Mischaracterizations, Dept.

“Traditionalists” (as they like to style themselves) are incapable of grasping the fact that high school math exists, and that most high school math teachers aren’t constructivists.

The above quote was from a blog written by a math teacher, and was a post about an article that Katharine Beals and I wrote which was published in online Atlantic in 2015. It caused a stir among those who don’t like what “traditionalists”  have to say about teaching math.

In fact, we do know that most high school math teachers do not teach in the inquiry-based manner. What we also know is that in K-6, much of math has been dominated by the math reform ideology as embodied in many textbooks. Constance Kamii’s belief that teaching the standard algorithms to young children does them harm by eclipsing understanding has set the stage for how math has been and is being taught in the lower grades. What has happened in K-6 math over the past 30 years or so, is in part an increase in inquiry-based, student-centered learning, but in larger part an obsession with understanding.

I have written about this in several places, but one place to start is here (as well as my book!)

What we “traditionalists” do notice about high school math is that many entering freshman do not know basic computation rules, and are dependent on calculators. In an eighth grade algebra class which I teach (and which is equivalent to 9th grade high school algebra), I had a student who had great difficulty multiplying two-digit numbers. He used a convoluted method that took up much time.  Thirty years ago, most entering high school freshmen had the mastery of such elementary procedures.

The blogger whose quote I posted also states that Beals and I do not believe that “math zombies” exist. By “math zombies” the blogger is referring to students who can reproduce procedures but lack the “understanding” to apply the concepts underlying the procedures to new or novel problems. Yes, such students exist. They are on the novice scale of learning; there are levels of understanding that accumulate with experience. Judging novices in terms of the expectation of expert performance seems to be the goal of those who are in the “understanding uber alles” mode.

Also,–based on my observations and those of people I know who teach mathematics in college–it is interesting that the so-called “math zombies” are generally the ones who don’t need remedial math in college.


5 thoughts on “Mischaracterizations, Dept.

  1. “Also, it is interesting that the so-called “math zombies” are–based on my observations and those people I know who teach mathematics in college–generally the ones who don’t need remedial math in college.”
    Interesting how the constructivists can talk all day about “understanding,” but when it comes down to “Can the person do math or not?” all we get is crickets.

    Liked by 1 person

    • Also, I went and checked out the blogger referenced.
      His issues with traditional math-teaching are valid, but he doesn’t understand that traditional math teachers are onto the whole “spit back with no understanding,” that we structure tests so that those who really understand will get the A, and the spit-back kid won’t. Maybe once he/she teaches an AP math class he/she will see this.


      • So even though the blog post is 5 years old, I went and left the following reply:
        Way to get people on your side, demeaning them and all. I am just wondering if you have ever taught an AP math class, because students need both procedural knowledge and conceptual knowledge, to get the 3 or higher, and it’s mostly conceptual now, at least to understand the question being asked, which requires procedural knowledge to answer. They go hand in hand, and thinking that one is better than the other is just foolish.
        Traditionalists get this and do both. Constructivists don’t seem to understand this. At least in my experience.


      • “we structure tests so that those who really understand will get the A, and the spit-back kid won’t. ”


        I’ve said before that when I gave math tests to my college algebra students, even slight variations from the homework tripped up those who only followed a process seen in a previous problem. No good teacher just changes the numbers for test questions. That’s just plain bad teaching. Homework sets are filled with different problem variations and my tests had different variations of those.

        There are different levels of “zombies” – those who somehow get past poor testing, and those (properly prepared) who all have some lack of full understanding, because, duh, there are different levels of understanding. Those who use the “zombies” term know they are wrong. They know that it’s not that simple. They know that they offer no alternative path that produces better results. They claim that their techniques will produce better STEM-prepared students (even if you account for slowing down), but they show no results.

        I would be their biggest fan and supporter if they could only show results. Nada.

        They could just say that their techniques work best for those who aren’t “math brains”, but they show no results. The College Board now offers its Pre-AP Algebra course in 9th grade (which is anything but inquiry-based) and one would expect to see these better understanding K-8 kids take off and accelerate to cover four years of math in three years. Do even THEY believe that’s possible? That would be tough for even properly prepared math brains.

        The College Board knows exactly what’s going on, but they are in complete denial mode. We parents now have to cover their asses at home and with tutors.

        I came across this the other day: https://www.youtube.com/watch?v=gUij8FCg0z8 about learning to play the trumpet. Being able to “express your emotions”, requires practice on “scales, scales, and more scales.” (and double and triple tounging!) We want students to become math users and not just appreciators, but appreciation is an end result, not a starting point. Understanding comes from skills, not the other way around.

        My son’s piano teacher once held his hand low and told him that he was trying to have too much fun “down here.” Then he raised his hand high. However, if you work really, really hard, you will have much more fun “up here.” K-6 educators can find ways to make the down low hard skill work easier and perhaps more fun, but they rather flip the process around and assume that the hard work will be driven by engagement and conceptual understanding. They are in dreamland and show absolutely no proof.

        Liked by 1 person

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