Standing Up to the Critics: Responses to Common Misconceptions and Misrepresentations
An interview with F. Joseph Merlino, LaSalle University ([email protected])
Mathematically Sane actively promotes rational discussion about issues related to the reform of school mathematics. Unfortunately, a number of misconceptions (at best) and misrepresentations (at worst) of efforts to improve mathematics education are being sold to parents and teachers across the nation. In the following interview, Joe Merlino discusses how he has handled some of these distortion in his role as Director of the Greater Philadelphia Secondary Mathematics Project. In the past eight years, he and his colleagues have trained over 1,000 secondary mathematics teachers in National Science Foundation sponsored mathematics curricula in the Greater Philadelphia area and New York City. More information about the results they have gotten can be found in the "Philadelphia Story" on this site.
Mathematically Sane (MS): Let's begin with your general perspectives on how to effectively stand up to what can become a nearly overwhelming barrage of opposition.
Joe Merlino (JM): You start and end with the facts and good data---not ideology. My personal beliefs aren't going to persuade those who are honestly skeptical. These skeptics are my target audience. That is where the battle is fought. On the other hand, I don't try to persuade those who are already so biased as to be immune from self-contradiction. They will eventually isolate themselves. It's about what I can show, and the questions the other guys can't answer. I don't try to say anything unless I can back it up. Then I turn it around and ask the same of the so-called and self-appointed critics. Just because you have a Ph.D. in math doesn't mean you know anything about how kids learn any more than a weightlifter is automatically qualified to design a fitness program for children. More than likely they'll wind up hurting the child by burdening them with too much weight, too fast.
MS: Many of the critics have characterized the reform effort as "new-new math," an attempt to link it with the "new math" reform effort of the 1960s, generally viewed by the public as a fiasco.
JM: This is just demagoguery. These programs bear no resemblance to the "new math" texts of decades ago.
MS: Other critics have called it "fuzzy math", charging that the reform curriculum programs lack mathematical depth and rigor. (Ed.: More information about the reform curricula, the center of much misinformation, can be found in the "NSF-funded Curriculum Projects" article on this site.)
JM: I don't get into an argument about math curriculum or textbooks with parents, or other potential critics. Instead, I focus the conversation on the teacher professional development. We do an unprecedented amount of professional development, some 240 hours per teacher over four to five years. If the material weren't rich, rigorous and robust we would not need to do so much training. Ultimately, it's still the teacher's job to teach, to bring out these 3 R's (e.g., rich, rigorous, and robust) for math.
In the new reform programs, more important content has been included in a four-year program that leads to both AP calculus and AP statistics than is included in traditional programs. SAT scores are higher and high school graduates are better prepared for college courses. For example, see data from the Philadelphia initiative at http://www.gphillymath.org/StudentAchievement and http://www.mathimp.org/research/.
MS: Other opponents claim that the reform curricula are based on a radical philosophy of teaching called "constructivism".
JM: These curricula are based on the national standards for school mathematics established by the National Council of Teachers of Mathematics [NCTM]. Read the first page of the "Overview of the NCTM's Principles and Standards for School Mathematics." The word "understanding" is used five times on that one page. I don't consider the goal of promoting understanding to be a radical philosophy. Children are not rodents or robots, and our goal should not be to induce them to do some mindless performances or routines. The Principles and Standards are based on 229 scholarly citations about how kids learn. That's what these curricula are based on.
MS: What about the charge that teachers are not allowed to teach the students correct mathematical rules and procedures? That students must "discover" everything on their own with little teacher assistance?
JM: I have to laugh at this one. How ridiculous! Do you mean that kids will have to rediscover the sigma notation for series all by themselves? This charge has little grounding in reality. When you teach a baby to eat, there comes a point where they no longer want to be spoon-fed. They want to feed themselves. But parents still buy the food. They still cook it. They still put the baby in the highchair.
Has a teenager ever learned how to drive a car without actually driving it? Certainly not. But does this mean the teenager can initially go anywhere he or she pleases or ignore the rules of the road? That is equally absurd. The goal is to develop independent, responsible drivers. In math education, our goal is to develop independent thinkers who can reason quantitatively. To do that, a teacher has to let them get behind the wheel of a math problem. They should not do their thinking for them, but guide them through artful questioning and judicious demonstrations. That is and always has been at the heart of good teaching.
