For those who are wondering what future math teachers learn in ed school, here is a concise summary:

**Traditional mathematical teaching has never worked and has failed thousands of students. **

The standard ed school catechism is that traditional math teaching is based on rote memorization with no understanding, and no connection between concepts. Another is that the conceptual underpinning of math procedures are not explained. According to ed school teaching, procedures are presented as a “bag of tricks” (such as “keep, change, flip” for dividing fractions). The evidence presented is simply that many adults do not remember how to solve certain problems. This stands as proof that the traditional methods are not effective–if they were, they would “stay with us.”

That people do not maintain proficiency in math as they age says less about traditional or reform math than about the way in which a population’s knowledge and skill base is maintained over a lifetime. It is not evidence of failure of traditional math. The results of not using math on a consistent basis can also be seen in a study conducted by OECD. In the study, people from ages 16-65 in over twenty countries, including the U.S., were given the same exam consisting of math computations and word problems. According to the study, “the percentage of U.S. adults between 55 and 65 years old who scored at the highest proficiency level (4/5) …was not significantly different than the international average for this age group. (Goodman, et al., 2013).” These findings can be used in tandem with the first argument above since people in the U.S. in the 55 to 65 age group learned math via traditional math teaching—and the differences in proficiencies between the U.S. and other countries is not significant.

This argument ignores that in countries doing well on such international tests, students learn math mainly via traditional means — and over the past two decades, increasing numbers of students in the U.S. have learned math using the reform-based methods. Reformers are quick to point out that Japan and perhaps other Asian countries actually use reform methods, ignoring the fact that many students are enrolled in “cram schools” (called Juku in Japan) which use the drilling techniques and memorization held in high disdain by reformers.

The argument also fails to consider that traditional math can also be taught poorly. There have always been good and bad teachers, as well as factors other than curriculum and pedagogy that influence the data. In order for such arguments to work, one would have to evaluate how achievement/scores vary when factors such as teaching, socioeconomic levels and other variables are held constant and when pedagogy or curriculum changes. Studies have been conducted that examine how math is taught in specific areas of North America, as well as looking at the common traits of high-performing systems across the world. They indicate that when both conventional and non-conventional (i.e., reform) math are taught by well-trained teachers, students learning under traditional mathematics instruction show much higher achievement than those learning under the reform math methodology. (Stokke, 2015; see here

**Traditional math failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval.**

This argument relies on the tracking argument, when many minority students (principally African Americans) were placed into lower level math classes in high school through courses such as business math. It goes something like this: “Most students did not go on in math beyond algebra, if that, and there were more than enough jobs that didn’t even require a high school diploma. Few went to college. Now most students must take advanced math, so opting out is not an option for them like it was for so many in the past.”

First, in light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place — the instructional method had nothing to do with it. Also, I don’t know that most students *must *take advanced math in order to enter the job market. And I don’t think that everyone needs to take Algebra 2 in order to be viable in the job market.

Secondly, while students only had to take two years of math to graduate, and algebra was not a requirement as it is now, many of today’s students entering high school are very weak with fractions, math facts and general problem solving techniques. Many are counting on their fingers to add and rely on calculators for the simplest of multiplication or division problems. In the days of tracking and weaker graduation requirements, more students entering high school than now had mastery of math facts and procedures including fractions, decimals and percents.

Some blame the “changing demographics” on the decrease in proficiency, but this overlooks variables like poor curriculum and the reform-based approach to math which views memorization “workarounds” as deep understanding. Also frequently overlooked is the fact that students in low income families who make up the “changing demographic” cited in such arguments do not have access to tutoring or learning centers, while students in more affluent areas are not held hostage — dare I say “tracked”? — to poor curricula and dubious pedagogical practices.

**Teachers use a combination of reform- and tradition-based methods so we are all saying the same things and there’s no point in making such distinctions.**

I do not think that I am alone in drawing a distinction between reform and traditional modes of math teaching. While traditional math can be taught properly as well as badly, I believe that poor teaching is inherent in most if not all reform math programs. I base this on having seen good teachers required to follow programs that present content poorly, lack a coherent logical sequence and rely on questionable pedagogies.

