Tag Archives: High school

Misunderstanding mathematics

Plots of quadratic equations with discriminant...

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Robert Lewis, a professor at Fordham University, has published this essay entitled “Mathematics: The Most Misunderstood Subject”. The source of the general public’s misunderstandings of math, he writes, is:

…the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student’s duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand “Quick, what’s the quadratic formula?” Or, “Hurry, I need to know the derivative of 3x^2 – 6x +1.” There are no such employers.

Prof. Lewis goes on to describe some ways in which this central misconception is worked out in our schools and in everyday thinking. The analogy between mathematics instruction and building construction, in which he compares current high school mathematics instruction to a building project where the scaffolding is constructed and then abandoned because we think the job is done, is pretty compelling. The whole essay is well worth reading.

I do think that it’s a bit too easy to lay the blame for the current state of mathematics instruction at the feet of American high schools, as Lewis does multiple times. Even if high schools do have flawed models of math instruction, certainly they are not alone in this. How many universities, even elite institutions like Fordham, have math classes or even entire curricula predicated on teaching math as rote mechanics? And what about the elementary math curricula? Pointing the finger at high schools is the natural thing to do for college professors, because we are getting students fresh from that venue and can see the flaws in their understanding, but let us not develop tunnel vision and think that fixing the high schools fixes everything. Laying blame on the right party is not what solves the problem.

Lewis brings up the point that we should be aiming for “genuine understanding of authentic mathematics” to students and not something superficial, and on that I think most people can agree. But what is this “authentic mathematics”, and how are we supposed to know if somebody “genuinely understands” it? What does it look like? Can it be systematized into a curriculum? Or does genuine understanding of mathematics — of anything — resist classification and institutionalization? Without a further discussion on the basic terms, I’m afraid arguments like Lewis’, no matter how important and well-constructed, are stuck in neutral.

Again coming back to higher education’s role in all this, we profs have work to do as well. If you asked most college professors questions like What is authentic mathematics?, the responses would probably come out as a laundry list of courses that students should pass. Authentic mathematics consists of three semesters of calculus, linear algebra, geometry, etc. And the proposed solution for getting students to genuinely understand mathematics would be to prescribe a series of courses to pass. There is a fundamentally mechanical way of conceiving of university-level mathematics education in which a lot of us in higher ed are stuck. Until we open ourselves up to serious thinking about how students learn (not just how we should teach) and ideas for creative change in curricula and instruction that conform to how students learn, the prospects for students don’t look much different than they looked 15 years ago.

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Filed under Education, High school, Higher ed, Liberal arts, Life in academia, Math, Teaching

Plagiarism in high school

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About two dozen seniors at Hamilton Southeastern High School in the affluent northern suburbs of Indianapolis have been caught plagiarizing in a dual-enrollment college course, thanks to turnitin.com. Full story with video here, and there’s an official statement from the HSE superintendent on this issue here (.DOC, 20KB).

This would be an ordinary, though disappointing, story about students getting caught cheating if it weren’t for some head-scratchers here. First, this bit from the superintendent’s statement:

We took immediate action because the end of the school year was rapidly approaching. Several students were in danger of not graduating on time. We found a teacher who was willing to step up and administer a complete but highly accelerated online version of a class that would replace the credit that was lost due to cheating. Each student who wishes to graduate on time and participate in commencement now has the opportunity to do so. [my emphasis]

It’s troublesome that the superintendent chooses to describe the teacher as “stepping up” to deliver an online makeup course. “Stepping up” is what you call it when there’s something that needs to be done and somebody agrees to get it done. But it seems to me that the school system here owes these students absolutely nothing. HSE, in conjunction with Indiana University, offered a legitimate college course with clearly-defined parameters for academic performance, and HSE did a particularly thorough job describing the boundaries of academic honesty. The students chose to violate that contract and cheat. The school system is therefore not obliged to offer an online makeup course, or indeed to offer anything to these students at all. To imply that HSE does owe the students a path to graduate on time is like saying that if someone gets caught shoplifting, the grocery store owes it to the shoplifter to find a way to help him buy his groceries.

Also, what is the teacher who “stepped up” being paid to run this online course? If the teacher is being paid from public school coffers for this, and if I lived in Hamilton County, I would have a big problem with my tax money being spent to offer online courses to students guilty of cheating just so they can graduate on time — especially when public school money is historically scarce right now. Let the students find their own way to graduate. It’s not like they were barred from graduating on time, fair and square, in the first place. Let the residents’ school money go to help the students who are working hard and doing things the right way instead. (If the teacher’s doing it for free, then other questions arise.) This is the way we’d do it in college, and this is a college course, right?

