# The Prevailing Caricature of Traditional Math, Dept.

In a recent article about math education that ballyhoos the “latest approach” in how to teach math, this statement was made:

“If the teachers are telling students how to solve a problem, and then that problem isn’t exactly what’s on the test, it creates this disequilibrium for a student,” said Beverly Velloff, the math and science curriculum coordinator for the University City School District.

There is nothing new about the so-called breakthrough ideas the article discusses. Moreover, this quote is representative of how traditionally taught math is mischaracterized.  The notion is that students are taught by rote, given a set of problems that are all exactly alike, and thus leaves them flummoxed when presented with a problem that is even slightly different. Such a caricature may be true for traditionally taught math done poorly, but it makes no allowance for it being done well.

Looking at an example from algebra: there are many varieties of distance/rate problems. There are problems in which two objects are going in opposite directions, going in the same direction (i.e., playing catch-up), round-trip problems, objects being influenced by wind or current, and so on.  At some point students are given basic instructions for solving these various types.  In opposite direction problems, students should be taught that we are dealing with two distances that are equal.  That is, if two people are driving towards each other, then the distance each travels before they meet up is equal to the initial distance of separation between them.

In well written textbooks, such problems are scaffolded so that initial problems are solved by following the worked example. But subsequent problems might have some small variation.  Instead of two cars coming towards each other at respective speeds of 60 and 40 miles per hour, we might be told the speed of one car, the distance between them initially, and the time it takes for them to meet. For example, two cars are 200 miles apart, and one car goes 60 mph. It takes 2 hours for the two cars to meet. What is the speed of the other car?  We know that in two hours the 60 mph car has travelled 120 miles. The distance of the second car added to 120 miles equals 200 miles. That’s the “distance = distance” relationship so that if x equals the speed of the other car, then 2x+120 =200. The speed of the second car is 40 mph.

Students will need some guidance in going through these problems, but after given practice with these types, they learn what to look for.

Math reformers may look at this as spoon feeding and rote. They would rather give students problems for which they have not been given specific instructions, and need to synthesize prior knowledge. Alternatively, they are expected to learn in a “just in time” manner what is needed to solve problems.  Thus, problems are given in a top down form in the belief that over time, students will develop a problem solving “schema”.

An article by Sweller et al (2011)  states that such notion is mistaken:

Recent “reform” curricula both ignore the absence of supporting data and completely misunderstand the role of problem solving in cognition. If, the argument goes, we are not really teaching people mathematics but are teaching them some form of general problem solving then mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general and that will make them good mathematicians able to discover novel solutions irrespective of the content.

We believe this argument ignores all the empirical evidence about mathematics learning. Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach was never the most effective or efficient way to learn mathematics.

Nevertheless, reform/progressive math ideas rule the roost in education. Articles such as Sweller’s are thought of as fluff, not proven, no evidence to back it up, or dismissed in light of arguments such as “It has worked in my classrooms”.

Students need instruction, worked examples, scaffolding, ramp-ups in difficulty of problems, guidance, and much practice. Reformers view such steps as “inauthentic math” that produce “math zombies” who do not have “deeper understanding”.  Ignored in all this is the fact that the so-called math zombies are the ones in college who by and large are not in need of remedial math classes.

## 10 thoughts on “The Prevailing Caricature of Traditional Math, Dept.”

1. Debbie Mauk

This article is refreshing to read ! I totally agree. I love math. However, when I have been in “discover” instructional workshops, as a learner I hated the process. If I had learned this way in school, I would never have grown up loving math.

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2. Your articles provide more and more evidence that , for whatever reason, some people do NOT want children to learn, math or other subjects. “New Math’ has been a failure since it came out! Just like teaching reading without phonics is a failure. And not teaching actual history is producing little Socialist/Fascists. Call it conspiracy theory or stupidity, but it is the only logical conclusion.

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3. Chester Draws

Math314 Tries To Help Students Do More Than Memorize Math Equations

Who in Hell teaches kids to memorise Maths equations?

The reporter is an idiot, since memorising “equations” would be worthless. I think he means either formulas or algorithms. That he cannot get even that right makes me suspect that the reporter has mischaracterised the whole approach. I wouldn’t trust him to report anything — the Maths314 might be entirely traditional for all he knows.

I am a fully paid up member of the traditional Maths teaching cabal. I *never* teach kids to memorise formulas. (I do every year have to try and stop students from just doing it that way — because that’s the most natural way for students who aren’t very good at Maths.)

It helps that in my jurisdiction the students are provided formula sheets. All the US has to do is provide a formula sheet at the start of each test if they want to limit memorisation.

I do sometimes teach them a set of steps when approaching a problem, but even then you cannot make it firm and fast, because problems differ. They key is generally making sure they recognise what type of problem they are facing — and after that the rest is more or less whether they’ve done enough practice.

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• There are some formulas I require students to know: quadratic formula, area of a circle, circumference of a circle, area of rectangle, triangle.

I agree that the reporter doesn’t know what they’re talking about, but that person isn’t the only one. I start with a quote from Beverly Velloff, the math and science curriculum coordinator for the University City School District who talks about teachers telling students how to solve a problem, etc.

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• Chester Draws

There are some formulas I require students to know: quadratic formula, area of a circle, circumference of a circle, area of rectangle, triangle.

