"A day without math is like a day without sunshine." -- my friend V's mom

[mrissa]

The state of Minnesota is having another go-round with its high school graduation standards, particularly in the area of math. We set up a math test people would have to pass to graduate high school. Surprise! People didn't pass it. Lots of people didn't pass it. Surprise again! There was a great deal of uproar, because what were we to do with these high school juniors who didn't pass?

High school graduation the supposed skill certification and high school graduation the social ritual have become inextricably intertwined in our culture. So it's no surprise that people are up in arms and saying things like, "These kids ought to be able to graduate." I even agree with them, but not in the way they think: I don't think high school graduation is most useful when it's a certificate of attendance. But I do think that if you don't know enough or can't do enough to meet graduation requirements, you should be getting feedback to that effect. The idea that you would have passed all your classes and yet not know the things they feel you should know at that point seems like something has gone wrong, and I doubt that there would be this much uproar if we were talking about kids who hadn't passed their classes--for whatever reason, we are culturally on board with the idea that if you fail math, you don't graduate. But these kids are failing at learning math, and they're not failing math, and that, to me, is a big problem. Sure, we're not talking about students who are passionately committed to mathematics here; not every student is or should be. But we are talking about students whose best indications on whether they know an acceptable level of math for a high school student is that they do, and those best indications are, apparently, wrong.

I'm sure there are people who are totally okay with a math test but not with this math test. But that's not what we hear every time this issue comes around. It starts to boil down to, "But math is hard! You don't really need math! And it's hard!" And at that point, well, what do you really need from a high school education? What can't you work around? There's not a heck of a lot, on the level they're talking about here. If having to calculate the area of a room from its dimensions is too much to ask of high school graduates, I'm starting to think that the people constructing these arguments are, in fact, arguing for a high school diplomat to be a certificate of attendance, a verification of age.

One of the things we are not willing to say in this discussion is that people who can't do math are missing out. They're missing out on ways of protecting themselves, sure, on a measure of independence that comes from being able to do some rough calculations yourself. But they're also missing out on something wonderful. Something beautiful. I know I'm talking to some of you about yourselves, and yes, I'm sorry: you're missing out. That dimension of understanding is worth cultivating. It is worth having. Some of you can't do math the way a person who is completely tone-deaf from birth can't learn to identify a piece of music upon hearing it, but the vast majority of you who can't do math are more like someone who doesn't know any songs because no one ever taught you any. It doesn't make you a worse person. It doesn't make you an unintelligent person. But it's still a damned shame to induce disabilities in people who don't have them to begin with.

I believe that math-related learning disabilities are real. I absolutely do. I do not believe that irremediable math-related learning disabilities are as prevalent as people who were taught math very, very badly, often by people who did not themselves know how to do math.

I don't really know what to do about that. Saying, "Yes, fine, go on ahead and get out of here; it's not like we have any real preparation to teach you math from here anyway," seems practical in the short-term but distinctly suboptimal in the long-term. It treats the problem as one of what to tell the students--yes, you are a high school graduate, or no, you are not--rather than what to do to fix a system that "should have" done something but did not.

It allows us to keep on with math education the way we have been. And on the one hand, we sort of have to. And on the other hand, we sort of can't.

Comments

[mmerriam.livejournal.com]

I do not believe that irremediable math-related learning disabilities are as prevalent as people who were taught math very, very badly, often by people who did not themselves know how to do math.

Seems like I was involved in a conversation about this just a few days ago...

[akitrom.livejournal.com]

Do you have a link to the particulars of the high-stakes math test in question?

I'm looking to return to the secondary math teaching field, after a couple of years as a Test Development Associate, working on End of Course tests at ACT. And I can tell you, there are good reasons to have an EoC test that's independent of, and not scored by, the classroom teacher, but that independence makes it very difficult to test curriculum-specific skills well.

Added to all the normal issues (The kid is sick on test day, or her parents had a fight the night before...) are issues like notation (a kid might know how to answer a question, if only it were phrased in the terminology that his textbook and teacher use), time (the EoC test from ACT comprised 38 multiple-choice questions, and we expected students to complete the test within a 45-minute class, which pretty much messed up students who were used to carefully solving problems and showing all their work), and a half-dozen other variables.

[mrissa.livejournal.com]

Here (http://www.startribune.com/local/stpaul/47047072.html?elr=KArksUUUU) is the most recent Strib article, others available for the Googling or the link-following.

To my way of thinking, when we're talking about a test for leaving high school, if students know math if it's phrased very narrowly, they don't know math. The outside world will not be nearly so careful with phrasing.

