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Cited by 6Mathematics Education and Teaching TechniquesTeacher Education and Leadership StudiesScience Education and Pedagogy

Preservice teachers' mathematical knowledge for teaching: Focus on lesson planning and reflection

Open access
Sapkota, Bima, Hayes, Amanda Huffman·2024·School Science and Mathematics
DOI 10.1111/ssm.12644
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Abstract

Abstract Mathematics teacher educators have suggested preservice mathematics teachers' (PMTs') practices provide evidence of their mathematical knowledge for teaching (MKT). PMTs engage in approximations of practice to rehearse teaching. To explore connections between MKT and PMTs’ practices, we operationalized six MKT subdomains in terms of approximations of practice. We used that operationalization to investigate, which subdomains were evidenced in PMTs’ lesson plans and their lesson plan reflections. Eleven secondary PMTs participated in our study. Using a qualitative content analysis method, we found Knowledge of Content and Teaching most evidenced, suggesting that the PMTs demonstrated MKT when they described mathematical tasks and associated instructional activities. Also, the PMTs demonstrated Knowledge of Content and Students when they described how their students would engage with proposed mathematical tasks. However, the PMTs struggled to plan for activities by building on students’ potential mathematical thinking and reasoning. Only limited instances of Horizon Content Knowledge suggest that the PMTs had limited opportunities to demonstrate how their advanced mathematical knowledge connected with their mathematics teaching in the approximations of practice provided. We propose alternative forms of approximations of practice to optimize PMTs’ opportunities to demonstrate and develop MKT.

Integrity signals

Registry

Crossref · 10.1111/ssm.12644

Cites

46 of 46 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 36 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    MKT subdomains within pedagogical content knowledge were more evidenced in the PMTs' lesson plans than descriptors of subject matter knowledge.

    “suggesting that the MKT subdomains within pedagogical content knowledge were more evidenced in the PMTs' lesson plans than the descriptors of subject matter knowledge.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.
  • Findingown data

    KCT was primarily identified when PMTs described selecting and describing mathematical tasks, formulating, sequencing, and posing questions, identifying affordances of methods or procedures, and identifying strategies to assess students' understanding.

    “This subdomain was primarily identified when the PMTs described the following teacher actions: (a) selecting and describing mathematical tasks, (b) formulating, sequencing, and posing questions, (c) identifying the affordances of particular methods or procedures, and (d) identifying strategies to assess students' understanding.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.
  • Findingown data

    The shift in frequency from lesson plan I to II was more evident in pedagogical content knowledge than subject matter knowledge, with 22 subject matter knowledge instances in lesson plan I versus 29 in lesson plan II.

    “The table also shows that the shift in frequency from the PMTs' lesson plan I to II was more evident in pedagogical content knowledge than subject matter knowledge with 22 instances of subject matter knowledge in lesson plan I versus 29 instances of those subdomains in lesson plan II.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.
    • Rests on an assumption: Lesson plan I and lesson plan II are comparable enough that differences in MKT subdomain frequencies indicate shifts in the PMTs' MKT.
  • Findingown data

    KCT was also evidenced when the PMTs formulated mathematics-specific questions and presented procedures to pose those questions.

    “KCT was also evidenced when the PMTs formulated mathematics-specific questions and presented procedures to pose those questions.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.
  • Findingown data

    PMTs demonstrated Knowledge of Content and Students when they described how their students would engage with proposed mathematical tasks.

    “Also, the PMTs demonstrated Knowledge of Content and Students when they described how their students would engage with proposed mathematical tasks.”
    Quote in the paperSubstantial claim
  • May combine two claims. 18
  • The quote only partly states this claim. 4
  • Classified as probably not a substantial claim. 18

  • May combine two claims.

    Findingown data

    KCT was the most highlighted subdomain in both lesson plans, including 87 instances in lesson plan I and 102 instances in lesson plan II.

    “KCT was the most highlighted subdomain in both lesson plans, including 87 instances in lesson plan I and 102 instances in lesson plan II.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.

  • May combine two claims.

    Findingown data

    There was a shift from lesson plan I to lesson plan II, with more instances of math-specific student actions in lesson plan II than in lesson plan I.

    “Our findings indicate a shift from lesson plan I to II with more instances of math-specific student actions in lesson plan II than in lesson plan I.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The PMTs' written lesson plans and reflections accurately reflect the mathematical knowledge for teaching they hold and would use in instruction.