I presume you require them to know them because they aren’t given. But why aren’t they given? What useful comes of having to memorise the quadratic formula?

I’ve taught both ways, and I’m fully on board being given stuff that otherwise clogs their brain — the most annoying being students who have memorised the formula dutifully but have no idea how to use it.

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4. SteveH

We keep looking for some angle that will get through all of their fluff.

This is one of their goals:

“OUTCOMES ORIENTED We care about results. Your student results become our student outcomes. We hold ourselves accountable to measuring the impact of our work together.”

“To achieve this end • Teachers remove barriers in order to provide equitable access to math learning • Every student is engaged in high level cognitive tasks • Students are empowered as mathematicians • Students are capable and confident problem solvers • Students have procedural fluency, enabling them to apply appropriate procedures to new problems • Teachers create an environment where students authentically share different solution pathways and engage in discourse to deepen their conceptual understanding • Levels of proficiency on standardized tests improve over time”

“Levels of proficiency on standardized tests improve over time”

Statistics. So much for individuals or increasing STEM degree readiness, since “proficiency” on CC tests only means the probability of needing no remediation if you end up taking a college algebra course. “Distinguished” only means that you have a 75% likelihood of passing the course. I really should have saved that link I read years ago. It was deep inside a PDF of our state CC test provider.

In high school, subject-certified teachers (many with real world experience), AP/IB classes, and students and parents drive the agenda and allow individual drive and ability to bloom. In K-8, with a wider range of willingness and ability, age-tracking academic courses and low CC expectations make that impossible. Knowledgeable parents make up for it at home or with tutors, and K-8 non-subject expert educators live in a fairyland of denial.

I no longer have any hope that these educators will understand the realities of mathematical education when they see right before their eyes the success of AP/IB math curricula in their own high schools. They don’t even seem to want to look at individual success cases to see how they can duplicate them. They see that success in a traditional algebra course in 8th grade is a key to success, but have no interest in digging deeper to see how that happens. It’s all guess (what they hope for) and check (badly and with low expectation statistics).

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5. Stephanie Sawyer

Agree with all comments made so far, but I think we forget the obvious: good “traditional” math teachers know what has passed and what is coming. We know why we teach the curriculum we do, because we know what’s coming in the next year or 2 or 3 years. We also know why we should be able to teach our level because of what has supposedly come before.

I think I am probably the end of an era, where my Catholic school teachers all lived together and could literally vertical team over the dinner table every night. I can just hear it.

Sister Gerarda: Get this, Sister Margaret, ALL of my first-graders can add with carrying! You are so set up for next year!

Sister Margaret: Well that’s just great, Sister Gerarda, have you had them add decimals yet? Because of money? They don’t need to know why it works because Sister Carol will break down the decimal place values in 4th grade, but it sure would help if they were already familiar with adding money.

I have this weird feeling that teachers are so caught up in their own “curriculum” because they have to be that they lose sight of the “down the road.”

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• Absolutely right. When I teach Math 7, I’m getting them prepared for algebra: either Math 8 algebra, or Algebra 1 in 8th grade. For Math 8, I teach more algebra than is currently in the CC curriculum so that when they take Algebra 1 in 9th grade, they are familiar with things like factoring, algebraic fractions and various types of word problems.

This tends to be forgotten nowadays. High schools used to tell elementary schools what they wanted to see in terms of things students should have mastered. Same with colleges talking with high schools. Now it’s all “Think like a mathematician” type crap.

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• SteveH

“High schools used to tell elementary schools what they wanted to see in terms of things students should have mastered. Same with colleges talking with high schools.”

Colleges and admissions competition drive reality down through high schools via AP and IB curricula. That’s when students and parents really begin to pay attention and little of their pedagogical silliness sticks, or more likely, is ignored. Seventh and eighth are the transition battleground grades where reality begins to sink in for many students and parents. Some recover, but many don’t. Clearly, the low CC slope to no remediation for college algebra has to change to a proper STEM AP/IB slope, and that has to start in 7th grade. Everyone knows this, but many CC lovers never seem to want to make that transition clear. They desperately try to make lower expectations sound like better math understanding and results.

When my son was in K-6 the word was that the high school math teachers trashed the average mastery level of the incoming freshmen, but they never said anything to the lower grades. That was when our school used Everyday Math and then CMP for 7th and 8th grades. I once emailed and asked the head of the high school math department about this and her response was only that they wanted to see fewer “do-overs.” Cleary, they didn’t want to step on the curriculum and pedagogy toes of K-8.

The only reason our K-8 schools got rid of CMP and replaced it with proper Glencoe textbooks was because students and parents complained that CMP did not do a proper curriculum job of preparing students for Geometry as a freshman. However, that just pushed the fairlyland/reality wall back to the end of 6th grade. Now, everything is based on the CC, but the school website does talk about an accelerated CC 7th and 8th grade coverage in 7th grade to prepare for Algebra I in 8th grade. However, the site says nothing about what textbooks, if any, they use or how formal the algebra class is.

High school math teachers know there is a problem in K-8 math, but they know that telling them (themselves?) the truth is not possible. However, enough of us parents make up the difference at home and with tutors so that things seem the same to them. It’s not the same for us parents.

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