Correcting the Corrective Math problem, II
By Guest blogger on Apr 13, 2010 01:02 PM
This week's guest blog is the second installment in a threepart 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 lowperforming 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 higherlevel thinking and problemsolving. 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 problemsolve 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: problemsolving, 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 testprep 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 handson materials or visuals to make the content interesting or relevant.

The Corrective Math program uses specific nonconventional terms that are mathematically incorrect and problematic when you get to higherlevel 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 lowestperforming 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 atrisk students are receiving the most disconnected and illconceived 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 afterschool programs receive yet a fourth instructional program called VMath.
The end result of this effort to remediate is that students in Empowerment Schools are getting less than 45 minutes of exposure to highquality math curriculum, and often none at all since the more recent mandates to implement Voyager test preparation lessons, while their peers in nonEmpowerment Schools are receiving 90 minutes of meaningful higherlevel instruction a day. Similarly, teachers in Empowerment Schools are being asked to forgo what little support they had for teaching a highquality 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 longdivision algorithm. Whereas students had understood the process of division and had an accurate and efficient method for dividing multidigit numbers, they are now confused by long division and learning a complicated procedure that they did not need.
This is just to point out that the first item in this article contains a logical error.
"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."
Let C and H denote "Computation skills" and "Higher level thinking."
The first sentence says that a faulty assumption is H => C. By the common terminology, C is necessary for H whereas H is sufficient for C.
The decades of research showed that it is not true that C => H.
It must be observed that the falsity of the latter implication does not imply either verity or falsity of the former.
Other than that, your thesis is quite convincing.
The analysis of the fallacious presentation was on point in the post above.
Speaking to the veracity of H => C: even if computational skill isn't prerequisite to higherlevel thinking (a statement which I think wrongly assumes a dichotomy), it makes higherlevel thinking a pain in the neck.
For instance, you can understand enough algebraic concepts and routines to solve an equation like 2(x+9)  4x = 3(10x) + 2x  5 without knowing your times tables, but it involves several steps and is painfully slow if you have to reach for a calculator at each step.
It sounds to me that this is the official voice of power scapegoating Corrective Math. There are placement problems, teacher problems, implementation problems, etc. Instead of addressing some of the details involved in fixing the machinations of the system, we read about the philosophical differences between the writer and Direct Instruction. The writer seems to have a competent understanding of her own beliefs and how to appeal to authority (e.g. NCTM) to justify them, but she doesn't have a good understanding of DI.
Good luck, Philadelphia. I wish you well. I hope you find your way out.
Liping Ma, in "Knowing and Teaching Elementary Mathematics" contends that procedural fluency leads to understanding. In early grades, students proceed from learning the algorithms and procedures for solving problems to learning the conceptual underpinning later. I don't know what research the blog author is referring to, to support her thesis but it might be fun if she enlightened us. HungHsi Wu, a mathematician from Berkeley, makes the same point in an article he wrote called "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy" found here: http://archive.aft.org/pubsreports/american_educator/fall99/wu.pdf
Please read both, then get back to me.
>>Whereas students had understood the process of division and had an accurate and efficient method for dividing multidigit numbers, they are now confused by long division and learning a complicated procedure that they did not need.
If the students truly did understand the concept of division and place value, then long division would make sense. If they're having difficulty, they are missing mastery of a prerequisite skill, or the long division is being taught as a rote procedure, rather than as a way of accounting for the division. Some will also have problems with the lack of workspace that the typical school worksheet gives, as compared to the space the student actually needs.
Singapore Primary Math has a good way of teaching long division, for those of you who would like to do professional development on your own.
That is exactly the pointit IS being taught as a rote procedure only in Corrective Math and the Voyager materials. Teachers in empowerment schools are not allowed to deviate from the script or help students make sense of the procedure. They cannot use the Singapore math approach even if they wanted to. The issue is not about which approach is betterwhat's more important is that the district is mandating two very different programs in low performing schools that are conflicting rather than complementary, and teachers can 't stop to help students make sense of the different approaches. Empowerment Schools are monitored in district walkthroughs on how well they are following the corrective math and reading scripts and whether or not they are keeping to the pacing guide. I'm sure some other teachers in Empowerment Schools can testify to the confusion that this is causing for students.
Again, the problem is Direct Instruction or the intervention programs. The problem is that not all interventions work for all children.
Let's not forget that teachers were told to abandon direct instruction and use more discovery learning, cooperative groups, and more "teacher as facilitator" instructional models. During observations by the myriad of people parading through our classrooms, if a teacher was involved in a DI lesson (which we do and need to use), the teacher was told you are "talking too much." Now, apparently, all we're supposed to do is talk as long as what we say in written in blue.
Students learn best (as evidenced by research) when information is presented in a variety of ways, including DI. After the teacher provides the information then other methods of presentation should be used. It's called the gradual release of responsibility model and it works for everyone  I show you how (DI), we do it together (shared lesson), you do it alone (independent lesson). Students who need additional instruction meet in small groups or meet with a specialist.
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I don't know about all these problems, I've never had a look at how these sessions go, but I do think that, if done properly, these can be very good for the students. I needed a lot of help in math and I couldn't get it in school. Now I'm very sorry for that because there are a lot of careers that I would like, but I lack the math skills for them.
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