One can to some extent explain the distributive law for positive numbers. But our extension of the distributive law to negative numbers is just a convenient convention.
There is a second sense in which mathematics is hard to explain. Everything (other than axioms) has a logical explanation, but the process of finding that logical explanation doesn't seem at all logical. In science (at the pre-college level), it is almost always easy to take a statement and figure out what experiment or observations would be needed to verify the statement. In math, though the steps to take are logical, figuring out which of the many possible logical steps will actually help in solving the problem requires intuition and experience.
I do think that some math principles are much more clearly and easily demonstrated with examples, and that in some of these cases, the underlying principles will become apparent. But that doesn't mean that a good explanation from first principles isn't to be preferred. I don't think I agree that math is intrinsically harder to explain than science -- a lot of science is very difficult to explain. If it were so easy, science wouldn't be considered a hard subject. Both science and math can be rendered with elegant explanations.
ReplyDeleteI also don't agree that the best explanation for the positive product of two negative numbers is that you need that to make the distributive law work. That's totally backwards. The outcome is such *because* the distributive law works, and it is *not* a "convenient convention" to have it apply to negative numbers. If you posit the existence of negative numbers, then the distributive law must apply to them because they are an additive entity, just like positive numbers, and the distributive law derives from the fundamental nature of addition.
Dear quasi human, as a mathematician I would love to hear more details of why the product of two negative numbers is positive is to make the distributive property work. Oh, and how is it that the distributive property for negative numbers is a convention?
ReplyDeleteThese results will have pride of place on my wall next to the geometric "proof" that pi is rational and my all time favorite "proof" of the Riemann hypothesis.
FSP, I just noticed "quasihumanist" was autocorrected. If you do post that last comment, please, if possible, correct the salutation. I didn't mean to be insulting there.
ReplyDeleteAccording to student evaluations of teaching (for what they are worth) and a few other indicators, the math professors and other math instructors are by far the worst of any department at my university. Perhaps some of that relates to the fact that many students don't like math and don't want to be taking a math class but it may also indicate that many mathematicians are bad at explaining math (by examples or not) and some don't even try.
ReplyDeleteWow, I was your daughter when I was in HS! My rural public school didn't offer very much acceleration but wouldn't, at the time, let me out to take more advanced college classes. So my family worked it out with the school that I would take math tests with the rest of the class but sit in the back (or elsewhere in the school) and work at my own speed on problems assigned by my dad (a math professor), which I would then go over with my dad.
ReplyDelete