• Allan J. Mucerino

REVERSING THE VARIABLE AND THE CONSTANT

Updated: Mar 22, 2018

OK, I’m starting this essay with two operational definitions to frame my argument that improving proficiency in mathematics requires more than simply varying the time students have to learn it. Like Humpty Dumpty said in Through the Looking Glass, "When I use a word, it means just what I choose it to mean, neither more nor less." The two words: variable and constant. Variable is a property associated with a concept that varies when measured (such as learning). If a property does not vary, it is a constant (such as the teaching time period).

In the most successful schools, time is a variable and learning is a constant. In the least successful schools, it’s the opposite: students learn mathematics, for example, for 50 minutes a day (time remains constant). Some learn it well, some do not learn it as well, and some do not learn it at all (learning varies). For students who are somewhere on the not learning it well spectrum, a targeted intervention, AKA Tier 2 support, is necessary. The instructional principles of an effective math intervention are elusive. Math anxiety and avoidance behavior further exacerbate the problem. Experts agree that math makes some students very anxious, starting in elementary school. By the time the student gets to high school anxiety develops into fear. Math anxiety might be prompted both by genuine concerns such as skills deficits and by social cues that subtly convey the message that math should be feared.

Experts also agree that the more capable students become, the better they feel about their math abilities. As their confidence increases they become more motivated to persevere, which increases their chances of success. Alleviating the learning challenge is a fundamental priniciple of instructional design and the foundation of Tier 2 math intervention.

A recent joint study conducted by Texas A&M University and the Harvard Kennedy School examined the time variable as it related to freshmen Algebra I learners in an urban school district. Since Math SAT scores predict higher earnings among adults (compared to verbal SAT scores which do not predict higher earning among adults), educators continue to search for a potion to help students find their math mojo (according to Merriam-Webster, mojo is a power that may seem magical and that allows someone to be very effective and successful. As a math teacher myself I have concluded that building a student’s mojo – or call it confidence - is a critical first step to building a successful math student). The researchers found that students who were provided a double-dose of algebra had higher college entrance exam scores, higher high school graduation rates, and higher college enrollment rates. Interestingly, the benefits of double-dose algebra were largest for students with decent math skills but below-average reading skills. This is significant because the double-dose amounted to an intervention focused on written expression of mathematical concepts.

While time did vary (relative to students not in double-dose algebra) it was not simply offering students twice as much time to do algebra that was identified as the difference. Instead, it was the intervention’s focus on reading and writing skills in the context of learning Algebra I that made a difference.

Studying best practices related to successful Algebra I interventions more closely, including this study, reveal careful instructional design practices. Instructional design for intervention is the process of analyzing learning needs and goals of students and developing a system of delivery to meet those needs. It includes development of instructional materials and activities and evaluation of the effectiveness of those materials and activities. Promising treatments focus on identifying cognitive difficulties and developing pedagogical solutions and targeted approaches for the most knotty areas of Algebra I that commonly trip students up. Two requirements for effective instructional design practices are an ongoing formative assessment cycle using data protocol and time to collaborate using an inquiry process to solve problems of practice, respectively.


In conversations with teachers participating in UCLA’s Center X Common Core Math professional development initiative, most agree that it is unrealistic to expect students in a standard class period to learn skills and procedures in multiple math domains and do project-based learning. But given the Common Core’s focus on a deeper understanding (known as the shift from “how” to “why”) that is exactly what is being asked of students. Teachers, on the other hand, are being asked to do more coaching and less lecturing. Which, in some cases is a radical departure from past practices, and in most cases a sizable shift. For teachers in school districts that have already abandoned the traditional sequence of Algebra I, followed by Geometry, then Algebra II in favor of how math is taught in most high performing nations (Integrated math: Mathematics I, II, and III or International I, II and III) there is a deep concern that the structures in place (bell schedules, intervention programs, collaboration periods, etc…) do not support either the curricular or pedagogical shift to Common Core. I share their concern. That's putting the cart before the horse. Or, as a math colleague of mine used to say, "That's putting Descarte before the horse."

Not only do students suffer from math anxiety. Teachers do too. I share the anxiety that many teachers are experiencing related to the Common Core State Standards in general. Implementation plans were underfunded and rushed and required significantly more reflection than it has been afforded. The same can be said of the LCAP, with templates and evaluation rubrics being developed on the fly. It takes time to implement with fidelity. And it takes support too. Depending on a school’s or district’s starting point, the California Common Core State Standards Professional Learning Modules provide teachers accustomed to working as high-functioning Professional Learning Community™ subject-alike teams with a solid foundation for professional learning. Teachers who are not engaged in a high-functioning collaborative inquiry process, however, will require significantly more external support.


Adults learn by solving problems. If the problem is that all students are not learning (the variable) Algebra I (or any other learning goal) during the prescribed time period (constant), than the solution is regularly assembling all of the teachers who teach that subject to study the problem. Principals and other leaders should work closely with teams to ensure high functionality.


Back to Humpty Dumpty. In response to his aforementioned comment, “When I use a word, it means just what I choose it to mean, neither more nor less," Alice quipped, “The question is whether you can make words mean so many different things."


I hope we can all agree that learning should not mean so many different things for different students. Learning should not be a variable – at least not for the common core of State standards that all students are expected to be proficient in upon graduation. Beyond that it might vary some, I’ll give you that.

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