Algebra without Formal Operations?

[Shaw Qin]

I am working in Mrs. Talbot’s 8th-grade math class in Northfield Middle School. I have only been to two middle schools in the U.S., so I cannot comment on how typical or unique the school is. However, its boxy buildings, enclosed corridors, and brightly lit classrooms do fit my imagination and experience of a stereotypical middle school. The decorations within Mrs. Talbot’s classroom, however, look more like an art classroom than a math classroom (see image below). Bright and colorful art and posters filled most sides of the walls, blending into a board of blank worksheets and homework printed on colored paper for students who missed a class.

Mrs. Talbot, like her classroom, radiates energy even when this is the last period of the day. As for the students, there might be less than 20 students for the days I was there, but Mrs. Talbot told me that many students were missing those days. Even then, the class size is much smaller than what I was used to in the school system in Beijing, China. The first thing that surprised me about the students was the proportion of racial and ethnic minority students. I have tutored classrooms in two elementary schools in Northfield and one middle school math class during my Northfield Reads and Counts time, but my memory of a class has been more white than what I see in this 7th-period math class. Maybe this period has a disproportionate concentration of racial and ethnic minorities for some reason (Mrs. Talbot did write in the description of her tutor requirement that there is an opportunity to work with EL students in her class). I am also familiar with the subject, algebra, from both my education and my NRC tutor experience. Students are introduced to abstract thinking of manipulating positives, negatives, and zero, while still practicing concrete operations of numbers. I will go into detail on this subject later when connecting it to Piaget.

One thing I noticed about the “pandemic hangover” in my classroom is the arrangement of outdoor classes. On Wednesday, the day I wasn’t at the school, Mrs. Talbot took the students outdoors for the period, doing a practice test to review for the exam on Friday. Mrs. Talbot told me that both she and her students were excited to take off their masks for some time. On Thursday, I noticed that many students were disappointed that they wouldn’t have another outdoor period that day. However, I suspected that a significant part of the reason was the freedom (and physical space) outdoors, instead of the prospect of taking off their masks. Mrs. Talbot did say that some students were enjoying their time outdoors without getting much work done on their practice test the other day. Indeed, I noticed some students who were much more behind on their first practice test. Those students, additionally, were also talking to their friends a lot during the second practice test, which was in the classroom, with me trying to direct them back on their task multiple times.

According to Piaget’s stage theory, the students in the 8th-grade class are probably around 13 years old, typically at the beginning of the formal operational period. Of course, with its unknown variable and operations on those variables, algebra requires at least some level of formal operational skills. However, Mrs. Talbot has emphasized strategies to turn abstract problems into concrete ones, specifically by plugging in the correct kinds of numbers for an abstract problem. For example, to determine whether an expression “P-N” (a positive number subtracted by a negative number) is positive, negative, or zero, a student should formulate a hypothesis and plug in different positive and negative numbers to test their hypothesis. I found it interesting that, by making abstract problems concrete, thus facilitating the transition from concrete to formal operational period, Mrs. Talbot also introduces (or perhaps reinforces) the logic of the hypothetico-deductive scientific method or reasoning. Therefore, the students are transitioning from the concrete to formal operational period from a pedagogical perspective. From the students’ perspective, I observed various individual differences across the concrete and formal operational periods. Some students were using the plug-in strategy in their heads, demonstrating familiarity and mastery of the technique. Other students, however, needed further guidance on which numbers to plug in and whether a problem (such as my example) requires different sets of numbers. In general, my observations do not challenge Piaget’s stage theory on the structures and characteristics of the concrete and formal operational periods. However, most of the 13-year-old students struggled with some concrete operational level problems. Very few exhibited much fluency in formal operations without turning the question into a concrete one. Hence, I suspect most of the students are at best at the beginning of the transition into the formal operational period. My suspicion may call into question whether Piaget’s rough estimation that children transition into the formal operational period at 11 is too early.

After discussing the teacher’s “equalness” and mutual respect with their students in our last class, I reflected on my observations in the 8th-grade classroom. I found out that Mrs. Talbot used a language that expressed respect toward her students when displaying authority over them. For example, before the class started, she referred to some students as “gentlemen” when redirecting them from playful fights at the back of the classroom. This behavior displays a dynamic balance between respecting the students and fulfilling her unique and authoritative responsibilities. In the example, the students listened. However, as an outside observer, I wondered whether addressing the students as “gentlemen” (and perhaps “ladies” or some other terms I didn’t witness) sounds superficial. I also recognize that I did not have the opportunity to observe the weeks of relationship- and respect-building between Mrs. Talbot and her students, so I could not comment based on the incomplete picture I saw.

• This topic was modified 3 years, 6 months ago by Shaw Qin.
• This topic was modified 3 years, 6 months ago by Shaw Qin.

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