A Math Paradox: The Widening Gap Between High School and College Math

By Joseph Ganem

<snip>


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.

<snip>

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.

http://www.aps.org/publications/apsnews/200910/backpage.cfm?renderforprint=1

His "3 problems" are the same I've been talking about for years, from a K-8 perspective.

For example, back in elementary school I was forced to learn to 3 different methods of multiplication and division (even though I could barely do either.) The only time I ever succeeded in math was when my teacher forced me to learn the orthodox method and to practice it for hours on end. This was about 10 years ago, so it might have changed. However, from what I am hearing it does not sound like it changed for the better.

argues against the rote memorization of fact and procedure, and for making sure students understand why they are following a procedure, why it works, and when it's the effective choice to solve a problem.

problem. Math is an exact science. An answer cannot be just "close" it has to be right on the dot. When my children were taking Algebra in the 1970s, I went to help them because they weren't being taught the basic rules that go with Algebra, and that was my introduction to the textbook. I went to look up a definition to help them better understand the problem they were working and the definition was no good. It used the term I was attempting to define in the definition. Sooo I taught them the way it should have been in the book.
I found the same thing when helping my grandchildren. They had no textbook for the most part, just sheets of problems with no explanation to help them figure out how to do the problems. On top of that, many problems required a guesstimate..not the real solution to the problem. Almost like they are being set up to not know math when they leave high school.

the only "guesstimating" I've found is in the CAT6 for math, the standardized test used in California. There were some problems in that test for elementary kids that asked them "which of the following is closest," or something like that. It's been 4 years since I gave it, so I'm not sure about the wording.

It's obviously supposed to measure how well students can estimate, which is a valuable skill when used in appropriate circumstances.

I haven't experienced, in my, or my kids, or my grandson's, math curriculum, "guesstimating" as a replacement for finding an answer.

The problem introduced by calculators is that kids bang in numbers, make a keystroke error, and are blissfully unaware that their answer is off by orders of magnitude.

The point of introducing guesstimating was to have them develop a sense of when their use of the calculator has produced a tremendously wrong result. It didn't need to be a "taught skill" in the 70's to that extent.

However, at least by the time they are doing algebra, it's simpler just to have them plug their results back into the problem to check consistency, which is much more important in the long run than a gut sense of basic calculation error, IMHO, but I'm not a math educator.

Any course which is called a "math" course should not require, or allow, calculators.

Learning to make approximations is an extremely valuable skill, it's just not real mathematics. Both skills need to be learned to a substantial extent to prepare students for either a career or college -- or to fluorish in a technologically advanced society.

in the 1960's, when I was in high school, called UICSM (University of Illinois Committee on School Mathematics) which was the most amazing program ever. I'm yet to find anyone who I didn't go to h.s. with who's even heard of it.

What was so good about it was that we discovered everything more or less on our own. As a consequence I was able some thirty years later to test directly into Algebra II when I needed to take math for a program I wanted at my junior college. And I'd sit in class while information bubbled up from the darkest depths of my brain.

I've never understood why the program didn't become standard. Possibly it would have required a much greater commitment on the part of the math teacher to lead the students along the path of discovery.]\


I also recall that around the time I was in junior high school it had become popular in elementary math to teach kids to work with base-5 or base-12 or other non base-10 systems. I always thought that was a real waste of time. My own kids didn't get that, so maybe it disappeared.

as I recall, screwed up a whole generation of kids, most of whom now vote GOP because they can't add or subtract and have to go with their guts, which are invariably wrong.

I hated tutoring math when I was in nursing school because not only did the high school grads coming in think they were just going to sit there without participating, they honestly couldn't do the most basic arithmetic. There I was trying to teach them college level algebra and they couldn't even figure out percentages, multiply fractions, or do any basic tasks. Most couldn't do basic functions without a calculator.

I was expected to fix them without their active participation.

On the one hand, kids from an affluent loving families are not learning their times tables. Without their calculators, they cannot compute a thing.

And then, on the other hand, once the kid arrives in 7th grade, the material is so far ahead of them due to overly ambitious school boards that even I am stuck. Not only have I been stuck - I had to call a friend who for years taught statistics at a university to help me. And she was stumped by some of ht material.

So it seems that a cohesive intelligent protocol for instructing kids in math has not yet been uncovered. In my humble opinion, learning the math tables helps your brain wire itself and serves to develop your thinking processes. And pretending that kids who only know how to use calculators will suddenly know how to jump from sixth grade to college level statistic course work is overly ambitious.

at the dependence on calculators. Not that I don't think kids should learn to use them, like any other tool. But to depend on them for simple calculations?

I was always stronger in math concepts than calculation, and the mass introduction of calculators, around the time I started engineering, was really liberating. I can handle complex differential equations, but do NOT ask me to tote up a golf score card unless you have a lot of time on your hands.

I can tote up columns of figures in my head, but struggled with higher math throughout middle school and high school. It wasn't until I got to college that I realized why.

I think knowing elementary things like addition and multiplication facts is important, but it's not the MOST important. Understanding concepts is equally important. When math is taught by rote, not every student grasps the underlying concepts.

a disservice to the children they claim to be helping."

