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Why Our Kids Don't Get Math

CJ Westerberg, November 1, 2011 6:15 PM

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"So if eighth graders are taught math at the level of a college sophomore
why are graduating seniors struggling?
-Joe Ganem
 

A Math Paradox:
The Widening Gap Between High School and College Math
Three surprising reasons why
By Joseph Ganem, Ph.D.

We are in the midst of paradox in math education. As more states strive to improve math curricula and raise standardized test scores, more students show up to college unprepared for college-level math. The failure of pre-college math education has profound implications for the future of physics programs in the United States. A recent article in my local paper, the Baltimore Sun: "A Failing Grade for Maryland Math," highlighted this problem that I believe is not unique to Maryland. It prompted me to reflect on the causes.

The newspaper article explained that the math taught in Maryland high schools is deemed insufficient by many colleges. According to the article 49% of high school graduates in Maryland take non-credit remedial math courses in college before they can take math courses for credit. In many cases incoming college students cannot do basic arithmetic even after passing all the high school math tests. The problem appears to be worsening and students are unaware of their lack of math understanding. The article reported that students are actually shocked when they are placed into remedial math.

The article did not shock me. It described my observations exactly. In recent years I've witnessed first hand the disconnect between the high school and college math curricula. As a parent of three children with current ages 14, 17, and 20, I've done my share of tutoring for middle school and high school math and I know how little understanding is conveyed in those math classes. Ironically much of the problem arises from a blind focus on raising math standards.

For example, the problems assigned to my children have become progressively more difficult through the years to the point of being bizarre. My wife keeps shaking her head at how parents without my level of math expertise assist their children. My eighth-grade daughter asked me one evening how to perform matrix inversions. I teach matrix inversion in my sophomore-level mathematical methods course for physics majors. It is difficult for me to do matrix inversions off the top of my head. I needed to refresh my memory by pulling Boas' book: Mathematical Methods in the Physical Sciences off my shelf. Not exactly eighth grade reading material.

On another night my eighth-grader brought home a word problem that read: If John can complete the same work in 2 hours and that it takes Mary 5 hours to complete, how much time will it take to complete the work if John and Mary work together? That's an easy problem if you know about rate equations. Add the reciprocals of 2 and 5 and reciprocate back to get the total time. However it took me a lot of thought to arrive at an explanation of my method comprehensible to an eighth-grader.

My other daughter struggled through a high-school trigonometry course filled with problems that I might assign to my upper-class physics majors. I certainly wouldn't assign problems at such a high level to college freshmen. I kept asking her how she was taught to do the problems. I wondered if the teacher knew special techniques unknown to me that made solving them much easier. Alas no such techniques ever materialized. The problems were as difficult as I judged. At least I could solve the problems, a feat the teacher couldn't manage in a number of cases.

For example one problem involved proving a complicated trigonometric identity. My daughter brought it to me saying she had tried but couldn't find a solution. I saw immediately that the textbook had an error that rendered the problem meaningless. One side of the problem had a combination of trigonometric functions with odd symmetry and for the other side the symmetry was clearly even. I told her it was not an identity and that fact could be proven with a simple numerical substitution on each side. If it is an identity the equality condition must hold for all values of the angle. A single numerical counter example proves that it is not an identity. It only took one try to find a counter example.

The next day she reported to me that the teacher couldn't solve the problem.

"Did you tell him that it is impossible?" I asked.

"I told him it was not an identity and if he put numbers in he would find that out. He didn't believe me. He just said 'We'll see'."

The teacher never talked about that problem again. He did teach the class about the symmetry properties of trigonometric functions but evidently he didn't understand the usefulness of that knowledge.

At the same time I work the summer orientation sessions at Loyola College registering incoming freshmen for classes. Time and again students cannot pass the placement exam for college calculus. Many students cannot pass the exam for pre-calculus and that saddles them with a non-credit remedial math course--the problem described in the newspaper article. Without the ability to take college-level math the choices students have for majors are severely limited. No college-level math course means not majoring in any of the sciences, engineering, computer, business, or social science programs.

