Abstract
This study examines the association between mathematical knowledge for teaching and instructional quality in a sample of firstyear elementary school teachers. Ten teachers completed the mathematical knowledge for teaching (MKT) survey at the end of teacher preparation. Three mathematics lessons taught during their first year of teaching were videotaped and scored using the Mathematical Quality of Instruction. Findings replicate prior studies that were conducted with more experienced teachers. A strong, positive and statistically significant association was found between teacher knowledge and the mathematics is clear and not distorted dimension of instructional quality. In addition, associations of moderate strength were found between MKT and other dimensions of instructional quality centered on the mathematics taught in the lesson. Analyses also revealed individual differences among teachers and raised the question of what other factors might impact instructional quality. Three cases studies highlight the role of lesson design, mathematics tasks, and participation structures that support or inhibit instructional quality and the use of knowledge during teaching. Conclusions suggest that preparation and induction programs should include a focus on individual teachers’ mathematical knowledge for teaching, the development of a studentcentered vision of mathematics instruction, and tailored support during the first year of teaching.
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Acknowledgements
The authors thank the teachers for their participation and for opening their classroom door and making their instruction public. In addition, they are grateful to Janet Mercado and Cathery Yeh who assisted with data collection and organization; to Rosalind Alicia Ball who scored lesson videos for interrater reliability; and to several undergraduate research assistants who completed lesson transcriptions.
Funding
This research was supported by the National Science Foundation (REESE program) under Grant DRL0953038. Any opinions, findings, and conclusions expressed in this material are those of the authors and do not necessarily reflect the views of the funding agency. Previous versions of this paper were presented at the 2017 biannual meeting of the Korea Society of Educational Studies in Mathematics (KSESM), Korea National University of Education, Cheongju, South Korea and the 2018 annual meeting of the Association of Mathematics Teacher Education, Houston, TX.
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Appendix A
Appendix A
Sample items from Hill et al. (2004) are below:
1. Imagine that you are working with your class on multiplying large numbers. Among your students’ papers, you notice that some have displayed their work in the following ways:
Which of these students would you judge to be using a method that could be used to multiply any two whole numbers?
Method would work for all whole numbers  Method would NOT work for all whole numbers  I’m not sure  

(a) Method A  1  2  3 
(b) Method B  1  2  3 
(c) Method C  1  2  3 
2. Takeem’s teacher asks him to make a drawing to compare \(\frac{3}{4}\) and \(\frac{5}{6}\). He draws the following:
and claims that \(\frac{3}{4}\) and \(\frac{5}{6}\) are the same amount. What is the most likely explanation for Takeem’s answer? (Mark ONE answer.)

(a)
Takeem is noticing that each figure leaves one square unshaded.

(b)
Takeem has not yet learned the procedure for finding common denominators.

(c)
Takeem is adding 2 to both the numerator and denominator of \(\frac{3}{4}\), and he sees that that equals \(\frac{5}{6}\).

(d)
All of the above are equally likely.
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Santagata, R., Lee, J. Mathematical knowledge for teaching and the mathematical quality of instruction: a study of novice elementary school teachers. J Math Teacher Educ 24, 33–60 (2021). https://doi.org/10.1007/s1085701909447y
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Keywords
 Mathematical knowledge for teaching
 Instructional quality
 Novice teachers
 Elementary school teachers
 Video
 Mixed methods
 Case studies