Part 1: Some thoughts on thinking about mathematical knowledge

for secondary mathematics teachers


We began our consideration of the question in terms of goals and thoughts about those goals.


Goal: To characterize the mathematical knowledge that is needed by secondary mathematics teachers (SMTs).


Characterization of mathematical knowledge for SMTs requires responses to each of the following:


What are the ways of thinking about mathematics with which SMTs should be familiar?

What are the mathematical understandings, skills, and dispositions with which SMTs should be familiar?

What are the understandings of the nature of mathematics that SMTs should have?

[What are the ways in which SMTs should understand their students’ mathematical thinking? – is this mathematical knowledge?]


One way to think about responses to questions like these is to think about the mathematical knowledge required in the work in which SMTs engage.


How will SMTs be called upon to draw on mathematical knowledge? Following are some of the ways:


Š      Deciding on what mathematics is important for students to learn– wrt the goals of an entire mathematics program – wrt the possible futures of their students. This requires (among other things):


o     Identifying mathematical ideas that cut across school mathematics


Š      Developing units and lessons that sequence and develop important mathematics – taking into consideration the connections among topics. This requires (among other things):


o     Identifying appropriate ways in which school mathematics can be structured to facilitate its use in further endeavors (work?) and its development in later mathematics


Š      Responding to students’ mathematical questions:


o     Choosing from among a large range of possible representations and explanations. This requires (among other things):

§      Seeing the mathematical objects in external representations

§      Understanding what various representations of a mathematical object conceal and what they reveal


o     Capitalizing on opportunities to help students see connections across mathematical areas. This requires (among other things):

§      Understanding the generalizations and principles that connect mathematical ideas in the school mathematics curricula


Š      Constructing assessments of student understanding.


How might we as a field begin to answer questions like these[1]? In order to make progress on understanding the mathematical knowledge that is needed by secondary mathematics teachers (SMTs), we need to draw from both mathematics and practice. We want teachers to be flexible enough in their mathematical knowledge to recognize those classroom opportunities to make connections to other mathematics, to create foreshadowing of ideas to come, and to identify and build toward “big ideas” in mathematics. We have developed a rough plan to start on the road to developing those ideas. The plan builds from vignettes based on classroom practice. We have two sketches of examples of vignettes.


Sample Vignette 1:


The first example of a vignette arose from talking about Pat’s recent observation in a high school classroom. The student teacher Pat was observing had just completed a discussion of special right triangles, including (of most interest here) discussion of the 30°-60°-90° triangle. Discussion then turned to right triangle trigonometry, with the teacher showing the students how to calculate the sine of various angles using the calculator. A student then asked how he could calculate sin (32°) if he did not have a calculator. The student teacher was stymied. Pat’s observation was that the student teacher could have made a good estimate of sin (32°) by returning to what he had just taught about the 30°-60°-90° triangle. The point is that there are lots of mathematically appropriate responses to this student question, and that teachers ought to be able to generate those responses and choose from among them. In this case, the student teacher, under some circumstance, might use this as an occasion to begin discussion of the trig functions as defined on a unit circle. With a carefully drawn (also large and legible) unit circle, the student could find an estimate of the sine of any angle. The student could be drawn into thinking about the unit circle as a trig “calculator,” and of the sine as a function defined on a continuous interval (not those words , but that idea). And so on…


Sample Vignette 2:


A second example comes from Rose Zbiek’s work with parametric GSP drawings. This example, appearing in CAS-Intensive Mathematics (Heid and Zbiek, 2004)[2], was inspired by a student mistakenly grabbing points representing both parameters (A and B in f(x) = Ax + B) and dragging them simultaneously (the difference in value between A and B stays constant). This generated a family of functions that coincided in one point. Interestingly, no matter how far apart A and B were initially, if grabbed and moved together, they always coincided on the line x = -1. See Figure 1 for a screen dump after A and B have been simultaneously dragged.  Rose then thought to generate an extension to quadratic functions (which also appears in CAS-Intensive Mathematics).  Because of her mathematical understanding, Rose was able to recognize the potential of this accident for making mathematical connections – first from the graphical phenomenon to a symbolic proof and then to extend this exploration to a polynomial of higher degree (which generated another interesting relationship along with its proof). In this case, GSP was a vehicle that brought mathematical relationships to the fore. Seeing the phenomenon is not enough – teachers need to recognize the potential for mathematics in the patterns they see.



Figure 1. Screen dump showing trace of f(x) = Ax + B after A and B have been dragged simultaneously.


These two examples are limited in several ways: they both arise from unanticipated student actions, and they are not necessarily common occurrences in the mathematics classroom. The set of examples could be expanded in several ways. First, the examples might include instances of situations in which teachers need to call on their mathematical knowledge without being prompted by a student action. For example, teachers might need to call on their mathematical knowledge in creating new ways to develop a lesson or unit or in creating assessments or assignments for their students. Second, the examples might include more common classroom occurrences. Examples were provided by a seminar of University of Georgia doctoral students. Ones by Bob Allen and Dennis Hembree follow.

Example 1: A teacher is teaching about factoring perfect square trinomials and has just gone over a number of examples (e.g., ). Students have developed the impression that they need only check that the first and last terms of a trinomial are perfect squares in order to decide how to factor it. They are developing the impression that the middle term is irrelevant so that   no matter what the term in the box. The teacher needed to construct a counterexample on the spot, and he wanted one whose terms had no common factor besides 1. What mathematical knowledge did he did to accomplish his task?

