Uncommon Descent Serving The Intelligent Design Community

# Why Teach Algebra, Pt. 2

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Denyse was kind enough to post a link to the first part of my article on why Algebra is an important subject for high-school and college students. Part 2 of my article is now available as well.

You can read Part 2 here.

From the article (I bolded the part especially important for the ID community):

Many of the people who argue against teaching algebra recommend, instead, courses in practical mathematics. They point out that the way that mathematics is actually used in the real world is usually very field-specific. They recommend skipping the abstractions of algebra and focusing on very concrete problems and how to solve them.

There is nothing wrong with teaching practical math. The problem is that, absent the more abstract reasoning that undergirds it, what you wind up with are people who know how to do tricks but don’t understand what they are doing. The goal of mathematics is to train students in reasoning abilities, not do tricks. Sometimes we need to teach tricks, but the goal is the reasoning abilities.

The problem with teaching tricks is that it prevents people from being able to analyze what other people tell them. If we teach students to reason, then if someone tells them something unreasonable, they have the tools and skills to analyze it. If we just teach students tricks to do their job, then they are at the mercy of everyone else. They only know partial facts fed them through external authorities, without the ability to judge their truth or falsity.

polistra,
No. The Russians are not manipulating us. Our own intel agencies (CIA, NSA, Google, Apple) are manipulating us into thinking that their manipulation is Russian. You need to look at the Wikileaks exposures of NSA tricks.
The two propositions are not mutually exclusive. A lot of parties are working to manipulate us.
THIS HAS BEEN GOING ON SINCE 1946. The fact that many Americans still fall for this transparent manipulation is the BEST PROOF that we need more critical reasoning.
This is where it gets difficult. I don't think the problem in this particular case is lack of critical reasoning, but rather lack of reliable data. I don't have time to personally fact-check the """information""" that the media report, so I can rarely be very certain of anything that occurs outside my immediate vicinity. IMO, both hypotheses "the Russians are manipulating us" and "the Russians are not manipulating us, it's the CIA etc." are more or less reasonable (that is, neither is completely absurd). We could ask which hypothesis fits the rest of our experience of the world better, and for me, "the Russians are manipulating us" wins. Of course my experience of the world has itself probably been tainted through manipulation by others. Who has the motive to engage in this manipulation, focusing on the matters around the election in particular? I could understand why a rival world power would want to destabilize the USA and erode our faith in the democratic process. Is there some reason that the CIA, NSA, erm, Google and Apple all would prefer a less stable USA? I can't think of any plausible motives. daveS
"Yes, and speaking as a USA citizen, the fact that the Russians are apparently employing increasingly sophisticated methods to manipulate us is very concerning. " No. The Russians are not manipulating us. Our own intel agencies (CIA, NSA, Google, Apple) are manipulating us into thinking that their manipulation is Russian. You need to look at the Wikileaks exposures of NSA tricks. THIS HAS BEEN GOING ON SINCE 1946. The fact that many Americans still fall for this transparent manipulation is the BEST PROOF that we need more critical reasoning. polistra
It tells me everything I need to know that there is even a discussion about why we should teach algebra. High schools in Ontario in the 1960s (where I attended) featured some of the dumbest bunnies in the universe at that time. Many believed themselves smart. Algebra taught us that we had to BE smart, which is somewhat different but much more useful. News
jdk,
And obviously, since people reasonably well-trained in reasoning often disagree strongly about what is reasonable or unreasonable, skill in reasoning can be said to be necessary but certainly not a sufficient source of determining which is which.
Yes, and speaking as a USA citizen, the fact that the Russians are apparently employing increasingly sophisticated methods to manipulate us is very concerning. Young people are growing up in a hostile "information environment", where it's very difficult to tell fact from fiction. They're going to need all the intellectual self-defense they can muster. As a side note, I would mention the urban/rural divide we have in the USA. I have heard from educator friends that in some urban areas in my region, students are beginning calculus at the freshman high school level. Meanwhile, in rural areas, they have a hard time attracting and retaining basic algebra teachers, so that students often can't take any math beyond algebra II (if that). daveS
