By Conor Cameron

The Common Core State Standards demand that students of mathematics, “Construct viable arguments and critique the reasoning of others.” In a vacuum, that requirement probably seems more descriptive of what might occur in an English or social studies classroom. Many math teachers probably think to themselves, “Students may do that sort of work in their Geometry classes, not so much in mine.”

As an Algebra teacher, I embraced that philosophy longer than I should have. I can best describe my skepticism with the following list of questions: What sorts of arguments should emerge in an Algebra class? Given all of the standards I have to cover, how will I find time to teach argumentation? Is there a way to use argument to teach existing content instead of adding something brand new to the list of lessons that we barely managed to cover last year? How does being able to construct arguments help my students? Do I teach argumentation as a procedure, or a concept, or a problem-solving idea?

While I do not have answers to most of those questions, I have committed to taking at least a step toward increasing the extent to which my students “construct arguments and critique the reasoning of others.” This step has come in the form of a tool familiar to geometry teachers: “Sometimes, Always, Never” (SAN) Questions.

For example, consider the statement, “If it rained, the ground will be wet.” At first glance, one might conclude that the statement is true: When it rains, stuff gets wet. That gut reaction is why the first answer of many of my students is that the statement is “Always true.” But “always” is a strong word. It poses questions of the sort, “Are there exceptions to this observation?” and “Does ALL of the ground get wet?” It is the answer to those questions that leads students to conclude that the statement is only “Sometimes true.”

In the first 8 weeks of this school year, I have been trying to integrate the opportunity to discuss when something is true into my instruction. While the experiment is still new, what follows are my thoughts on the types of student struggles this technique can address, what it has looked like so far, and what challenges and potential modifications will be required to maximize the success of this trial in argumentation going forward.

Struggle 1: Mathematical Definition

My students tend to struggle to define mathematical objects. Part of the reason is that I have never held them accountable for defining vocabulary words. As long as they can figure out what a question is asking (which they often do by looking at the problem, not the directions), they are fine.

Another part of the reason is that they are unused to the precision of mathematical definitions. If I were to ask my students to define a word like “plate,” they’d probably tell me that it is something round on which food is placed. Take them to a fancy restaurant with rectangular plates and they might recognize that the “round” requirement is not absolute. Ask them to remember grandma’s fine china that is displayed but never used and they might admit that the “on which food is placed” requirement is also a bit fuzzy. But so long as they can distinguish a plate from a frisbee, the imprecision of their definition is unlikely to impact them in any important way.

Similarly, if I were to ask my students to define a word like, “number,” all but the most astute students would probably start listing off {1,2,3,4,5,…}. When they think numbers, they think of positive integers OR counting numbers. That interpretation is fine until they encounter an equation like (x – 7)² = 25 and the answer they need is 2. Many of my students will struggle because when asked, “What number times itself is 25.” They forget that negative numbers are within the realm of possible answers. For struggling students, 7 – 12 = 5 might also come from the fact that they forget that negative numbers are numbers too.

But unlike the plate example, a student’s inability to precisely define a concept in a math classroom makes it difficult if not impossible to think about the answers to certain type of questions. In order to make my students comfortable with the idea that not all numbers are counting numbers, I have thrown questions of the following sort at them:

SAN: If the product of two numbers is even, then at least one of the numbers must be even.

Which is true in the instance of 3 * 2 = 6. However, it is not true in the instance of 5 * 0.8 = 4. As such, the statement is only sometimes true. However, a student would not be able to arrive at the correct conclusion until s/he began thinking about numbers that are not positive counting numbers. I have found this type of question to be an effective way in which to “remind” students that “number” and “counting number” are not the same idea. I think part of its effectiveness derives from the fact that it does not require that students merely memorize definitions but rather requires that that apply the nuances of those definitions.

For students who can arrive at the correct answer quickly, it is interesting to ask them how they would have to change the statement to make it either always true or never true. This extension pushes student reasoning and differentiating.

Struggle 2: Generalizability / Applicability

Even with significant practice, students often forget mathematical procedures. I therefore try to select procedures that work in as many cases as possible, and that make sense or can be recreated by remembering a big framing question. For example, when teaching polynomials this year, we focused on algebra tiles and rectangle models, in part because my students already know how to find the area of a rectangle.

But even the most robust instruction still requires that students be able to make good decisions with regard to when to select which procedure and how procedures can be adapted for related but novel situations.