MS: Along these same lines, critics often charge that students are limited to using primitive methods (such as pictures and physical materials) and are never taught standard algorithms to do arithmetic with whole numbers or fractions.
JM: If I give you something to eat that you never saw or tasted before, what do you do with it? You first look it over, maybe smell it, and then take a little bite of it. Then you look at the remaining piece in your hand, nibble a little more, chew what is in your mouth. If it tastes good, then you swallow it and maybe take a larger bite. If it doesn't taste good, you spit it out. If you ask my daughter how much she likes traditionally-taught math, she'll say, "About as much as I do spinach." Most kids hate spinach, and they spit it out. All joyful knowing starts with the primitive, the intuitive, the physical in play, and then proceeds to the formal and the abstract. We don't eat recipes. We eat food. So, kids first have got to know the food, then we can teach them the various recipes of math.
Solutions must still be correct. But just like there are different recipes for the same dish, there are often multiple ways of looking at and representing a problem and multiple paths to the correct solution(s). Indeed, this is certainly true of problems the students will face in the real world. We train teachers to ask probing questions to elicit thinking and to develop mathematical understanding through students' active engagement with complex problems. Small group discussions are organized and guided by the teacher, resulting in deeper student understanding and better retention of learned material. Teachers do model correct procedures.
MS: What about the role of memorization and practice?
JM: Practice is good. But practice what, and how? If you practice lifting weights everyday, your muscles actually get worn down and weaker, and that's how you get hurt. So you have to practice smart. Good athletes know this, which is why there is a science to training and conditioning programs. Still, the training program has to be enjoyable, or at least not so dreadfully boring that no one short of a masochist will want to practice.
Now about memory. How do you retain things in your long-term memory? Isn't that the goal, to develop memorable mathematics, not just the kind where you cram before a test and then forget everything the next month? So how do you do that? The critics don't tell you. That's the realm of cognitive psychologists and neurophysiologists. But common experience also tells you that things done in context with an abundant amount of interconnecting associations, together with some personal affect linked to a meaningful experience tends to be retained in long-term memory. That's why we remember passages from stories on a single reading but frequently don't retain the pages from an inorganic chemistry text that we have read over and over again. In the same way, practice in math must be done in a meaningful manner, spaced over time and done in multiple formats.
MS: And what about their multiplication tables? Will they learn their multiplication tables? That seems to be a top priority for so many of the critics.
JM: Well, if that is their top priority we certainly won't be getting very far. Isn't that 3rd grade material?
MS: And what about long division?
JM: I'm more interested in having kids understand simple short division and orders of magnitude. Once they really understand that, a long division algorithm is a mere computational detail. In the reform programs, division concepts and skills are carefully developed. As a result, students in these programs learn to compute in remarkably flexible and efficient ways in a variety of problem solving contexts.
MS: They make much the same argument about algebra, that students in reform programs never develop proficiency in algebraic manipulation.
JM: That's not borne out by the data, or by teachers' experience. Even so, algebraic manipulation, of and by itself, is a limited goal in our day and age. The TI-92 graphing calculator can do all of the algebraic manipulation found in traditional high school algebra texts. No one gets paid for sitting all day at a cubicle doing factoring problems. People get paid for thinking about meaningful real-world problems and how to solve them. They get paid for innovation. They get paid for doing things in a different, better, non-traditional way that soon becomes the new tradition. Again, correct procedures and answers are absolutely necessary, but there is often more than one correct method of solving a problem. If that were not the case, our computers would not become outdated every six months.
MS: The focus on writing and communications has often been an easy target for the critics who want to argue that the reform programs promote watered-down mathematics.
JM: If Euclid didn't do any writing, we actually would have to rediscover his geometry. Not only Euclid, but also all of mathematics and science. This criticism solidly reveals the ignorance of the person making it.
MS: The critics are making a big deal of a letter opposing reform signed by over 200 leading mathematicians and scientists, including Nobel laureates and Fields Medal winners. This letter appeared in a paid advertisement in the Washington Post in January of 2000.
JM: Did those mathematicians and scientists pay for it? Actually, no. One large donor paid $67,000. I wonder how many would have signed it if they each had to pay $335 of their own money to put it in the Post. The truth of the matter is that many university mathematicians have been included on the national advisory boards of all of the reform projects, and each project included mathematicians on their writing teams. Many organizations have voiced the need for reform, such as the National Research Council, the National Science Foundation, the American Association for the Advancement of Science, and the National Council of Teachers of Mathematics. All of these organizations include strong representation by university level mathematicians.
