Davis and Simmt: Mathematics-for-teaching

Davis and Simmt propose a theoretical approach to teach mathematics using mathematics for
teaching. The article argues that the coarse separation of  "teachers' subject matter knowledge" and "formal disciplinary knowledge" is inherently problematic. Rather, they argue that mathematics-for-teaching is likely neither a matter of "more of" nor "to a greater depth than" of an in-service session, but rather a "branch of mathematics". They use classroom multiplication to illustrate four categories of elements involved in  teachers’ mathematics-for-teaching, namely, “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”.

1. In the discussion of teaching multiplication, the authors assert that the understanding multiplication  is not about how many figurative aspects of multiplication are taught but physical actions out of which multiplication arises. First of all, if this is the sudden conclusion of 24 experienced teachers after lengthy discussion of teaching multiplication in classroom, I would be disappointed at the quality of the mathematics courses which qualify those teachers. Strangely, I have to agree with the authors that we do need a special "branch of mathematics" for math teachers.

2. I found that the following illustration in the article was very intriguing especially when I realized that the stem splits to symmetrical halves: stable knowledge and dynamic knowledge.






























Question: Compared to mathematicians, math teachers' expertise lies in the fact that their math is done through interactions with others. Should we value more their knowledge or their ability to engage students in meaningful discussion?

Books exhibited by Susan

I took a photo of the books our professor recommended in class. Will check some out later.

Romeo and Mari

In “ASSESSING UNDERSTANDING IN MATHEMATICS: STEPS TOWARDS AN

OPERATIVE MODEL”, Romero and Mari use an interesting example of a multiplication algorithm to demonstrate the complexity of diagnosis and assessment of understanding in mathematics for middle to high school students. By gathering explanations and interviews on the students’ work, a “multi-faceted assessment strategy” is proposed featuring the use of different tasks with an intention to obtain indicators of students’ understanding of particular mathematics aspects.

The authors assert that “to achieve this, it is important to make the learner feel involved in the situation, so it is convenient for the situations to be specific, simple to understand, without distracting elements and with controlled answers.”  One challenge is always to get students “involved” in the assessment procedure when the assessment is too general or too specific. More studies need to be put into the dimensions and scope of such assessment.

The study provides three categories of assessment when studying student’s use of algorithms and I find the Analytical category to be very innovative. According to the authors, it includes situations that require a reflexive use of the algorithm to lead to a solution rather than simply applying the knowledge. It requires an intentional analysis of the “External structure” as explained in the example of “The product of two consecutive natural numbers is 132. Find them.” This question is used as an analytical assessment of students' understanding of the algorithm, which goes beyond memorization from the original procedure (it is more like a reversed or decomposed procedure that the student is required to reconstruct), and makes it possible to identify analytical understating. 


Question: Are there other ways to assess analytical understanding of math?