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The preparation experiences of elementary mathematics specialists: examining influences on beliefs, content knowledge, and teaching practices

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Abstract

Many in the field of mathematics education call for elementary schools to have elementary mathematics specialists (EMSs) who provide needed mathematical expertise and support for children and teachers. EMSs serve as a reasonable, immediate alternative to the challenges generated by elementary teachers needing improved mathematical knowledge for teaching in the classroom. However, limited inquiry has explored how to best prepare EMSs and how program features and learning activities influence their development. This mixed-method study identifies some of the interrelated benefits from a K-5 Mathematics Endorsement Program designed to prepare EMSs through examining changes in mathematical beliefs, specialized content knowledge (SCK), and classroom teaching practices during the program. Data (n = 32) were collected over the 2-semester program via belief surveys, a content knowledge assessment, observations of teaching practices, and individual interviews from elementary teachers participating in the program. The findings show some changes in beliefs can be made relatively quickly, other shifts in beliefs take more time and continued support, and changes in SCK and adoption of various aspects of standard-based pedagogy require considerably greater opportunities to learn. The described program features and learning experiences provided a context for these changes and offer considerations for EMS preparation programs.

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Correspondence to Susan L. Swars.

Appendices

Appendix 1: Example worthwhile mathematical task collection and rationales assignment

The NCTM’s Principles to Actions (2014) and Professional Standards for Teaching Mathematics (1991) emphasize the posing of learning activities often called worthwhile mathematical tasks. These tasks are to be based on:

  • Sound and significant mathematics;

  • Knowledge of students’ understandings, interests, and experiences;

  • Knowledge of the range of ways that diverse students learn mathematics;

And these tasks are intended to:

  • Engage students’ intellect

  • Develop students’ mathematical understandings and skills; stimulate students to make connections; and develop a coherent framework for mathematical ideas;

  • Call for problem formulation, problem-solving, and mathematical reasoning;

  • Promote communication about mathematics;

  • Represent mathematics as an ongoing human activity;

  • Display sensitivity to, and draw on, students’ diverse background experiences and dispositions;

  • Promote the development of all students’ dispositions to do mathematics. (1991, p. 25)

Stein, Smith, Henningsen, and Silver (2009) encourage the analysis of mathematics instructional tasks for “the level and kind of thinking in which students engage” (p. 1) in order to successfully solve the task. Their analysis of cognitive demands divides mathematical tasks into two general categories, each of which are divided further into two subcategories: Lower-Level Demands (including Memorization Tasks and Procedures without Connections Tasks) and Higher-Level Demands (including Procedures with Connections Tasks and Doing Mathematics Tasks). Worthwhile mathematical tasks align with those characterized as Procedures with Connections Tasks and Doing Mathematics Tasks.

For this assignment, select, adapt, or generate (and organize) ten (10) worthwhile mathematical tasks across grades P-5 focusing on developing understanding of the major concepts of number and operations. For each of the tasks in the collection, provide a complete solution strategy of your own work. Following your solution, explain in writing your thinking used to complete the task. For each task collected, provide a rationale/cover page that identifies the following (refer to Stein, Smith, Henningsen, and Silver):

  • Anticipated students (age, grade level, and prior knowledge/experience);

  • Goals for student learning (from CCSS-Mathematics or NCTM Standards);

  • Mathematical features of the task, including what students are asked to do, in what context, with what tools (including the impact of the use of calculators or other technology), etc.;

  • Level of cognitive demands (kinds of thinking required by the task);

  • Rationale for the categorization of cognitive demands

Attach the worthwhile mathematical task immediately following the cover page. Then, attach your evidence of solving the worthwhile mathematical task and your explanation of mathematical thinking to complete the task.

Appendix 2: Example CGI addition and subtraction interview assignment

  1. a.