I would like to see studies conducted to document how U.S. students who do well in math and science and pursue STEM majors and careers are learning math. The chances are fairly good that such investigations would show that in K-8, many students are getting support at home, from tutors, or from the many learning centers that are springing up all over the U.S. at rapid rates. Since tutors and learning centers (and parents) tend to use traditional methods for teaching math, I somehow doubt that the clientele are exceptions to some ill-defined rule. In my view, as well as the view of many parents and teachers I’ve met, there are few exceptions to the educational damage reform math programs have caused, even when such programs are taught “well.”

**In summary…**

People may choose to use the information I’ve presented here — or persist in ignoring it. I don’t expect that I’ve changed anyone’s mind about anything, but I am always hopeful. that there are some exceptions.

“You will (rote) learn all the strategies for analysing (a + b) x (c+d) because one of them may be on the test”.

Something wrong somewhere !

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If you’re saying that’s traditional math, and that traditional math is always taught without understanding, I would say it’s traditional math done poorly. Such mischaracterizations really don’t advance any dialogue but just feed into misconceptions. I am very traditional, but I don’t teach multiplication of binomials that way, nor do the older textbooks approach it in such manner. There is an explanation of why (a+b)(c+d) is an application of the distributive property; i.e., think of (c+d) as Y and then students can see that (a+b)(Y) = aY + bY. Now substitute c+d for Y and one obtains a(c+d) + b(c+d). It is then a short hop to the (shunned) FOIL method for multiplying two binomials. In teaching FOIL, some students don’t find much difference between the first method and FOIL and some choose to do it the first way, while others choose to use FOIL. For problems like (a+b+c)(d+e+f), they know to use the first method.

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I think I have jumped over the actual test item.

The aim was to achieve the solution to an arithmetical 2 digit multiplication by the “area strategy”, and the student was expected to follow the strategy exactly, with a sequence of steps.

Marks: 1 to succeed totally, 0 for inadequate explanation,

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That’s just plain bad teaching and you offer no description or indication of anything better. The traditional AP Calculus (or IB) track in high school is the only success path for proper STEM careers. Other “integrated” high school math options have lost the battle and are not what colleges are looking for. My son will be graduating from college in May and one of his degrees is in abstract math. He went through a proper AP Calculus track in high school and he now sits next to the best math students from around the world. They offer no understanding advantage (everything they do is proof-based) and they all went through “traditional” math programs. You can’t just throw out a pejorative complaint without backup. If you offer a process or pedagogy that works better than the traditional math curriculum track, I might be your biggest supporter. I won’t hold my breath.

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Reblogged this on Nonpartisan Education Group.

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Hey Howard, I’m familiar with that critique, but I fail to see the problem. First, it is not an adequate characterization of “traditional math”. As Barry points out there is a tendency to shove traditional instruction into a small box, and that only a caricature of the real thing. But there’s also a tendency to isolate a single aspect and denigrate it without explanation as if it were obvious that some grave error were taking place.

For context, I teach university math. Well … more precisely, I’m a professional mathematician and teaching courses, from Year 1 through Graduate school is part of my job (the rest being research, and professional service to the community). We make mathematicians. I mean we take people still struggling out of high school and we ground them in genuine understanding and appreciation of the subject, and often inspire them to become future mathematical professionals. For this we have to mentor them through levels of competence that lead to PhD work, in which a student must solve problems previously solved by nobody. They are genuinely creating new mathematics, which become part of the growing lexicon of the discipline. At that point students must be so independent, you can tell them, “here’s a problem the smartest people in the world have knocked themselves apart trying to solve … and failed. Take a couple of years and develop an original solution.” At that point a student must have incredibly deep understanding. And technical competence. And confidence. And broad knowledge of standard tools. And a pretty good idea of what HASN’T been tried — that is, by knowing what HAS been tried.

It is remarkably frequent that these students succeed. They don’t usually solve the original exact problem posed — nobody fully expects that. But they generally find a new inroads and establish a new and revolutionary outpost along the frontier. It is the main way in which our field advances.

I tell you all that to ask … how do we get those students to that point?

With lots of drillwork. Lots of rote (and other kinds of) learning. And lots of extension exercises, stretching them based on things they have been taught, to come to more and more advanced conclusions of their own.

None of this happens without the students having a huge toolbox of standard knowledge and skills.

Which brings us to your question, supposedly in a middle-school classroom, the day before a unit test:

“You will (rote) learn all the strategies for analysing (a + b) x (c+d) because one of them may be on the test”.

First I’m going to frown on “all the strategies”. There is only one useful strategy for this — it is the distributive rule. They should know that rule. They should learn it by rote (which means “by repetition” — lots of practice). They should have been through different sorts of lessons that help connect it to prior knowledge, so that it is a coherent idea and it has been “proven” at the level appropriate for that stage of learner.

If by “strategies” you mean can they draw rectangle diagrams? Can they describe word problems in which two groups of a and b students each walk c miles and bike d mails … how many miles were travelled altogether? And so on?

Those are practically worthless things to examine, but if you as a teacher want to do so … go ahead. But please, please please … test their COMPETENCE at performing the standard procedure based on a simple application of the distributive law. And vary the question a few ways so that they have to use the skill enough different ways that it is certain that they are performing it correctly and have the mental “muscle memory” to automatically see how it will apply when the expressions have numbers, or numbers and symbols, or more than 2 terms, and so on.

This obsession with “strategies” does not confer understanding. Testing strategies just means you’re testing yourself — you’re testing whether you taught those strategies. Nobody cares later on. A student well-versed in the distributive law will have no problem with grasping how a rectangle diagram models this expression. But students who can draw those pretty pictures may well be technically incompetent. Or if they have some competence, but it derives solely from the diagram, then they are forever hobbled into doing mathematics with crutches that will impede them from any reasonable level of fluency. Test what matters for moving on. Test the knowledge and skill that students will need going forward. They will NOT need algetiles next year (or they’d better not!). They won’t need the pretty diagrams. Those are about pedagogy — they are not about the finished, take-home skill these students will need for life and later education.

Then there is the silliness of saying ‘rote’ in the same sentence as ‘strategies’. Uh … yes, I know students learn stuff for tests by rote. And why not? It’s an effective learning strategy. If I told you I was going to test you on some newly learned mathematical skill tomorrow and you considered the outcome of this test to be really important you bet you’d go home and do lots of examples until you knew you had it down pat. In other words, you’d engage in rote (by repetition) learning. To fail to do so would just be stupid.

Finally, I presume you have a problem with telling students the content of tomorrow’s test. Something they can practice to perfection. Why do you have a problem with this? I, for one, think students with a good work ethic SHOULD do well on tests, and they should routinely, all through their education, experience the reward of succeeding through systematically learning what they are supposed to learn.

And good teachers will inform their students what they will be tested on, and expect the students to prepare. It is one of our best teaching strategies for ensuring students learn what they should be learning. TELL them what they need to know for the test. Then test them on that. It shouldn’t be a guessing game. Students need to have clear expectations and understand what is required of them to achieve success. That’s true for the adult learners we teach. But the need for clarity of expectations is even greater for younger students.

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I used to tell my college math students that if they did all of the homework well, they would get an ‘A’ in the course. Why? Because homework/p-sets are not rote and had lots of variations. I suppose you could be a really bad test making teacher who used the exact same problems, but just changed a number. We students were prepared for that but it rarely ever happened. There were problem variations that strayed from the assigned homework set, so some of us figured out that it helped to do other p-set variations that were not assigned. This was not rote in the tiniest little bit because you never see all of the possible variations. It just lowers the transference leap, which is important on timed tests.

