**A Math Paradox:**

**The Widening Gap Between High School and College Math**

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.

3. Teaching concepts that are developmentally inappropriate.

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

I teach Computer Science in a Maryland Public School (a Montgomery County High School to be precise). I came to teaching as a second career, having spent twenty years or so in all aspects of the science, from academic research to commercial software development.

I consider myself fortunate to be in a profession where I can continue learning; presently, I'm pursuing a PhD in Mathematics Education at UMD because I'm very interested in these kinds of questions. Moreover, I see the results of so-called "accelerated mathematics curriculum" on a daily basis. Bad is perhaps an understatement. I believe that the original writer and the follow-up from the Mathematician above make some excellent points. Perhaps the most salient, for me anyway, if the failure of students to concentrate on anything for more than 7 seconds. I picked 7 seconds because that's about the average amount of time a person spends on a web-page.

A lot needs to change to rectify a failing situation. In my classes, I teach a fair amount of set theory, combinatorics, and algebraic structures---all in the context of functional programming first, so-called object-oriented or imperative models later. I have been doing this for 8 years with good success on the standard "measures" that preoccupy schools and the public---i.e., AP tests and the like.

Now, here's what I see on that front: The quality of the AP exams in the Computing Sciences has dramatically diminished. I'm told that similar things have happened on some of the calculus exams as well.

One problem might be that Public Schools are allowing ETS to drive curriculum, but ETS is interested in making money by ensuring that more students take these exams. Ultimately, it's the Department of Education and the ridiculous No Child Left Alive law that has brought us to what I fear is an irretrievable situation... it will take years to turn this around because we have spent years getting here. In the interim, I believe that the agenda belongs to those nations who are educating their youth while the United States has co-opted theirs as the next marketing possibility.

Oh, before I forget: I spend much more time teaching mathematics than computer programming. Seems that once they are comfortable with logic and mathematics, I cannot STOP them from programming, and programming well.

For the mathematicians in the audience: think of programming as theorem proving, program constructs are axioms, programs are proofs, etc.

Here are two solutions of quadratic equations from homework by senior college mathematics majors.

1) bx^2 + cx + a = 0 has solutions x = (-b +- sqrt(b^2 - 4 a c)) / (2a).

2) ax^2 + bx + c = d has solutions x = (-b += sqrt(b^2 - 4 a c))/ (2a).

In both cases the quadratic formula had been memorized, but in a meaningless format. When I inquired about why the error, in each case the answer was "we had that a long time ago, so we don't remember it."

But ask them to diagonalize a matrix or solve a second order linear differential equation, and they will get it right. They took Linear Algebra and Differential Equations this semester.

One of my mentors helped me to understand this phenomenon. He said: You never understand Course N until you have taken Course N+1 which uses Course N.

--gbc

I consult for several small and medium size manufacturing businesses and am amazed of the number of applicants with high school diplomas that cannot pass the basic math tests given by these companies. For example, asking an an applicant to convert 1/4 to a decimal and getting .4 as a response is not uncommon. The issue of math education deficiencies is much more basic than preparation for a college calculus course!!

I'm on this website because I'm trying to figure out how both my sons could be at this point in college - they both did well in math in high school and one son passed AP Calculus; took two more calculus courses in college and fell on his face. The other son took up to Trig in highschool, took the math placement test in college and wound up taking remedial courses in math and is now almost ready for calculus. They both want to be engineers, and both had great gpa's in highschool and had no trouble getting into engineering courses at a local polytechnic university. They learn quickly and both have build RC cars, go-carts and even restored engines in classic cars - no laziness or short attention spans here. They shine in anything english and reading comprehension because I got in front of a sorry teaching profession by teaching them to read at a very young age and encouraging reading from the time they were able to look at a book. But I feel I failed them in some way because I did not predict this problem. Neither of them even like math much at this point and are somewhat afraid of the subject.

This isn't just their mom talking - these are 'bright' kids and both my husband and I are engineers also so there's no mystery to the profession.

I don't remember struggling with math as much as they do now.

Something is wrong with the way the subject is taught. Kids coming from other countries have strong math skills entering highschool - but not the kids from here - something is going wrong at a very elementary level in school to wind up with so many kids turned off by math in highschool and so many more obviously unprepared for college level courses.

If some of you are math instructors, are you the problem or do you have too much to work with, too many bad habits to undo, what? by the time these kids come to you in college? Lastly, is there any way to turn this around? What would you insist that the schools do from the beginning of the math teaching at the elementary level to preparation for college?

I want to help my sons - what can I do at this point to help them get through their college level math classes and do so with confidence and hopefully enjoy the process?

Thanks for sharing - your story is, sadly, not unusual. You may want to check out:

http://www.thedailyriff.com/2010/05/a-mathematicians-lament---part-1-of-3---the-dream.php

We highly recommend the book but the excerpt Paul shares is quite extensive.

Also, you may want to check out the "Singapore Math Demystified!" series on this site. It, too, offers valuable insights.

We look forward to hearing some comments from other math educators.

Dr. Ganem's heart is in the right place, and I agree with the fact that so many students that have passed state tests saying they know high school mathematics, in fact do not. But I think his daughters' experiences were somewhat atypical, and he has extrapolated incorrectly from them. I am a Ph.D. mathematician who has taught mathematics to K-12 teachers in a school of education for the past 6 years, so I feel I am qualified to speak.

Dr. Ganem makes 3 points, which have varying degrees of validity, in my opinion.

(1) Confusing difficulty with rigor and (3) Teaching concepts that are developmentally inappropriate seem to me to be essentially the same point. I don't see this happening very much. Much material that I learned in a public high school circa 1960 is simply not taught any more, and has been replaced with material that is easier, not harder. This phenomena of teaching advanced topics at too early an age seems to exist primarily in upscale suburbs where parents demand that students take AP classes in Calculus and other subjects to help them gain spots in ridiculously competitive high-prestige universities.

(2) Mistaking process for understanding. The math education reform movement of the last 30 years (spearheaded by NCTM) has assiduously worked to overcome the overemphasis on process, and has come under heavy fire from the traditionalists in the so-called math wars. I am very glad to see Dr. Ganem make this point, because many in the University community have sided with the traditionalists and complained that their students should have spent their K-12 years learning to perform basic algorithms. In my view, both process and understanding are important.

I agree that the fact that students can pass a math test for high school graduation (such as the MCAS in Massachusetts, where I live) and still not be prepared to take an introductory Calculus course is a scandal. There is plenty of blame to go around. We in the Schools of Education are not producing enough teachers who are good enough to teach mathematics with understanding, schools are having their feet placed to the fire by legislatures and so are teaching to the test that their students have to pass, students of the ipod/twitter generation who lack the ability to concentrate for more than a few seconds at a time.