HSE might think it’s doing right by the students in “allowing each student to work his or her way back toward the proper path so they can graduate on time, continue their educations [sic] and understand the benefits of making good choices” (quote from the superintendent’s statement). But isn’t this really illustrating the benefits of making bad choices — as in, go ahead and cheat, because the school will find a way to let you graduate on time anyway? Other than potentially not getting into IU, what consequences are these students having to face, exactly, other than sacrificing a bit of their summer to retake a course at taxpayer expense? (By the way, if this course is dual-credit, whose rules about academic dishonesty are supposed to be followed? IU’s appear to be more strict that Hamilton Southeastern’s.)

This bit from a fellow student is equally disturbing:

“If you’re going to do something dishonorable, there’s going to be consequences for it,” said [a fellow student, not part of the plagiarizing group]. But she says she sympathizes with her friends who were caught cheating. She claims students have been cheating for years, but this is the first year teachers have used the software system that gives them the ability to easily catch cheaters. She believes this incident likely serves as a lesson for students for years to come.

So, it’s about the consequences, not so much the act itself. The sympathy didn’t show up until turnitin.com caught them. Until we stop “sympathizing” with plagiarists and start treating plagiarism on the same level as lying and stealing — which it is both — this problem isn’t going to go away.

What’s your take on all this? Is HSE acting honorably or just enabling future plagiarism? What’s the best way to punish teenage plagiarists on the one hand but really help them make better choices on the other?

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Filed under Academic honesty, Education, High school, Higher ed, Life in academia

You can’t become an expert in college

Cover of "Outliers: The Story of Success&...

Cover of Outliers: The Story of Success

Here’s something of an epiphany I had at the ICTCM while listening to Dave Pritchard‘s keynote, which had a lot to do with the differences between novice and expert behaviors in problem-solving.

Malcolm Gladwell, in his book Outliers, puts forth a now-famous theory that it takes at least 10,000 hours to become a true expert in a particular area, at the top of one’s game in a particular pursuit. That’s 10,000 hours of concentrated work in studying, practicing, and performing in some particular area. When we talk about “expert behavior”, we mean the kinds of behaviors that people who have put in their 10,000 hours exercise as second nature.

Clearly high school or college students who are in an introductory course — even Dave Pritchard’s physics students at MIT, who are likely several levels above the typical college undergrad — are not there yet, and so there’s not a uniform showing of expert behavior. There are more hours to be put in. But: How many more?

On the one hand, if a person spends 40 hours a week working at this activity, for 50 weeks out of the year, then it will take 5 years to reach this level of expertise:

(10000 hours) x (1 week/40 hours) x (1 year/50 weeks) = 5 years

But on the other hand, a typical college student will carry a 16 credit hour load, which means 16 hours of courses per week. If the student does this over a 14-week semester, and if the student takes the standard advice of spending 2 hours outside of class for every hour inside of class, and if the student undergoes two semesters of classes every calendar year, how long does it take to get to 10000 hours?

10000 hours x (1 week/48 hours) x (1 semester/14 weeks) x (1 year/2 semesters) = 7.44 years

That’s fairly close to double the usual time it takes for people to earn a bachelor’s degree. And it assumes that all that coursework is concentrated into one area, which of course it isn’t.

So there’s an important truth here: Nobody can become an expert on something just by going to college. College might add the finishing touches on expertise that was begun in childhood — for example, with kids who start playing music or programming computers at age 6 — but there’s just not enough time in college to start from zero and become an expert.

This has implications for college coursework. Many of us profs have “expertise” in mind as the primary instructional objective of our courses, but this is quite possibly an unreachable goal for most students. Instead, along with reasonable levels of mastery on core subject content, college courses should focus on what students need for the remaining hours they need to get to 10,000. We should be teaching not only content in the here and now, but also processing skills and broad intellectual tools that set students up for success in continuing towards expertise after college is over.

We can’t make students experts in the time we have with them, probably, but we can put them in position to become experts later. Ironically, the harder we try to make experts out of everyone, the less we stress broad intellectual skills, and the less likely they are to become experts later. How are students supposed to continue to learn, practice, and perform to get to that top level if nobody teaches them how to think and learn on their own?

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Filed under Education, High school, Higher ed, Life in academia, Problem Solving, Study hacks, Teaching

Geometry first

Jackie at Continuities is wondering whether the usual path through high school mathematics — Algebra I, then Geometry, then Algebra II, etc. — is out of order, and whether geometry ought to come first:

As far as I can tell the only difference between Alg II and Pre-Calc is that trig is taught during Pre-Calc and Pre-Calc introduces the concept of the limit. Functions are developed a bit more rigorously too.

The first semester of Algebra II is mostly a repeat of Algebra I as they’ve forgotten it with the year “off” during Geometry.

Why not then teach Geometry first? I’m talking about plane and solid geometry with an emphasis on reasoning, and right angle trig. Obviously there would need to be some supplementing needed (work with radicals, solving equations). Most students have “seen” the solving of equations in 8th grade (Have they mastered it? No, of course not).

