Is Math Education Too Abstract?

| September 12, 2011 | 18 Comments
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Tim Geers

“There is widespread alarm in the United States about the state of our math education,” wrote Sol Garfunkel and David Mumford in an op-ed in The New York Times last month. It led me to remember the opening salvo in a pitch for re-thinking math education — that of Dan Meyer’s TED Talk last year: “Math class needs a makeover: I sell a product to a market that doesn’t want it but is forced by law to buy it.” Students’ attitudes are, indeed, alarming, particularly when we use students’ performance in math to gauge (to praise, to lambast) the success of American education.

But according to Garfunkel and Mumford, our concerns over math education are misplaced. Instead of worrying about it in terms of whether or not test scores show American students are keeping pace with their peers globally, we should be focused on rethinking what math education actually looks like. “Today,” they write, “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.”

The authors contend that the “traditional” math curriculum focuses too much on abstract reasoning and abstract skills. “Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering,” they suggest.

In some ways, that’s a very appealing, if not highly practical solution: teach kids the math they need, and as such, be ever ready to answer their age-old question “When am I ever going to use this when I grow up?” Give students a grounding in math that is “applied” so that they can understand the “why” as part of the “how” it works. Contextualize math so that it makes sense — practically and conceptually. Contextualize math so that it means something. Give students a reason to want, to need to solve a math problem — a want or need that doesn’t include “This will go down on your permanent record.”

But does thinking about “applied mathematics” mean necessarily that we have to steer clear of algebra and calculus as their own units in math education? Does putting math in context mean that we cannot teach math in abstract?

In a response to the editorial, Al Cuoco, the director of the Center for Mathematics Education at EDC, takes issue with the assumption that contextualization is such a simple task:

“Many of the students in my high school classes came from situations that many of us would find hard to imagine; the last thing they cared about was how to balance a checkbook or figure the balance on a savings account. But they loved solving problems. For another thing, reality is relative. The authors claim that ‘it is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now,’ and I agree. But the best mathematicians and scientists I know, and the students in my classes who really got it (and these were not necessarily the ‘good students’)—who saw the power and satisfaction one can derive from doing mathematics—all see mathematics as part of their real world.”

In some ways, the debate between abstract and contextualized, “pure” or “applied” math belie the real problem: one that Dan Meyer makes so clear in his TED Talk: students simply aren’t engaged. Will offering more context make for better math education? (And will it make for better future mathematicians?)

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  • http://www.facebook.com/d.sudhakr S Dayal Shri

    Mathematics is a lot simpler than what people think.  Its more of a mental Barrier than the real one.  Just for an example if there are 1024 Players for a Badminton Single tournament and “Only one” person need to be declared as a winner, on knock out basis.
    How many Games will be played ?

    Since only one player need to be declared as a winner (1024 -1) Games are to be played.

  • Anonymous

    I never really understood math until I took math courses in college. Going to a engineering and technology school, most of math I was taught was tied to some application. When you give math an application you can visualize the operations and understand their usefulness. Abstractness should only follow applied math.

  • http://twitter.com/davidwees davidwees

    I’d like to see more balance in our teaching of math between real contexts the kids understand, which should be culturally situated and mathematics which is highly interesting, but also fairly abstract.

    If we focused only on abstract reasoning, most kids would struggle to find ways to use the mathematics they learn, and if we focused only on practical math, we’d see less kids get to the stage when they get to invent and discover new mathematics.

    Just to clarify: those horrible word problems which show up in textbooks as examples of mathematics are neither relevant (since they are often situated in cultures much different than that of the students, and using dated examples) nor interesting.

  • http://www.facebook.com/profile.php?id=723779329 Alex Becker

    Perhaps my personal interest in mathematics means that I am too close to this issue to judge it fairly. However, as a keen math student and recent high school graduate, I feel I have some standing to weigh in on this.

    I took math in high school through pre-calculus (which in my district was in 10th grade for kids on the accelerated track) and though I generally did pretty well, I hated it. The problem was that mathematics was treated as the memorization of a set of rules and techniques to apply in a particular order to particular strings of symbols. Then I took a calculus course at the local university and began reading “Modern Algebra: an Introduction”, a book given to me be my father. My world changed completely, and I realized that mathematics is profoundly different from what I had been taught. I have since taken additional undergraduate and graduate mathematics courses and researched algebra and dynamical systems outside of class.

    I certainly agree that public school math classes are terrible, but for different reasons. The thing is, I never learned anything in those courses that I couldn’t have asked WolframAlpha or a calculator. While teaching these things made sense before we had those things, we have them now and it is thus simply a waste of time. When I point this out to people, I usually get one of two reactions. The first is “but then they wouldn’t know how what they’re doing works”, to which I respond that the classes I was in never addressed that and most of classmates didn’t know how these things worked either (often I found teachers unable or unwilling to explain as well). The second is “but then what is the point of learning math?”, which I answer by saying that that isn’t what math is. This flusters lots of folks. “What do you mean that isn’t what math is? That’s the math I learned!” is a typical response. What I find hardest to convey is that most people have absolutely no idea what math is (or rather completely the wrong idea).