And sure, there will always be slow workers and kids who are sick. But 40% failure is no longer looking like, "A is very methodical and B was coughing up a lung on test day." It's looking a great deal more like, "Many of these kids cannot pass this test."

Do you feel that the overwhelming majority of people who graduate from high school know as much math as you would consider a good, reasonable idea for that stage of life/education?

[aamcnamara.livejournal.com]

As a completely irrelevant note to the topic of better math education (which is definitely needed), I'm fairly certain that they didn't have a good baseline for this test--my class was the try-it-out class, last year; we were told we had to take it, but as we'd taken the MBSTs and passing the MCA didn't have anything to do with graduating, a lot of people skipped school/doodled pictures instead of answering questions/filled in random bubbles/turned in blank sheets of paper. And I would believe someone who told me that my school was pretty typical in that regard.

However, since I know nothing about test-judging standards, I may be completely off base here. Maybe they adjusted for that.

And certainly there are many and numerous problems with how a lot of math classes are run. (Though I would argue that standardized testing is also not the best way to test abilities for a lot of students.)

... this comment sounds like I am disagreeing with you. I am not. I am agreeing with you. Just... sideways agreement.

[mrissa.livejournal.com]

I'm not convinced that standardized tests are The One True Way or even A Pretty Good Way, either. But I'm wary of people (not you!) who go from that to, "And therefore the way we're teaching math is probably fine." Because it mostly really isn't.

[stillnotbored.livejournal.com]

Having crashed my computer trying to read the article you linked to (the site's fault, not yours) and googling but failing to find what I wanted, could you please tell me what is on this math test? Is it basic math or is it advanced math?

[snickelish.livejournal.com]

High school graduation the supposed skill certification and high school graduation the social ritual have become inextricably intertwined in our culture.

This, to me, seems the key to the entire debate. The skill certification and the social ritual are put in conflict, and many people prioritize the social ritual.

[cissa.livejournal.com]

I know that when my daughter was in secondary school, and refusing to do any work, we BEGGED the teachers to flunk her for not doing the work. And they wouldn't. So she learned that doing the work is, in effect, optional.

And she can't do math, even though both of us her parents like math and use it a lot.

I think if her math teachers had given her the F she had earned, her pride would have been tweaked and she might have learned some math. Maybe not- but they sure didn't help, passing her- with good grades even!- when she knew NOTHING.

I'm a metalsmith, and I use math daily- couldn't work without it. When I was teaching, it was really amazing how many people freaked out about having to do even simple calculations that had been spelled out on a worksheet, and to help with which they had a calculator. Honestly- not hard... and pretty important.

Math is not only beautiful in and of itself- but it's key to everything from household budgeting (at one level) and being able to enter some technical fields; it's a shame that so many people cut themselves off, and are cut off, from basic competence.

I know many of my elementary school teachers had little grasp of math or science, and that made them boring. I'm reading now that teachers can be certified as competent to teach- including math and science- without answering one math question correctly on their exams. I think this shows, and it needs to change.

[redbird]

On the other hand, a lot of standardized tests will, if anything, make it look as though the students know more math than they do, because they're multiple choice with no penalty for wrong answers. So the students are, reasonably, encouraged to guess.

(Hints like "if there's no penalty for a wrong answer, guess" and suggestions on ways to guess well don't even count as teaching to the test.)

Conversely, there's no way on a multiple-choice test to give partial credit for someone who shows their work and clearly understood most of it, but absent-mindedly turned a plus sign into a minus sign when copying from line 3 to line 4; it looks the same as someone who had no idea of what's being asked.

[I'll spare everyone the bit about "real world" problems that nobody in the real world ever addresses except if they're interested in numbers for their own sake, like the surface area of the leftover three slices from a medium pizza. [at this point, the bailiff stuffs redbird into a sack, head-first, and deletes the rant about pizza area before posting the comment.])

[fidelioscabinet.livejournal.com]

I did usually did well in math classes, and enjoyed them, but did very badly as a high school freshman, because the woman teaching the class was very bad at teaching a subject she loved and was knowledgeable about. There are so many reasons why math and the sciences are handled badly in our schools (although I have to say that I think we handle English, foreign languages and history just as badly at times--but math and the sciences are harder to fake your way through, and harder to pick up on your own.)

