Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress
Abstract
In both the United States and Europe, concerns have been raised about whether preservice and in-service training succeeds in equipping teachers with the professional knowledge they need to deliver consistently high-quality instruction. This article investigates the significance of teachers’ content knowledge and pedagogical content knowledge for high-quality instruction and student progress in secondary-level mathematics. It reports findings from a 1-year study conducted in Germany with a representative sample of Grade 10 classes and their mathematics teachers. Teachers’ pedagogical content knowledge was theoretically and empirically distinguishable from their content knowledge. Multilevel structural equation models revealed a substantial positive effect of pedagogical content knowledge on students’ learning gains that was mediated by the provision of cognitive activation and individual learning support.
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Figure and Tables
Figure 1 Hierarchical linear model to test the significance of teachers’ pedagogical content knowledge (PCK) and content knowledge (CK) for instructional quality and student learning.
Note. SES = socioeconomic status; - - - = no relation expected.
| Variables | Total | Nonacademic Tracks | Academic Track | Δtracks | ||||
|---|---|---|---|---|---|---|---|---|
| M | SD | M | SD | M | SD | t | p | |
| Mathematics achievement, end of Grade 10 | 0.05 | 0.98 | –0.31 | 0.90 | 0.53 | 0.86 | –30.3 | .000 |
| Mathematical literacy, end of Grade 9 | 0.04 | 0.97 | –0.32 | 0.92 | 0.53 | 0.87 | –31.6 | .000 |
| Reading literacy, end of Grade 9 | 0.06 | 0.96 | –0.25 | 0.92 | 0.47 | 0.85 | –19.3 | .000 |
| Mental ability, end of Grade 9 | 0.05 | 0.96 | –0.22 | 0.96 | 0.43 | 0.81 | –23.4 | .000 |
| Age, years | 15.7 | 0.66 | 15.8 | 0.71 | 15.6 | 0.67 | 10.4 | .000 |
| Socioeconomic statusa | 53.4 | 16.0 | 48.6 | 14.6 | 59.9 | 15.4 | –24.4 | .000 |
| % | % | % | χ2, (df = 1) | p | ||||
| Parents with university degree | 32.3 | 19.1 | 50.7 | 40.9 | .000 | |||
| Immigrant background | 19.3 | 21.4 | 16.6 | 5.7 | .000 | |||
a
As operationalized by the International Socio-Economic Index. Specifically, the analysis used the higher-status occupation between the two parents.
| Constructs and Variables | Total | Nonacademic Tracks | Academic Track | Δtracks | ||||
|---|---|---|---|---|---|---|---|---|
| Ma | SD | Ma | SD | Ma | SD | t | p | |
| Class Level | ||||||||
| Cognitive level of tasks | ||||||||
| Type of mathematical taskb | 1.58 | 0.25 | 1.57 | 0.24 | 1.59 | 0.26 | –0.68 | .50 |
| Level of mathematical argumentationc | 0.07 | 0.10 | 0.04 | 0.06 | 0.12 | 0.13 | –5.50 | .000 |
| Innermathematical translationc | 0.38 | 0.30 | 0.38 | 0.34 | 0.38 | 0.30 | 0.02 | .98 |
| Curricular level of tasks | ||||||||
| Alignment to grade 10 curriculumb | 2.72 | 0.19 | 2.68 | 0.20 | 2.78 | 0.15 | –3.38 | .000 |
| Individual learning support | ||||||||
| Adaptive explanationsd | 2.82 | 0.43 | 2.84 | 0.43 | 2.78 | 0.44 | 0.95 | .34 |
| Constructive response to errorsd | 2.94 | 0.44 | 2.91 | 0.44 | 2.97 | 0.43 | –0.84 | .40 |
| Patienced | 2.76 | 0.53 | 2.76 | 0.50 | 2.75 | 0.58 | 0.07 | .95 |
| Adaptive pacingd | 2.20 | 0.44 | 2.20 | 0.44 | 2.21 | 0.44 | –0.21 | .84 |
| Respectful treatment of studentsd | 3.22 | 0.49 | 3.16 | 0.50 | 3.31 | 0.47 | –2.10 | .04 |
| Caring ethosd | 2.66 | 0.50 | 2.69 | 0.50 | 2.61 | 0.48 | 1.20 | .23 |
| Quality of classroom management | ||||||||
| Prevention of disruptiond | 2.53 | 0.62 | 2.52 | 0.61 | 2.56 | 0.63 | –0.47 | .64 |
| Effective use of timed | 2.66 | 0.58 | 2.66 | 0.56 | 2.65 | 0.60 | 0.21 | .84 |
| Teacher Level | ||||||||
| Age (in years) | 48.15 | 8.0 | 48.51 | 7.8 | 47.46 | 8.4 | 0.79 | .43 |
| Gender (male) | 52.1% | 49.5% | 57.1% | χ2 = 0.85 | .36 | |||
| Years of service | 22.1 | 9.4 | 22.7 | 9.5 | 20.9 | 9.3 | 1.13 | .26 |
| Grade point averagee | Mdn = 2.6 | Mdn = 2.8 | Mdn = 2.4 | 5.66 | .35 | |||
a
Unless noted otherwise.
b
1 = low, 3 = high.
c
0 = low, 3 = high.
d
1 = low, 4 = high.
e
1 = highest, 6 = lowest.