And I agree with virtually everything kf said! :-) jdk
Folks, My thought is, Mathematics at core is the logic of structure and quantity. Where, geometry (with trigs) and algebra then first steps calculus are gateways. I'd also suggest that boolean algebra including the effect of digital feedback may also be helpful; though this last is maybe better done as part of the expanded math focus -- computing. I suspect basic physics or physical science will bring out much of the applications being highlighted. Where a good part was invented to do physics. Probability, distributions, statistics and process quality may also be a useful focus; just now I saw a real world case where someone suggested sizing a facility on averages rather than looking at the distribution and likelihood of service unavailability. We likely need to be selective and take a longer term, broader view but across primary and secondary schooling we need more not less math. In a multimedia age there is no reason why it cannot be better taught; though Khan Academy should not be despised. Could we get a "Math street" effort? KF kairosfocus
Johnny introduced his post by writing that "he bolded the part especially important for the ID community:
If we teach students to reason, then if someone tells them something unreasonable, they have the tools and skills to analyze it.
But this is especially important for all communities, and it is as true for when someone tells you something is reasonable as it is for when someone tells you something is unreasonable. And obviously, since people reasonably well-trained in reasoning often disagree strongly about what is reasonable or unreasonable, skill in reasoning can be said to be necessary but certainly not a sufficient source of determining which is which. jdk
I agree, Dave. Learning math teaches one that there are logical steps that connect the given problem to the solution, and that one needs to be able to persevere to find those steps rather than accepting a solution for which the steps can not be shown. Also, looking back, I do object a bit to Johnny's use of the word "tricks" to describe what should perhaps be more generously called techniques, However I also understand that if one is taught how to use a specific technique for a limited specific situation, without understanding either the broader theory or why that technique applies to the situation, that one could be said to just be teaching tricks. jdk
johnnyb,
If we teach students to reason, then if someone tells them something unreasonable, they have the tools and skills to analyze it.
I like this sentence. I think our ultimate goal is to help young people achieve intellectual independence, and certainly algebra is an essential ingredient in that process. That said, it's often hard for even those well-trained in philosophy, logic, mathematics (and liberal arts in general) to discern clearly what is reasonable and what is not. daveS
P.S. I just skimmed, very quickly, through Johnny B's article, and am in general agreement with that all that he says. And math education does need to change: for example, I don't think we need to teach finding the roots of polynomials using synthetic division (even though my local junior college still does) any more than we need to teach the algorithm for finding a square root. (I can still do that, by the way, but I bet not many people can). I'm OK with students knowing how to factor certain very easy cubics, for instance, but I think finding the roots using a calculator or graphing program is probably a more valuable practical skill. But, and I think this is the key point, I think the bigger abstract point, separate from actually finding the roots, is understanding the theory of polynomial roots: such things as every polynomial of degree n has n roots, and thus n factors of the form (x - a(n)), if you count complex and multiple roots. Students should understand the theory (including that of the associated graphs) as well as practical tools for working with specific polynomials in specific real-world situations. I explicitly taught my students that we always learned at three levels: theory, mechanical skills, and applications. I strongly disagree with those that say we should drop this approach, although I also think that too much math teaching is spent on just mechanical skills without tying them to either bigger abstract ideas or practical uses. [/math curriculum soapbox spiel] jdk
As I wrote in the first thread, it is very possible to build real-world applications into an academic curriculum so that students see how math is used in the real world and see some of the reasons why having the abstract skills is crucial to using math, in addition to its intrinsic value for building logical reasoning skills. This should not be an either/or issue. My experience was that the real-world applications helped convince students that the abstract skills were worth learning, even if we didn't do a real-world application for every skill. I don't know who the people are being referred to in the first paragraph of the quote from the article above (starting "Many of the people ..."), but I think those people are wrong. jdk