This was written by Timothy S. Norfolk, who is the interim chair of the Department of Mathematics at The University of Akron, where he has been teaching since 1984.
While most of what I have to say below concentrates on mathematics (my area of specialty), much applies equally to the sciences and engineering.
It is crucial to our economic survival that we do better in mathematics and science. Global competition requires manufacturing to be made more efficient and more automated. Incredibly sophisticated mathematical tools are used in the analysis of large data sets in medical research and economics, and in forming government policy on energy and other resources.
There was an effort to correct this weakness decades ago, when it was perceived as a military issue. The launch of Sputnik in 1957 scared us so much that we embarked on a program to strengthen the system. That momentum carried us to 1975, when the end of the Vietnam War, cutbacks at NASA and economic problems allowed our national attention to drift again.
Indisputably, the U.S. K-12 system continues to fail in mathematics: multiple international studies show that our performance lags below the median. A study two years ago indicated that only 15% of 12th-grade US students were academically ready to take a college-level course, which means that they have not mastered high school Algebra I. Thus, 70-80% of college-bound students require expensive (and largely ineffective) remediation. And this learning lag is not confined to the average student: the top 10% ofU.S. students achieve at only the median level for their South Korean counterparts.
In the past, we have made up for the deficiency by importing talent from elsewhere (which explains why the typical university science department looks like the United Nations). From the 1930s to 1960, it was political refugees fromEurope. In the 1960s and 1970s, it was economic refugees from theUnited Kingdom. In the 1980s and 1990s, they came from the former Soviet Union andChina. That flood of talent has slowed to a trickle, as most return home for better opportunities. So we are now thrown back on our own, home-grown resources, and their inadequacy is painfully exposed.
One important factor inhibiting our progress is the negative attitude to mathematics that is common in our culture, even in university circles. Recently, a professor of English in “Academe” magazine wrote that the technical fields might impart useful skills, but “real” critical thinking takes place in the liberal arts. I have heard similar comments throughout my 37 years in higher education. Yet studies show that mathematics achievement is the most robust predictor of college graduation rates, independent of academic major. I would argue that its place is at the very core of higher learning.
But it is a hard core, in two senses: both strong, and demanding. So the solutions proposed over the years have involved attempts to avoid facing up to the difficulty. Some have tried to pin the blame on a group of people (lazy teachers, bad students, or the government). Others look for a quick fix, emphasizing pedagogy and technology, to make the learning easier; this is doomed to defeat, since the intellectual challenge is inherent in the subject.
My experience leads me to believe that the most effective methods involve knowledgeable teachers with a passion for the material, and a set of expectations which are as hard as possible, while attainable.
This requires time (which is running short), resources and most importantly, the will to deploy them. We have pulled ourselves up by our bootstraps at least twice before (during World War II and the Cold War), and I believe we can do it again in the face of our latest challenge.
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –Let me make one thing clear: I am not a math teacher. I am a science teacher. Of course, my students and I frequently used math in science class: in the collection of data, in measurement, in analysis, in graphing tasks, in counting, in timing, in calculations, in interpretation, in modeling, etc. And, I taught math for 17 years as a third grade teacher, and for one year as a fifth grade teacher of math and science.
Now, here’s the kicker. I am lousy at math. However, I was very good at teaching arithmetic and elementary mathematics. I am not delusional. I was very skilled at communicating basic concepts of arithmetic, mathematical thinking, logical processes, and the appropriate pedagogy for kids ages 7-8-9-10-11. I was good at figuring out what they knew and what they did not know or understand. I was good at analyzing individual student’s misconceptions and coming up with alternatives for instruction in order to get them back on track. I was good at helping them CONSTRUCT math understandings.
Luckily, I do know my limitations. I would not be any good at math instruction above 5th grade. I was fully competent at integrating the necessary math into the 6th grade science curriculum and giving refresher instruction about things like calculating speed, analyzing data using mean-mode-median, or creating proper graphs from the data students had collected. My own children’s homework got to be too challenging for Mom in 7th grade. Good thing that my two were very competent math students. Phew!
So, what’s wrong with the way we teach math to younger students in grades 2-6? Here are my own observations–in no particular order.
> Too much may be crammed into any one year. This is not the way that it has always been. Pressures from testing and efforts to advance all students has played a big role in this over-stuffing of the math landscape. It is easy for someone in charge to imagine that some of what cannot be completed in 6th grade could be pushed down to 5th. Some 4th grade math could be pushed down to 3rd. Math curriculum is huge these days. Plenty of what used to be middle school math is now included in 4th, 5th, and 6th grades. This may work for some bits, but what tends to happen in elementary school is that you get a little bit about a lot of math–broad and expansive with not much depth. There is a great deal to cover. So, the dilemma becomes: do I try to teach an in-depth program based on mastery of key concepts or do I push ahead just to get it all done? Remember–there’s a test on the horizon and I need to have “covered” everything prior to that test. I would like to suggest that less may be more when it comes to elementary math instruction.
