By Barry Garelick
In the never-ending dialogue about math education that has come to be known as the “math wars”, proponents of reform-based math tend to characterize math as it was taught in the 60’s (and prior) as “skills-based”. The term connotes a teaching of math that focused almost exclusively on procedures and facts in isolation to the conceptual underpinning that holds math together. The “skills-based” appellation also suggests that those students who may have mastered their math courses in K-12 were missing the conceptual basis of mathematics and were taught the subject as a means to do computation, rather than explore the wonders of mathematics for its own sake.
Without delving too far into the math wars, I and others have written that while traditional math may sometimes have been taught poorly, it also was taught properly. In fact, a view of the textbooks in use at that time reveal that they provided both procedures and concept. Missing perhaps were more challenging problems, but also missing from the reformers’ arguments is the fact that not only are procedures and concepts taught in tandem but that computational fluency leads to conceptual understanding. (See http://www.psy.cmu.edu/~
John Woodward, currently Dean of the School of Education at the University of Puget Sound, is one such person who refers to traditional math teaching as “skills-based” in various papers he has written (as well as in a personal communication to me). I was therefore interested to learn that he chaired the panel that wrote “Improving Mathematical Problem Solving in Grades 4 through 8” which was published by the Department of Education’s “What Works Clearinghouse”.  (http://ies.ed.gov/ncee/wwc/pdf/practice_guides/mps_pg_052212.pdf) Upon going through the guidance, I was heartened to see that the panel recommends whole class instruction, defining terms so that students are not thrown off by unfamiliar vocabulary, and helping students recognize and articulate mathematical concepts and notation.
The recommendation of whole class instruction is admittedly a step in the traditional direction, as opposed to reform methods such as problem-based learning in small groups, facilitated by a teacher who refrains from direct/explicit instruction. As if to ensure that such a step is not interpreted as advocating a purely “skills-based” approach to teaching math, the report is careful to recommend that whole class instruction include presentation of non-routine as well as routine problems. Non-routine problems are those for which there are not predictable approaches suggested by the problem, or worked-out examples that apply to them.
There is no argument from me or others in the traditional camp that students benefit by being given both routine and non-routine problems. It is important to recognize, however, that routine problems are prerequisite for solving the non-routine ones. And while students certainly should be given challenging non-routine problems, they must be able to be solved using prior knowledge of skills and procedures.
The necessity of prior knowledge is something that reformers tend to dismiss. A prevalent belief among math reformers is that just as students develop problem solving habits for routine problems, a similar “habit of mind” development occurs for solving non-routine problems. And in fact, it appears that based on an example of a non-routine problem included in their report Woodward and the other panel members are thinking along the “habits of mind” route. In the problem, the student is asked to find the value of an angle as shown below:
The problem is described as “likely non-routine for a student who has only studied simple geometry problems involving parallel lines and a transversal.” This is true but the authors fail to completely characterize why students would find it non-routine. The problem is solved by drawing in a line that is not shown, called a “supplemental line”. If the students have had no prior knowledge in supplemental lines and how they are used in proofs, the problem is non-routine not because of its newness, but because they lack the prior knowledge and skills needed to solve the problem.
The figure below shows how drawing in a supplemental line to extend an existing one creates a transversal where there wasn’t one before. At the top parallel line, the supplementary angle to 155 is easily calculated as 25. The transversal now makes it obvious that the supplemental angle of 70 is an alternate interior angle and is the second angle in the triangle formed by the supplemental line. Since angle x is an exterior angle to the triangle, it is the sum of the two remote angles 70 and 25, or 95.
The report does not make clear for what grade level the non-routine problem is being presented. I assume that since the report is for math taught in grades 4-8, that this problem would be for eighth graders. While an appropriate way to introduce how to use supplementary lines in proofs and solving problems (followed by explicit and systematic instruction in the technique) the report makes no mention of using it in this fashion. Without the knowledge of drawing supplemental lines, students are at a significant disadvantage in trying to solve the problem. Teachers guiding the student would ultimately give hints about supplemental lines, and would provide the needed knowledge in a “just in time” basis. The new knowledge acquired in such fashion may show the student how to proceed, but does not develop any kind of habit of mind.
In another chapter of the report (on how teachers can provide prompts to help students solve problems), they give an example of a problem in which, again, students do not have the proper tools to solve it efficiently. In particular, they pose the following problem: Find five different numbers whose average is 15. They then give an example of the type of “prompts” teachers can give students to help them solve it.
They describe a student who is picking numbers, adding them and dividing by 5. The teacher notices that the student has some numbers bigger than 15 and some smaller and through questioning, gets the student to observe that they can’t all be greater than 15, nor all smaller than 15 because then the average would be greater than or smaller than 15. The student says “Some have to be bigger and some smaller. I guess that is why I tried the five numbers I did.” The teacher responds: “That’s what I guess, too. So, the next step is to think about how much bigger some have to be, and how much smaller the others have to be. Okay?”
In essence the teacher is helping the student develop a more efficient way to do guess and check which is an inherently inefficient process. The problem would be a good one for a pre-algebra or algebra class in which students have had some instruction in expressing words algebraically. Rather than present this problem to students who lack algebraic knowledge or skills, it could be presented to pre-algebra and algebra students. Then, rather than prompting the student to do an inefficient method efficiently, the teacher could prompt the student by asking what is an average, and whether the problem tells us what the sum of the five numbers is. Since the problem does not provide the sum, the student can be prompted to express the unknown sum as “x”, thus setting up a way to express the average using algebraic symbols. Since the sum is divided by how many numbers are summed, an equation of x/5 = 15 is obtained. Early students of algebra know how to solve the one-step equation to obtain 75. Now it is much easier to then find five different numbers that average 15, since the student now only needs to find 5 different numbers that sum to 75.
People may object to my criticisms here by saying that the recommendations of including non-routine problems and of guiding students via prompts are very reasonable and sound. I agree; they are. But despite the authors’ willingness to enter into discussions of traditional modes of teaching where the edu-establishment has been reluctant to go before, the examples I have discussed here belie a general cautiousness. It is as if they are afraid of an outcome that will be their recurring nightmare: Skills-based math. And so they fall back on their conceptions of “habits of mind”. The authors probably believe that they have taken significant steps to meet the traditionalists half way. It is probably more accurate to say that their best intentions are driven by an agenda that will continue to teach math in a “just in time” manner, and will foster bad habits of mind. They have obsessed over the simplest good ideas to the point that they become bad ones.
 The entire panel is as follows: John Woodward (Chair) University of Puget Sound, Sybilla Beckmann University of Georgia, Mark Driscoll Education Development Center, Megan Franke University of California, Los Angeles, Patricia Herzig, Independent Math Consultant, Asha Jitendra University of Minnesota, Kenneth R. Koedinger Carnegie Mellon University, Philip Ogbuehi, Los Angeles Unified School District.