#### Title

### Examining Mathematical Knowledge for Teaching: An Exploratory Study of Prospective Teachers’ Transition of Knowledge to Practice in Clinical Simulations

#### Date of Award

May 2017

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Teaching and Leadership

#### Advisor(s)

Benjamin H. Dotger

#### Keywords

Clinical Simulations, Instructional Practices, Mathematical Content Knowledge, Mathematical Knowledge for Teaching, Prospective teachers, Transition of knowledge to practice

#### Subject Categories

Education

#### Abstract

In this study, I examine prospective teachers’ mathematical knowledge for teaching (MKT) with a focus on their transition of knowledge to practice, within two clinical simulations (Algebraic Equations and Negative – Zero). The two clinical simulations presented problems of practice that challenged prospective teachers to enact their teacher knowledge to support students learning of mathematical concepts. The Algebraic Equations simulation focused on the translation of words in a sentence into algebraic equations. The Negative – Zero simulation focused on two mathematical axioms: (a) why a negative number times a negative number equals a positive number, and (b) why division by zero is undefined.

Using an exploratory case-study design, I used secondary data in relation to the two clinical simulations to examine prospective teachers’ (PTs) mathematical content knowledge, instructional practices, and quality of explaining mathematical concepts contained within the two clinical simulations. PTs’ mathematical content knowledge was evidenced through the instructional moves that they enacted to guide standardized students toward understanding concepts, noticing (and addressing) mathematical errors made by standardized students and areas of possible misconceptions, and appropriate use of the mathematical register. Evidence of PTs’ instructional practices included the use of decompressing and bridging in the Algebraic Equations simulation, contextual examples in the Negative – Zero simulation, and asking questions and using representations across both clinical simulations. With regard to the quality of explaining mathematical concepts, PTs with a superior quality of explanation score demonstrated conceptual understanding of mathematical concepts by sharing multiple instructional moves and maintaining the responsibility of learning on standardized students through indirect instructions. PTs with a basic quality of explanation score demonstrated conceptual understanding of mathematical concepts, but used direct instructions. PTs with a low quality of explanation score demonstrated procedural understanding characterized by rote memorization of mathematical concepts.

Findings in this study suggest that PTs with conceptual understanding of mathematical concepts are likely to have the knowledge they need to facilitate students’ learning of mathematical content, and that teaching experience explained variations in the quality of explanations scores. In addition to supporting the diffusion of clinical simulations in teacher preparation, this research suggests possibilities afforded in engaging clinical simulations as a platform for capturing teachers' thinking about mathematical concepts and instructional practices within an instructional context.

#### Access

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#### Recommended Citation

Njoroge, John Mungai, "Examining Mathematical Knowledge for Teaching: An Exploratory Study of Prospective Teachers’ Transition of Knowledge to Practice in Clinical Simulations" (2017). *Dissertations - ALL*. 746.

https://surface.syr.edu/etd/746