I completely agree. It seems to me that the reason Geometry gets sandwiched between Algebra I and Algebra II is that people want to use algebra concepts in geometry. But I think that doesn’t necessarily have to be the case. If you look at the source — Euclid’s Elements — you will not find a drop of algebra in it. All the concepts that we, today, would label as being algebra or number theory or what-have-you are just latter-day retrofittings of Euclid’s ideas. Euclid himself phrased everything in terms of geometry, with the algebra and number theory done in terms of commensurable lengths and other geometric terminology. I wouldn’t go so far as to say Euclid knew nothing of algebra or number theory, but if you follow Euclid you don’t need algebra, as we know it, at all in your geometry.

That would leave a geometry course that is mainly about logical reasoning, cogent organization of facts, objective deductions from data, and clear exposition of an argument. One might add to this list the art/craft of forming conjectures from experimentation and then writing an argument in favor of your conjectures, which is astoundingly simple these days thanks to Geometers Sketchpad and other fun, low-cost dynamic geometry software packages. (My students who use Sketchpad in their student teaching report, to a person, that students really turn on when they use Sketchpad and do some very good mathematics, for 8th-9th graders.) This sounds like precisely the kind of foundation, and buffer zone, that students need to acquire before tackling algebra with a view towards understanding how it works rather than just memorizing facts. (Indeed, memorizing facts in algebra is quite hard unless you understand why the facts work.)

Of course, if you ask ten people whether they liked their geometry class in school, eight will probably say “no” and seven of those eight will say it was because of “proofs”. But I wonder what that really means. Perhaps, having gotten a taste of equation solving in algebra and therefore acquiring the “there’s only one right answer and I have 30 seconds to find it” mentality about mathematics, they are spoiled for ever encountering mathematics as it really is (which is something that geometry is a lot closer to than algebra I). Perhaps they had a geometry teacher who was not really good at, trained in, or interested in math at all — or someone who was like so many teachers out there who “just love kids” but who choose not to translate that love into teaching their kids how to think well.

But I think if you put a geometry class like what I described above into the hands of a competent, mathematically astute teacher with a mind to help his/her students become excellent thinkers, a year of that could very well change a generation of kids.


Filed under Geometers Sketchpad, Geometry, High school, Math, Teaching

Valedictorians by the wayside?

School districts in Boulder, Colorado will no longer be awarding valedictorian status to their top high school graduates, starting in 2010:

The district’s high schools used grade-point averages to determine the honor, but the top students were sometimes separated by just hundredths of a grade point, leading to complaints. Officials also worried students were focusing on heavyweight academic classes at the expense of arts and other electives.

Let that sink in: They’re worried that students are focusing too much on academic courses.

“We have a responsibility and a goal of educating the whole child and not just coming up with this race for tenths of a percentage,” said school board President Helayne Jones. “High school is supposed to be a time to try things out.”

Boulder Valley had previously abolished class rankings to reduce “unhealthy competition,” and the committee said keeping the valedictorian system no longer made sense.

Under the system recommended by the committee, the top 20 percent of students will get honors—with the top 3 percent earning summa cum laude, the next 7 percent magna cum laude and the remaining 10 percent cum laude.

“This honors more kids for academic achievement,” said Fairview High School Principal Don Stensrud, who co-chaired the committee. “It gives kids something to strive for.”

So, striving is OK but competition is not OK. I await word from the Boulder school distrcts about their dismantling of the athletics and band programs too, since those also encourage competition.

This is all obviously nonsensical hand-wringing on the part of the Boulder schools, which apparently would rather students take more art rather than more math and science and not compete with each other in hopes of some vague and homogeneous commune-like happiness among its kids.

But Boulder may yet have a point here. The current trend in high schools is that more and more students are clustering at the top of the class rankings, resulting in absurd numbers of “top students”, thereby perpetuating the Lake Wobegon Effect. One Seattle-area high school crowned 44 out of 406 seniors as the valedictorian — a 44-way tie for first place. The valedictorian award is supposed to distinguish the top-performing student at a high school; if that award is shared by 11% of the graduating class, then it really doesn’t mean anything any more.

[Hat tip: Homeschool 2.0]


Filed under Education, High school

Dual enrollment in the blogosphere

The other day I linked to this article on dual enrollment and briefly explained why I support dual enrollment programs. It turns out that there’s at least one blog entirely devoted to dual enrollment. The name — “Concrurrent Enrollment” — isn’t very catchy, but the content seems good (and that’s better than the other way around). High school students who are dually enrolled, or thinking about it, and their parents might find it worthwhile to check out. There’s quite a bit of controversy and/or politics involved and it’s often not as easy as it could be.


Filed under Education, High school