Quality of teaching mathematics and learning achievement gains: evidence from primary schools in Kenya

Abstract

This paper examines the contribution of quality mathematics teaching to student achievement gains. Quality of mathematics teaching is assessed through teacher demonstration of the five strands of mathematical proficiency, the level of cognitive task demands, and teacher mathematical knowledge. Data is based on 1907 grade 6 students who sat for the same test twice over an interval of about 10 months. The students were drawn from a random selection of 72 low- and high-performing primary schools. Multi-level regression shows the effects of quality mathematics teaching at both individual and school levels, while controlling for other variables that influence achievement. Results show that students in low-performing schools gained more by 6 % when mathematics instruction involved high-level cognitive task demands, with two thirds of all the lessons observed demonstrating the strands of mathematics proficiency during instruction. The implication to education is that quality of mathematics instruction is more critical in improving learning gains among low-performing students.

Appendices

Appendix 1: sample items

• (a) Pedagogical knowledge item (teacher test)

  

• (b) Pedagogical content knowledge item (teacher test)

  Mr Godana is teaching his grade 6 class about the relative sizes of fractions.

  He tells them a story about a birthday party where Namwamba eats ½ of one cake, and Nyagaka eats 5/8 of another cake of the same size. Who ate the most? Which of the following children is correct?

  • A. Salim says eighths are very small pieces because there are so many, but halves are bigger because there are only two. Therefore, Namwamba ate the most because halves are bigger than eighths.

  • B. Lawrence says Nyagaka ate the most because there are 5 eighths and only 1 half.

  • C. Mariam says ½ is the same as 4/8 which is smaller than 5/8. Therefore, Nyagaka ate the most.

  • D. Jane says that Namwamba and Nyagaka ate the same amount of cake.

• (c) Examples of classroom mathematics tasks classified as having weak, moderate, and strong cognitive demand.

  The questions were asked by the teacher during instruction.

  • Weak

    • 1. 12 is less than 16, isn’t it?

    • 2. 1 take away 1?

    • 3. 10 take away 5?

  • Moderate

    • 4. What is the area of the rectangle?

    • 5. Which number do you multiply with 16 to get 128?

    • 6. So we have 17 cm 6 mm, this is equivalent to how many millimeters?

  • Strong

    • 7. 50 m by 180 m into hectares, what is the first step?

    • 8. These are two oranges divided by 3. How many thirds are there?

    • 9. What time will it be at 5 pm in 12 h in the 24 h clock?

• (d) Examples of student mathematics test items and levels of cognitive demand.

  

Appendix 2: model specification

Following the framework shown in Fig. 1, we estimated multi-level linear regression following the value addition models derived from a basic educational production function (EPF). In theory, student learning achievement is determined by an EPF:

$$ A=f\left(H,\;I,\;S,\;\alpha \right) $$

(1)

where achievement A is a product of home or social economic background (H), individual characteristics (I), school resource inputs (S), and an efficiency parameter measuring capacity utilization in the school (α) (Marshall, 2009). This general EPF does not specify the effects of levels of the determinants of learning achievement. Showing the effects levels is relevant to policy since it enhances the understanding of the learning achievement dynamics. According to Glewwe (2002), if the independent variables do not change much over time, the analysis of levels will return similar results to that of a general EPF. Three models are estimated: (i) the overall model that includes all schools, (ii) one model for the high-performing schools, and (iii) one model for the low-performing schools.

The multivariate model assumes that all students have the same or varying number of repeated IRT measurements taken at identical points in time (Verbeke & Molenberghs, 2000). In the analysis, we consider the repeated IRT measurements for all the students and schools computed from the same mathematics test administered in rounds 1and 2 over an interval of 10 months. Let y _ij1 and y _ij2 be the IRT scores for rounds 1 and 2 for the jth student in the ith school where j = 1, 2, …n _i and i = 1,2,… N. The two IRT scores can be grouped together in a vector y _ij  = [y _ij1, y _ij2]. The student’s scores y _ij in the ith school can be clustered into a vector ${\mathbf{Y}}_i=\left[{\mathbf{y}}_{i1},{\mathbf{y}}_{i2},\dots .,{\mathbf{y}}_{i{n}_i}\right]$ i = 1,2,… N. The general multivariate model assumes that the repeated measurements in Y _i satisfy a regression model given by:

$$ {\mathbf{Y}}_i={\mathbf{X}}_i\boldsymbol{\upbeta} +{\boldsymbol{\upvarepsilon}}_i\;i=1,2,\dots N $$

(2)

where ε _i is a vector of error components and ε _i  ∼ N (0,∑). The response vector for the ith student Y _i has a multivariate normal density

$$ {\mathbf{Y}}_i\sim N\left({\mathbf{X}}_i\boldsymbol{\upbeta}, {\displaystyle \sum}\right) $$

where β is a vector of fixed effects and ∑ is the covariance matrix. Since the study has two time points (rounds 1 and 2), we adopt an unstructured covariance matrix for the covariance structure∑. The unstructured covariance matrix offers the most generalized structure that does not assume any prior knowledge of the relationship between the variables of interest.

Appendix 3

ᅟ

Table 3 Mean IRT score gain based on student, teacher, and school characteristics

Appendix 4

ᅟ

Table 4 Multiple linear regression model based on all schools and top and bottom performance schools