Example 2: A teacher  is intending to illustrate how to solve three linear equations in three unknowns and has written  the following on the board: . Before the teacher can write the third equation, the student asks, “What if you have only two equations?” What options does the teacher have to respond to the student’s question and upon what mathematical knowledge would the teacher have to draw in order to respond in those ways?


Vignettes like these can help us think about the mathematical knowledge needed by secondary mathematics teachers. Here are steps of a potential plan for creating and using these vignettes to assist us to understanding the nature of the mathematics for secondary teachers.

1.             Activity:          Generate some good sample vignettes. We need to create some and send them back and forth for feedback to refine them. One question is who would generate them–perhaps groups at each of the institutions could generate some. Whatever the process, we could coordinate that production and feedback cycle once we find out who is interested. One by-product of this process could be the development of guidelines for the creation of vignettes. [Should they be descriptions of the incident or problem being addressed? Should they include development of the mathematical responses and of the mathematical knowledge needed to create those responses?] Resources that might provide some ideas for these vignettes include Mathematical questions from the classroom by Richard Crouse and Cliff Sloyer and Situations in Teaching, by Alan Bishop and Richard Whitfield.


Timeline:       To occur between now and the end of May. This will require sending drafts back and forth until we get a consensus on what we want these vignettes to look like.


2.            Activity:       a.           Generate vignettes at the Penn State and Georgia sites thought groups of faculty, grad students, student teachers. Communicate with each other throughout their development. Convene a group of teachers to expand the set of vignettes. The vignettes would be based on an event that actually happened for which the teacher needed to draw on their mathematical knowledge instantaneously and creatively. The writer should say what happened and describe the range of possible mathematical responses. Enough would have to be included to allow the reader to figure out the facilitative mathematical knowledge needed. The members of the group to generate these vignettes would include secondary mathematics teachers with a known ability to think creatively and well about school mathematics.

Timeline:       To occur during Summer 2005. An early venue for feedback could be a session at the Mid-Atlantic Center research conference. Later in the summer we could convene a group of about 10 outstanding teacher-leaders who are especially suited for this task. The goal would be to generate a total of 250 vignettes, 50 of which would survive the final cut as we are developing the framework in step  .


Activity:       b.           Have the writing group and others expand on and generate rich sets of mathematical responses for each of these vignettes. These sets of mathematical responses would outline possible mathematical pathways that teachers could take in response to the situation presented in the vignette and the mathematical knowledge required of a teacher in order to pursue that path.  They would probably include extensions, connections, generalizations, and abstractions. They would be possibilities, not to be considered an exhaustive set or a penultimate one.




Timeline:       To occur during Summer 2005, some of it simultaneously with step a .


3.            Activity:          Analyze the vignettes and responses with respect to the big ideas that teachers would need. This analysis might be done with research groups at Penn State and U of Georgia working separately and then together in weekly(?) Polycom sessions.


Timeline:       To occur during Fall 2005. We could simultaneously get some reaction from “friends” who would be willing to tell us things like which 5 out of 20 vignettes are the best.



4.            Activity:         a.          This analysis would be formed into a draft framework and a document about the work of establishing the framework. The framework could serve two purposes: they could provide guidance in doing practice-based innovative professional development , they could help guide the field in thinking about research on the mathematical knowledge of secondary teachers, they could help inform the development of mathematics teacher educators, and they could inform mathematics department about mathematics goals for secondary teachers . The framework could also reveal the types of mathematical knowledge (e.g., representations) that is needed frequently by teachers.

Timeline:       To occur during early Spring 2005.


Activity:         b.         As framework is being constructed, create test items to illustrate the framework. These would be intended as prototypical instruments. They could be in the form of polished vignette to which teachers would need to respond.

Timeline:       To occur during early Spring 2005.


5.             Activity:          Commission experts in different areas to conduct reviews of the framework and vignette packets. Each of these experts would be given a particular question about the framework to address. They would be paid to produce papers that would be used in the conference in step 6. 

Timeline:       To occur during late Spring 2006 to early summer.


6.            Activity:          A conference would be held to provide feedback from the field on the framework and on the items created to illustrate the framework. Feedback would be used to modify the framework and the items. A range of people would be invited to the conference. Mathematicians could help identify big ideas that were missing. Experienced teachers would generate instances that might not fit the framework. Curriculum developers could identify big curriculum ideas that were missing. Researchers who have studied secondary mathematics could provide feedback on the viability of the framework for providing a direction for research in the area. Some of these individuals would be commissioned to write reviews prior to the conference.  The attendees could also indicate which of the vignettes were most helpful in characterizing the mathematical knowledge of secondary teachers.

Timeline:       To occur during late Summer 2006.


7.             Activity:          Subsequent work would test the framework in observational settings [Is this the best way of testing the framework?]. Once it was felt that the framework adequately captured the mathematical knowledge that secondary teachers need, assessment instruments could be developed to use in research.

Timeline:       To be determined .


8.            Activity:          A product would be a small book with the framework and the illustrative items. Timeline:       To be determined .


9.            Activity:          Connect with MAA and NCTM to explore ways to use the framework in professional development and in the work of mathematics departments.

Timeline:       To be determined .




[1] This process should be informed by the work that Deborah Ball and others have done in constructing their understanding of the Mathematical Knowledge for Teaching needed to teach mathematics at the elementary level.


[2] Heid, M. K. & Zbiek, R. M. (2004). The CAS-Intensive Mathematics Project. NSF Grant No. TPE 96-18029