| No. | Citation | Inclusion | Country | Genre | Causal | Sample | Instrument | Level | Teachers | Problem | Lesson | Cited by |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | cite:&alex2019 | 1 | RSA | oth | sta | md2(40) | national math exam | all | fut | What teachers know? | Lack of knowledge | 7 |
| 2 | cite:&bansilal2014 | 1 | RSA | oth | sta | lrg(253) | national math exam (adjusted) | sec | prc | What teachers know? | Lack of knowledge | 71 |
| 3 | cite:&bowie2019 | 1 | RSA | oth | sta | lrg(770) | math content test | prm | fut | What teachers know? | Lack of knowledge | 16 |
| 4 | cite:&chikiwa2019 | 1 | RSA | nat | qal | sma(1) | none | prm | exp | What relationships? | KCT is foundational | 7 |
| 5 | cite:&defreitas2019 | 1 | RSA | nat | sta | lrg(93) | TPACK | mid | prc | What relationships? | Attending to aspects of knowledge important | 19 |
| 6 | cite:&feza2016 | 1 | RSA | oth | qal | md1(17) | scenario based questionnaire | prm | prc | What teachers know? | Lack of knowledge | 11 |
| 7 | cite:&feza2018 | 1 | RSA | dev | qal | md1(14) | COEMET | prm | prc | What PD? | Lack of knowledge | 1 |
| 8 | cite:&fonseca2015 | 1 | RSA | dev | sta | lrg(108) | math content test | mid | fut | What PD? | Lack of knowledge | 9 |
| 9 | cite:&fonseca2021 | 1 | RSA | dev | qal | md2(62) | self-report | prm | fut | What TE? | Attending to aspects of knowledge important | 0 |
| 10 | cite:&helmbold2021 | 1 | RSA | dev | qal | sma(6) | questionnaire (self-report) | prm | prc | What PD? | LS influenced knowledge | 0 |
| 11 | cite:&jacinto2020 | 1 | MAW | oth | qal | sma(6) | questionnaire (self-report) | prm | fut | What teachers know? | PSTs emphasise particular knowledge | 6 |
| 12 | cite:&jita2019 | 1 | RSA | dev | qal | lrg(125) | portfolio (self-report) | all | prc | What PD? | LS influenced knowledge | 3 |
| 13 | cite:&kalobo2016 | 1 | RSA | oth | sta | md2(66) | questionnaire (perceptions) | sec | prc | What teachers know? | Lack of knowledge | 11 |
| 14 | cite:&makonye2020b | 1 | RSA | nat | qal | md1(20) | questionnaire/interview (test) | ter | exp | What is MKT? | Lack of knowledge (account for cultural background) | 6 |
| 15 | cite:&msimango2020 | 1 | RSA | dev | qal | md1(12) | interviews (self-report) | prm | fut | What TE? | Mentors can hinder knowledge development | 0 |
| 16 | cite:&mwadzaangati2019 | 1 | MAW | nat | qal | sma(2) | observation/interview | sec | exp | What is MKT? | Supporting knowledge development is complex | 0 |
| 17 | cite:&pournara2015 | 1 | RSA | inf | sta | md1(21) | math test (for learners!) | sec | prc | Contribute to student learning? | MKT influences student learning | 91 |
| 18 | cite:&setoromo2018 | 1 | LES | oth | sta | md2(48) | questionnaire (test) | prm | prc | What teachers know? | Lack of knowledge | 4 |
| 19 | cite:&sibanda2021 | 1 | RSA | dev | sta | lrg(203) | survey | all | prc | What PD? | Mentors can influence knowledge development | 2 |
| 20 | cite:&siyepui2021 | 1 | RSA | oth | qal | md2(30) | math content test | all | fut | What teachers know? | Tasks can influence knowledge development | 0 |
| 21 | cite:&spangenberg2021 | 1 | RSA | inf | qal | md1(12) | observation/interview/documentation | sec | prc | What contributes to practice | PCK influences practice | 1 |
| 22 | cite:&ubah2018 | 1 | RSA | oth | qal | md2(60) | interviews | prm | fut | What teachers know? | Lack of knowledge (deep understanding) | 14 |
| 23 | cite:&venkat2015 | 1 | RSA | dev | qal | sma(1) | observations, interviews | prm | prc | How MKT develops? | Development is possible | 3 |
| 24 | cite:&vermeulen2017 | 1 | RSA | oth | qal | sma(3) | math content test | prm | prc | What teachers know? | Lack of knowledge | 30 |
After a full coding of the 30 articles that were considered for inclusion, I ended up with a final sample of 23 studies that were included. Below are overviews of each code.
Citations were found by looking up each individual article on Google Scholar. This was done on <2022-07-29 Fri>.
The overwhelming majority of studies were conducted in South Africa (20 out of 23 studies). Two studies were conducted in Malawi, and one was from Lesotho.
| Country | No. |
|---|---|
| South Africa | 21 |
| Malawi | 2 |
| Lesotho | 1 |
Almost half of the studies (n=10) had a focus on “other”, which in these cases indicated that they focused on evaluating teachers’ knowledge. The underlying problem in these studies was: What teachers know?
| Genre | No. |
|---|---|
| Other | 10 |
| Development | 8 |
| Nature | 4 |
| Influence | 2 |
| Type of causal design | No. |
|---|---|
| Qualitative | 15 |
| Statistical | 9 |
There was an even distribution of studies in terms of their sample size.
| Sample | No. |
|---|---|
| Small-scale (<10) | 6 |
| Medium 1 (10–29) | 6 |
| Medium 2 (30–70) | 6 |
| Large (>70) | 6 |
Whereas Hoover et al. (2016) highlighted the promise of measures, it is striking to see that almost no studies from the African context used standardized measures to study mathematical knowledge for teaching.
Some studies used items from national math certificate tests for teachers citep:&alex2019;&bansilal2014, whereas others developed their own test – often based on existing measures cite:&bowie2019.
Only a couple of studies used existing instruments, like TPACK citep:&defreitas2019 and COEMET citep:&feza2018, which is an observation protocol.
After having grappled with how to code instruments, I decided to consider instruments in relation to the problem of the study.
| No. | Citation | Instrument | Problem |
|---|---|---|---|
| 1 | cite:&alex2019 | national math exam | What teachers know? |
| 2 | cite:&bansilal2014 | national math exam (adjusted) | What teachers know? |
| 3 | cite:&bowie2019 | math content test | What teachers know? |
| 6 | cite:&feza2016 | scenario based questionnaire | What teachers know? |
| 11 | cite:&jacinto2020 | questionnaire (self-report) | What teachers know? |
| 13 | cite:&kalobo2016 | questionnaire (perceptions) | What teachers know? |
| 18 | cite:&setoromo2018 | questionnaire (test) | What teachers know? |
| 20 | cite:&siyepui2021 | math content test | What teachers know? |
| 24 | cite:&vermeulen2017 | math content test | What teachers know? |
| 21 | cite:&ubah2018 | interviews | What teachers know? |
| 4 | cite:&chikiwa2019 | none | What relationships? |
| 5 | cite:&defreitas2019 | TPACK | What relationships? |
| 7 | cite:&feza2018 | COEMET | What PD? |
| 8 | cite:&fonseca2015 | math content test | What PD? |
| 10 | cite:&helmbold2021 | questionnaire (self-report) | What PD? |
| 12 | cite:&jita2019 | portfolio (self-report) | What PD? |
| 19 | cite:&sibanda2021 | survey | What PD? |
| 9 | cite:&fonseca2021 | self-report | What TE? |
| 15 | cite:&msimango2020 | interviews (self-report) | What TE? |
| 14 | cite:&makonye2020b | questionnaire/interview (test) | What is MKT? |
| 16 | cite:&mwadzaangati2019 | observation/interview | What is MKT? |
| 17 | cite:&pournara2015 | math test (for learners!) | Contribute to student learning? |
| 21 | cite:&spangenberg2021 | observation/interview/documentation | What contributes to practice |
| 23 | cite:&venkat2015 | observations, interviews | How MKT develops? |
Among the studies of what teachers know, most included some kind of mathematics content test. A couple of these were drawing upon some kind of national mathematics certificate test for teachers citep:&alex2019;&bansilal2014, whereas many developed their own tests, often based on existing measures or frameworks citep:&bowie2019. Whereas many instruments had a primary focus on mathematical content, a few included some kind of scenario or teaching context citep:&feza2016. Although most studies in this category involved some kind of content test, there was also an example that included a self-report questionnaire citep:&jacinto2020, and one that involved a questionnaire that focused on participants’ perceptions citep:&kalobo2016.
Six studies explored what PD or TE influence mathematical knowledge for teaching. Most of these studies used instruments that involved some kind of self-report, either through interviews citep:&msimango2020, questionnaires citep:&helmbold2021, portfolios citep:&jita2019 or self-report journals citep:&fonseca2021. One study applied a standardized observation protocol (COEMET) to assess practice citep:&feza2018, and one study used a mathematics content test citep:&fonseca2015.
The two studies that explored what mathematical knowledge for teaching is both used a combination of interviews with questionnaires or observations citep:&makonye2020b;&mwadzaangati2019. Amont the two studies that investigated relationships between different aspects of knowledge, one of these used a standardized TPACK instrument citep:&defreitas2019. A study of what contributes to student learning used mathematics learner tests citep:&pournara2015, a study of what contributes to practice used a combination of observations, interviews and other documentation like lesson plans citep:&spangenberg2021, whereas a study of how mathematical knowledge for teaching developes used a combination of observations of interviews citep:&venkat2015.