    Prepare a script of 11 addition and subtraction word problems to pose to a child. Include one of each type of problem identified in the CGI framework of addition and subtraction problem types. Vary the names of participants, objects, and numbers used in the collection of problems. Provide for your selection of alternative number sizes during the interview, depending on the as yet unknown needs of the child. The problems must make sense with all of the alternate number sizes. Use realistic contexts for all problems, but make the problems as simple in context and syntax as possible. The goal is for the problems to be engaging yet easily understandable. Further, the problems should be sequenced from least to greatest difficulty as identified by the CGI framework.

  2. b.

    Interview one child with the purpose of coming to know what that child understands about solving addition and subtraction word problems. Provide a collection of appropriate physical materials as well as paper and pencil for the child to use in solving the problems. Begin by asking one of the easier problems from your script and record in as much detail as possible what the child does and says in trying to solve the problem. On the basis of the child’s strategy and success in solving the first problem, sequence additional problems that will explore the extent of the child’s strategies and understanding while continuing to encourage and support the child’s success in solving the problems you pose.

  3. c.

    Write a report that lists the problem you posed, identifies the problem type from the CGI framework (e.g., JRU for Join Result Unknown), describes the child’s response as completely as possible, and analyzes the child’s response on the basis of the CGI framework for solution strategies. Repeat this process (problem as posed, CGI problem type, child’s response, and CGI analysis) for each of the problems that you posed. At the end of this report, write one paragraph that summarizes what you learned about the child’s understanding of addition and subtraction, the types of problems the child successfully solved and struggled with, the range of numbers with which the child was familiar, and the types of strategies the child demonstrated. Also include a good next problem for this student, identifying the problem type, and justifying the decision with evidence from the report.

Appendix 3: Interview protocol

  1. 1.

    Do you believe you can teach math effectively? Why or why not?

  2. 2.

    What do you believe should be the teacher’s role during math instruction?

  3. 3.

    How confident are you in your skills and abilities to fulfill this role? Why do you feel this way?

  4. 4.

    Did the math endorsement courses affect your confidence in your skills and abilities to teach mathematics effectively? If so, how? If not, why?

  5. 5.

    Have your beliefs about effective math instruction changed as a result of the mathematics endorsement courses? If so, how? If not, why?

  6. 6.

    Have your teaching practices in math changed as a result of the math endorsement courses? If so, how? If not, why?

  7. 7.

    After taking these math endorsement courses, do you feel confident that your math content knowledge is sufficient to understand PreK-5 mathematics? Why or why not?

  8. 8.

    Do you find what you learned in the math endorsement courses is useful in your teaching? If so, how? If not, why?

  9. 9.

    What do you apply that you learned in these courses? What are some of the challenges you have encountered in applying what you learned in these math endorsement courses? What is easy? Why?

  10. 10.

    Do you believe what you learned in the math endorsement courses will impact your students’ learning? If so, how? If not, why?

Appendix 4: SBLEOP included classroom events

CE1. The lesson provided opportunities for students to make conjectures about mathematical ideas:

  • A conjecture is a claim, proposition, or inference that something is true. Students commonly make three types of mathematical conjectures: (1) claims that previously used solution strategies will work for new but similar problems; (2) claims that specific mathematical statements are true; and (3) claims that mathematical processes or properties are always true, always work, or are true for specific numbers (i.e., generalizations). Students need opportunities to make, explore, and validate conjectures.

Scores 1–3 are as follows:

  1. 1.

    Students had few, if any, opportunities to make, explore, and/or validate conjectures in this lesson. The teacher generally did not solicit or encourage conjectures.

  2. 2.

    Students had some opportunity and/or encouragement to make, explore, and/or validate conjectures. When observed, they were either prompted by the teacher or offered by students, but with minimal follow-up or discussion.

  3. 3.

    Conjectures of at least one of the three types described provided a meaningful portion of the lesson activities. The lesson discussion involved significant follow-up on these conjectures.

CE.2. The lesson fostered the development of conceptual understanding.

