This article describes a collaboration among four mathematics teacher educators who teach mathematics content courses for prospective elementary teachers (PSTs) at different universities. As members of the Hoosier Association of Mathematics Teacher Educators, we shared our experiences with addition and subtraction of single- and multi-digit whole numbers and believed our collaboration on this topic would help us improve our courses.

Jansen et al. (2009) highlight the importance of collaboration around targeted and shared learning goals in the building of a knowledge base for teacher education. Targeted learning goals are “sufficiently well specified and fine grained to suggest interventions for supporting learners in achieving them and to indicate the types of evidence needed to determine if the goals have been achieved” (p. 526). Shared learning goals are “mutually understood and committed to by all participants in the knowledge-building process” (p. 526). These authors also note:

It is not realistic to expect different teacher education programs to adopt the exact same set of learning goals, as local contexts and populations have specific needs and work under varying constraints. It is more plausible to consider developing cross-site communities of knowledge builders who commit to a small set of related learning goals. We call these small sets of related learning goals a learning progression. (p. 533)

Designing the Learning Progression

Our efforts to design a learning progression for PSTs were guided by the essential ideas for elementary teachers identified by the Conference Board of Mathematical Sciences (CBMS, 2012), informed by our experiences and instructional resources, and enhanced with connections to the development of children’s mathematical thinking.

We call attention to children’s mathematical thinking for two reasons. First, PST knowledge of children’s mathematical thinking is deemed essential by the Association of Mathematics Teacher Educators (AMTE, 2017). Second, focusing on children’s mathematical thinking can provide PSTs with increased motivation for learning mathematics (Philipp et al., 2002).

The developmental nature of children’s learning also can provide a pathway for the development of PST learning. For example, Van de Walle et al. (2019) describe a three-phase instructional sequence for developing children’s understanding of multi-digit computation – starting with direct modeling, moving to invented strategies, and culminating in standard algorithms. Because we believe it is also important for PSTs to engage with these strategies as learners, we used the three phases to inform our learning progression.

The learning progression in Table 1 is our effort to articulate a set of targeted and shared goals for PSTs’ learning of addition and subtraction of single-digit and multi-digit whole numbers.

Table 1. PST Learning Progression for Addition and Subtraction

Implementing the Learning Progression

Developing a knowledge base also involves creating sequences of learning tasks, or hypothetical learning trajectories (Simon, 1995), as well as local instructional theories (Gravemeijer, 2004) that provide hypotheses about the effectiveness of the tasks for supporting PSTs’ achievement of the learning goals. Jansen et al. (2009) note that there can be multiple hypothetical learning trajectories and local instructional theories for the same learning progression. We took this approach in our collaboration. As we examined and revised our learning progression, we discussed and shared instructional resources. We committed ourselves to paying attention to our individual work through teaching journals or other means. Although we do not have space here to share the specific details of that work, we highlight some of the general features of our various approaches and what we learned from our collaboration. A summary of our instructional resources appears in Table 2.

Table 2. Instructional Resources

Sheryl. I valued this collaboration as an opportunity to more intentionally design sequences of tasks, to engage PSTs in examining their own and others’ mathematical thinking, and to focus their attention on specific details of the mathematics they will be teaching. My co-authors helped me imagine new ways to explore models and algorithms and to place more emphasis on children’s thinking.

Barbara. Discussing the learning progression with my co-authors and evaluating my own implementation helped me recognize the value of the number talks I use in my classroom in eliciting and explaining informal addition and subtraction strategies. The collaboration and the shared resources led me to develop more ways for PSTs to analyze children’s reasoning strategies and to make more specific connections between the details of the standard algorithms and base blocks and drawings.

Brooke. Designing this learning progression brought awareness to parts of the progression that I perceived as receiving sufficient attention (e.g., analyzing children’s strategies) and parts that did not receive sufficient attention (e.g., the standard algorithm). Listening to my co-authors’ approaches gave me strategies and confidence to better support the overall learning progression, particularly within the standard algorithm and its connection to place value and the base-ten blocks.

Clay. Having no colleagues in my department with a specialty in mathematics education, I benefited from this collaboration by seeing commonalities and differences in our approaches and developing a common learning progression we could all get behind. The peer interaction gave me confidence in the learning progression, and I have since added more experiences for students using base blocks in conjunction with the standard algorithms.

Next Steps

Reflecting on our collaboration, we all agree that it was a valuable professional learning experience. Our next steps in building a knowledge base include gathering evidence of PST learning to determine if our goals have been reached. A collaborative focus on the assessment of mathematical knowledge for teaching, specifically addition and subtraction of single-digit and multi-digit numbers, can provide insights into the knowledge of teacher candidates as well as the effectiveness of our mathematics teacher preparation programs. Learning from and with other programs may accelerate the improvement process (AMTE, 2017). 

References

Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. https://amte.net/standards

Beckmann, S. (2022). Mathematics for elementary and middle school teachers: With activities (6^th edition). Hoboken, NJ: Pearson.

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Carpenter, T. P., Fennema, E., Franke, M. L. Levi, L. & Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2^nd edition). Portsmouth, NH: Heinemann.

Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. Washington, DC: The American Mathematical Society.

Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6, 105–128.

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Jansen, A., Bartell, T., & Berk, D. (2009). The role of learning goals in building a knowledge base for elementary mathematics teacher education. The Elementary School Journal, 109, 525-536.

Philipp, R. A., Thanheiser, E., & Clement, L. (2002). The role of a children’s mathematical thinking experience in the preparation of prospective elementary school teachers. International Journal of Educational Research, 37, 195-210.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.

Sowder, J., Sowder, L., & Nickerson, S. (2017). Reconceptualizing mathematics for elementary school teachers (3^rd ed.). New York: Macmillan Learning.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2019). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson.