Response to " Problem Solving as a Basis for Reform in Curriculum..."

The article argues that reform in math curriculum and instruction should be based on providing opportunities to problematize the subject. The principles of the proposed approach are different from many historic and current views of problem solving.  In contrast of frustrating students with overly difficult tasks,  the article use “problematic” in a sense  to problematize what they study and “ allowing students to wonder why the things are, to inquiry, to search for solutions and resolve incongruities.” A specific example is presented from a 2^nd grade classroom where students were asked to find the difference between 62 and 37 and they were able to solve a seemingly routine math question by using alternative methods. The article concludes that the achievement is due to the choice of problematizing task over conventional teaching. The article argues that widely endorsed application-based approaches cannot fully resolve the misconception caused by a distinction between knowledge acquisition and application. The article draws on John Dewey’s notion of reflective inquiry to establish an alternative view of problem solving approach which advocates practices of problematizing the subject. The article further examines how the approach fits both views of mathematics understanding, functional and structural as well.  

STOP:

The notion of classroom culture is very clever. Given the current trend of advocating problem solving over computation in the classroom, many obstacles need to be well considered in order to provide the promised outcomes. In general, problem solving activities seem to offer fertile ground for interaction between students and instructors in classrooms, but there is not much proof showing that it will teach core concepts effectively and efficiently across a variety of settings. As a teacher, I always find myself needing to make compromises: giving students a challenge produces chaos, but telling students exactly what to do at every step produces boredom and little learning.

Question:

According to the article, real-life problems provide a legitimate context for problematizing mathematics. Will word problems provide a similar environment for engagement?

 

Response to "The Effects of Stereotype Threat.." by Keena Arbuthnot

In this article, Arbuthnot presents the findings that involve two studies that attempt to explain how stereotype threat affects standardized test performance in mathematics. The study concerns black eighth-grade students who are considered influenced by stereotype threat defined as “a social-psychological threat that arises when one is in a situation or doing something for which a negative stereotype about their group applies. According to Steele (1997) (cited by Arbuthnot) “this predicament threatens the individual: he or she fears being negatively stereotyped, judged or treated stereotypically, or inadvertently conforming to the stereotype."

 In the first study, the author presents findings from an analysis of black students’ test performances on standardized mathematics exam questions that do and do not include differential item functioning (DIF) which the author claims to signify a gap that exists between particular groups. The second study explores how stereotype threat influences students’ strategy choices in standardized exams. Findings from the two studies suggest that stereotype threat may have a negative impact on black adolescent students’ test-taking strategies and achievement on standardized mathematics tests.

Stops: 

According to the author, stereotype threat can negatively affect the intellectual performance of African Americans taking the math test, due to the stereotype that African Americans are less intelligent than other groups. This made me wonder what would occur to groups who are under the effect of stereotype promise, the opposite of stereotype threat. Such a group should seemingly benefit from the situation. However, I recall a different study concluding that positive stereotypes form a considerable burden and therefore adversely affect performance in the stereotyped domain. In Arbuthnot’s study, the entire sample consisted of 257 eighth-grade students: 36 Asian American, 159 black non-Hispanic, 40 white, and other participants. According to the author, although all of these students participated in the experiment, only black students are featured in this paper. The study could be more comprehensive and convincing if further analysis was done for other ethnicity groups.  

Question:

There many conditions under which people are likely to apply stereotypes. As teachers, stereotypes cause more problems than benefits, if any. What are the common stereotypes that we experience in K-12 settings?

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?