This week's guest blog is the second installment in a three-part series about Corrective Math from Dr. Caroline Ebby, a lecturer at the University of Pennsylvania Graduate School of Education. Last week we learned about the math curricula the District uses. This week focuses on issues with Corrective Math. Next week, we'll learn about alternative options.
Students at Empowerment Schools receive 45 minutes of Corrective Math a day. While the intent is to provide much needed remediation for low-performing students, there are numerous reasons why it may actually do more harm than good. Consider the following about the program:
The Corrective Math program is based on the faulty assumption that computational skill is a necessary prerequisite for higher-level thinking and problem-solving. In the past few decades research and NAEP results have shown that knowing how to compute does not guarantee that students know what operation or strategy to use to solve problems.
What today’s employers value most is people who can problem-solve and deal with quantitative information intelligently and flexibly - computational tasks can be accomplished much more efficiently with calculators and computers.
Even if one takes the position that it is increasing PSSA test scores that matter most, beefing up computational skill isn’t going to have much effect: in grades four and above, students are permitted to use calculators on all but a few questions on the PSSA.
Corrective Math does not address underlying concepts, understanding of number or operations, connections between topics, or four of the five main strands of the PA standards (measurement, geometry, probability and statistics, and algebra). Furthermore, it does not reflect any of the processes critical to learning mathematics: problem-solving, reasoning and proof, communication, connections, or representation. As the National Council of Teachers of Mathematics states, “Without facility with these critical processes, a student’s mathematical knowledge is likely to be fragile and limited in its usefulness.”
The way students learn concepts and strategies in Corrective Math may actually hinder their progress in the core curriculum rather than support it. In Everyday Mathematics students “learn to understand the mathematics behind the problems they solve.” Students are introduced to alternative algorithms for each of the basic operations and encouraged to make sense of both the quantities and the operation.
In Corrective Math students are not permitted to use the alternative algorithms they learn in Everyday Math and are instead taught to follow and memorize a series of discrete steps to carry out calculations. Similarly, in the Voyager test prep program, standard algorithms that are not developed in Everyday Math are taught. While making sense and understanding are central goals of the core curriculum, they are explicitly discouraged in the remedial and test-prep programs.
Corrective Math is not based on what we know about how students learn mathematics. A recent research review from Johns Hopkins University concludes that “the most successful mathematics programs involve student interaction.” In Everyday Mathematics, students are encouraged to “explain and discuss their mathematical thinking” on a regular basis. The scripted nature of the Corrective Math program means that students cannot ask questions and teachers cannot deviate from the script to help them make connections, explain something in a different way, or use hands-on materials or visuals to make the content interesting or relevant.
The Corrective Math program uses specific non-conventional terms that are mathematically incorrect and problematic when you get to higher-level mathematics. For example, teachers are explicitly told not to use the correct terms factor, product, divisor, dividend, and quotient. Instead students are taught that in a multiplication problem “when you multiply the two small numbers, you end up with the big number,” a concept that becomes quite problematic when fractions and decimals are introduced. (When you multiply by a fraction, the answer gets smaller.) Many middle and high school students are taking these remedial modules on multiplication and division at the same time that they are learning about fractions, decimals, and ratios in the core curriculum.
The implementation of Corrective Math by the School District ignores federal requirements that all students be taught by highly qualified teachers. Since 2006 Philadelphia has enforced the NCLB requirement that all middle school teachers be certified in the subject areas that they teach. However, in order to roster the many sections of Corrective Math (there are seven levels) and fit Voyager lessons into the school day, teachers are teaching outside of their certification areas and students are being taught by teachers who are not qualified to teach mathematics. Thus the right to highly qualified teachers is being ignored in the lowest-performing schools. Moreover, the teachers who are highly qualified to teach mathematics are spending 90 minutes a day teaching Corrective Reading, wasting a limited resource.
The most at-risk students are receiving the most disconnected and ill-conceived instructional program. Corrective Math, Voyager, and Everyday Math/Math in Context have very different approaches to teaching particular concepts and rest on very different assumptions about how mathematics is learned. There has been no attempt by the central office to bring coherence to these different approaches, nor is there any time in the day or room in the scripts for teachers to do so. In 8th grade, Empowerment schools are told to teach 45 minutes of Corrective Math each day; 45 minutes of Math in Context on Monday, Tuesday, and Wednesday; and 45 minutes of Algebra 8 on Thursday and Friday. It’s hard to come up with any rationale that would support learning Algebra two periods a week. Students in after-school programs receive yet a fourth instructional program called V-Math.
The end result of this effort to remediate is that students in Empowerment Schools are getting less than 45 minutes of exposure to high-quality math curriculum, and often none at all since the more recent mandates to implement Voyager test preparation lessons, while their peers in non-Empowerment Schools are receiving 90 minutes of meaningful higher-level instruction a day. Similarly, teachers in Empowerment Schools are being asked to forgo what little support they had for teaching a high-quality mathematics program in order to chant verbatim from a script. Are we really narrowing the gap or are we in fact widening it?
Fortunately there are other options that the School District can consider, which will be outlined next week.
* A fourth grade teacher remarked that her students had been learning, practicing, and making sense of an alternative algorithm for division all year in Everyday Math as recommended by the core curriculum. Now, three weeks before the PSSA, the Voyager program has them learning the traditional long-division algorithm. Whereas students had understood the process of division and had an accurate and efficient method for dividing multi-digit numbers, they are now confused by long division and learning a complicated procedure that they did not need.