After having reviewed studies of what teachers know, it struck me that most of them seemed to emphasize CCK. I decided to check out this hypothesis further.
cite:&alex2019 used the SLDCK test, which focused on pure mathematics domains like:
- financial maths
- probability
- data handling
- analytical geometry
- trigonometry
- euclidean geometry
They do not include any sample items, but they disclose that “SLDCK talks about teachers’ own understanding of the content they are expected to teach at a particular grade level” (p. 5), which indicates that the focus is on CCK.
cite:&bansilal2014 used a shortened version of the “National Senior Certificate March 2010 supplementary examination” (p. 39). Items focused on:
- quadratic equations and inequalities
- patterns
- hyperbolic function
- parabolic function
- finding derivatives using rules
- optimisation
- linear programming
In the text, they illustrate that items involve regular mathematics problems like solving inequalities like
cite:&bowie2019 had items on:
- numbers and operations
- rational numbers
- patterns, functions and algebra
- geometry
- measurement
Again, sample items that are given indicate a focus on CCK. Some examples of items are to calculate
cite:&setoromo2018 refer directly to the categories of CCK, SCK, and KCT. Example items of CCK include: “List/Write down the even numbers that are between 1–20 in the space below” (p. 5) and “Write the following in number symbol: two hundred and fifty five” (p. 6). A follow up of the latter involved division with space to show their calculations. Example of an item that was categorized as SCK was: “What would you do when you have planned an activity for learners to count 5 objects and you discover that some learners find the task too easy?” (p. 7). An example KCT item is (p. 8):
For the statement below circle the response which you think is correct. When learners are given objects to count, the following common errors are likely to occur.
A. Learners will count without arranging objects sequentially. B. They wil not skip some objects. C. They may not be able to tell the total number of the objects. D. They may continue counting endlessly. E. They may repeatedly point at the same objects when counting.
cite:&siyepui2021 included items that were intended to “assess students’ understanding in developing conjectures for the total surface area of a cylinder” (p. 4), “assess pre-service teachers’ understanding of the application of the formula for the total surface area of a cylinder” (p. 4), “assess pre-service teachers’ understanding of basic concepts in solids” (p. 5), “assess pre-service student teachers’ understanding of the formulae for calculating the areas of triangles and rectangles” (p. 5), and “assess pre-service student teachers’ understanding of key concepts that determine formulae for the area of a cone …” (p. 6). Although such items (in particular the last few) could be SCK items, the information given in the text indicates that all of these items were mostly CCK items. For instance, the authors report that several students understood how the area of a cone could be represented by a certain formula, but most students “could not calculate the total surface area of a cone correctly” (p. 11). This indicates that the items were focusing on CCK more than on SCK.
cite:&vermeulen2017 also seem to include mostly CCK items. An example is (p. 144):
Alan started a problem with a one-digit number. He multiplied the number by 3, added 8, divided by 2 and subtracted 6, and got the same number he started with. What was the number Alan started with? (Show all working out)
a. 2 b. 4 c. 6 d. 8
| Level | No. |
|---|---|
| Primary | 12 |
| Middle school | 2 |
| Secondary | 5 |
| Tertiary | 1 |
| All | 4 |
| Teachers | No. |
| Future teachers | 8 |
| Practicing teachers | 13 |
| Experienced teachers | 3 |
Almost half of the studies were studies on what teachers know, and these studies tended to report on insufficient knowledge among teachers.
| Problem | No. |
|---|---|
| What teachers know? | 10 |
| What PD? | 5 |
| What TE? | 2 |
| What relationships? | 2 |
| What is MKT? | 2 |
| Contribute to student learning? | 1 |
| What contributes to practice? | 1 |
| How MKT develops? | 1 |
After having completed coding, I decided to consider what theoretical framework or foundations that were used in studies.
Below, I use 0 or 1 to indicate if studies have unclear or theoretical framework (on MKT), and I then use 1 to indicate if a particular framework is primary, and 2 to indicate if it is a secondary framework (mentioned).