    I can easily answer the final question in the article “And will it make for better future mathematicians?” with an unconditional NO. Why? Because what the article is advocating teaching is entirely different from what mathematicians do, to the point where it bears little more resemblance to actual mathematics than does English or History. Many math professors I have talked to lament that the students who enter college wanting to major in math a precisely those students who were good at “math” in high school, and that often these students are simply bad at actual math. As bad if not worse is the fact that many potentially talented mathematicians lose interest in the subject because they are not good at high school “math”. To make an analogy to sports, high school math classes teach dribbling in place only, whereas mathematicians play the game of basketball; clearly how well one dribbles in place is at best a very poor indicator of how well one actually plays basketball.

    If it were up to me, I’d start by teaching elementary school kids elementary set theory and boolean logic. It isn’t difficult, but it teaches kids how to think logically about the world (which I consider one of the most important aspects of education) and gives them a foundation for learning math. Of course, this has been tried before (in so called “New Math” programs) and parents revolted because they couldn’t understand their kids math homework. Simply put, if we are not willing to teach math that parents can’t do, then we can’t teach math at all because most parents have absolutely no knowledge of mathematics.

    I apologize if this seems like an offensive and long-winded diatribe. I don’t mean to say that mathematics is something most people cannot grasp, but simply something they were never taught. If I seem angry, it is because the material that passes for math in public schools nearly turned me off mathematics forever, and had it done so I would never have realized the sublime beauty of the subject and never felt the peace and joy that has come with understanding it.

    • http://twitter.com/kwassinkEPS Kip Wassink

      As a new high school math teacher I would love to hear more of your ideas. Find me on twitter @kwassinkEPS:twitter 

    • M Double H

      Great Article! I will take your suggestion about how to teach Elementary school math under advisement!

    • corrolary

      Thanks, Alex! A very decent and thoughtful answer! I am a mathematician and a math educator, a passionate of history and philosophy of math, who had in her school years some of the revelations you articulate here. I wish you take your thoughts to further depths  and put them into a sort of an article that can be published for a larger audience to read. There are some elements you point out here that are missing from the arena of general discussions around math ed reforms-and I believe it should play an important role in thinking about “how we go” about educating in general, not only in math and science. For example this bit you touch on here: “And will it make for better future mathematicians?” with an unconditional NO. Why? Because what the article is advocating teaching is entirely different from what mathematicians do, to the point where it bears little more resemblance to actual mathematics than does English or History.” 

    • http://profile.yahoo.com/HH42G4PI5V56XCDE5NA5Y4EIZE Michael G

      Just had your post here pointed out to me by your father, Alex. Here’s a blog piece I did in response to Mumford and Garfunkel. http://bit.ly/qxoV7s

      I agree with your comments about needing to go past what parents know and can help with, but having a bit more mileage on my meter than you do, it’s hardly that simple. Not only parents but teachers and students are part of a vast American culture of education in general and mathematics education in particular that makes for some very ingrained opposition to your ideas. Some of that resistance comes from very conservative people in the university mathematics, science, and engineering communities as well as those outside academia who use high-end math in their jobs. They don’t like seeing what they view as inviolable tradition messed with. If something was good (or bad) enough for them, it’s good enough for EVERYONE, EVERYWHERE, FOR EVER. Hence, the last two decades of the Math Wars. 

      Take a look at the Dolciani series of high school math books. That is what most people take as “THE New Math,” but in fact there was never truly a single thing that earned that name. Rather there were multiple projects going on throughout the math communities of the country, very few of which ever resulted in published texts. Dolciani’s books, by default, became THE books and the reaction was, well, generally not very good. On the other hand, there are some of those conservative STEM types who adore those books yet say they hate the New Math and the New-New Math. Go figure. 

      Personally, I’m not a big fan of Dolciani, but now that I have learned math, I could go back and read them with ease. And therein lies the problem. They’re okay for some kids, but hardly for most. Anyone who wants that approach is certainly welcome to it, but there are other ways to get to rigor that are just as mathematically sound but perhaps appeal to other sorts of people. You might note that some pretty accomplished folks have gone well ahead of what K-12 had to offer in the way of education by just using the public library, long before there was an Internet to make things even easier (albeit more dangerous in terms of poor materials readily available free). 

      We certainly can do MUCH better, but going the route of extreme formalism, a la what France tried to do with K-12 education under the sway of the Bourbaki mathematicians is unlikely to be successful with very many people, in my view. I’ll happily lend out my various copies of books by Serge Lang to any truly motivated reader. ;)

  • Thecathancegroup

    I understand that a finance course would not be abstract, but the writer hasn’t shown me that geometry or algebra is? A few years back I took algebra at night school because I wanted to see how it was taught and learned relative to what I got 50 years earlier. There were drill exercises, and the rest [maybe 40%] real-world applications. I didn’t think I was stuck in an abstract exercise.

