The Frontal Cortex

Algebra Education and John Dewey

It’s long been recognized that American kids suck at math, at least when compared to kids in Singapore, Finland, etc. What’s less well known is that the steep decline in proficiency only starts when kids are taught algebra. That, at least, is the conclusion of a new government report:

“The sharp falloff in mathematics achievement in the U.S. begins as students reach late middle school, where, for more and more students, algebra course work begins,” said the report of the National Mathematics Advisory Panel, appointed two years ago by President Bush. “Students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation.”

The problem with algebra in America actually extends far beyond the international comparisons. More kids drop out of high school because they fail introductory algebra than for any other academic reason. In the fall of 2004, 48,000 ninth-graders took beginning algebra in the Los Angeles Unified School District. Forty-four percent of students flunked the class, nearly twice the failure rate of English. An additional seventeen percent finished with Ds. The vast majority of these students will never graduate. They will fail algebra again and again, and then they will give up.

Why is algebra so hard to learn? Because it’s so abstract. No other high school subject is as disconnected from the real world. When students open their algebra textbook, they enter into a world of pure ideas, with page after page of elusive equations and intangible theories. In fact, supporters of mandatory algebra classes tout this as one of the subject’s benefits: it is often a student’s only introduction to abstract thinking.

But this abstraction, especially when it is taught abstractly, comes at a steep cost. Algebra uses an idiom of symbols and variables to express mathematical relationships. Unfortunately, students rarely understand how these relationships map onto everyday experience. Although algebra was invented to solve practical problems, struggling students rarely grasp what problems algebra can actually help them solve. A study commissioned by the Los Angeles Unified School District concluded that the single biggest problem in algebra instruction was its “systematic failure to teach algebraic concepts which students could then knowingly apply.”

The solution, I think, lies in the pedagogy of John Dewey. While Dewey’s educational philosophy goes in and out fashion, I think he got a basic fact of learning right, which is that the brain learns by doing. Abstract concepts, untethered to experience, are never internalized by our neurons.

In November 1894, John Dewey described his educational philosophy in an excited letter to his wife, “When you think of the thousands & thousands of young ‘uns who are practically being ruined by the Chicago schools every year, it is enough to make you go out & howl on the street on the street corners. There is an image of a school growing up in my mind all the time; a school where some actual and literal constructive activity shall be the centre & source of the whole thing, & from which the work should be always constructive industry.”

A little more than a year later Dewey opened the University Elementary School at the University of Chicago, where he was already employed as a philosophy professor. For Dewey, the school wasn’t just about practicing a new way of teaching. It was also “a laboratory of applied psychology”. In fact, its official name was “The Laboratory School.” As Dewey wrote in a planning letter, “This school is a place to work out in the concrete, instead of merely in the head or on paper, a theory of the unity of knowledge.” The students were his metaphysical guinea pigs.

But how could a small private school demonstrate the “unity of knowledge”? Dewey’s insight was that primary education was the ideal place to collapse what he called “the invidious distinction between learning and doing.” At the Laboratory School, knowledge was viewed as a by-product of activity. Of course, this is a radically unconventional way of teaching children. Traditionally, things worth knowing are handed down from teacher to pupil as a disembodied encyclopedia of information. The job of a student is to memorize as much of this information as possible.

Dewey thought this approach was not only a terrible way to teach children (“its primary effect is boredom,” he once remarked) but was rooted in a false theory of thinking. Even worse, it reinforced this same erroneous distinction in its students, who grew up believing that learning and doing were separate activities. As a result, education became a useless exercise, nothing but the memorization of “truths ready made”.

At the Laboratory School, Dewey was determined to make knowing and doing part of the same learning process. His mission was to “reinstate experience into education”. The Laboratory students spent most of their day outside of the classroom, engaging in activities like sewing, carpentry and cooking. (Unlike other schools at the time, Dewey made boys and girls participate in activities together. Girls learned how to use a hammer and boys learned how to bake.) But these activities weren’t simply exercises in manual labor. Rather, they were demonstrations of “active learning”. “If a child realizes the motive for acquiring a skill,” Dewey argued, “he is helped in large measure to secure the skill. Books, the ability to read and bookish knowledge are, therefore, regarded as tools.”