> There is limited time to practice what is taught. Not the kind of practice where the kid gets 35 virtually identical 3-place addition problems to slog through, but too little time during the class session to have direct instruction on adding fractions, followed by guided practice, followed by discussion and refinement, followed by more guided practice, followed by discussion and reflection, followed by independent practice while the teacher is around. Real instruction takes time. Assigning work is a breeze! Teachers may be afraid to spend too much time–the push is to move on–move ahead–catch up. Pacing charts and tight schedules do not help.
> Some practice is unnecessarily repetitive. Too much homework just for the sake of homework, for example. Too many similar problems, as noted above. If Johnny can do five problems correctly, does he really need to do the calculations 25 times? What does this prove?
> In most classrooms, there is not enough conversation involved in math instruction–not just question and answer, but deeper discussion and debate and explanation and production of evidence. This requires a pedagogical understanding that has not been afforded to all of our math teachers. Modeling, practice, implementation, and reflection on best practices has been the reality in too few schools and districts.
We have come to realize that there’s more than one way to get an answer. The old algorithms and lock-step manner of working through math may not be useful, efficient, or effective. So, we gotta talk. Talk takes time.
> Textbook designers and curriculum specialists have lost sight of the value of knowing (memorization) basic +, -, x, / facts. Just like in the revision of reading instruction about twenty years ago that created the PHONICS vs. WHOLE LANGUAGE wars, math has experienced some of the same struggles. A balanced approach is what is needed. We moved so far over to connected math and real-world math and “math my way” that we may have left some of the basics behind. Kids need to KNOW those facts so that they become automatic, like breathing. This goal can be worked on and met while we still move forward with applying those skills to more advanced math.
> Some time ago, we got our hands justifiably smacked for limiting kids’ access to math and to moving forward in math instruction. In some schools, kids could not go on to more advanced math until they had mastered a proscribed portion of the curriculum–thus, widening the achievement gaps–an unintended consequence if I ever saw one.
> We do not do a good job of bringing and keeping all math teachers up to par on best practices in basic math instruction. Effective teaching involves much more than just being really good at math oneself. It is a fine balance between content and pedagogy–the information itself and the skills and strategies necessary to communicate effectively about the topic and help another person LEARN IT. Math teachers need regular, frequent, and in-depth training to become better and better math teachers. In many cases, math teachers know what they need. Whether they get what they need when they need it is another question.
> We still act like math is hard, mysterious, boring, and nerdy. That math is for others who are smarter, and that what we do here in math class is disassociated from the real world. Math suffers from some of the same PR problems as science.
> We still allow kids to think and talk about themselves as if they are math impaired or dumb. Kids, and their parents, say, “Oh, I’m not good at math.” In dismissing the possibility that the child can learn, they are helping create roadblocks to learning.
I know all about this problem area. I struggled with arithmetic–starting in 4th grade. In the 50’s, math instruction was very segmented: you did addition and subtraction in 1st, 2nd, and 3rd grades. In 4th grade, you did multiplication. In 5th grade, you did division and fractions. That was it.
I was OK until grade 4. I got slammed by multiplication. I could not get it. It never made sense. Long division was a nightmare. My father was an engineer, so to have a child who struggled so with math made no sense, and he was helpless in helping me. He yelled and I cried. My mother used to send both of us to our rooms. My mother tried to help by saying things like: “I’m not good at math so maybe you’re not good at math.” and “It’s OK–girls don’t need to be good at math.” I became so certain that math was HARD–impossibly confusing and mysterious–and that I was dumb when it came to math, that I just took myself out of the math equation. I avoided math at all costs.
High school algebra and geometry were terrifying. At the same time, I was studying and acing three world languages: Franch, Spanish, and German–I was the only kid at AIHS studying three langauages at one time. I loved language classes and excelled. French and German were my majors in college for a while. How is it that a girl who can master three new languages could not do basic mathematics? My math phobia was pretty extreme back then.
NCTM–the National Council for Teachers of Mathematics–could certainly shed more light on this. They are a great group of professionals. They produce an excellent publication.
One other thing. Why, oh why, have we all been so reluctant to allow younger students to utilize the marvelous technology of calculators? It’s not cheating–it’s smart. Using a calculator will not tell you if you need to multiply or divide at a certain juncture of a problem, but it will allow you to handily and speedily get a correct answer. I continue to be amazed by this.
As a teacher colleague once said–and had made up into T-shirts: LIFE IS TOO SHORT FOR LONG DIVISION. She is so right.