A few weeks ago, I discussed the notion of “high-quality instructional materials” with Alex Baron. Well, Barry Garelick, a veteran 7th and 8th grade math teacher, sent me a thoughtful note on the exchange. Garelick is the author of several books, most recently Traditional Math: An Effective Strategy that Teachers Feel Guilty Using, and publishes a math-themed Substack. He has previously contributed to RHSU regarding reform-induced problems in math instruction and how partisanship undermines math education. I thought his take was interesting and worth sharing. Here’s what he had to say.
—Rick
Rick,
I enjoyed reading your recent discussion with Alex Baron, “How Much Autonomy Should Teachers Have Over Instructional Materials?” In particular, I was struck by your skepticism about whether “high-quality instructional materials” are always high quality. I have a lot of sympathy for your observation that: “A lot of these determinations seem to depend heavily on whether materials check certain boxes or are assembled by curriculum designers who know the tricks of the trade. That’s not necessarily a recipe for rich, rigorous instruction.”
Those who have read my book Out on Good Behavior know that in my 8th grade algebra classes, I used Mary P. Dolciani’s 1965 Modern Algebra: Structure and Method in place of the official textbook the school was using for a few reasons: I didn’t care for the topic sequence of the required textbook. There was a dearth of word problems and a notable lack of instruction on how to solve them. Finally, the exercises did not scaffold problems very well. Algebra wasn’t the only class where I was dissatisfied with the school’s preferred textbook. For my nonalgebra classes, I used the approved text, but I supplemented it with materials from other books.
Having spoken with other teachers, I know I am not alone in using external materials. Many teachers decide to use materials of their own liking based on their experience or knowledge of the research on how students learn. I look at such decisions as a form of civil disobedience. Teachers who are doing good work shouldn’t have to pivot to the latest “shiny new thing.”
However, there is a larger issue. Many of the math textbooks that are considered to be “high-quality instructional materials” adhere to the math standards of a particular state. In most cases, such standards are either the Common Core math standards verbatim or a slightly morphed version of the same. The problem is that in either case, as Tom Loveless has described it, embedded within the Common Core-derived standards are the “dog whistles” to reform math. By reform math, I mean educational progressives’ pedagogical ideas that emphasize “conceptual understanding” at the expense of procedural fluency.
Probably the most significant example of this is how the Common Core treats standard algorithms. Consider the standard algorithm for multidigit addition and subtraction, which makes its appearance in Grade 4 of the Common Core math standards. In earlier grades, the standards call for teachers to teach “place-value strategies” for adding and subtracting. An example is the “10s first method” for addition. The problem 57 + 64 can be thought of as 50 + 60 + 7 + 4, which results in 110 + 11, or 121. The rationale behind introducing alternative methods before covering the standard algorithm is the reform-infused belief that the standard algorithm eclipses any understanding of how place-value figures into the procedure.
Having students learn alternative strategies before finally being taught the standard algorithm is thought to provide the conceptual understanding of how place value works in the standard algorithm. In the meantime, students are presented with a smorgasbord of strategies and are told in the end: “Pick the one that you find easiest.” The standard algorithm, when students are finally presented with it, is just another side dish in an endless line of them. As a consequence, the fastest, most efficient way of doing arithmetic—the standard algorithm—gets brushed aside as just another confusing procedure.
In fact, the Common Core standards don’t prohibit teaching the standard algorithm in earlier grades. They simply require that students learn the standard algorithm for multidigit addition/subtraction no later than 4th grade. Both Jason Zimba and Bill McCallum, co-authors of the Common Core math standards, have noted in public forums that the standard algorithm can be taught earlier. However, this information isn’t stated anywhere on the Common Core website or in its guidance document for textbook publishers. As a result, many textbooks delay teaching the standard algorithm in the name of “conceptual understanding.”
Now, to be fair, place-value strategies are not new. They were in the older textbooks I was taught from, including those from the 1950s and 1960s. However, the difference is that the standard algorithms were taught and mastered first, before alternative methods that spotlighted underlying concepts were introduced.
A second issue is that textbooks are often considered HQIM if they are given a glowing review by EdReports or similar entities. This is a concern because EdReports gives textbooks grades based on alignment with the Common Core, not on proven efficacy. Rather than evaluate a textbook or program on its alignment with dubious—and often misinterpreted—standards, why not look at how it fits with the way students actually learn?
The problem is that the cadre of teachers emerging from various schools of education have been taught that the reform ideologies and pedagogy work. They are also taught that traditional math pedagogy has failed thousands of students and is nothing more than “rote memorization” without conceptual understanding. As such, the Common Core-based textbooks are in alignment with what many teachers and administrators believe to be effective. The result is that the methods considered as HQIM often do not incorporate what the research actually shows about how students learn.
Research provides evidence of effective practices both in pedagogy and presentation of content. Such material has been written by Carl Hendrick and Paul Kirschner in How Learning Happens and How Teaching Happens, in Tom Sherrington’s Rosenshine’s Principles in Action, and in Zach Groshell’s Just Tell Them. These books provide essential techniques that align with how students best learn new material.
For example, new information should be provided in chunks that students are able to master rather than overloading a lesson with multiple approaches and topics. Practice problems should start off following a worked example that students use to guide them. Subsequent problems should then vary from the initial example, gradually increasing in complexity while allowing students to apply what they have learned and giving them confidence in the process. To prevent students from forgetting earlier material, practice should include related problems from previous lessons to keep techniques and concepts fresh. Finally, and perhaps most importantly, standard algorithms should be taught first and mastered to act as an anchor, before presenting students with alternative strategies.
In my opinion, and in the opinion of the teachers who I’ve worked with, “high quality” should refer exclusively to the materials that best build student skills and fit with how students learn in practice. Changing how we assess classroom materials might take some time and effort, but the result would be a far more effective way of ensuring that students actually have the tools they need to learn.
