Pedagogy is difficult – especially math

My pre-retirement career went back and forth about three times between teaching and educational publishing, the latter as an editor, a managing editor, and a technology director for a publisher with imprints both in the U.S. and Great Britain. After over 30 years combined in those related fields and I am still no expert on this professional concept called pedagogy, but I think I know more than Florida’s governor Ron DeSantis. DeSantis sees grand conspiracy everywhere in math textbooks and furry toons. But real pedagogy, which is the profession of teaching young people and developing curricula for them, is a very difficult skill to master.

Mathematics pedagogy, a very public focus of DeSantis’ ire, is especially hard, because we are basically teaching a second language, and one that is far from intuitive at that. Mathematics is perhaps the only necessary life skill that grown people commonly brag about not knowing. “Hey, I don’t DO math!” Therein lies the problem.

I did very well in most math courses, but I will say that most of my generation’s math books were much like DeSantis wants, and they were terrible. Two major pedagogical sins are found in those old books:

  1. The mathematics, whether it be algebra, geometry, calculus, or statistics, were typically presented either with zero context, or at best using very unrealistic “story problems” at the end of chapters. They never answered the question, “Why should I be learning this?”
  2. To a text, these books gave no reason why even I, who could do the math, should like the math. Indeed, most of these books reinforced to the math phobic that they could not learn math, and because of the lack of context, that there was no need to learn math. Since math is often dependent on additive learning, once you fall behind, you are typically dead in not only that class, but also any beyond it. The last time I taught a quantitative required course to college juniors, at least of half the class did not have the very basic “9th grade math” skills required to do the work.

Over 70 elementary and high school math teachers, college professors, principals and math curriculum coordinators reviewed 132 Florida math texts, but only a handful of these reviewers found objectionable material, causing the governor to ban most of the books from Florida classroom use. Upon review, it appears that pictures of Black mathematicians as role models were enough for a few reviewers to cry “Critical Race Theory!” Some reviewers objected to using statistics to teach the analysis of data showing that we are in the midst of climate change, which of course is “fake news” in a state that is already dealing with “King tides” of ocean water in Florida coastal streets for months at a time.

Black mathematician role model

A Black mathematician role model in a rejected Florida math textbook.

Learning math the old way

During our 18th century Colonial period, a man (remember that we usually did not waste education on women or slaves back then) was not considered educated unless he had worked his way through the thirteen volumes of the Elements, attributed to the Greek mathematician Euclid (c. 300 BCE). This was the dominant text on geometry and mathematical logic in the western world for 2000 years with little change except translation into English.

Elements taught a very demanding pedagogy, where you worked your way from simple geometry definitions (“A line is breadthless length”) up through increasingly complex postulates and propositions (“The sum of the opposite angles of quadrilaterals in circles equals two right angles” — and we are only halfway through Book III here). If you had the tenacity to stick with Elements, you not only learned a lot of geometry, but also the basic skill of using logic to get to increasingly complex understandings in any field. Have at it, Governor!

The past 200+ years of American education can be characterized as a lot of creative people trying different ways to simplify and broaden math pedagogy, as well as other basic skills like reading itself, in order to reach past the privileged and tutored aristocrats of the Colonial era and make basic literacy universal. It has been a difficult road.

While Florida’s governor was busy looking for Black faces and names in math books so he could cry (incorrectly) “Critical Race Theory!” he should know that all pedagogy is unavoidably presented in a cultural context. I learned to read during the 1950s, for instance, with the help of the best-selling “Dick and Jane” books, which were steeped in images of a well-off white suburban traditional family with a cat and a dog. The pedagogy was basic rote “see and repeat.”

Dick-and-Jane

Getting past Dick and Jane

Teachers of reading have long known that this approach neither instills a love of reading, nor does it bridge the cultural and biological learning barriers for millions of children. It was a World Book Encyclopedia at home and lots of Hardy Boys mystery books (banned by our elementary school librarian) that encouraged my early reading. We have been through a host of approaches in the decades since, such as phonics, differentiation of learning styles, games, to the more recent, but newly-controversial “social-emotional learning.”

All of these new approaches work sometimes (and are heavily tested before publication) yet don’t work other times. That is what makes pedagogy so hard. American teachers often have large classrooms full of students of widely varying learning abilities and socioeconomic backgrounds. No curriculum is “one size fits all,” but one curriculum may be all that is available to the harried teacher.