  • Keg

    The authors will have to swim against the flood of the Common Core State Standards on this one…  Good Luck!

  • Jordana J.

    I am all for a more applied approach to math.  I was pulling Cs and Ds in math classes since I was in grade school while getting As in science and winning major national and international awards for my science fair projects in high school.  I understood statistics (even if I didn’t do the math longhand), used parametric and non-parametric statistical tests to figure out the significance of my results, and, with the help of an engineer I did some systems engineering and some rather complex math to figure out the perfect balance of a lot of different variables for a semi-enclosed sustainable farming method incorporating crops, farm animals and fish.

    I went on to college totally unprepared for the difficult math classes I needed to take.  I took remedial precalc and managed a B, but then went on to honors calculus and at that point I realized I could never, ever be a professional scientist because I was completely out of my depth on day 1.

    Part of the problem was my attitude… there was nothing I hated more than doing the same thing over and over… but I could not see how calculus was going to play into the research I wanted to do.  I didn’t have the foundation skills I needed, and ultimately, I dropped out of school.

    I’m about to finish my English degree ten years later… I’ve always been a better writer than a scientist, but all the same I wonder what might have been if I had experienced an applied approach to math.  Not to say that we don’t need the abstract stuff… but I think it should be optional for the students who develop an interest, rather than required.

    • Lisa Starrfield

      Until you were willing to put for the energy to actually practice, even a practical approach will not help.  You need repetition to develop fluency and exposure to a wide variety of problems to be able to develop the background knowledge to know when to apply a skill or concept.

  • Nikihayes

    I have two suggestions about “choice” and “attitude”: 

    1)  Two tracks in high school would let students take “relevant” courses for the world they think will be real for them, and others could take the traditional track that leads to degrees in science (including medicine, psychology, etc) and mathematics (the language of science). Unfortunately, too many teenagers don’t have the background knowledge or adult support to make such powerful decisions that can impact the rest of their lives. Nonetheless, let them, or their parent figures, make the choice in 9th grade. They will likely be working for those from the second (traditional) group.

    2)  Grow up and learn that some fields of study are hard and aren’t meant to offer entertainment and soul satisfaction when they are being mastered. After mastery takes place (it doesn’t just occur) in any activity that requires practice and dedication, great feelings are immeasurable. In Malcolm Gladwell’s book, The Outliers, he has a chapter on “10,000 hours of practice.” It points out that the greatest of champions in any field have spent 10,000 hours of practice, whether it is a classical pianist, golfer, emergency personnel (including military), etc. It is drill and practice that provide the foundation that allows someone to grow so freely in creativity and nuance  (another of Gladwell’s points).

  • Lisa Starrfield

    Seriously?  Our current math scope and sequence was put in place to prepare more students for science and engineering careers.    Going to more ‘practical’ math will simply put American further behind the rest of the world.

  • Natstahl

    I think the problem is not abstraction but too much formalization too soon. The common thread in reform efforts is not so much one of contextualization but of starting from more intuitive beginnings. I agree with the rebuttal, I have seen more students get totally engaged in solving an interesting number theory problem than some contrived ”real world” example. However, I agree with garfunkel and mumford that there are tons of legit applications that should be taught. I just don’t think that’s THE answer.

  • kkeiter

    As a teacher of elementary, I believe we go to abstraction too young without developing conceptualization. If in the early years, we develop concepts the problems many are confessing to as barriers later will be surmised. Additionally, relevance needs to be taught along with any subject in order to build better global citizenship. Math in a vacuum is rarely found in any career, so integration should be presented, even in the more abstract and higher level math classes. A good mathematician should always be able to explain WHY.

  • macleanriyadh2013

    When I was in 9th grade, anytime there was an “X” in an equation I would always write in 24. X is the 24th number in the alphabet. I couldn’t understand why I was failing algebra. I was put in a figurative corner, and I was made to feel as though I was a complete loser. Fast forward to college when I was taking chemistry. I had the experience of needing to solve a problem for which the solution was unknown. It didn’t dawn on me until then that X was simply a placeholder. I’m happy to say that I majored in Biology with a minor in mathematics. I teach now. I feel my experience makes me a much more effective teacher.
    I agree with the idea that what we teach needs to be more relevant. How about teaching programming? There are so many applications where you can teach problem solving without the context of algebra. The important thing is to help students think rationally. There is more than one way to skin the cat.

  • disqus_c8UFxf14eh

    the problem in the UK is that many subjects are not taught properly. the students are spoon fed the information. They are not taught how to think and critique the information. then as soon as the pass rates drop, they water it down again to suit the students…