Teaching for Understanding

Silver et al's article "Teaching Mathematics for Understanding: An Analysis of Lessons Submitted by Teachers Seeking NBPTS Certification" presents an analysis of a random collection of 32 portfolios submitted by mathematics teachers of upper elementary and middle grades who wish to be certified by the National Board for Professional Teaching Standards (NBPTS) in 1998. The authors argue that although there was a variety of mathematics topics covered within the portfolios, there was a lack of cognitively demanding tasks for both assessing and developing understanding. Moreover, the authors found that fewer than half of the portfolios analyzed included student generated explanations. These findings were significant in that compared to previous research, significantly more of the teachers in this study included cognitively demanding mathematical tasks within their lessons. Although the inclusion of such tasks signifies good teaching practices, one must remember that these teachers were most likely submitting what they consider some of their best work to receive the certification. Thus, particularly with the authors' generous definition of "demanding task" in mind, the fact that just under half of the proposals contained no cognitively demanding tasks could be considered disappointing.

One of the most significant points in this paper for me was the authors' claim that teachers need "additional support to learn to solicit mathematical explanations as a tool in developing and assessing students' mathematical understanding." Although the teachers excelled in all of their pedagogical practices, student generated explanations seemed to be lacking. The authors suspect that this may be due to different definitions of explanation, but with mathematical justification being a central feature of the NCTM standards, this is still a surprising finding. Collaborative work and student centered learning was at the forefront for many, if not most, of these teachers. How is it then, that student explanations and justifications fall to the wayside? Why is this so much "harder" to do than integrating technology into the class?

High Stakes Testing

This week, I read Suzanne Lane's 2004 NCME presidential address, which addressed high-stakes testing and questioned whether or not students are participating in complex mathematical thinking. The article, written 3 years after the establishment of the No Child Left Behind Act (NCLB), does not dismiss NCLB, but rather argues that there is a lack of cohesion in the system. Perhaps no child is left behind, but is this keeping some children from moving ahead? Lane contends that high-stakes testing has become more of an accountability system and less of an opportunity for students and teachers to engage with complex mathematical thinking. Lane addresses the misalignment of state standards and assessment, what is meant by proficiency, and the impact of large-scale assessment and instruction.

Lane's mention of the imbalance of classroom assessment and high-stakes assessment was the most significant point in the article for me. Even at the university, we often speak of the imbalance between homework assignments and what students encounter exams. Instructors remark that they want students to see conceptually heavy content and problems in their homework, but at the same time, produce exams and midterms primarily testing procedural knowledge. Is it reasonable for instructors to expect that all students will spend the time to learn particular material if their "most important" form of assessment does not consider such knowledge? Within the elementary and secondary system, Lane expresses that a "balanced assessment system is needed with a focus not only on quality large-scale assessments, but also on quality classroom assessments that reflect the content standards and are designed to enhance student learning" (p. 13). She emphasizes that assessments can be opportunities for learning, not simply an analysis of.

If classroom assessment is based off of content standards, why aren't large scale assessments? Why is there such an imbalance between two educational practices, which presumably, were developed by the same group of people? As a teacher in such a system, how might you navigate such unsteady waters?

Mathematics-For-Teaching, Davis and Simmt

This week, I read the article "Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know" by Davis and Simmt. The article develops a theoretical description of what mathematics teachers (need to) know from various workshops they conducted with in-service mathematics teachers. The teacher workshops were conducted to help teachers gain a more nuanced understanding of the mathematical knowledge at stake within the mathematics curriculum, but also acted as a research opportunity for the authors, to see how the teachers learned "new" mathematics. The authors used complexity theory to interpret their results, although the discussion of this theory as a method of interpretation in research would require a completely different discussion.

The authors identified four major aspects of teachers' mathematics-for teaching. Namely, "mathematical objects," "curriculum structures," "classroom collectivity," and "subjective understanding." (Sidenote: It was unclear to me how these aspects were uncovered and why there were not any others. I was generally unsatisfied in the authors' description of how these themes emerged in their research.) In the discussion surrounding "mathematical objects" as an aspect of mathematics-for-teaching, the authors note that many of the teachers' were unaware of a figurative notion of multiplication. That is, multiplication as something besides "repeated addition" or "groups of." They were simply unaware that alternative images and metaphors for a core concept of the mathematics curriculum existed.

My question is, how are teachers expected to come to know this? Is making such connections not a key component of building mathematical understanding? Should teacher education programs include deep, mathematical discussions of core concepts?

Teahers as Mentors by David Tall

David Tall's article "Teachers as Mentors to encourage both power and simplicity in active mathematical learning" argues that mathematics teachers should mentor their students into becoming independent mathematical thinkers. He argues that teaching mathematics sequentially, as is traditionally done, leaves students doing mathematics, but not necessarily thinking about mathematics. The majority of the article is spent explaining what is meant by procedure, process, concept, and procept, and how a teacher might introduce particular topics in a manner that allows students to gain meaning from symbol manipulation.

After reading Tall's article, I was left wondering what he meant by mentor. He begins by stating that "teachers need to act as mentors to encourage their students to build thinkable concepts that link together in coherent ways," but it is unclear to me why he chose the word "mentor." Are good teachers necessarily mentors? Are mentors necessarily good teachers? There was obviously some importance for teachers as mentors, but I'm unsure as to how Tall intended to use it.

Tall's recommendations for teaching the various concepts mentioned within the article are very engaging ways of interacting with the concepts. A great deal of creativity went into some of the examples, particularly the example of a difference of squares, where he uses a rearrangement of geometric shapes. Tall is a well known mathematics education researcher with a strong background in mathematics, and although I see great benefit in utilizing the teaching practices he mentions, I question how the majority of teachers would react to non-sequential mathematics. What if the teachers themselves think of mathematics sequentially? How could one expect to teach mathematics in terms of "thinkable concepts" when they do not think of mathematics in this way? What if the teacher needs to be a mentee?


There is an enormous amount of research regarding the insufficient practices of teachers in mathematics classrooms around the world, as mentioned by Jo Boaler in the video last night. If we want to change classroom practice, it is my opinion that we need to start with teachers who have not yet entered a classroom. True, one could provide future teachers with a list of neat, embodied ways to understand factorization, but isn't this just as bad as having one way to factor it? Although there is more variety, are we not still encouraging teachers to have a toolbox of ways to do and understand things? How might we encourage teachers and students to come up with their own concept images, rather than relying on ones predetermined by someone on the outside?

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