Take cooking. At the Laboratory School, the children were often responsible for preparing their own lunch. Dewey’s insight was to build into this activity a wealth of related academics. Before students could boil an egg, they had to conduct experiments to determine the proper temperature at which to cook the egg. When they graduated to the preparation of more complicated dishes, the students had to weigh and measure the ingredients (arithmetic), understand the process of digestion (biology), analyze the process of cooking (chemistry and physics), and so on.

The secret, of course, was to sneak in the science. The knowledge had to seem indivisible from the lunch. “Absolutely no separation is made between the ‘social’ side of the work, its concern with people’s activities and their mutual dependencies, and the ‘science,’ regard for physical facts and forces,” Dewey wrote in 1899, in his best-selling pamphlet The School and Society. If the teaching was done right, the children wouldn’t even realize they were being taught.

I wrote about Bob Moses’ Algebra Project – a modern version of the Laboratory School – a few years ago in Seed. For more on Dewey, check out The Metaphysical Club, by Louis Menand.

Comments

  1. #1 Tony Jeremiah
    March 14, 2008

    Seems like Dewey’s ideas are consistent with the Montesorri Method. Also, the general idea of making mathematics less abstract as a means for increasing comprehension is a premise related to Dave Munger’s article (Children learn and retain math better using manipulatives) on CogDaily.

  2. #2 Julie Stahlhut
    March 14, 2008

    The sad part is that algebra is something that we really do use in everyday life. Anyone who has ever doubled a recipe, figured out the savings on a bulk purchase, or estimated how much farther to drive before filling the gas tank has done algebra on the fly.

    So, how do we sneak the math in?

  3. #3 jb
    March 14, 2008

    How sad that the LA schools have forgotten the lessons taught by Jaime “Kimo” Escalante, a math teacher in eastern LA, who was immortalized in the 1988 movie “Stand and Deliver”. Edward James Olmos won an Oscar nomination for the Best Actor in 1989 in the starring role. Escalante took gang members and potential drop outs and trained them to pass the AP Caluclus exam in their senior year. There’s more but you should watch his methods and the movie. A true story. Great flick.

  4. #4 T. Bruce McNeely
    March 14, 2008

    When I was in school in BC, a lot of our algebra assignments were in the form of practical problems which required solving by algebraic methods. Math was always my weakest subject, but as long as I kept up with my assignments, I was able to achieve reasonable grades.
    I would have liked to given some examples, but this was a while ago, and the clay tablets have since crumbled into dust…

  5. #5 Tony Jeremiah
    March 14, 2008

    To sneak the math in, I’d imagine one would have to create a learning environment whereby the everyday situations mentioned are directly connected to the mathematics involved.

    As an example, if a school is near traffic, a teacher could give students a stopwatch and measuring tape, and have them figure out how fast (on average) vehicles are going in a particular area.

  6. #6 Ted
    March 14, 2008

    Do you, or does anyone, know what is different about the teaching methods in Finland or Singapore that makes their success rate greater? Are they using the methods you describe? Or do they offer algebra in a different grade? I remember having a very hard time with algebra, which was given before geometry. I thought it might make more sense that after learning the basics through elementary school, one could go on to geometry – applying knowledge of math to spaces and shapes and the concept of proofs. And then go on to more abstract applications like algebra and trig. But as far as I know, that is still the order in which they are taught.

  7. #7 APR
    March 14, 2008

    Maybe, in addition to crappy books (and crappy algebra teachers… oh, the stories this very good HS algebra student [because of my dad, the physicist] could tell…), the problem is that so little abstract thinking is taught prior to high school.

    Algebra needs desperately to be taught better but poor algebra books, teaching and performance is certainly as much a symptom of something larger as it is a problem in and of itself… and the important focus on making abstract materials concrete only produces genuinely good students if concrete materials are also situated as meaningful because of the metacategorical/abstract phenomena they represent.

  8. #8 Tristram Brelstaff
    March 14, 2008

    Anyone interested in mathematics education should read A Mathematician’s Lament by Paul Lockhart.

