A Student Teacher's Curriculum

Gwen Lloyd, Ph.D., is a former faculty member at my alma mater, Virginia Tech. She was my advisor when I initially transferred into the Mathematics Education program here. Her article explores one student teacher's interaction with materials used in a kindergarten mathematics curriculum. Anne, the student teacher participant, had used two different approaches to her use of curriculum, but in both cases, each use was adaptive (p. 63).

I had used this article as one of my resources for my Curriculum Principle Project. The most relevant sections of the article were not those of Lloyd's methods and results, but of the background research. Lloyd talks about the history of views of teachers' curriculum use. Over time, researchers have had a great contrast of views, ranging from the view that teachers see the textbook as a fixed source of any and all information that is to be delivered to the students in a linear manner; to the polar opposite view that teachers are interpreters of information, changing the curriculum to suit their own classes' needs. Further, two specific, independent studies had analogous findings: they each demonstrated that these contrasting views are extremes of a linear spectrum, with any kind of teacher interaction with curriculum falling on any point on the spectrum.

The motive for Lloyd's study was reasonable. As I've stated in my Curriculum project, I think it's important to learn more about how teachers interact with their curriculum because we can use that information, cross-referenced with data about students' responses, to see what works and what doesn't. Knowing this, we can change and develop, and know how to change and develop, a curriculum that fits students' needs.

After reading the research questions, I found out that Anne was using two different sets of materials (abbreviated EM and MTW (p. 71)) in her kindergarten mathematics instruction. What I wanted to know was whether she used these different sets in the same class, or across different class. If the latter is true, I wonder how big of an effect differential Anne will have on her students' lives at such a young age. If she is using different methods on different classes would there be a butterfly effect? This question applies all the time, when we consider different teachers of the same course. I found out later in the Data Collection section that she, and another teacher, were using both of these materials on the same class, with alternate chunks of time (2 to 4 consecutive days) devoted to each set. It turns out that Anne's class is receiving instruction from both material sets.

Another data collection discussion I found interesting was that among the four kindergarten teachers in this class, each teacher saw one-fourth of students each day, and rotated stations. That way each teacher would have to teach the same lesson each day for 4 days in a row, to a different group of students. Lloyd also pointed out that in this system, the students got to participate with each of the teachers, but none of the teachers were able to observe each other.

I think the Findings section was very in-depth and complete. Lloyd covered Anne's use of each set of materials and the Curriculum Design and Curriculum Construction in each set.

The Discussion talked about how Anne fell on the spectrum from the two studies. She initially lay on the middle of the spectrum but her alterations were leaning her to the right (towards the more deviant extreme). The rest of the Discussion was about what factors could have been an influence in Anne's curriculum use. I think this is a major subject to talk about for any kind of data collection. There are so many variables that we must account for when collecting data, and we need to consider how much and what kind of an effect they have on the data. This also helps contribute to suggestions for future research: to conduct a wider, less in-depth, study and to minimize the variables.

Lloyd suggests that future research examine teachers at different levels of experience and us[e] different types of mathematics curriculum materials and textbooks (p. 91). Basically, she suggests that future researchers broaden the scope to get a better understanding of how Anne's case can be generalized. After all, the findings from Anne's class are particular and specific to her class only. Teachers who are reading this study must be careful when interpreting its results.

References

• Lloyd, G. M. (2008). Curriculum use while learning to teach: One student teacher's appropriation of mathematics curriculum materials. Journal for Research in Mathematics Education, 39(1), 63–94.
• Vennebush, G. P., Marquez, E., & Larsen, J. (2005). Embedding algebraic thinking throughout the mathematics curriculum. Mathematics Teaching in the Middle School, 11(2), 86–93.

Why Two Wrongs Make a Right

Rapke (2008) explores the reasons why $(-1)(-1)=1$ and what knowledge must be known to explain it. I liked how she divided her findings into two sections: Pseduoreasoning, and formal explanations. In order to rigorously prove $(-1)(-1)=1$, background information on the distributive property and other axioms (but not really) of the real numbers is required.

The point Rapke makes is that when teaching children mathematical axioms, they must be taught why. Teaching a student the reason why a property holds will help them build skills as recognizing patterns. If we say, believe for now then the student will just accept the fact without question, and will only regurgitate that information on a test.

There were many different approaches to the explanation of why $(-1)(-1)=1$, including word problems, but Rapke says that these word problems are inaccurate and may be falsely logical. It cannot help the student identify a similar situation and use the same concept. Another method is that students may use patterns to predict solutions. Although this is not a formal proof, it can provide for some great intuitive reasoning and even lead to inductive reasoning. I never realized, however, that it is not ideal because because sometimes the pattern may not work. For example, when trying to find $0^0$, the student may write this:

$$
(-2)^0=1\\
(-1)^0=1\\
0^0=?\\
1^0=1\\
2^0=1\\
\therefore 0^0=1
$$

In other words, $\lim_{x\to 0} x^{0} = 1$. Following this pattern, students were led to believe that $0^{0}=1$. Since $0^0 \neq 1$, the patten fails. Sometimes using patterns does not lead to correct results, regardless of the intuitive benefits they provide.

In addition, students may develop a pattern for the wrong reason when trying to solve a problem. Using patterns to explain concepts is not ideal because teachers must explain which patterns are okay to use, which are not, and which have exceptions.

Rapke says that the most ideal method of proving the $(-1)(-1)=1$ problem is to use the distributive property. Furthermore, all algebraic properties should be used, because all algebraic properties hold across the number systems real numbers, which is the number system used in 99% of high school mathematics.

References

• Rapke, T. (2008). Thoughts on why (-1)(-1)=+1. Mathematics Teacher, 102(5), 374–376.