A colleague in the engineering department who also works summer orientation complained to me that many students who wanted to major in engineering could not place into calculus. The engineering program is structured so that no calculus means no physics freshman year and no physics means no engineering courses until it's too late to complete the program in four years. For all practical purposes readiness for calculus as an entering freshman determines choice of major and career. The math placement test given to incoming freshmen at orientation has much higher stakes than any test given in high school. But, the placement test has no course grade or teacher evaluation associated with it. No one but the student has any responsibility for or stake in its outcome.

Through the years I've found it discouraging as a faculty member to see so many high aspirations dashed at orientation before classes even begin. I tell students with poor math placement scores to go home, review high school math over the summer and take the test again. But, few take my advice. Most students with poor placement scores switch to majors that do not have significant math requirements.

So if eighth graders are taught math at the level of a college sophomore why are graduating seniors struggling? How can students who have studied college level math for years need remedial math when they finally arrive at college? From my knowledge of both curricula I see three problems.

1. Confusing difficulty with rigor. It appears to me that the creators of the grade school math curricula believe that "rigor" means pushing students to do ever more difficult problems at a younger age. It's like teaching difficult concerti to novice musicians before they master the basics of their instruments. Rigor-defined by the dictionary in the context of mathematics as a "scrupulous or inflexible accuracy"-is best obtained by learning age-appropriate concepts and techniques. Attempting difficult problems without the proper foundation is actually an impediment to developing rigor.

Rigor is critical to math and science because it allows practitioners to navigate novel problems and still arrive at a correct answer. But if the novel problems are so difficult that a higher authority must always be consulted, rigorous thinking will never develop. The student will see mathematical reasoning as a mysterious process that only experts with advanced degrees consulting books filled with incomprehensible hieroglyphics can fathom. Students need to be challenged but in such a way that they learn independent thinking. Pushing problems that are always beyond their ability to comprehend teaches dependence-the opposite of what is needed to develop rigor.

2. Mistaking process for understanding. Just because a student can perform a technique that solves a difficult problem doesn't mean that he or she understands the problem. There is a delightful story recounted by Richard Feynman in his book: Surely You're Joking, Mr. Feynman!: Adventures of a Curious Character, that recounts an arithmetic competition between him and an abacus salesman. (The incident happened in the 1950's before the invention of calculators.)

The salesman came into a bar and wanted to demonstrate the superiority of his device to the proprietors through a timed competition on various kinds of arithmetic problems. Feynman was asked to do the pencil and paper arithmetic so that the salesman could demonstrate that his method was much faster. Feynman lost when the problems were simple addition. But he was very competitive at multiplication and won easily at the apparently impossible task of finding a cubed root. The salesman was totally bewildered by the outcome and left completely discouraged. How could Feynman have a comparative advantage at hard problems when he lagged far behind at the easy ones?
 
Months later the salesman met Feynman at a different bar and asked him how he could do the cubed root so quickly. But when Feynman tried to explain his reasoning he discovered the salesman had no understanding of arithmetic. All he did was move beads on an abacus. It was not possible for Feynman to teach the salesman additional mathematics because despite appearances he understood absolutely nothing. The salesman left even more discouraged than before.

This is the problem with teaching eighth-graders techniques such as matrix inversion. The arithmetic steps can be memorized but it will be a long time, if ever, before the concept and motivation for the process is understood. That raises the question of what exactly is being accomplished with such a curricula? Learning techniques without understanding them does no good in preparing students for college. At the college level emphasis is on understanding, not memorization and computational prowess.

3. Teaching concepts that are developmentally inappropriate.
Teaching advanced algebra in middle school pushes concepts on students that are beyond normal development at that age. Walking is not taught to six-month olds and reading is not taught to two-year olds because children are not developmentally ready at those ages for those skills. When it comes to math, all teachers dream of arriving at a crystal clear explanation of a concept that will cause an immediate "aha" moment for the student. But those flashes of insight cannot happen until the student is developmentally ready. Because math involves knowledge and understanding of symbolic representations for abstract concepts it is extremely difficult to short cut development.

When I tutored my other daughter in seventh grade algebra, in her words she "found it creepy" that I knew how to do every single problem in her rather large textbook. When I related the remark to a fellow physicist he said: "But its algebra. There are only three or four things you have to know." Yes, but it took me years of development before I understood there were only a few things you had to know to do algebra. I can't tell my seventh grader or anyone else without the proper developmental background the few things you have to know for algebra and send them off to do every problem in the book.