  9. #9 PUZZLED MATH LURKER
    March 14, 2008

    This reminds me of my highschool/middle school experience with algebra.
    I switched school systems in the middle of the year. In the first, I was excited about algebra. We learned by FIRST solving equations, THEN going to the more abstract theory behind them. It was fun – sort of playing with puzzles.
    In the second, the theory was introduced first – we sometimes NEVER got to play around with related numbers, but spent a lot of time “proving” things and showing work.
    I got a B in the second class, but never really felt I understood it. I thought that math just wasn’t for me. That discouraged me from taking anything but geometry (the only required course) in high school. I enjoyed the geometry, but was more concerned with GPA than challenge to take more advanced math.(Being female in the 60s sort of added to the push away from math)
    I studied anthropology and linguistics in college
    And wound up as a computer programmer eventually anyway. I probably would have gotten there sooner with a less abstract algebra experience!

  10. #10 jb
    March 14, 2008

    It was with gritted teeth and consternation that I finally stooped to taking education courses in order to teach middle school science in the public schools of Maryland… I who had done graduate work in two sciences, who had worked as a scientist, and had taught at the college level. Imagine my surprise when I was turned on by Paiget’s 4 Stages of Cognitive Development and the necessity of teaching to the appropriate stage of your students with plenty of hands-on experiential lessons. This must precede moving into the Formal(abstract thinking)stage which happens in high school though there are many adults who never get there.
    I am also indebted to a behavioral psychology course in which I learned how to properly motivate students to learn. I find myself still using this knowledge to teach adults most of whom aren’t at the Formal stage and who are too busy to do homework unless you can motivate them. The best book on motivation is “Don’t Shoot the Dog” by Karen Pryor now in it’s third revised edition. For a sense of what this is about google “What Shamu Taught Me About A Happy Marriage” ,the most popular 2006 online story of the NYT. What works with motivating animals, works in the classroom, at home, and with oneself (the most difficult animal).
    It is the rare teacher, like Escalante above, who knows his/her students, uses the correct techniques and motivators, and is willing to put in the hours. Beyond that there are the differences in families and societies as to how much the education is valued and how useful it is. The answer to Ted’s questions lie there.

  11. #11 6EQUJ5
    March 14, 2008

    Our high school algebra, geometry, and trigonometry were taught by rote. In college I went into calculus with no mathematical insight whatsoever and had to switch out of the college of science because I couldn’t memorize my way through the material.

    After completing my formal education, I discovered algebra could be used to solve real-world problems. I taught myself geometry and trig in order to solve more kinds of problems.

    One place I worked, it seemed our gasoline vendor was shorting us, but we couldn’t prove it. I taught myself enough calculus to solve the problem (liquid depth in a right circular cylinder laying on its side) and from there on we could measure the volume in the underground tank. When the vendor heard about our ‘computer program’, suddenly our losses stopped.

    I ended up in aerospace, using mathematics to solve practical problems, such as proving a secondary signal appearing in the ground receivers tracking a satellite in a polar lunar orbit were actually the reflection off the lunar surface of the satellite’s downlink through its toroidal antenna.

    Yeah, math (and physics) rules. So why didn’t I learn this in public education?

  12. #12 load of nuts
    March 14, 2008

    6EQUJ5, your story about the gasoline vendor is great. I called some guy named “Bob” at the 800 number listed on the pump to report that one station was overcharging. “Bob” told me the gauge in my car was broken, harhar.

    I took these Math classes back in the 70’s when nobody cared about learning anything but humanities and personal problems. The teachers were awful. Math was one of my best subjects and the only reason for that was because I didn’t mind plugging numbers into formulas for the hell of it. I begged the teacher for a practical explanation and he couldn’t give it so I quit and went into the arts.

  13. #13 Cherish
    March 14, 2008

    I homeschooled my son using the Singapore math books. I found that they tried to give a very intuitive feel for what was going on in each type of problem. There were lots of examples, lots of pictures, etc. They also teach you some nifty little tricks for calculation so that you don’t spend all your time on calculating things. However, these tips are often not accessible until you have an intuitive feeling for what is going on.

