Conceptual understanding in math has served as a dividing line between those who teach in a conventional or traditional manner, and those who advocate for progressive techniques. Among other things, the progressivists argue that understanding of a procedure or algorithm must precede the procedure/algorithm itself. I am a middle school math teacher and am in the former camp. Arguments people have made in support of this statement not only border on the ridiculous but often cross it. For example, a teacher stressing over how to assess understanding with students learning to add and subtract told me: "When "your work" consists of counting and adding and subtracting, there isn't a whole lot of work "to show". At these basic levels there has to be another way to ascertain whether a student understands these basic concepts in a meaningful way." So if a student consistently gets addition and subtraction problems correct and applies them to solve problems, what basic concepts does she feel the student lacks? Apart from such an extreme, one of the most popular arguments that has emerged as the poster child for the reform math movement’s push for understanding is the “invert and multiply” rule for fractional division. They argue that students know how to use the rule but have no idea why it works and that such conceptual understanding always helps students to solve problems. What is frequently left out of such discussions is that the teaching of fraction operations is not devoid of understanding; fractional division is presented in the context of what such operation represents, and what types of problems can be solved using it.
Excellent post. I imagine if we asked surgeons why they make an incision in a particular place many would answer “that’s how this procedure is done”, others might give a more specific reason like “to give access to this particular area” or “this is the most efficient way to reach this” but few would be able to give the reasons in terms of physics or fluid dynamics. Nonetheless they likely all perform the procedure very well. Why we put this extra layer on to elementary math makes no sense to me. If they can adequately understand to perform the math and have heard of the reasoning from a teacher do they need to have the ability to explain it? Are they lesser students if they can’t? The test should be can they progress to the next level with adequate skills to learn the next concept. Also much of our “understanding” comes from doing a variety of problems that allow us to explore the skill at its various limits:bigger numbers versus smaller, negatives versus positives etc. As we develop proficiency we see the concept in greater depth. By focusing on understanding and deemphasizing practice it makes it harder for many students to gain proficiency.
Beyond their fetish over "conceptual understanding" is there fetish for equity (equality of outcomes) rather than equality (equality of opportunity). The combination of the two is absolutely impossible since equity alone is impossible without watering down expectations to a point where everybody - achieves "success" even if they don't bother to come to school very often, much less learn anything of substance needed for admission to almost all STEM majors in college. They are DOA while having been promised reality of opportunity. They had been given a lie. The only way to "achieve" it is to drop requirements for the SAT or ACT. Much less an AP calculus score of 3 or better. These are now being dropped for college acceptance - even at prestigious schools - as they are allowed in only to be allowed to flunk out without massive intervention in order to qualify for beginning courses, most notably a solid calculus base.
Excellent post. I imagine if we asked surgeons why they make an incision in a particular place many would answer “that’s how this procedure is done”, others might give a more specific reason like “to give access to this particular area” or “this is the most efficient way to reach this” but few would be able to give the reasons in terms of physics or fluid dynamics. Nonetheless they likely all perform the procedure very well. Why we put this extra layer on to elementary math makes no sense to me. If they can adequately understand to perform the math and have heard of the reasoning from a teacher do they need to have the ability to explain it? Are they lesser students if they can’t? The test should be can they progress to the next level with adequate skills to learn the next concept. Also much of our “understanding” comes from doing a variety of problems that allow us to explore the skill at its various limits:bigger numbers versus smaller, negatives versus positives etc. As we develop proficiency we see the concept in greater depth. By focusing on understanding and deemphasizing practice it makes it harder for many students to gain proficiency.
Beyond their fetish over "conceptual understanding" is there fetish for equity (equality of outcomes) rather than equality (equality of opportunity). The combination of the two is absolutely impossible since equity alone is impossible without watering down expectations to a point where everybody - achieves "success" even if they don't bother to come to school very often, much less learn anything of substance needed for admission to almost all STEM majors in college. They are DOA while having been promised reality of opportunity. They had been given a lie. The only way to "achieve" it is to drop requirements for the SAT or ACT. Much less an AP calculus score of 3 or better. These are now being dropped for college acceptance - even at prestigious schools - as they are allowed in only to be allowed to flunk out without massive intervention in order to qualify for beginning courses, most notably a solid calculus base.