Exactly. I teach remedial math at a community college. The students I work with have never actually been taught arithmetic . . . they've been trained to do various tricks well enough to pass the next test -- and then those tricks are promptly forgotten.

Once someone takes the time to explain the concept they are able to grasp it quite readily (and are quite appreciative). Too bad they wasted a half a dozen or more years of math classes making test manufacturers richer and not really learning any math.

*edit to add quotation marks*

back when I was tutoring high school math students. I was in college at the time. One of my HS students showed me a problem that she had gotten wrong on a test. It was a statistics problem that appeared simple enough at first glance, but I quickly realized that I couldn't solve it. I took it to a friend of mine in the math department, who was a genius on scholarship from Russia, working on a graduate degree in Combinatorics. It turned out that he couldn't come up with a solution, either. The problem got passed around the math department until one of the big wig professors spent a weekend working on it. I got it back with a 4 page solution and a note that said the problem was much too hard for a high school student and the answer the teacher had given was obviously completely wrong.

I gave the solution to my student, but I think she was too afraid to confront her teacher about it. What's the point of trying to teach kids higher level math when you're only confusing them? So many of my students were very bright but lacked any confidence because they didn't feel they understood the material at a fundamental level. They were only able to solve specific problems if they had worked through a similar problem in class. I often felt like I had to retrain them from the ground up, which is not the role of a tutor.

and dysfunction to decide that students who can perform an operation or solve a problem by following an example are ready to move up the ladder.

It's especially common when teaching the gifted, who can race through the generally low-level activities happening in their class. Too often, instead of taking them deeper, they are rushed forward before they should be.

That's a problem with math education that has been around for generations. There is more emphasis on memorizing facts and rote procedures than on understanding math at that fundamental level. It, like everything else, is exacerbated in the "standards and accountability" culture that "raises standards" by dropping curriculum down grade levels, instead of adding that emphasis on concepts, and then pouncing with high-stakes tests.

meaning I haven't read the article, only the excerpt. I read it yesterday, and it stuck in my mind:

"As more states strive to improve math curricula and raise standardized test scores, more students show up to college unprepared for college-level math."

If true, then what more proof is needed that changes made during this unspecified time period ("strive to improve") aren't working?

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

Obviously, that's the con, or sales, job, for the taxpayers benefit; that K-12 graders have been prepared. Instead, it seems, their time has been wasted by the state in stupid lessons. Why it is happening is less important than the fact that it is happening. What is important is that the state has gone on record as saying the 'intent' is to prepare for college, while failing that task.

However, there's another issue. What about those that won't be going to and completing college? Isn't that about 74-75% of everyone? In those cases, how has K-12 not been, overall, an authoritarian punishment? Jump through these hoops, because we told you to jump through them, the sales lie is we're preparing you for something you don't need or want, and we're not going to compensate you one dime for your time that we forced you to spend doing what we told you to do.

In this latter case, what good does it do to tell these folks that their time was wasted? That their educations were worthless? Isn't that the same schtick that employers (college grads) constantly drum into their employees? You're worth-less or worth less than we're paying you.

The pattern of the two suggests the problem is rooted in money, and whether we have the right to determine our own lives by making our own choices, or have to live out the lives of drudgery and punishment the state and associates have outlined for us by doing their will, their stupid, under threat of their constant punishment.

Like it or not, the hoops we all must jump through in school become punishment when those tasks fail us.

---

I read an article online the otherday, I can't find the link right now, the basic premise was that things are the way they are because that's how our rulers want them in spite of what they constantly tell us to get elected. It seems everything we've been exposed to is a great big con job.

So, while the ultimate purpose of deliberately failing to teach 3/4 of us seems to be about and rooted in money, the secondary purpose is likely to keep us divided and calling each other names such as "stupid" and "unprepared".

It's essentially the division of the haves and the have nots, and the college grads stuck in the middle between the two, executing the will of the rulers against the employees for their own financial benefit, hoping against all hope that their dream of education really was and is for everyone and that everyone had just as much opportunity as they did and that any failures were personal instead of systemic.

To acknowledge that the problems are systemic risks questioning one's own success, for what is success in a sick system but sickness itself?

to connect the links.

It's a logical consequence to high-stakes testing and politically-motivated demands for "higher standards" imposed by politicians and others who are not educators; we spend our time on mandates, striving to avoid the AYP lash of failure, and students don't learn as well.

They test better, but then, educators have always known that raising standardized test scores doesn't necessarily indicate more learning.

That means giving teachers the freedom to make subjective evaluations, and to modify their lessons interactively with their students. It's absurd to believe that every, or even many, aspects of intellectual development can be reduced to directly, quantitatively measurable outcomes, or that pushing children to reach highly specific milestones guarantees that development has followed an appropriate path.

Really grasping the material requires more contextualization, not more testing.

NCLB is not just 180 degrees in the wrong direction, it is a skew vector to any desirable trajectory. Only authoritarians want it.

"authoritarians." That's what the whole standards and accountability movement is about, and it doesn't benefit students.