One of my teaching mentors described his own approach as “creative redundancy” — keep trying different approaches to the same content until one sticks or you run out of options. That takes lots of time and preparation, and it may take the teacher afield of the approved materials or allotted time for a topic. When every school day becomes “test prep” in some states, flexibility and “creative redundancy” go out the window.

The teaching of mathematics has gone through its own set of new pedagogies over the years, often to the frustration of parents who only learned multiplication through the rote memorization of “times tables.” Even with basic skills like multiplication, different countries often teach the skills in different ways, and so there is not just “one way to do it.” There are “creatively redundant” approaches that may well be more successful in some children than in others. The best curricula for math hit the same problem from different angles. Some of these may entail solving techniques unfamiliar to parents or may show a picture of a child of unfamiliar ethnicity in the parents’ town.

China vs Japan multiplication

Alternative ways to do multiplication. Source: RISHABH-FACTS

And don’t get me started on science books. Especially in Texas, publishers have walked a fine line for years in what they can say about evolution and other well-proven science, lest fundamentalist Christians be offended and ban their books.

The Trump wannabe governor of Florida has been on an ideological tear recently, hiring a conspiracy-mongering surgeon general and gutting the state Department of Health of people pushing a public health response to the Covid pandemic. He has used his hold on the legislature to squash minority voting districts, punish Disney for acknowledging that a significant portion of their employee population is LGBTQ+ and of minority heritage, and to ban textbooks for teaching students American history. A white, fundamentalist, dominionist, Christian (and ignorant) theocracy is being implemented in the state as we watch.

Teaching is a hard, underpaid profession in the United States. Countries like Finland, where good teachers are well-trained and well-compensated, are seeing far better results in student preparedness and overall cultural happiness than our schools are currently putting out. Let me suggest that guns in our schools are a far greater threat to our students than pictures of Black mathematicians. There are no grand conspiracies here, Governor; just teachers trying to do their job with too many heads butting in.

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2 thoughts on “Pedagogy is difficult – especially math

  1. Bruce Lindgren

    In many respects a lot of math, beyond the simple functions of addition, subtraction, etc., is an exercise in logic. Thus, the frustrating exhortation of math teachers to, “Show your work.” That is, the answer is not just “72;” the answer is “the path to 72.” I studied theology and philosophy in graduate school despite having little background in either during my undergraduate years. I attribute the logical skills from my math courses as the reason I was able to make my way through this “new land.” Math is a process of getting from a known “here” to an unknown “there.” One problem that DeSantis and his ilk have is that the forcefully define a “here” and then simply apparate to a pre-determined “there.” Depending on a math-phobic public, they present a solution that solves no one’s problem and declare victory. It will only be stopped when the public says, “Can you show us your work?”

    Reply
  2. Lisa

    Applied math: in April 2022, the University of Colorado released some of attorney John Eastman’s emails sent/received using the university’s email system, including some advice to a GOP representative in PA. “(L)ate-received” in Eastman’s email refers to ballots received after 11/3/2022. SCOTUS ultimately ruled that ballots postmarked on 11/3/2022, as well as ballots with no postmark but presumed to have been mailed on Election Day (unless there was “strong” evidence showing otherwise), and received by 5 pm on 11/6/2022, could be counted.

    From Eastman’s email:
    “For example, depending on how many ballots were counted that were received after the statutory deadline
    (say 10,000 for example’s purpose), those 10,000 votes need to be discarded, and you can take the absentee
    ballot ratio for each candidate in the counties were (sic) late-received ballots were illegally counted and deduct the
    pro-rated amount from each candidate’s total.

    For the signature verification violation (and perhaps the banning of observers), you could take the difference
    between the 4% historical rejection rate and the .34% rejection rate done under the illegal procedures, and
    similarly discount each candidates’ totals by a prorated amount based on the absentee percentage those
    candidates otherwise received.

    Then, having done that math, you’d be left with a significant Trump lead that would bolster the argument for
    the Legislature adopting a slate of Trump electors – perfectly within your authority to do anyway, but now
    bolstered by the untainted popular vote. That would help provide some cover.” (Page 23 in the PDF linked below.)

    https://www.politico.com/f/?id=00000180-b081-d3ee-a392-b19b26750000

    Reply

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