The purpose of this study is to describe teachers’ opportunities to learn mathematics subject matter, pedagogical content knowledge for mathematics topics and practices, and curricular knowledge related to introduction to variable and geometric transformations in middle school mathematics curriculum materials by investigating the content and voice of teachers’ guides.

Framework & Method

            For this study I developed a coding scheme adapted from Beyer, Delgado, Davis, & Krajcik (2009) to analyze the content of teachers' guides. This scheme included the types of knowledge needed for teaching conceptualized by Shulman (1986) and the types of supports (i.e., implementation, rationale) identified by Beyer et. al (2009). I coded each sentence in the teachers' guides and recorded the location of supports. To analyze the voice of the textbook I drew on Halliday's (1985) interpersonal metafunction of language. Like Herbel-Eisenman (2007), I examined the use of personal pronouns and you-forms (i.e., imperatives, you+modal verb) by counting the number and types.

            I analyzed one unit focused on the introduction to variable and one on geometric transformations from four middle school curricular series. I chose the curricular series using the following criteria: a) U.S. market share and b) varied design principles. The series were Connected Mathematics Project (CMP), Math Connects (Glencoe), Mathematics in Content (MiC), and Transition Mathematics (UCSMP).

Results

Content

Support Types. Figure 1 illustrates the percentages of support for each knowledge type.

 

Figure 1.

Percentages of Support for Each Knowledge Type by Unit and Curriculum

            The most prevalent type of support for CMP, Glencoe, and UCSMP was PCK for Practices, accounting for at least 37%.. These supports included those that helped teachers engage students in questioning, reasoning, and using appropriate participation structures. In the MiC variable unit, PCK for Topics was the most prevalent, whereas in the transformations unit there was an almost even split between PCK for Practices and Curricular Knowledge. PCK for Topics included supports for engaging students with mathematical experiences, some of which may be problematic for students, using representations and tools, and anticipating and using student thinking. Curricular Knowledge included supports for developing an understanding of the curricular features, philosophy, and storyline.

            Similar to results found by Beyer et al. (2009), these curriculum materials provided less rationale support (why approaches are appropriate) than implementation support None of the four teachers’ guides devoted more than 6% of their support to rationale guidance.

Support Locations. The location of educative supports may impact whether teachers use the supports (Schneider, 2006; Schneider & Krajcik, 2002). Table 1 illustrates the location of supports.

Table 1

Percentage of Support by Location

	Variable			Transformations
	Unit	Section	Lesson	Unit	Section	Lesson
CMP	14	28	58	48	5	47
Glencoe	15		85	12		88
MiC	24	14	62	16	11	72
UCSMP	19		81	18		82

 

For the variables units a majority of support was at the Lesson Level. For the transformations units Glencoe, MiC, and UCSMP included most of their support at the Lesson Level, whereas CMP had a relatively even split between support that appeared at the Unit and Lesson Levels.

Frequent Educative Supports. Common supports for both units across all curricula included Content Support, Implementation Guidance for Questioning (with suggested answers), and Implementation Guidance for Engaging Students in Mathematical Experiences). These supports accounted for much of the focus in CMP as questioning support accounted for over 40% for both units. In addition, common supports in both units for Glencoe and MiC included Implementation Guidance for Working with Potentially Problematic Mathematical Experiences, whereas common in both units for UCSMP was Implementation Guidance for Developing Mathematical Vocabulary.  There were differences with respect to supports a particular curriculum used for each of its units; however, this analysis is in progress.

Voice

Personal pronouns and you-forms. The pronoun we may indicate the authors' involvement in the activity in the text (Morgan, 1996) whereas you may implicate the reader. You was the most common personal pronoun and was used by all four curricula, whereas CMP and UCSMP were the only curricula to use the pronoun we. You was used implicitly by the use of imperatives and explicitly with the use of modal verbs.

            Imperative Form. According to Martin & Rose (2007), unlike the other grammatical forms for realizing commands, the imperative assumes a position of authority and does not open up possibilities of negotiation. Grammatically this form does not require a subject, but the pronoun you is implied. Figure 2 illustrates the percentage of sentences that were imperatives form for each unit in all four curricula.

Figure 2.

Percentage of Sentences in the Imperative Form by Unit and Curriculum

 

            For both units Glencoe had the greatest percentage of imperatives. In three curricula there was a greater percentage of imperatives in the variables unit, although in CMP the difference was minimal. UCSMP was the only curriculum that had a larger percentage of imperatives in the transformations unit. 

You-modal Form. Modal verbs used with the pronoun you included could, should, may, might, and will. Figure 3 includes the frequency of these modals by curriculum.

Figure 3.

Percentage of Modal Verbs Used by Unit and Curriculum

The most frequent modal verb used with the pronoun you in both units for all four curricula was may/might, accounting for at least 65% in each curriculum. Each curriculum used the form you could/can, which accounted for between 4-19%, but only CMP and UCSMP used the form you should.

Significance

This study not only provides a description of the opportunities for teacher learning related to variable and geometric transformations, but also provides curriculum authors and researchers with valuable information as to the content and the voice in a sample of commonly used, diverse curriculum materials. The results indicated that the types of supports available to teachers included mostly implementation guidance and that teachers were often being directed to do certain things as evidenced by the number of imperatives The results from this study can help curriculum authors develop teachers' guides that speak to teachers, rather than through teachers (Remillard, 2000) and include the types of supports they need.

References

Beyer, C.J., Delgado, C., Davis, E.A., & Krajcik, J. (2009). Investigating teacher learning supports in high school biology curricular programs to inform the design of educative curriculum materials. Journal of Research in Science Teaching, 46, 977-998.

Day, R., Frey, P., Howard, A.C., Hutchens, D.A., Luchin, B., McClain, K. et al. (2009). Math connects: Concepts, skills, and problem solving (Michigan Edition). Columbus, OH: The McGrall-Hill Companies, Inc.

Davis, E., A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34 (3), 3-14.

Halliday, M.A.K. (1985). An introduction to functional grammar. London: Edward Arnold.

Herbel-Eisenmann, B. (2007). From intended curriculum to written curriculum: Examining the “voice” of a mathematics textbook. Journal for Research in Mathematics Education, 38, 344-369.

Lappan, G., Fey, J., Fitzgerald, W., Friel, S., & Phillips, E. D. (2006). Connected Mathematics 2. Boston: Pearson- Prentice Hall.

Martin, J.R., & Rose, D. (2007). Working with discourse: Meaning beyond the clause. London: Continuum.

Morgan, C. (1998).Writing mathematically: The discourse of investigation. London: Falmer Press.

Remillard, J. T. (2000). Can curriculum materials support teachers' learning? Two fourth-grade teachers' use of a new mathematics text. The Elementary School Journal, 100, 331-350.

Schneider, R. M. (2006). Supporting science teacher thinking through curriculum materials. In S. A. Barab, K. E. Hay & D. T. Hickey (Eds.), Proceedings of the 7th International Conference on the Learning Sciences (pp. 674-680). Bloomington, IN: International Society of the Learning Sciences.

Schneider, R. M., & Krajcik, J. (2002). Supporting science teacher learning: The role of educative curriculum materials. Journal of Science Teacher Education, 13, 221-245.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Viktora, S.S., Cheung, E., Highstone, V., Capuzzi, C.R., Heeres, D., Metcalf, N.A., et al. (2008). Transition mathematics. Chicago: Wright Group/Mc-Graw Hill.

Wisconsin Center for Education & the Freudenthal Institute (2010). Mathematics in Context. Chicago: Encyclopedia Britannica, Inc.