(Second in a series.)
A few years ago we visited old friends. At dinner our conversation migrated to education, a topic in which they have a strong interest because they and several of their grown children – also present during our visit – are public school teachers. At one point I suggested that the public schools had “gone off the rails” with respect to teaching math. They became defensive because each felt he/she was putting forth a good effort and working very hard. They admitted that the system had “some problems,” but thought they were mostly “administrative.” None was a mathematically trained teacher, however.
I know these folks to be intelligent, capable people, and I know their children are, too. Obviously there are many fine teachers in our public schools. But it would be disingenuous to claim that every teacher is capable, for all is not well. We have a serious problem with math and science teaching. And if we don’t figure out what it is, our country is going to be in a lot of trouble. This won’t show up at Starbucks Coffee or the mall, but at the far deeper level of engineering, new product development, manufacturing, well-paid jobs, and international trade – all critical to our country’s long-term strength.
It’s not just a few old fogies (like yours truly) blowing off steam. A study conducted by the Third International Mathematics & Science Study (TIMSS) in 1996 showed that American high school students ranked 15th out of 16 nations in mathematics, and dead last in physics. Said Bruce Alberts, president of the National Academy of Sciences:
“Our students fare poorly on the largest, most comprehensive and most rigorous international comparison of education ever undertaken. This simply is not acceptable. It is our responsibility to prepare our youth for the next century, and we are failing them.” 
Thus, my friends’ contention that they were working hard and putting in a “good effort” is not persuasive. Many business people work hard and put in a good effort, yet their businesses fail because their skill level is not great enough or their product was of insufficient quality or their business/marketing plan was flawed. A “good effort” doesn’t cut it in the marketplace (or with a sports team). Why do educators think it should in their “marketplace?” (Especially when the results are so poor.) Even Hottentots in Africa know that failing schools get increased funding. What other business gets that?
The TIMSS study showed that American fourth-graders scored competitively, but that scores had declined significantly by the eighth grade. This begins to indicate where the problem might lie. Researchers also found that the highest-performing students study the most rigorous subjects under well qualified teachers who majored or minored in the subject in college. Students receiving less demanding studies, under less qualified teachers, don’t fare as well. And now – as we saw in last week’s article – some educators are blaming students for not learning under bizarre methods of instruction. Later TIMSS studies have not been much more encouraging about American students’ capabilities.
I first saw that something was amiss with mathematics education when I tutored a young math student in 1962, near the close of my sophomore year of college. The student, trying to complete the 9th grade, had been enrolled in an experimental mathematics program. It featured nine workbooks to be completed across the year at the student’s own pace.
Unfortunately, the program’s designers had either failed to comprehend – or had forgotten – that 14-year-olds rarely have an accurate sense of the pace needed to complete a large task across an entire year. The teacher had not monitored progress, except near year-end. By then, my pupil had completed only two of the nine required workbooks. He faced failure in the course, which would have sentenced him to summer school if he wished to start high school in the fall.
In a madcap three-week effort I helped tutor him through the remaining books at something approaching warp speed. The max-cram effort let him squeak through with a passing grade and avoid summer school. Maybe the program worked for some, but in this case a teacher forgot that he still had to teach and monitor his students’ progress. Although fully capable, intellectually, my student never recovered from his poor math foundation. He should have repeated 9th-grade math, but that didn’t happen.
After completing college, I tutored some junior high school students who were weak in math. Invariably, my students were trying to do algebra without having mastered the multiplication tables. I was astonished that their teachers had not noticed this deficiency. I explained to their parents that a student who must wrack his brain to solve 9x8 or 5x7 cannot effectively study algebra or any higher math. It’s a foundational issue. You can’t do algebra without facility with fractions, and you can’t work fractions or parse a number into factors without an absolute mastery of the multiplication facts.
I spent hours drilling those kids on the multiplication facts, making them write the tables over and over – as my 5th grade teacher did – until they could say 9x8=72 and 5x7=35 without pausing to think. Then we could finally catch up on the skills and concepts of algebra. Eventually, I worked myself out of a job with each one. I have often wondered if any went on to higher math.
Doing a complex thing – baseball or music or math – always starts with mastery of certain fundamentals. In baseball, you have to be able to throw and run and catch and hit. These are the basic skills of the game. Players who aspire to advanced levels must do them very well. Playing an instrument or singing also involves basics. All this is well known and seems obvious.
Over the last 50 years, however, educators have experimented with “new” ways of teaching math. They theorized that students could be taught to “think mathematically” without having to go through the hard labor of learning the fundamentals of the discipline. This has led to disastrous results onto which platoons of educrats have piled increasingly bizarre theories.
When we lived in New Jersey, during the late ‘90s, I read a news article which described New Jersey’s top math educator’s plans for future public school math-teaching. Notably, he wanted to eliminate the proof-based teaching of geometry. He claimed that this “turned off” many students. His stated aim was to make math-teaching more “intuitive” in order to reach the 80% of students who typically opt out of higher mathematics.