  • Conceptual understanding involves making sense of big ideas of and building a network of connections and relationships between these ideas and prior knowledge/experience. The development of this knowledge is fundamentally different from the typical teaching of procedures, skills, and definitions, which can be memorized in isolation without understanding.

Scores 1–3 are as follows:

  1. 1.

    The general focus of the lesson was on the development of procedural knowledge, skills, or definitions, with little, if any, attention to the development of conceptual understanding.

  2. 2.

    The general focus of the lesson was on the development of a mathematical concept or relationship; however, the teacher did not actively engage students in building connections between these ideas and their prior knowledge.

  3. 3.

    A significant portion of the lesson focused on building conceptual understanding and making connections to students’ prior knowledge, and the teacher actively engaged students in explaining their understanding and/or thinking about these relationships.

CE.5. Students explained their responses or solution strategies.

  • Students engage in mathematical processes of communication and justification/proof as they explain their thinking, elaborate their solutions, justify their approach to a problem, or support their results. Simply stating answers overemphasizes the importance of the result and relies on the teacher as the authority for correctness.

Scores 1–3 are as follows:

  1. 1.

    The teacher generally did not encourage students to elaborate on answers or solution strategies. Rather, students simply stated answers to problems or questions posed by the teacher and the teacher accepted these answers without further probing.

  2. 2.

    The teacher sometimes encouraged students to orally explain how they arrived at an answer, but these explanations generally focused on the execution of procedures rather than on elaboration of thinking or problem-solving strategies.

  3. 3.

    The teacher generally encouraged students to explain their responses or solution strategies, justify their approach to a problem, explain their thinking, or support their results, either orally or in writing.

CE.6. Multiple perspectives/strategies were encouraged and valued.

  • Student-centered instruction encourages and values a variety of student perspectives and strategies for solving problems and communicating understanding. As students come to understand other students’ perspectives and strategies, they are able to develop greater flexibility in thinking and adopt more efficient strategies with understanding and confidence.

Scores 1–3 are as follows:

  1. 1.

    The teacher did not generally encourage students to offer different perspectives and/or strategies to solving problems. Generally, if a correct solution was offered by a student, the teacher accepted it and moved on.

  2. 2.

    Different perspectives or strategies were occasionally elicited from students or mentioned by the teacher. However, the teacher seemed to be searching for or emphasizing a standard procedure or perspective.

  3. 3.

    The teacher encouraged students to view problems or mathematical situations from multiple perspectives and to learn from each other’s viewpoints, strategies, and/or thinking.

CE.7. The teacher valued students’ mathematical statements and used them to build discussion or develop shared understanding.

  • Teachers can add importance to students’ statements by inviting students to listen carefully to each other, to ask each other clarifying questions, and to compare other students’ strategies, thinking, and understanding with their own. Discourse about students’ statements provides the opportunity to develop common understandings or to explore important mathematics deeply and thoroughly. The teacher may encourage this type of discussion by asking questions such as: “Does everyone agree with this?” or “Would anyone like to comment on this?”

Scores 1–3 are as follows:

  1. 1.

    The teacher seemed interested primarily in correct answers. The majority of the teacher’s remarks about student responses were short comments such as “Okay,” “All right,” or “Fine.” No attempt was made to use students’ statements to initiate further discussion.

  2. 2.

    The teacher established a dialogue with one or more students about student thinking processes or solution strategies, but did not use this discussion to develop common understandings or deepen students’ understandings of the mathematics.

  3. 3.

    The teacher valued students’ statements about mathematics by using them to orchestrate a discussion about the mathematics or to deepen students’ understandings of the mathematics.

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Swars, S.L., Smith, S.Z., Smith, M.E. et al. The preparation experiences of elementary mathematics specialists: examining influences on beliefs, content knowledge, and teaching practices. J Math Teacher Educ 21, 123–145 (2018). https://doi.org/10.1007/s10857-016-9354-y

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