I went to a high school in a small town that had a science and engineering-focused university in it, and so, thanks to parental pressure, these subjects were taught fairly well. Public demand has a definite effect on how well things are taught, and people who have the expectation "My kid should be able to learn these topics well enough to get into MIT" are in a really good position to shape the argument when dealing with their local schools. Part of the problem is that enough parents aren't really sure why their children need to learn these things well, and so aren't in a position to make useful arguments about why the schools aren't doing this as well as they could, and how to do it better.

[mrissa.livejournal.com]

It's what I would consider basic algebra and geometry. No trig, no calc: the idea is that this is the minimum for high school graduates, not the minimum for college prep or for someone who considers herself/himself pre-engineering or anything extravagant like that. It was the level of math I had mastered by seventh grade at the latest.

[mrissa.livejournal.com]

Yes. "My friend V's mom," the person I quoted in the subject line, is a grade school teacher who loves math and teaches it well, but she's very rare, as far as I can tell. I know that all of my grade school teachers were of the "this is the formula and you follow it" approach to math and couldn't explain anything beyond that. Secondary school teachers were somewhat better, as they had to actually choose to be math teachers and not some other kind of teachers. But by then a lot of people were already terrified of math, and were already behind in it.

[mrissa.livejournal.com]

Picking up on your own is something I've watched people try to do. It's possible with math. But it's not always easy, particularly when people have learned a fear of math as well as not learning how to do it. I once spent half an hour with a student who could not understand that if T - W = 0, T does not in general = -W. The basic algebraic steps were beyond him. He was trying to learn physics (granted, in a class for the non-major, but still). I was glad to be a gatekeeper preventing him from going farther in pre-med until/unless he learned math, because that level of math ignorance is not okay in a doctor. But it was ridiculous that he'd gotten that far in the first place.

[akitrom.livejournal.com]

Do you feel that the overwhelming majority of people who graduate from high school know as much math as you would consider a good, reasonable idea for that stage of life/education?


Oh, Mrissa; oh, oh, Mrissa.

Let me provide my take, which I understand makes me terribly unpopular with certain of my peers.

On a day-by-day basis, the amount of calculation most people need is taught by the time a student completes 8th-grade Pre-Algebra. The "overwhelming majority" of secondary students -- could have that proficiency by somewhere in their freshman year. Now, high school mathematics ought to be focused on helping students be more than arithmetically proficient, but for most students the curriculum should not comprise techniques that devolve into intellectual exercises.

I define "algebra" as the mathematics of patterns. Not solving-for-variables, but rather variables-as-placeholders to show relationships.

A good general-purpose high school selection of math courses should include:

• a year of basic algebra and geometry (if a school is using some sort of Integrated Math series (http://holtmcdougal.hmhco.com/hm/series.htm?level2Code=MSIB10010&level3Code=3_IM), that's terrific.)
• a semester of probability and statistics, and
• a year of "consumer math", modeling, logic, and problem-solving (incorporating algebra, geometry, and probability).

Two-and-a-half years, a bare-bones curriculum for every student. For a more rigorous curriculum, I wouldn't mind seeing an additional semester of algebra, geometry, and statistics, for a total of four years.

That's what normal people need, to be competent college students and fully-participatory members of modern society. And the proficiency test should be taken immediately after completing that coursework, instead of a year or so after those bare-bones students have stopped taking math courses.

The modern high-school curriculum --Algebra I, Geometry, Algebra II, Trigonometry and Pre-Calculus, leading on to Calculus for good measure-- can be an excellent, analysis-heavy regimen, designed and implemented during America's reaction to Sputnik, to prepare typical high school students for careers in engineering and the hard sciences. As a student, I was inclined to like that sort of stuff, and engineering professors in post-secondary institutions appreciate the fact that their incoming freshmen have seen a lot of analysis.

But seriously, normal people do not need to manipulate trigonometric ratios and employ the half-angle formulas. Normal people don't need to use Descartes' Rule of Signs. Normal people don't need to find the intersections of conic sections, or the determinant of a 3x3 matrix, or the quotient of complex numbers.

ASIDE: Those might be fair game for the course on modeling and logic, maybe, perhaps, but if so, that's how they should be taught. (For example, we have two ways to describe how "slanty" a straight line is: its slope, and the angle degree it makes with the horizon. Introduce a function that connects those two. So a 45-degree angle produces a slope of 1. A 60-degree angle produces a slope of sqrt(3), about 1.732. And we call that function, which relates two common-sense ideas, "tangent". )

Under ideal circumstances, secondary Social Studies courses should be teaching students critical thinking skills, as well as historical narrative, sociological data, and theories about politics and economics. That's stuff normal adults should have.

