Side Dish vs Main Dish, Dept.

My previous post about solving problems in multiple ways has been interpreted in multiple ways. In previous eras, students were taught the standard algorithm first. The standard algorithms for multi-digit addition and subtraction were explained via diagrams and other means to show what was really happening when we “carry” and “borrow” (now called “regrouping”). I.e., it was not taught without understanding.

After mastery of the standard algorithm, students were shown alternative methods such as “making tens” and other short-cuts, which spotlighted the conceptual underpinning behind the standard algorithm. Often, students discovered these methods themselves. Anchoring mastery with the standard algorithm first created a distinction and students could see what was the main dish versus the side dish.

Current procedure under textbook interpretations of Common Core, is to delay teaching the standard algorithm, and to teach the alternative methods first under the belief that the standard algorithms eclipse understanding. In so doing, the distinction between main and side dishes are obscured and students are often confused, sometimes being forced to solve problems by inefficient methods such as drawing pictures when they are clearly ready to move on.

Two quotations from Steve Wilson, math professor at Johns Hopkins come to mind in this regard from an article that appeared in Education Next.

There will always be the standard algorithm deniers, the first line of defense for those who are anti-standard algorithms being just deny they exist. Some seem to believe it is easier to teach “high-level critical thinking” than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared. Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older.

There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school, as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way.

Then there are those who say that “if only I had been taught why these algorithms work”, now seeing the math through an adult lens. They fail to see that their ability to understand may have come from being taught in a traditional method that did, in fact, teach the conceptual understanding, but without the same degree of obsession about it that now pervades math education.

Solving Problems in Multiple Ways, Dept.

There is a notion floating about in education land that teaching students multiple ways to solve particular mathematical problems builds flexible thinking, and reasoning skills. I have been looking for research studies that show this, but the closest I’ve found is a study by Rittle-Johnson et al (2016) conducted in algebra classes, that had students compare and discuss alternative methods. It does not address the effect of learning multiple methods for solving a problem. And it also does not definitively conclude that the comparison of methods increases flexible thinking and reasoning skills.

Flexible thinking comes up often in edu-land because it is associated with a nagging question—one that was articulated to me by my advisor when I was attending ed school:  “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?” 

This question has been addressed in part by cognitive scientist Dan Willingham who argues that if students fall short of solving novel problems “it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots. There is a broad middle-ground of understanding between rote knowledge and expertise.” Simply put, no one leaps directly from novice to expert.

 While there is no direct path to learning the thinking skills necessary to apply one’s knowledge and skills to unfamiliar territory, Willingham argues that one way to build a path from inflexible to flexible thinking is through worked examples. Students extend their knowledge along scaffolding built from examples—examples that fit over the underlying structure. Although it does not necessarily happen automatically, thinking becomes more flexible as more knowledge and experience are acquired. 

The current interpretation of the seventh grade Common Core Math Standards as it applies to ratios and proportion provides a case in point. One of the authors of the standards, Phil Daro, was apparently guided by an unmoving and unshakeable conviction that traditionally taught math was nothing more than “getting the answer”.  He has spoken about proportional reasoning and how it has been taught with no regard to process or conceptual understanding. I suspect that he is the main reason why proportional reasoning is now taught with multiple methods. 

To put this in perspective, those who were taught “the old way” remember problems that asked to solve problems like “If John can type 100 words in 2 minutes, how many words can he type in 6 minutes? Students then solved, using the equation 100/2 =x/6.

The problem could be done in two ways. The first was cross multiplication, obtaining 2x = 600, and x = 300 words.  The other way was multiplying 100/2 by 3/3 to get the equivalent fraction 300/6, which immediately revealed that 300 words could be typed in 6 minutes.

Cross multiplication, in the eyes of Daro and others with similar reform math inclinations, is viewed as a “trick” that obscures the conceptual understanding, even though the process is based on sound mathematical principles. That is, if a/b = c/d, it is easy to see that multiplying both sides by the common denominator of bd, results in ad=bc, thus explaining why cross-multiplication works.   And, I may add, that those principles are taught to students, (usually using numbers instead of letters to cut down on abstractness). Students tune this out, in general; they are more interested in doing the problem. Despite the resulting student confidence in their problem solving, cross multiplication is still looked upon as a “trick” and “rote procedure.”

To thus counteract what is perceived as rote memorization, students are now taught that they can solve the problem by finding the unit rate first, and then multiplying. In the above problem, then, the unit rate is 50 words in one minute. Multiply by six to find the number of words typed in six minutes.

Having taught this method to seventh graders, I see some students confused: “Which way do we do it” and “When do we use unit rate and when do we use the other way with cross multiplication?”  But the purveyors of multiple methods have thought of this, so they have extended it even further. Let “w” equal words typed and “m” equal minutes. Then students are taught to express “w” divided by “m”, or w/m as the unit rate. In the above problem, we would have w/m=50.  Solve for “w” to obtain w = 50m, and voila! A formula! Now we can find out the words typed for any number of minutes by plugging into a formula. And they don’t have to use “w” and “m”, they can use “y” and “x” which gets to the next extension of ratio and proportion: direct variation.

Now students learn that equations in this y = kx form are called “direct variation”. And they can be graphed! And the graph goes through zero, and is a straight line!Then they are taught what slope is, and taught that “k” is the slope, which is the same as “unit rate” which is also called “constant of proportionality”.

I have taught these lessons for several years now and can tell you that seventh graders are immensely confused. Those inclined toward progressive math approaches would say that they’re confused because I am teaching it wrong.

And I agree. I am teaching it wrong.

Because to teach it right, you should just teach the basic proportion equation with cross multiplication and leave it at that like it used to be done. Once a student has something that works every time and they have confidence, then they can branch out and explore other possibilities. In particular, when they take algebra later, they can build upon mastered foundations, adding richness through other representations as the contexts present themselves, such as tables, graphing and slope. In this manner, they are motivated to learn other ways of looking at a familiar problem.

I would agree that it makes good pedagogical sense in having students solve things in more than one way. Demanding it as a necessary element of instruction can cause cognitive overload, however. As math professor Rob Craigen says, “Overemphasis may lead not to an ability to think outside the box, but for the box to be lost.”

Reference: Rittle-Johnson et al., (2016) “Comparison and Explanation of Multiple Strategies: One Example of a Small Step Forward for Improving Mathematics Education” in Policy Insights from Education Research, Volume 3 Issue 2, October