All three of these problems are the result of the adult obsession with testing and the need to show year-to-year improvement in test scores. Age-appropriate development and understanding of mathematical concepts does not advance at a rate fast enough to please test-obsessed lawmakers. But adults using test scores to reward or punish other adults are doing a disservice to the children they claim to be helping.

It does not matter the exact age that you learned to walk. What matters is that you learned to walk at a developmentally appropriate time. To do my job as a physicist I need to know matrix inversion. It didn't hurt my career that I learned that technique in college rather than in eighth grade. What mattered was that I understood enough about math when I got to college that I could take calculus. Memorizing a long list of advanced techniques to appease test scorers does not constitute an understanding.

###

Joseph Ganem is a professor of physics at Loyola University Maryland and author of the award-winning book:  The Two Headed Quarter:  How to See Through Deceptive Numbers to Save Money on Everything You Buy, that teaches quantitative reasoning applied to financial decisions.

Previously Published by The Daily Riff 4/15/10
Originally published by APS Physics.org


Related post on The Daily Riff:
Why Other Countries Do Better in Math:  Should Parents "Race to the Tutor?"


  • MyTwoCents

    I'm finishing up an introductory statistics class. The instructor feels he must include everything in the book in 10 weeks of class, so he rushes through each class pointing out concepts in the chapter and then tells us 'it's easy' and to go home and read it. He is so busy showing everything in the book that he actually teaches very little. I wonder if this isn't the same thing that is happening in our high schools, with all the new ideas they are required to present, instead of teaching the main ideas more thoroughly to start with.

  • cjwesterberg

    Yes, it does. Whether by teacher choice or imposed mandate up the food chain (or a combo of both), students are often subject to the "gotta get through this material" regardless of whether there is true learning happening. Often "less is more". Deeper understanding of fewer concepts is often sabotaged by the" mile wide and inch deep" mentality of supposedly learning a lot. No wonder students remember little after the testing.

  • Sarah

    Our state testing scores would indicate that 40% of our students are only partially proficient in grade level math. Both of my children, despite them studying for countless hours still can't multiply, divide or do fractions with great accuracy . It is not without lack of effort either, I have consulted tutors, special educators, etc.

    So both of them have learned to use a calculator to aide them in processing their math problems. Man has always had a tool for math, remember the abacus?

    The other issues that come to mind is that not all students are meant to do college algebra. What happen to business math, everyday math, and things used in real life? I took college level math classes in high school, found calculus in college to be easy, etc. but I use almost NONE of those skills in everyday life.

    Of course many of the everyday use math I didn't learn until I was out on my own. I had to figure out the real sale price, the per unit price (which most places now list), how to balance a checkbook, etc. The math classes that taught those skills were eliminated from our school. I remember my mother taking me shopping and having to point out how to do those things when I was well out of school.

    Our kids have access to so many apps on phones, etc. that they really only need to know the concepts of WHY they are entering the numbers. I have to wonder if our kids are not excelling in math because of this.

    Many of the things taught in schools are useless or outdated or no longer apply... The other day I was reading a magazine from the late 1800's. They used many words that I had NEVER even seen. Does that mean I am stupid, less than or otherwise incompetent in reading? No, it just means that .. we don't do that anymore.

  • I have been the principal of an elementary school and a middle school and the breakdown starts to happen in the sixth grade when we transition from getting to the right answer to 'why is that the right answer'.

    because of the over reliance on test scores, primary grade teacher no longer teach students to have fun, or play with math. They don't teach then to think mathematically. The in sixth grade they hit the proportion and ratio wall. Or, the point that memorization of basic facts or operations isn't enough.

    The problem before is that our expectations weren't high enough. Now they are in the right place. But, because we get on primary teachers to get scores up, primary teachers are being penalized by taking the time to teach children to think mathematically.

    Example: We used to give third graders rules and ask them to measure everything around the room. Then compare sizes etc.... Now, we give them a 4 inch paper ruler, teach them to read the ruler and call that teaching measurement.

    Sorry, to say that I think it's going to get worse before it gets better at the college level.

  • Joye

    Everyday Math is the problem. A disastrous introduction to K-6 math.

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