    (I’ve also read a lot about learning styles. Most schools teach a very sequential style of learning, especially in things like algebra. This probably makes life pretty rough for people with other learning styles…which could be between 1/3 and 1/2 of the population.)

    My experience in algebra was very much the same as described here – disembodied concepts. The teachers were far more worried about calculation than making sure we understood what it all actually meant. I’m glad I know enough so that when my older son comes home from school and doesn’t understand his maths homework, I can explain it to him.

  14. #14 David Rock
    March 14, 2008

    Here’s something to keep US economists awake at night. Research by Dr Yiyuan Tang that was published in Nature a few years ago (and since repeated by others), shows that people from eastern societies do mathematics using different brain regions than westerners. The theory is that eastern brains spend years practicing picturing and combining 3,000 abstract ideas – the letters of their alphabet – as they learn to read and write. They end up with a strong ability to picture concepts and combine them, and this shows up in fMRI’s as increased capacity in some brain regions on average over westerners. The result? Eastern brains seem to be stronger with abstract ideas: think algebra, but also science generally, and of course, innovation. They are also more able to see relationships between concepts in some situations. And before I get yelled at…I am not for or against any particular culture, just presenting the science.

  15. #15 Janne
    March 14, 2008

    David Rock, only two “eastern cultures” actually use pictograms (Kanji/Hanzi): Japan and China. All other use syllabaries or alphabets. If true, you would see this effect only for Japanese and Chinese schoolchildren, but not Korean, Indian, Vietnamese… And you should see a fast drop in math ability among Korean schoolchildren from two generations ago up to now, corresponding to the gradual disappearance of pictograms in Korean and not correlate with other academic or societal changes (ie. if the education system or school-going population changes you’d expect changes also in physics or sociology knowledge).

  16. #16 Malcolm
    March 14, 2008

    When I was teaching algebra here in New Zealand, I found that the biggest problem came from kids misunderstanding the concept of an equation. Most kids thought of the equals sign as an operation.
    I found using scales as the best way of showing them simple examples. When they worked out that the equals sign was the equivalent of the scales themselves, the penny usually dropped. Without that understanding of what an equation is, algebra is almost impossible for most kids to comprehend.

  17. #17 Leigh
    March 15, 2008

    My eighth-grader is taking algebra this year. She’s always done well in math . . . top of the class . . . and her math teacher from last year told me that my daughter was always ahead of the others when new concepts were taught, so that the teacher had come to depend on her to help explain the conceptd to the other students.

    This year, algebra, and disaster. Suddenly my daughter is falling behind.

    The problem, at least from what I can see? An absolute mess of a math text. The thing is practically incoherent, and the teacher is trying to “improve” its organization by skipping around among chapters. I won the Senior Mathematics Award in my large high school, and *I* can’t understand this book.

    I’m going to tutor her this summer to ensure she has the foundation she’ll need — she wants to be an architect, for pete’s sake, so this is crucial.

    Does anybody have an opinion on the best algebra text currently available?

  18. #18 William Bennett
    March 15, 2008

    Students throughout the world study algebra and pass successfully. It is not an advanced topic, nor is it too difficult for mere mortal to understand.

    Very few math teachers in schools today have math related degrees, or more than a cursory understanding of mathematics. There is no understanding in the US of how to teach math. But teaching math is not difficult. The best solution would be to bring in teachers from overseas on H1B visas to teach math since american teachers have proven themselves incapable.

  19. #19 bad Jim
    March 15, 2008

    Here’s an interesting article in the LA Times about the Singapore math curriculum, which is starting to be used in California schools. I seem to recall that one of the problems schools encountered when introducing it is that the teachers didn’t always understand it.

    It may well be the case that too few teachers understand algebra well enough to teach it.

    Back in the Pleistocene, when I was in high school, I did a little tutoring. It felt like psychoanalysis – find out where the stumbling blocks are and work them through. A lot of kids had trouble with fractions, of course, but that wasn’t the only issue. Perhaps a lot of kids are going to need individual attention.