| Variables | Total | Type 1: Academic | Type 2: Nonacademic | Type 3: Integrateda | Δ(Type 1 – Type 2) | Δ(Type 1 – Type 3) | Δ(Type 2 – Type 3) |
|---|---|---|---|---|---|---|---|
| M (SD) | M (SD) | M (SD) | M (SD) | t (p) | t (p) | t (p) | |
| CK | –0.139 (0.99) | 0.737 (0.91) | –0.524 (0.82) | –0.446 (0.73) | 7.537 (.000) | 6.099 (.000) | –0.465 (.643) |
| PCK | –0.027 (0.99) | 0.427 (1.10) | –0.007 (0.83) | –0.596 (0.77) | 2.301 (.023) | 4.442 (.000) | 3.329 (.001) |
| PCK|CKb | –0.083 | –0.097 | 0.256 | –0.407 | –1.909 (.058) | 1.531 (.128) | 4.064 (.000) |
Note. Excluding the 14 teachers who completed a 5-year training program in the former German Democratic Republic. CK = content knowledge; PCK = pedagogical content knowledge.
a
Training in former German Democratic Republic.
b
PCK estimated after controlling for CK; PCK estimated at CK = −0.139.
| Variables | Total | Nonacademic Tracks | Academic Track | Δtracks | ||||
|---|---|---|---|---|---|---|---|---|
| M | SD | M | SD | M | SD | t | p | |
| CK | –0.11 | 1.00 | –0.58 | 0.84 | 0.73 | 0.68 | –10.2 | .000 |
| PCK | ||||||||
| Tasks | –0.04 | 0.97 | –0.24 | 0.89 | 0.31 | 1.00 | –3.46 | .001 |
| Students | –0.02 | 1.00 | –0.24 | 0.98 | 0.45 | 0.86 | 4.68 | .000 |
| Instruction | –0.03 | 0.97 | –0.31 | 0.93 | 0.48 | 0.85 | –5.37 | .000 |
| Overall | –0.05 | 0.98 | –0.33 | 0.96 | 0.49 | 0.76 | –5.65 | .000 |
Note. CK = content knowledge; PCK = pedagogical content knowledge.
| Predictor/Parameter | Individual | Dimensions of Instructional Quality | Black Box (PCK) | Mediation (PCK) | Black Box (CK) | Mediation (CK) | ||
|---|---|---|---|---|---|---|---|---|
| Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | Model 7 | Model 8 | |
| Individual Levela | ||||||||
| Mathematics achievement, end of Grade 10 on: | ||||||||
| Prior knowledge of mathematics, end of Grade 9 | .49 | .49 | .49 | .49 | .49 | .48 | .49 | .49 |
| Reading literacy, end of Grade 9 | .21 | .21 | .20 | .21 | .20 | .20 | .20 | .20 |
| Mental ability, end of Grade 9 | .24 | .23 | .24 | .24 | .24 | .24 | .24 | .24 |
| Socioeconomic statusb | .01 | .00 | .00 | .00 | .00 | –.01 | –.01 | –.01 |
| Parental education (6 dummies) | ||||||||
| Hauptschule, no apprenticeship (0/1) | –.02 | –.01 | –.02 | –.02 | –.02 | –.02 | –.02 | –.02 |
| Hauptschule and apprenticeship (0/1) | –.02 | –.02 | –.02 | –.02 | –.02 | –.02 | –.02 | –.02 |
| Realschule and apprenticeship (0/1) | –.03 | –.03 | –.03 | –.03 | –.03 | –.03 | –.03 | –.03 |
| Realschule and professional training (reference) | ||||||||
| Gymnasium (0/1) | –.02 | –.02 | –.02 | –.02 | –.02 | –.02 | –.02 | –.02 |
| University (0/1) | .03 | .04 | .02 | .03 | .02 | .02 | .02 | .02 |
| Immigration status (0/1) | –.03 | –.03 | –.03 | –.03 | –.03 | –.03 | –.03 | –.03 |
| R2 | .64 | .64 | .62 | .62 | .62 | .62 | .62 | .62 |
| Class Level | ||||||||
| Mathematics achievement, end of Grade 10 on: | ||||||||
| Track (nonacademic/academic) | .58 | .42 | .56 | .42 | .60 | |||
| PCK (CK) | .62 | .42 | .30 | |||||
| Cognitive level of tasks | .32 | .32 | .32 | .31 | ||||
| Curricular level of tasks | .30 | .17 | .17 | .18 | ||||
| Individual learning support | .08 | .11 | .10 | .10 | ||||
| Effective classroom management | .33 | .30 | .31 | .32 | ||||
| Cognitive level (MV1) on PCK (CK) | .24 | .01 | ||||||
| Curricular level (MV2) on PCK (CK) | .33 | .32 | ||||||
| Individual learning support (MV4) on PCK (CK) | .26 | –.06 | ||||||
| Effective classroom management (MV3) on PCK (CK) | .14 | –.01 | ||||||
| R2 | .37 | .68 | .39 | .54 | .69 | .44 | .65 | |
| χ2 | 25.4 | 787 | 819 | 31.8 | 33.6 | 882 | 31.5 | 874 |
| df | 9 | 190 | 200 | 15 | 18 | 253 | 14 | 240 |
| p | .003 | .00 | .00 | .007 | .07 | .00 | .005 | .00 |
| Comparative fit index | .99 | .97 | .97 | .99 | .99 | .96 | .99 | .96 |
| Root mean square error of approximation | .02 | .03 | .03 | .02 | .01 | .02 | .02 | .03 |
| Standardized root mean square residual between | .001 | .04 | .04 | .06 | .01 | .08 | .01 | .09 |
| Standardized root mean square residual within | .005 | .03 | .03 | .005 | .004 | .02 | .004 | .02 |
Note. Multilevel structural equation models with latent variables, completely standardized solutions; coefficients in bold are significant at least at the 5% level. CK = content knowledge; PCK = pedagogical content knowledge; MV = mediator variable.
a
All individual variables centered at the grand mean.
b
As operationalized by the International Socio-Economic Index. Specifically, the analysis used the higher-status occupation between the two parents.
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Article first published online: June 5, 2017
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