Recently, The New York Times published an op-ed calling for curricular changes in K-12 math education:
Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a "reform" version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.
For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a "group of transformations" or a "complex number"? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.
What the authors call for is an applied approach to teaching math:
A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. But there is a world of difference between teaching "pure" math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations. The former is how algebra courses currently proceed -- introducing the mysterious variable x, which many students struggle to understand. By contrast, a contextual approach, in the style of all working scientists, would introduce formulas using abbreviations for simple quantities -- for instance, Einstein's famous equation E=mc2, where E stands for energy, m for mass and c for the speed of light.Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers.
I have no idea if this would work. It sounds good, but I would like to see some data (I know similar approaches have been tried before, with mixed results). The other concern is that this approach has been used in Finland:
A plausible hypothesis stems from differences in the content of the two tests. The content of PISA is a better match with Finland's curriculum than is the TIMSS content. The objective of TIMSS is to assess what students have learned in school. Thus, the content of the test reflects topics in mathematics that are commonly taught in the world's school systems. Traditional domains of mathematics--algebra, geometry, operations with numbers--are well represented on TIMSS.
The objective of PISA, in contrast, is not to assess achievement "in relation to the teaching and learning of a body of knowledge." As noted above, that same objective motivates attaching the term "literacy" to otherwise universally recognized school subjects. Jan de Lange, the head of the mathematics expert group for PISA, explains, "Mathematics curricula have focused on school-based knowledge whereas mathematical literacy involves mathematics as it is used in the real world." PISA's Schleicher often draws a distinction between achievement tests (presumably including TIMSS) that "look back at what students were expected to have learned" and PISA, which "looks ahead to how well they can extrapolate from what they have learned and apply their knowledge and skills in novel settings."
The emphasis on learner-centered, collaborative instruction and a future oriented, relevant curriculum that focuses on creativity and problem solving has made PISA the international test for reformers promoting constructivist learning and 21st-century skills. Finland implemented reforms in the 1990s and early 2000s that embraced the tenets of these movements. Several education researchers from Finland have attributed their nation's strong showing to the compatibility of recent reforms with the content of PISA.
In other words, Finland does well on the PISA test because PISA reflects Finland's educational goals (interestingly, many Finnish mathematics university professors think those goals leave Finnish students woeful underprepared for college math, but that's a whole separate discussion).
Over 300 Finnish college mathematics professors signed a statement decrying the adoption of a 'constructivist' mathematics curriculum. I have no idea what percentage of Finnish mathematicians that is, but Finland's population is smaller than Massachusetts', so it's probably most of them.
Of course, the other thing to remember is that U.S. students, once we account for poverty, seem to do extremely well on the PISA exam, which tests precisely the sort of applied math approach the authors call for. Call me conservative, but I'm not sure this is the most pressing educational issue right now.
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To say in one sentence that we shouldn't teach algebra's "abstract 'x'" because it loses kids, and then say we should teach how computers are programmed? Supreme ignorance. Computer programming is *nothing but* "solve for X", and always has been.
As for e=mc^2, you could teach the young layman everything they could understand about it in 5 minutes (Niel deGrasse Tyson did exactly that in his PBS show). Doesn't mean they'll understand what it *means*, only that they can repeat a string of letters and numbers just as if they were still watching Sesame Street.
Most adults have forgotten the context of their algebra classes by the time they start complaining about them. High School Math classes are *full* of (semi-)practical real-world problems, they very problems that became the reason why we had to invent/discover the maths in the first place. Thing is, after working with the variables and formulas for so long, and then NOT working with them for even longer, we've (as adults) have forgotten the context that was included in the class, so we recall the formula and go "why the hell did I have to learn that" and think everything needs to be thrown away.
Now, that doesn't mean that I think that for those not going to university that they should still be on a calculus-track curriculum. In fact, many of the things he talks about used to be included in "home economics" classes back in the day, only those classes disappeared as more and more schools over-focused on meeting university-targeted graduation requirements.
*Either* solution alone (all home-ec, or all calculus-track) is wrong. Different types of kids have different needs, and it would be better for all if we found away to address that...but of course, the only way to address that is to throw "standards" out the window and stop treating kids like factory products and national symbols.
i wouldn't say that. i suppose a Turing machine (or lambda calculus, for that matter) might be entirely reduced to nothing but "solve for X", but that doesn't make them easier.
lambda calculus can also be reduced to INTERCAL --- or brainf*ck --- but those aren't improvements, either. computer programming is more an exercise in writing out obnoxiously specific descriptions of a problem or its stepwise solution process, or both. it uses a mode of thinking that's probably common to both it and mathemathics (and symbolic logic), but trying to do it by means of simple algebra isn't likely the most useful approach.
I don't like this. I do not think that you can be taught to apply something before you are taught the basic concepts. Like, with statistics, there is a reason you need to know algebra first, because to truly understand stats, you need to know algebra. Without the pure concepts it makes no bloody sense.
We want to change the entire curriculum so that we will do well on one test that a tiny country does well on? If you hadn't provided a link, I'd say you were making this up. Anyway, this old and experienced math teacher has been through the reform cycle a few times and I will repeat for the west coast audience:
a) Know your subject matter
b) Make sure your students know you have their best interests in mind in everything you do
c) Use every trick in the book to make things interesting (like real-world examples!)
Keep after them, Mike. I love your posts.
"For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a "group of transformations" or a "complex number"? "
As an adult, I have never needed to know about the French revolution, Enlightenment, the existence of two world wars or any of my countrys previous prime ministers or ruling parties. In fact, as a practical matter I have never needed any historical facts whatsoever, a situation I surely share with a large majority of contemporary citizens.