  • May combine two claims.

    Findingown data

    The PMTs could not articulate how their tasks or activities addressed students' current levels of understanding, because they did not engage with real students.

    “Unlike Chick (2009), our PMTs could not articulate how their tasks or activities addressed students' current levels of understanding as they did not engage with real students.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Chick (2009)

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Findingown data

    Overall, the approximations in the form of lesson plans and associated reflections provided opportunities for the PMTs to demonstrate KCT, KCS, KCC, CCK, and SCK, with some subdomains evidenced more than others.

    “Overall, the approximations in the form of lesson plans and associated reflections provided opportunities for the PMTs to demonstrate KCT, KCS, KCC, CCK, and SCK with some subdomains evidenced more than others.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The PMTs' written lesson plans and reflections accurately reflect the mathematical knowledge for teaching they hold and would use in instruction.

  • May combine two claims.

    Findingown data

    The descriptors for anticipating contextual factors that support or impede students' understanding and for planning strategies and questions based on students' possible mathematical thinking and responses were not identified.

    “We did not identify the descriptors,”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.

  • The quote only partly states this claim.

    Findingown data

    KCS was the second most evidenced MKT subdomain, with 14 instances in lesson plan I and 45 instances in lesson plan II.

    “We identified 14 and 45 instances of KCS in the PMTs' lesson plans I and II as well as in their reflections, respectively”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.

  • May combine two claims.

    Findingown data

    Kiara planned for what students were likely to do with the mathematical tasks, but planned to ask follow-up questions without mentioning what those follow-up questions would be in the summary section.

    “These quotes from lesson plan II indicated that Kiara planned for what students were likely to do with the mathematical tasks.However, she planned to ask followup questions without mentioning what those follow-up questions would be in the summary section.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    Some PMTs in lesson plan I formulated generic questions, while in lesson plan II they formulated more mathematics-specific questions.

    “For instance, some PMTs in lesson plan I formulated generic questions, while in lesson plan II, they formulated more mathematicsspecific questions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The PMTs presented mathematics-specific guiding questions but struggled to formulate follow-up questions, so few instances of instructional activities based on students' thinking were identified.

    “We noticed that the PMTs presented mathematicsspecific guiding questions but struggled to formulate follow-up questions for their students, thus we did not identify many instances in the PMTs' plans of instructional activities based on students' thinking.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    Many MKT subdomains were not identified in the summary section.

    “Thus, we did not identify many MKT subdomains in the summary section.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.

  • May combine two claims.

    Findingown data

    KCC was moderately evidenced in the PMTs' lesson plans and frequently discussed in their reflections.

    “KCC was moderately evidenced in the PMTs' lesson plans and frequently discussed in their reflections.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.
    • Rests on an assumption: The PMTs' reflections accurately describe the reasons they included particular activities in their lesson plans.

  • May combine two claims. The quote only partly states this claim.

    Findingown data

    In their reflections, all the PMTs discussed how their lesson built on prior lessons and how it prepares students for possible upcoming lessons.

    “In their reflections, all the PMTs discussed”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The PMTs' reflections accurately describe the reasons they included particular activities in their lesson plans.

  • May combine two claims.

    Premisecitation

    Anticipating students' mathematical thinking and selecting mathematical tasks to address student thinking are examples of MKT that could be evidenced through lesson plans.

    “Anticipating students' mathematical thinking and selecting mathematical tasks to address student thinking are examples of MKT that could be evidenced through lesson plans (Chick, 2009).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Chick, 2009

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim. May combine two claims.

    Study design and conductown data

    The first author compared her codes with the second author's codes to resolve discrepancies, and remaining discrepancies were discussed in meetings.

    “The first author compared her codes with the second authors' codes to resolve any discrepancies and any remaining discrepancies were discussed in meetings.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductown data

    The second author created new descriptors when she thought a different code would more accurately reflect the data.

    “The second author created new descriptors if she thought a different code would more accurately reflect the data.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Study design and conductown data

    Weekly meetings were conducted and discrepancies were resolved until a refined codebook was developed.

    “We then conducted weekly meetings and resolved any discrepancies until we finally developed a refined codebook.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given

    The second author collaborated with the first author to identify new descriptors that were added to the framework.

    “The second author collaborated with the first author to identify new descriptors which were added to the framework.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductcitation

    The study used the Simultaneous Coding method to identify all MKT descriptors evidenced in each coding unit.