| No. | Citation | Clear | MKT | COACTIV | TPACK | APOS | PCK | TEDS-M |
|---|---|---|---|---|---|---|---|---|
| 1 | cite:&alex2019 | 1 | 1 | 2 | ||||
| 2 | cite:&bansilal2014 | 1 | 2 | 1 | ||||
| 3 | cite:&bowie2019 | 1 | 1 | 2 | ||||
| 4 | cite:&chikiwa2019 | 1 | 1 | 2 | ||||
| 5 | cite:&defreitas2019 | 1 | 1 | 2 | ||||
| 6 | cite:&feza2016 | 0 | ||||||
| 7 | cite:&feza2018 | 0 | ||||||
| 8 | cite:&fonseca2015 | 0 | 2 | 2 | ||||
| 9 | cite:&fonseca2021 | 0 | ||||||
| 10 | cite:&helmbold2021 | 0 | ||||||
| 11 | cite:&jacinto2020 | 1 | 1 | 2 | ||||
| 12 | cite:&jita2019 | 0 | ||||||
| 13 | cite:&kalobo2016 | 0 | 3 | |||||
| 14 | cite:&makonye2020b | 1 | 2 | 1 | 2 | |||
| 15 | cite:&msimango2020 | 1 | 2 | 2 | 1 | 2 | ||
| 16 | cite:&mwadzaangati2019 | 0 | 2 | |||||
| 17 | cite:&pournara2015 | 0 | 2 | 2 | 2 | |||
| 18 | cite:&setoromo2018 | 1 | 1 | 2 | ||||
| 19 | cite:&sibanda2021 | 0 | ||||||
| 20 | cite:&siyepui2021 | 1 | 1 | 2 | ||||
| 21 | cite:&spangenberg2021 | 1 | 1 | 2 | ||||
| 22 | cite:&ubah2018 | 1 | 1 | |||||
| 23 | cite:&venkat2015 | 0 | ||||||
| 24 | cite:&vermeulen2017 | 1 | 1 | 2 |
In summary, 13 out of the 23 studies was specific about their theoretical framework, whereas 10 did not have a clear framework, or were unclear about how they conceptualized mathematical knowledge for teaching.
For those who presented a clear theoretical framework, below is an overview of the primary and secondary theoretical frameworks:
| Frameworks | Primary | Secondary |
|---|---|---|
| MKT | 8 | 6 |
| PCK | 1 | 11 |
| COACTIV | 1 | 2 |
| APOS | 2 | |
| TEDS-M | 2 | |
| TPACK | 1 |
In other words, the by far most studies in this literature review had the MKT framework of Ball et al. (2008) as their primary theoretical framework, and several also emphasized this framwork when presenting their theoretical framework, although it was not their primary framework. Only one study had PCK as a theoretical framework citep:&msimango2020, and they used the Clearly PCK framework by Chick et al. (2006), but almost half of the studies emphasized the conceptualization of PCK (mostly with reference to Shulman). Almost all of the studies that had MKT as a primary framework emphasized its connections with Shulman’s (1986, 1987) conceptualization of PCK. One study had COACTIV as their primary theoretical framework citep:&makonye2020b, whereas two referred to this. Two studies applied the APOS theory of Dubinsky (1991) as their primary theoretical framework citep:&bansilal2014;&ubah2018, whereas one used TPACK as their primary framework cite:&defreitas2019. Two studies gave prominence to the TEDS-M framework, but none of the studies in this review used it as a primary framework.
- Shulman (1986)
- Ball et al. (2008)
- Pasley (2011)
- Ball et al. (2008)
- Dubinsky, APOS theory (Trigueros & Martinez-Planell, 2010)
- Steinbring (1998)
- Shulman (1986)
- LMT, Ball et al. (2008)
- TEDS-M, Blomeke & Delaney (2012)
- Ma (1999)
- MKfT/MKT, Ball et al. (2008)
- Shulman (1986, 1987)
- TPACK, Mishra & Koehler (2006)
- Shulman (1986)
- Spaull & Kotze (2014)
- Venkat & Spaull (2015)
- Feza & Diko (2013)
- Venkat & Spaull (2015)
- MKT, Ball et al. (2008)
- Shulman (1986, 1987)
- Tatto et al. (2012)
- Quian & Youngs (2013)
- Chapman (2015)
- mathematics proficiency, Kilpatrick et al. (2001)
- Venkat & Spraull (2015) is references
- Ball et al. (2008)
- Shulman (1986)
- Venkat & Spraull (2015)
- Theory of peer observation process (Bell, 2002)
- Statistical literacy
- COACTIV, Baumert et al. (2010)
- Shulman (1986)
- Ball et al. (2008)
- Clearly PCK, Chick et al. (2006)
- TEDS-M, Tatto et al. (2008)
- Baumert et al. (2010)
- Ball et al. (2008)
- problem solving, Polya (1945)
- Stylianides & Ball (2008)
- Adler (2010)
Unclear conception of MKT, but literature review covers:
- Shulman (1986, 1987)
- Ball et al. (2008)
- Adler & Davis (2006)
- Krauss et al. (2008)
- etc.
- Shulman (1986)
- MKT, Ball et al. (2008)
- MKT, Ball et al. (2008)
- Shulman (1986, 1987)
- MKT, Hill et al. (2008)
- PCK, Shulman (1986, 1987)
- APOS theory, Dubinsky (1991)
- Mathematical practices
- MKT, Ball et al. (2008)
- Shulman (1987)
bibliographystyle:bath bibliography:mktafrica.bib