  20. #20 Nico
    March 15, 2008

    I’m in a university prep algebra class due to the ineffectual teaching methods from high school.

    My current instructor is utterly terrible at explaining anything and if you express any confusion, she usually states that maybe we just shouldn’t bother.

    The text is of her own making, cut and pasted and pulled from a variety of conflicting texts. The copying hasn’t really come through clearly so great swaths of it are virtually unintelligble.

    I can understand a mathematical concept if I’m given a logical linear progression through the topic, and applications for use but I’m finding now that a lot of people who teach math may understand it, but not how to teach it to others.

    I’ve found a couple more coherent texts that I rely on, ignoring the useless class text but it still leaves a lot unexplained.

  21. #21 Kevin
    March 15, 2008

    I struggled thru math in high school. Algebra and trig classes consisted of memorizing formulas and ways to solve them. There was no attempt to teach the logic behind the problems. I didn’t go to college till years later, and one of the courses I took was an algebra refresher. In the very first class the instructor explained the essence of algebra in a way that was clear, concise, and easily understandable. Suddenly something that had once seemed very difficult became very easy simply because my approach to it was based on a clear understanding of what i was trying to do, rather than just trying to determine what equation to use and what values corresponded to what elements of the equation.

    quality of instruction makes a world of difference

  22. #22 Dave
    March 17, 2008

    I agree that educational topics should be taught with application, but that theory doesn’t always make it easier for the kids. Give a student a simple algebra problem, and then give him a story problem and then ask him which one he thinks is harder. The story problem might give them a fuller, more long lasting understanding of the algebra problem, but it’s far more difficult for the student to get it right.

    On the other side of that topic, at some point in a students life they may HAVE to learn how to abstract problems away from real world applications in order to tackle some of the more complex mathematical problems. How do you find a real world application for the applied math problem to prove that the square root of two is not a real number? Or if you’re taking a computational linguistics course and you have to write a turing machine to accept a language.

    You can teach students how these questions come up, but at some point you have to learn to leave the story problems behind and tackle the formulas directly.

  23. #23 Morgan-LynnGriggs Lamberth
    March 18, 2008

    What is with this one size fits all? I never have recourse to any math beyond grammar school. I feel fine with that. Why was I made to suffer, because air heads think that math would make me better at thinking? It has long been known that math so helping is false. Behe is a mathematician yet he reasons less well than I.
    [And would help others,] yet I would not force languages on others.
    Why can we not focus on the individual than on what teachers find to their joy and wants? I study what I want and need rather than what would make me more psychologically disturbed! I study languages and the philosophy of religion. In the latter, I avoid modal logic without loss of comphrehension.
    My verbal IQ tested at 150 while the rest was low. Why had I to suffer so much when I knew my strengths and weaknesses better than the air heads ? I like history and geography but others are different.

  24. #24 Rhonda
    March 18, 2008

    Very simply, no child, teenager or adult of any age will find any subject, including math, easy and interesting unless they have an interest in it whether self induced or inspired by an instructor. If it ain’t fun to do then it’s a trudge. Personally I did very well in Algebra I and Geometry but just about failed Algebra II due to a teacher who was unable to communicate with the class. As a result, I did not continue to Trigonometry. Further, as a high school senior in 1969, I was forbidden to take physics as I was told all places in those classes were being held for boys “who would go on to college.” I was directed to take clerical courses even though I had high grades and fully intended to attend college. In the south, girls only were supposed to go to college to find husbands, not to learn anything. At least I think that has changed.

  25. #25 Frank Greco
    March 18, 2008

    Phantoms in the Classroom

    The solution to the current state of education, from instruction to student retention is brain-based learning.
    Teachers and other higher-ups should discover more about the brain in the classroom. Neural plasticity refers to the changes that neurons can undergo because of learning. Learning is one form of conscious activity that has been shown to influence neural reorganization. Teachers should understand basic principles of how the brain functions.
    The connectivity that is required in the classroom to develop dendrites requires the input of all students. Rather than have students in the class as gazed-out zombies, brain-based learning allows them to connect and get involved in the class.

    Frank Greco
    Teacher

  26. #26 Montez
    May 12, 2009

    Math

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