    “We then followed the Simultaneous Coding method (Saldaña, 2016) to identify all the MKT descriptors evidenced in each coding unit.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Saldaña, 2016

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning

    The absence of MKT subdomains in PMTs' lesson plans and reflections does not mean they have not developed that specific knowledge; rather, the approximations did not engage PMTs with those subdomains.

    “Thus, the absence of subdomains in the PMTs' lesson plans and reflections does not mean they have not developed that specific knowledge but the approximations we developed did not engage PMTs with those subdomains.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Study design and conductreasoning

    Theoretical validity was established by developing MKT descriptors from the MKT framework and related literature.

    “As described in the process above, we established”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites Ball et al., 2008

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites Hayashi et al., 2019 source

  • Classified as probably not a substantial claim. The quote only partly states this claim.

    Study design and conductcitation

    The reflection prompts were developed in alignment with prior work to invite PMTs to reflect on MKT in the six subdomains.

    “We developed the prompts in alignment with Gonz alez et al. (2020), Schilling and Hill (2007), and Özgün-Koca et al. (2020).”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites Gonz alez et al. (2020)

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites Schilling and Hill (2007) source
    • cites Özgün-Koca et al. (2020) source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Teachers' discipline-specific knowledge is dynamic.

    “Teachers' discipline-specific knowledge is dynamic (Baumert et al., 2010)”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Baumert et al., 2010 source

  • Classified as probably not a substantial claim. May combine two claims.

    Study design and conductnone given

    The coding scheme was developed through literature synthesis and then used to code the data.

    “We developed descriptors for the MKT subdomains through synthesis of the literature and then used these descriptors to code data.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductown data

    The first author used pilot study data to test initial codes and further develop the coding scheme.

    “The first author used the data from the pilot study (i.e., four PMTs' lesson plans and reflections collected in an earlier semester) to test the initial codes and further develop the coding scheme.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductreasoning

    The refined codebook was used to code the rest of the data.

    “We then used that refined codebook to code the rest of the data.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    In mathematics education, discipline-specific knowledge is commonly conceptualized as mathematical knowledge for teaching (MKT).

    “In mathematics education, discipline-specific knowledge is commonly conceptualized as mathematical knowledge for teaching (MKT; Ball et al., 2008).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Ball et al., 2008

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning

    The lesson plan template had prompts for the PMTs to describe teacher and student actions and not many options to demonstrate how their advanced mathematical knowledge connected with those actions, which might have influenced the findings.

    “The lesson plan template had prompts for the PMTs to describe teacher and student actions and not many options to demonstrate how their advanced mathematical knowledge connected with the teacher and student actions, which might have influenced our findings.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationnone given

    The MKT subdomain descriptors provide a foundation to design instructional activities that engage PMTs in both teacher knowledge and practices.

    “Our MKT subdomain descriptors provide a foundation to design instructional activities that engage PMTs in both teacher knowledge and practices.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Prior research has highlighted inservice and preservice teachers' reflections on teaching, teacher interviews, and observations of teachers' classrooms as a way to investigate MKT.

    “Prior research has highlighted inservice and PMTs' reflections on teaching, teacher interviews, and observations of teachers' classrooms (Edelman, 2017;Sudejamnong et al., 2014;Tepylo & Moss, 2011) as a way to investigate MKT.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites Edelman, 2017
    • cites Sudejamnong et al., 2014 source
    • cites Tepylo & Moss, 2011 source

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The few instances of HCK could be because the PMTs planned lessons to be implemented with their peers.

    “It could be because the PMTs planned lessons to be implemented with their peers.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    SCK could be evidenced when teachers recognize and build on students' vaguely expressed or partially formed ideas.

    “SCK could be evidenced when teachers recognize and build on students' vaguely expressed or partially formed ideas (Speer & Wagner, 2009).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Speer & Wagner, 2009 source

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The PMTs might not have identified the need for offering counterexamples to their peers.

    “Thus, they might not have identified the need for offering counterexamples to their peers.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductreasoning

    The PMTs did not experience how students might partially form mathematical reasoning and were not instructed to recognize students' partially formed or vaguely expressed ideas, possibly not providing them with opportunities to demonstrate SCK.

    “The PMTs did not experience how students might partially form mathematical reasoning; and were not instructed to recognize students' partially formed or vaguely expressed ideas, possibly not providing the PMTs with opportunities to demonstrate SCK.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    HCK could be evidenced when teachers examine the reasonings required to offer multiple examples and representations to present big mathematical ideas.