Under ideal circumstances, secondary Literature and Writing classes, and science classes, and gym and music and art classes, should all be addressing critical thinking skills as well as address real people's needs. Meanwhile, secondary math courses try to prepare students for college programs in math and engineering.

How many college majors require students to take a Statistics course? Versus how many require Calculus?

So, I suppose my answer depends on the school district's policies, the administration and staff, and the kind of environment the schools invoke, but generally speaking, schools don't teach the curriculum I'd consider to be math I would consider as good and reasonable.

[stillnotbored.livejournal.com]

Ah, for someone like me, any algebra and geometry--at all--are not basic math. Algebra, even at the most beginner level, does not stick in my head.

I tested off the charts in reading and language skills from 5th grade on. I could not do math beyond advanced arithmetic to save my life and barely tested out as average.

I got all As in natural science classes and bombed out of physical science because of the math. It took me two tries to get through Algebra I with a C-, with tutors and special help from the math department. I had a 4.0 in every class but math.

I would fail that test no matter how math was taught in school and I'm not stupid. And after I failed, I wouldn't be able to graduate.

Which is not to say that there aren't issues with the educational system, because there are. I'm just not a fan of one size fits all tests. People don't come with one size fits all brains.

[akitrom.livejournal.com]

Cissa, very rarely does a young woman go into elementary education because she loves math. Typically, it's because she loves working with kids, and secondarily because she likes dinosaurs and storybooks and caterpillars and art.

[aamcnamara.livejournal.com]

Very true. The MCAs do have free-response problems, though.

[aamcnamara.livejournal.com]

Very true. We still do need to improve the way we're teaching math. My point, I guess, was that while the situation is pretty bad, this particular test may not be even measuring the correct badness of the situation.

[cissa.livejournal.com]

Well, yeah- but the problem is, they're also teaching math, and doing it badly. And while teaching it badly, they're turning kids off to it as boring and/or incomprehensible. And THAT is a serious problem.

[cissa.livejournal.com]

While I pretty regularly had metals students that disliked figuring things out with numbers (even given a worksheet that spelled it all out for them), the really amazing one was the woman who refused to read a ruler, because she "didn't like math". I told her that then she could not do metals. (During the class, she got other people to do all her measuring... but thank goodness, she did not continue.)

[mrissa.livejournal.com]

But seriously, normal people do not need to manipulate trigonometric ratios and employ the half-angle formulas. Normal people don't need to use Descartes' Rule of Signs. Normal people don't need to find the intersections of conic sections, or the determinant of a 3x3 matrix, or the quotient of complex numbers.

Which is probably why none of those things are on this test.

You say that people could have calculational proficiency by their freshman year. I agree. The question is whether they do.

[mrissa.livejournal.com]

Note that I did say I believe that math-related learning disabilities are real. But I think that tutors and special help from the math department are the right answer here, rather than the school saying, "Oh, nobody should try to teach stillnotbored math and see if a different approach will work better."

I don't think that 40% of Minnesota students have those profound math-related learning disabilities. I really, really don't. I don't see any evidence that that's the case. When someone is blind, you make arrangements for their tests to go differently, including accounting for a difference in Braille reading speed or whatever other reading method the person is using. Same deal for a math-related learning disability--but not for the level of math difficulties that, with competent teaching, would be more like needing glasses than like being blind.

I don't believe that standardized tests hold all or even many of the answers. But I also am very wary of issuing them and then throwing them out only if we don't like the results.

[mrissa.livejournal.com]

Aaaaaagh.

I am a big meany meanhead as an instructor. I would not have let her get other people to do her measuring unless she had an actual disability that prevented her from reading numbers on a ruler.

(I have been presented with measuring devices where the markings were too fine for my corrected vision to discern--I literally could not see which was the correct measurement. This is not the same thing.)

[mkille.livejournal.com]

Apparently this problem (passing math classes without actually learning math, or at least, not retaining what is learned more than a month after the end of the year) is a massive headache for community colleges (http://suburbdad.blogspot.com/2009/01/remedial-sequences.html). Among the other many practical and moral implications.

[cissa.livejournal.com]

No, that was nothing like the same thing as refusing to even try to navigate a RULER. For heaven's sake!

Since I was teaching adult ed., I had no freedom to flunk anyone explicitly (alas!), and the best i could do was to, myself, make no concessions to her defiant stupidity. And it worked, in that she did NOT sign up for the intermediate class, at least. :P (She was a piece of work in other waysm too...)