The bigger issues came when the teacher tried to extend test problem variations to larger distance transference, as if those with full “understanding” can make the leap. Sometimes it worked, but often it did not. While preparing my son for possible SAT and AMC questions, I found that no amount of conceptual or higher-order knowledge helped with general transference. The best you could do was to teach basic classes of problems (like digit or odd/even problems or governing equations) and try to make the understanding variation leap smaller. In addition, I saw a number of problems that were basically tricks, like seeing the hidden radius or equation. Teachers have to be very careful not to test tricks or larger problem variations or transference. When I was in college engineering, I saw many of these attempts with the average grade in the 60’s. These tests were not effective or helpful.

It’s very difficult to test larger general transference with a limited time test. I hate math competitions and most problem solutions in real life are not a time issue. In one of my first (all-day) job interviews (at Ford), one of the interviewers asked me: “Are you a fast or slow programmer?” It was clearly a trick question, and I knew enough to say that I was a slow programmer. Actually I was, and that was after doing a lot of programming. The only way to deal with transference in an important timed test is to prepare and turn transference into a smaller variation. There are only a few general processes for improving larger transference to a timed test, such as finding m=n variables and equations. You have to prepare by studying all possible classes of problems to convert transference to variation. That’s what you do to prepare for the SAT or AMC tests – you laboriously study all of the past problems you can get your hands on. Dick Feynman studied large classes of trick problems so that his low variation quick answer would impress everyone. It did, but really annoyed many of his colleagues, who called him a “faker.” He was a faker. He faked out all of those who thought it was based on this amazing ability to transfer. His genius, however, was based on something else, and it wasn’t about timed transference.

True transference is a slow, non-test process.

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In all of my programming classes, tests were weighted very low and individual homework high. I also used this approach back when I taught college computer science courses. It’s impossible to test programming mastery and understanding with tests. This is also true with math tests for what many educational pedagogues call understanding. State tests (CCSS) are only useful for testing basic skills and to provide a last ditch yearly feedback. However, state tests are now being used for understanding assessment that is best left to the teachers who see the kids everyday – if they know what to look for.

Years ago, I was in a K-8 parent/teacher meeting where we talked about the results of the yearly state test. It said that our kids were low in “problem solving”, so they decided to spend more time on – you guessed it – problem solving. That’s not actionable. It would be better to have the yearly test be ONLY for mastery of basic skills, like fractions and percentages. That feedback is actionable on a school AND individual level. Why oh why do they think they can test understanding on a yearly basis with no daily (aware) teacher input and feedback? Schools and teachers are abdicating their roles and math skills (?) to avoid pushing and providing proper weekly feedback to students and parents. Waiting for yearly CCSS tests to provide proper feedback is vague, a year too late, and many tutoring dollars short.

The fundamental systemic flaw of K-8 education nowadays is that (with full inclusion), they trust the process and claim that education is natural (Everyday Math) and then ignore what we parents of their best students do at home. They have open houses and expect parents to help out at home, and then they have the audacity to claim some sort of higher ground of engagement and understanding – all while pushing only a CCSS slope to no remediation in college. You can’t increase the range of willingness and ability in common classrooms in K-6 with lower expectations while at the same time claiming to do a better job with skills or understanding or whatever.

I went thought this with my son in K-6. It’s something many of us parents talked about only in the grocery store and soccer sidelines. You can’t bring up full inclusion or question their fundamental assumptions. You just hide the help at home. Clearly, their social agenda (however nice) won over academic pushing and expectations. Instead of moving towards a full inclusion environment like high schools, their full inclusion classrooms (age tracking) are watered down and they ignore the increased need for help at home and with tutors. Then they have the audacity to claim a better job for understanding. It’s not them, but the students, parents, peers, society, or whatever. All they hope for is a two-generation+ solution. This is a statistical solution, not one based on offering the best individual educational opportunities. After K-8 gets through with the kids, those without help or push at home have few tracks left. They apparently just didn’t have enough engagement or grit.

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