This absurd proposal from an influential educator was the equivalent of a teacher insisting that the world is flat. (I think something snapped in me when I read it.) Did he not realize – I asked in a letter to the editor – that the precise difficulty with math is that once you get beyond using fingers and toes, much of math is counter-intuitive? Its abstract concepts cannot be grasped intuitively, but can be derived only by progressively applying more basic knowledge via formal proof.
Indeed, it seemed that New Jersey’s highest math-education official didn’t understand that proof-based learning is not only the key to math, but to all higher education. Beyond geometry class, one rarely needs to know that “when two parallel lines are intersected by a transversal, the alternate interior angles are equal.” But discovering new truth via rigorous proof – usually taught in the geometry curriculum – is an invaluable technique that carries a student through his entire education. How could an educator of such stature not comprehend this?
In a formal analysis of New Jersey mathematics teaching, Dr. William G. Quirk writes:
“…Progressives preach their gospel of ‘discovering math through problem solving.’ You may think this refers to the traditional process whereby teachers ask questions and present problems which have been carefully chosen to lead students to discover teacher-targeted math knowledge. Not so! Progressives preach open-ended ‘exploration,’ with no expectation that different kids will ‘discover’ the same thing.
“Forget about a careful step-by-step buildup of core math knowledge that all students learn to understand in the same correct way. Progressive educationists believe that each child must ‘construct [his] own meaning,’ with [his] own personal version of mathematical knowledge somehow emerging from attempts to solve complex, real-world problems, with the further complexity that the problems must be chosen by the students, based on their personal interests.
“Progressives don't believe it's right to pre-specify what kids should learn, and they don't believe that all kids should be required to learn the same content. This in turn forces them to redefine the meaning of ‘testing’ to equate it with ‘finding out what each kid has discovered,’ rather than identifying what each student has failed to learn.” 
Perhaps the most mind-boggling details of math-education theory documented by Dr. Quirk are the “progressive axioms” of the New Jersey Mathematics Curriculum Framework, which I list below. (Please note how often the word “belief” appears. Axioms are unsupported by data.)
- Belief that each child must be allowed to follow their [sic] own interests to personally discover the math knowledge they [sic] find interesting and relevant to their [sic] own lives.
Rejection of the concept of a common core of basic math knowledge that all children should learn during the K-12 years
- Belief that children must “construct” mathematical knowledge for themselves.
Rejection of teacher-directed knowledge transmission.
- Belief that all knowledge must be acquired as a byproduct of social interaction in real-world settings.
Rejection of classroom learning.
- Belief in the primary importance of general, content-independent “process” skills.
Rejection of the need to remember any specific math content.
- Belief that calculators have fundamentally changed the nature of math.
Rejection of the need to acquire traditional paper-and-pencil math skills.
- Belief that learning must always be an enjoyable, happy experience, with knowledge emerging naturally from games and group activities.
Rejection of any attempt to challenge a child to work harder.
Although the axioms suggest how far wrong math teaching has gone, there are undoubtedly far deeper issues involved in the story of how math education reached a point where such absurdity is accepted. Examination of these is beyond the scope of this short article. But I conclude with an obscure quotation that seems perfectly harmonious with modern approaches to teaching math:
“In its soundest application, education becomes a selective tool by which the student reinforces what he already knows to be the truth.” 
That almost-plausible sounding statement was penned by Adolph Hitler in 1924. I leave it as an exercise for the reader to calculate how far down that road we have gone, and whether there is any chance of getting back.
(As promised, solution of the problem posed in last week’s column is given below.) 
 “U.S. Teens Rank Low in World Tests;” Nanette Asimov; San Francisco Chronicle, February 25, 1998.
 “The Truth About the New Jersey Math Standards – An Analysis of the New Jersey Math Standards (NJMS);” Wm. G. Quirk, Ph.D., 1997-2002. (http://www.wgquirk.com/NJmathst.html)
 Mein Kampf; Adolph Hitler, 1924.
 Problem: There are two fractions which have the same denominator. If 1 be subtracted from the numerator of the smaller, its value will be 1/3 of the larger fraction; but if 1 be subtracted from the numerator of the larger, its value will be twice that of the smaller. The difference between the fractions is 1/3. What are the fractions?
Solution: Let x/a be the smaller fraction; y/a the larger fraction.
Then the equations are:
- (x-1)/a = y/3a
- (y-1)/a = 2x/a
- y/a – x/a = 1/3
- 3x - y = 3
- –2x + y = 1
- –3x + 3y = a
Adding (1) and (2), we find x=4, and we obtain y=9 from (1) or (2).
Using these solutions in (3), we find a=15.
Thus, the fractions are 4/15 and 9/15.