MS: That also challenges the charge that the developers of these programs do not know math.
JM: Well, if mathematicians are making this charge, then I guess they are implicating their university math colleagues who co-wrote the books. Look, just because a person has a Ph.D. in mathematicians behind his or her name doesn't made him or her immune from being a human being, with all the pettiness and arrogance that can go with that. My focus is on what children understand about mathematics and how to grow them into more effective thinkers.
MS: There are also claims that many teachers involved with reform curriculum have become disenchanted and oppose their further adoption.
JM: Not many, but some. This is at least partially true. Here in Pennsylvania, we still have people who use horse and buggies. They have a point of view and I respect that. But it is also true that in my experience of training over 1,000 teachers over the past eight years, the vast majority of teachers, even those who were resistant to reform, are now accepting of it, and indeed many have become very strong supporters. The proof of the pudding for them is in their own classrooms.
MS: What about the perception that parents are quite upset about these new programs? In fact, the critics generally highlight disgruntled parents.
JM: The National Publishers Clearinghouse also highlights the million dollar winners, as do all the state lotteries. They never feature the person who has lost all his or her savings in an attempt to win the big jackpot. In much the same way, the critics always highlight the parents who are unhappy, no matter how statistically rare of an occurrence this may be. This just goes to show you the vast popular naivete about statistics and probability-which is, incidentally, an educational topic all the reform programs and Principles and Standards features. In fact, I have found that the vast majority of parents have had a very favorable response to these new programs once they had an opportunity to actually do the math for themselves.
MS: What about the claims that these programs are unproven, that they are a risky national experiment?
JM: None of these programs is experimental. They have all been extensively field-tested and have a successful history of use. (Ed.: Some of this data can be found in the "Research on Reform Curricula" article on this site.)
MS: And of the charges that these programs have been dropped in California because of plummeting test scores and mushrooming enrollment in remedial programs?
JM: Again, this is an absurd assertion. For example, see http://www.mathimp.org/research/college.html. The California test scores actually declined for students who were taught traditionally. The number of remedial students in colleges rose in response to higher math standards reflected in new college entrance tests and to the increased number of students applying to colleges, not due to changes in the instructional program. In fact, students in standards based-programs actually did better when carefully matched against a comparable group of traditionally taught students. See, for example, http://www.mathimp.org/research/SylviaTurnerArticle.pdf.
MS: There have also been attempts to paint the reform effort as much weaker than the programs used in high-performing countries in international comparisons.
JM: Many of these programs were created in order to address the results of studies showing that US students are not performing well internationally.
MS: How about the assertion that immigrants and other ESL students will not succeed in these programs because of their lack of textbooks.
JM: All of these programs include textbooks or workbooks. The problems of English language learners are actually helped by the new programs through an emphasis on writing and reading in mathematics, which also helps such students in other subjects.
MS: As we bring this interview to a close, is there anything you would like to add?
JM: There are some critics who are well meaning but naïve, ill-informed or misguided in some respects. In point of fact, some of these critics raise valid points. They can be engaged in a constructive dialogue. However, other critics seem to have agendas that have little to do with the welfare of other people's children, which is the purpose of public education. They are driven by their own prejudices or vanities. For such people, there is no persuasion, no reasoning together, no appeal or obedience to the evidence, no sense of proportion or balance of values. The fact that ten of millions of children in the 20th century have grown up to hate and despise math, and to have minimal quantitative reasoning skills to boot, seems to be irrelevant to these critics. Much like the researchers employed by the tobacco companies, who refused to repudiate their preconceived conclusions that smoking is harmless no matter how many documented cancer related deaths, so these critics will never accept the data that shows the effectiveness of reform mathematics.
We face a truly formidable task. Just as it took decades to turn around both popular and professional opinion on smoking, so truly instilling mathematical understanding in children, and adults, will be a long-term struggle. In the meantime, we need to stay focused on the task at hand and not be turned aside by those with an agenda of their own. We need to focus our attention on the reasonable critics who are teachable, and who can in fact teach us.
MS: Thanks for sharing your insights with us. Best wishes to you as you continue to promote better mathematics for the children of Philadelphia and our country.