During our polynomials unit, my students struggled particularly with the difference between adding polynomials and multiplying polynomials. One of the ways that I attempted to distinguish the two procedures was to force them to think about the possible outcomes. Some questions I posed:

SAN: The sum of two binomials is a binomial.

SAN: The product of two binomials is a trinomial.

SAN: The sum of two quadratic polynomials is quadratic.

SAN: The product of two linear polynomials is quadratic.

The answer to most of these questions is “Sometimes.” Arriving at the answer forces students to understand both the nomenclature of classifying polynomials, and also to think about how the operation affects the result. This strategy seemed more effective than constantly repeating the need to be careful with the operation. Here again, the need to apply their knowledge to answer a difficult question seemed to make their understanding more robust. And for students who finished quickly, asking them questions like, “under what specific conditions is the statement true,” seemed particularly thought-provoking.

When asked to factor x² + 7x + 12, most students realize that the trinomial can be written as the product of two binomials. That is, when factored the trinomial will be of the form (x + __ ) * ( x + ___ ). While my students struggled a little more, when asked to factor 4x + 12, they were generally able to recognize that the binomial can be written as the product of a monomial and a binomial. That is, when factored the binomial will be of the form ___ ( x + ___ ). Finally, in a vacuum, they were reasonably successful when asked to factor x² – 25.

However, when I threw all three of those problems into the same problem set and did not provide specific directions, they were significantly less successful. They were not comfortable selecting a procedure to follow for factoring those objects when the objects were heterogeneous OR required “distinct” procedures. This struggle led to the following question:

SAN: If a binomial can be factored, it can be written as the product of a monomial and a binomial.

The answer could be best described as “usually.” But x² – 25 is a very special exception, making the correct answer “sometimes.” Most of my students struggled because when I said binomial, they only thought of x + 7 and its very close relatives, namely polynomials of the form of 2x – 8. They were not thinking of 3x² + 18x, or x + y or even 5 – x. Their inability to generate a representative list of binomials was what made this question challenging. And expanding the types of polynomials they thought of when they heard “binomial” illuminated a blind spot they had when factoring special cases like 4x² – 81.

Implementation and Potential Modifications

First, I began implementation of my SAN reasoning questions with real world examples like the wet ground example above. I strongly recommend plenty of these real-world examples to help acclimate students to the tool.

A note on tool familiarity: When justifying their answers, my students struggle to recognize that in order to prove a statement sometimes true, they only need an example in which it is true and an example in which it is false. In order to prove a statement always true or never true, examples are insufficient. Modeling is especially important to prove the absolutes (i.e., always and never).

Second, I have posed these questions to students in a variety of ways: I have given them homework assignments like this one. Especially at the beginning of the year, I think it is important to ask these questions judiciously, spend significant time on each question, and model what a good answer looks like. Thus far, unit exams have included two or three of these questions. The quality of responses has varied, but inclusion on unit exams forces kids to think deeply when we discuss these questions in class because they know they will be held accountable for this style of thinking later.

Several of my “Do Now” (i.e., “bell-ringer”) activities have involved one or two of these questions. We have then had subsequent discussions during the main portion of the class. I have also used these questions to power a philosophical chairs activity. Given a statement, students go to the corner of the room that corresponds to their answer. While in their corner, students can discuss with each other how to justify their choice. After such discussions, the different corners can argue with each other, and in the process, persuade students to move corners.

The error most students make in reasoning through SAN questions is forgetting an exception or not generating a counterexample. But once one student provides such example, the rest of the class can stop thinking. I need to find a way to involve more of the class in thinking toward these diverse examples. I have had some success with the aforementioned extension questions: “Under exactly which conditions is the statement true?” and “How would you have to change the statement to make it always true.”

As an Algebra teacher, I do not think I demand that my students “construct viable arguments and critique the reasoning of others” sufficiently often. However, even this experiment with “Sometimes, Always, Never” (SAN) questions has motivated me to include it more regularly in my instruction. Among other advantages, it deepens student understanding, improves their ability to use precise vocabulary to express themselves, and it makes their application of procedures more robust. For my current students, I hope that this high-quality thinking benefits them in my class and their subsequent math classes. And I hope to find ways to expand this experiment to integrate more argumentation into my Algebra instruction

Conor Cameron teaches Algebra at Solorio Academy High School in Chicago. He also runs the school’s competitive debate program.