I have also never needed to know about Shakespeare, Balsac, Kafka or any other canon literature; I am perfectly capable of finding and reading things on my own, and that is mostly, even wholly, not from those old literary giants. Again, I am far from alone in this.
The fact that the earth orbits the Sun is interesting of course, but even as a researcher I have never actually needed to use that fact in any way. Most people could cheerfully do without 95% of all science and never miss any of it.
So, with the same reasoning as above, why should we have any kind of general education in the humanities or sciences at all?
We recently tutored two high school students in geometry for a term, so we got a good look at an applied geometry curriculum. One of the most important things we learned in geometry is that one can start with a small number of simple statements and build a complex structure based on them by the sheer power of logic, and that just as the simple statements make sense in the real world, this whole complex structure makes sense as well. Geometry is the subject in which the raw power of mathematics and its "unreasonable effectiveness" is introduced.
This applied course, and it was an honors course, not the regular geometry course, played down the power of mathematical reasoning in favor of applications. Yes, there were a few proofs here and there, but no real sense of a structure flowing from simple roots. Instead there were rules, heuristics, and formulas, an entire hodge podge of them. There was an awful lot of rote mechanics of solution, unfortunately aided by the use of modern calculators which make it so much easier to assign a dozen or two dozen problems requiring sines, cosines and the like.
We actually did deviate from the curriculum now and then and did an actual series of proofs, and it was pleasant to see that glow of insight which was missing once we had gotten past the first few problems in a battery.
I think real world examples can be useful, but the whole point is that mathematics is about abstractions. Sure, you need to know how to make change, but it also helps to have an actual understanding of subtraction.
The three real world examples, about mortgages, programming and bayesian statistics, are all perfectly good things to teach, but they'd be a lot more useful if taught as examples of algebraic series, algorithms, and statistical reasoning, rather as just a grab bag of tips and tricks with no theoretical basis. Unfortunately, getting to that theoretical basis requires abstraction.
If you're familiar with the Common Core Standards, the New York Times article comes off as rather strange. In high school (which is what the NYT article discusses), the Common Core has much more about applications than the traditional US curriculum. For example, it's got standards on statistics and probability, and for modeling. It doesn't have standards for the vast majority of topics in calculus.
This can be seen by an examination of the Common Core Standards which are here: http://www.corestandards.org/the-standards/mathematics
This article really resonated to me as a high school student. Geometry class is challenging enough, from its numerous formulas to the dreaded proofs. However, whatâs even more brutal to me is the idea that in a few years, none of the material will pertain to my life. When the article questions, âHow often do most adults encounter a situation in which they need to solve a quadratic equation?â I couldnât help but cringe inside. With our ever-changing, modern world, I feel that America needs to take a step forward in the educational system. If our system was based more off of reality, I think that not only would this help prepare students for the future, but it also would engage more students in school. Students would see a purpose in schooling and would realize the benefits of instruction. Thereâs much more to life than knowing how to solve for X and Y, which is why I believe that this new type of curriculum would be highly beneficial in the United States.
As a high school student, I found this article to be very true. I spend hours on my math homework every day and I am tormented by the thought that this will all be unrelatable when I grow up. Personally, I would much rather be learning topics that I can actually use in the real world, such as finance, data, and buisness. I feel as though it is more important for the American school system to focus on real life problems because the current students are the future of America and if they are not prepared to lead the world, who can we depend on? Therefore, I believe that the math curriculum needs to change, in order to make a better future for all Americans.
Maggie, any education on "real world" stuff would be useless by the time you leave school, so would be pointless anyway.
Therefore your requirement would be "no school".
I gave a much longer answer, but Mike ate it and I can't be bothered to type it all in again.
If there was one thing I would like to see, is an increase in the number of years someone has to attend school. Our world is becoming more and more complex, with many more things to learn and yet, the number of years someone has to attend school has not increased, so what is taght, has to be compressed more and more into the time available to teach it. There will always be some people whom just don't want to learn more, or want to earn a living, to whom a range of 'trades'/skills could be taught, giving them an even better foothold on their first rung of the ladder of life.
would like to see, is an increase in the number of years someone has to attend school.
300'den fazla Finlandiyalı kolej matematik profesörleri 'yapılandırmacı' matematik müfredatı kabulü decrying bir bildiri imzaladı. Yüzde kaçı Fin matematikçiler hiçbir fikrim yok, ama bu yüzden muhtemelen bunlardan en Finlandiya'nın nüfusu Massachusetts', küçük.
I'm not a doctor, teacher, or professor, however, i think some of the things being taught to our children is more harmful than good. I see them teaching to the "test" in Florida. There is no understanding or reasoning of the knowledge. We must teach the whole subject with all the parts included not just pick and choose those the tests want to look for.
I cannot for the life of me think there is a good reason to "estimate" the answers instead of actually getting the right answer. If you can do the math in your head and get the right answer, you still must put in an incorrect answer just to appease the gods of math. Math was one of my favorite subjects it was not subjected to the opinions of others. I was asked to tutor classmates because they understood what I was saying better. I honestly don't know what I was say that was different from the teacher, but they got it and that was all that mattered.
I taught my son at home how to do math the way I was taught and told him to turn in the crap they wanted to just make sure his grades were up. We needed to purchase a ladder to reach the roof safely. I asked my husband what he thought and my son as well and neither one knew how to apply a^2 + b^2 = c^2 to find the hypotenuse or ladder's safe length. Though if asked what Pythagorean's theory was they could rattle off the formula.
Just a thought we need both the test answers but we also need to know when to apply them in life or why bother in the first place.
I await your flaming.