    “Researchers (e.g., Jakobsen et al., 2013;Wasserman & Stockton, 2013) suggested HCK could be evidenced when teachers examine the reasonings required to offer multiple examples and representations to present big mathematical ideas.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites Jakobsen et al., 2013

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites Wasserman & Stockton, 2013

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

34 of 34

  • The paper relies on this without support.

    When PMTs consider their peers as students during peer lesson planning, the way PMTs situate content for peers could be different from the ways they situate content for their students.

    “When PMTs consider their peers as students during peer lesson planning, the way PMTs situate content for peers could be different from the ways they situate content for their students.”

  • The paper relies on this without support.

    The reflection prompts could lead PMTs to unknowingly reflect on their MKT in the six MKT subdomains.

    “With these prompts, we aimed that the PMTs could unknowingly reflect on their MKT in the six MKT subdomains.”

  • The paper relies on this without support.

    The frequency of MKT subdomains in written lesson plans and reflections reflects the PMTs' mathematical knowledge for teaching.

    “We first report the frequency of the MKT subdomains that appeared in the PMTs' lesson plans I and II.”

  • The paper relies on this without support.

    Lesson plan I and lesson plan II are comparable enough that differences in MKT subdomain frequencies indicate shifts in the PMTs' MKT.

    “Finally, we report the shift in the MKT subdomains evidenced in lesson plan I and II.”

  • The paper relies on this without support.

    The PMTs' reflections accurately describe the reasons they included particular activities in their lesson plans.

    “We use the PMTs' lesson plan reflections to provide the context for and descriptions of potential reasons that the PMTs included particular activities in their lesson plans throughout the findings.”

  • The paper relies on this without support.

    The PMTs' written lesson plans and reflections accurately reflect the mathematical knowledge for teaching they hold and would use in instruction.

    “Overall, the approximations in the form of lesson plans and associated reflections provided opportunities for the PMTs to demonstrate KCT, KCS, KCC, CCK, and SCK with some subdomains evidenced more than others.”
6 of 6

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Contextual-factor descriptor is both absent and present

    Minor

    The findings say no instances of anticipating contextual factors were identified, but a later section lists that same descriptor among the actions PMTs began planning in lesson plan II.

  • Nearly identical observation is attributed to two PMTs

    Minor

    An observation about students having worked only with equations is attributed first to Kate and later to Jacob. The near-identical wording makes the attribution inconsistent.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 7

  1. Ball D., 1999, Teaching as a learning profession, 3
  2. Ball D. L., 2000, Multiple perspectives on the teaching and learning of mathematics, 83
  3. CPM Educational Program, 2022, More math for more people
  4. Jakobsen A., 2013, Proceedings of CERME 8, 3125
  5. Milkova S., 2012, Strategies for effective lesson planning
  6. Ribeiro C. M., 2013, Proceedings of the 37th conference of the International Group for the Psychology of mathematics education, 89
  7. Saldaña J., 2016, The coding manual for qualitative researchers

Cited by

Sorted by citations

Related

Authors
Sapkota, B., & Hayes, A. H.
Affiliations
  • The University of Texas Rio Grande Valley (US)
  • Purdue University West Lafayette (US)
Corresponding author
Sapkota, Bima
ORCID
Venue
School Science and Mathematics
Publisher
Wiley
Published
2024-02-22
Volume
124
Issue
5
Pages
323-339
ISSN
  • 0036-6803 (ISSN-L)
  • 1949-8594
DOI
10.1111/ssm.12644
OpenAlex
W4392039031
Cited by
6
References
45
Citations by year
2026: 2 · 2025: 2 · 2024: 2
FWCI
7.49
Type
journal article
Language
en
Open access
Licence
onlinelibrary.wiley.com/termsAndConditions#vor (vor)
Locations
  • School Science and Mathematics (publishedVersion)
  • ScholarWorks @ UTRGV (The University of Texas Rio Grande Valley) (submittedVersion)
Topics
  • Social Sciences › Education › Mathematics Education and Teaching Techniques
  • Social Sciences › Education › Teacher Education and Leadership Studies
  • Social Sciences › Education › Science Education and Pedagogy
Keywords
mathematical knowledge for teaching, pre-service mathematics teachers, lesson planning, secondary mathematics teacher education, student mathematical thinking

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