A few years ago we visited old friends. At dinner our conversation migrated to education. They have a strong interest in it 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” – particularly with respect to teaching mathematics. 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.”
I know all of these folks to be intelligent, capable people. Obviously there are many fine teachers in our public schools. But it would be disingenuous to maintain that all are excellent, for all is not well. There is 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 the hamburger stand, or Starbucks Coffee, or the mall, but at the far deeper level of engineering, manufacturing, new product development, high-income jobs, and international trade – all critical to the long-term strength of our country.
This is not just a few old fogies (like yours truly) thumping our canes and talking through our hats. 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.”1
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 not of sufficient quality, or their business/marketing plan was inadequate. A “good effort” doesn’t cut it in the marketplace, or in the NFL. Why do educators think it should in their “marketplace”? (Especially when the results are so poor.)
The TIMSS study showed that American fourth-graders scored competitively, but that scores had declined by the eighth grade. This begins to indicate where the problem might lie. Researchers found that the highest-performing students study the most rigorous subjects under well qualified teachers who majored or minored in the subject. 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, soon after I graduated college. The student, trying to complete the 9th grade, had been enrolled in an experimental mathematics program. It featured nine workbooks to be completed by the student, across the year at his own pace.
Unfortunately, the program’s designers had 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 failed to monitor 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 sentence him to summer school if he wished to start high school in the fall.
In a three-week effort I tutored him through the remaining books at something approaching warp speed. The cram-effort helped 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.
Later I tutored some other junior high school students who were weak in math. Invariably, they were trying to do algebra without knowing the multiplication tables. Somehow their teachers had not noticed this deficiency. (Maybe they thought it wasn’t their job.) I explained to the students’ parents that anyone who has to wrack his brain to solve 9×8 or 5×7 cannot effectively study algebra or any higher math. It’s absolutely fundamental. You must have facility with fractions to do algebra, and you can’t work fractions without mastery of the multiplication facts.
I spent hours drilling those kids on the multiplication facts, and made them write the tables over and over – as my 5th grade teacher did – until they could say 9×8=72 and 5×7=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 well. Playing an instrument or singing also involves basics. All this is well known and seems obvious.
Over the last 60 years, however, educators have experimented with new ways of teaching math. They have 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 “educrats” have piled increasingly bizarre theories.
When we lived in New Jersey, during the late ‘90s, I read a news article which reported what New Jersey’s top state math educator planned for future math-teaching. At the heart of his proposals was elimination of the proof-based teaching of geometry. He claimed that this construct “turned off so many students.” His stated aim was to make the teaching of math 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 scientist insisting that the world is flat. Didn’t he understand (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 learned only by progressively deriving more basic knowledge via formal proof.
It seemed that New Jersey’s highest math-education official did not understand that proof-based learning is not only the key to math, but to all higher education. Beyond the study of geometry, one very seldom needs to know (or prove) that “when two parallel lines are intersected by a transversal, the alternate interior angles are equal.” But discovering new truth by means of rigorous proof – as taught in geometry – 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 writes2:
“…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. (Note how often the word “belief” appears. Axioms are unsupported by data.)
The axioms suggest how far wrong math teaching has gone, but even more extreme developments are occurring in some schools. In California and other liberal-leaning states, moves are afoot to abandon advanced-placement mathematics courses so “equity” can be achieved across the entire student population. “Woke” educators argue that it’s “unfair” for some students to gain higher levels of mathematical capability. Evidently they believe a shared universal ignorance is the preferable educational result. Across the country parents are awakening to the danger, as their schools reach the point where such an absurd objective is actually accepted.
Last year author and mathematician James Lindsay reported: “Oxford University has revealed plans to ‘decolonise’ its math and science degrees and will allow students of any subject who have been affected by the Black Lives Matter furor to seek lenient marking.”
A BLM “spokesperson” said the “entire white supremacist mathematical construct of 2 + 2 = 4” needs to be re-examined and rethought. Another denounced the scientific method as a “racist construct.”
Finally, here is an obscure quotation that seems to fit 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.”3
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. In an earlier article I wrote:
“Education is one of America’s foundational pillars. Its soundness and goodness are absolutely essential to our continuance as a free society and a strong, vigorous nation. If our institutions of education become so corrupted that our young people are no longer receiving sound instruction – not just in the fundamentals, but in understanding how we became what we are – then there is reason to doubt that we can continue as a great nation. These are perilous times.”
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This content was originally published here.
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