Towards curricular coherence in integers and fractions: a study of the efficacy of a lesson sequence that uses the number line as the principal representational context

Abstract

This paper reports a study of the efficacy of Learning Mathematics through Representations (LMR), an innovative curriculum unit designed to support upper elementary students’ understandings of integers and fractions. The unit supports an integrated treatment of integers and fractions through (a) the use of the number line as a cross-domain representational context, and (b) the building of mathematical definitions in classroom communities that become resources to support student argumentation, generalization, and problem solving. In the efficacy study, fourth and fifth grade teachers employing the same district curriculum (Everyday Mathematics) were matched on background indicators and then assigned to either the LMR experimental classrooms (n = 11) or the comparison group (n = 8 with 10 classrooms). During the fall semester, LMR teachers implemented the LMR unit on 19 days and district curriculum on other days of mathematics instruction. HLM analyses documented greater achievement for LMR students than Comparison students on both the end-of-unit and the end-of year assessments of integers and fractions knowledge; the growth rates of LMR students were similar regardless of entering ability level, and gains for LMR students occurred on item types that included number line representations and those that did not. The findings point to the efficacy of the LMR sequence in supporting teaching and learning in the domains of integers and fractions.

Appendix

1.1 Measurement model

The measurement model used to obtain student ability estimates is given by:

$$ {\text{logit}}\left( {X_{sti} = 1\left| {{\varvec{\theta}},{\varvec{\delta}}} \right.} \right) = \theta_{st} - \delta_{i} $$

where $X_{sti}$ indicates the response of student s at time t is for item i, $\theta_{st}$ is the person and time specific ability, and $\delta_{i}$ is the item difficulty. The $\varvec{\theta}_{s}$ are assumed to follow a multivariate normal distribution with unrestricted covariance matrix. All measurement models were fit using ConQuest (Wu, Adams, Wilson, & Haldane, 2007). Table 5 contains the estimated correlation matrix between the assessments at the four occasions for the overall scale.

Table 5 Correlations between the assessments (from the estimated covariance matrix of the item response model)

1.2 Longitudinal growth models

Three longitudinal growth models were estimated using Stata12 (StataCorp. 2011) and graphed using R (R Development Corp Team, 2012).

1.2.1 Model 1

The first model, used for the overall comparison between LMR and Comparison students, is specified in three levels as follows:

Level 1:

$$ {\text{wle}}_{tsc} = \pi_{0sc} + \pi_{1sc} {\text{inter}}_{tsc} + \pi_{2sc} {\text{post}}_{tsc} + \pi_{3sc} {\text{final}}_{tsc} + \varepsilon_{tsc} $$

Level 2:

$$ \begin{aligned} \pi_{psc} &= \beta_{p0c} + \zeta_{psc} \quad {\text{for}}\quad p = 0,2,3 \\ \pi_{1sc} &= \beta_{10c} \\ \end{aligned} $$

Level 3:

$$ \begin{aligned} \beta_{00c} &= \gamma_{000} + \gamma_{001} {\text{comp}}_{c} + \zeta_{c} \\ \beta_{10c} & = \gamma_{100} \\ \beta_{p0c} & = \gamma_{p00} + \gamma_{p01} {\text{comp}}_{c} \quad {\text{for}}\quad p = 2,3 \\ \end{aligned} $$

$wle_{tsc}$ is the WLE estimate of overall student ability for student s in classroom c at time t and $inter_{tsc}$, $post_{tsc}$, and $final_{tsc}$ are indicators for test occasion. $comp_{c}$ is an indicator for classroom c being in the comparison group. $\zeta_{c}$ is a random classroom-level effect assumed to follow a normal distribution. The $\zeta_{sc}$ are student-level random effects assumed to follow a multivariate normal distribution with unrestricted covariance matrix.⁷ $\varepsilon_{tsc}$ is a time-specific error assumed to follow a normal distribution.

Table 6 gives the estimated variance components from fitting this model to the data. Table 1 in the text gives a subset of the estimated parameters (and their linear combinations). Figure 9 was drawn using estimates from this model.

Table 6 Estimated variance components from Model 1

1.2.2 Model 2

The second model extends the first model to include ability estimates for all subscales as the response variable and indicator variables for the four subscales (with the overall scale as the reference category), as follows:

Level 1:

$$ \begin{aligned} {\text{wle}}_{tsc} &= \pi_{0sc} + \pi_{1sc} {\text{inter}}_{tsc} + \pi_{2sc} {\text{post}}_{tsc} + \pi_{3sc} {\text{final}}_{tsc} \\ & \quad + \pi_{4sc} {\text{integers}}_{tsc} + \pi_{5sc} {\text{fractions}}_{tsc} + \pi_{6sc} {\text{line}}_{tsc} + \pi_{7sc} {\text{noline}}_{tsc} \\ & \quad + \pi_{8sc} {\text{inter}}_{tsc} \times {\text{integers}}_{tsc} + \pi_{9sc} {\text{inter}}_{tsc} \times {\text{fractions}}_{tsc} + \pi_{10sc} {\text{inter}}_{tsc} \times {\text{line}}_{tsc} + \pi_{11sc} {\text{inter}}_{tsc} \times {\text{noline}}_{tsc} \\ & \quad + \pi_{12sc} {\text{post}}_{tsc} \times {\text{integers}}_{tsc} + \pi_{13sc} {\text{post}}_{tsc} \times {\text{fractions}}_{tsc} + \pi_{14sc} {\text{post}}_{tsc} \times {\text{line}}_{tsc} + \pi_{15sc} {\text{post}}_{tsc} \times {\text{noline}}_{tsc} \\ & \quad + \pi_{16sc} {\text{final}}_{tsc} \times {\text{integers}}_{tsc} + \pi_{17sc} {\text{final}}_{tsc} \times {\text{fractions}}_{tsc} + \pi_{18sc} {\text{final}}_{tsc} \times {\text{line}}_{tsc} + \pi_{19sc} {\text{final}}_{tsc} \times {\text{noline}}_{tsc} \\ & \quad + \varepsilon_{tsc} \\ \end{aligned} $$

Level 2:

$$ \begin{aligned} \pi_{psc} &= \beta_{p0c} + \zeta_{psc} \quad {\text{for}}\quad p = 0,1,2,3 \\ \pi_{psc} & = \beta_{p0c} \quad \quad \quad \;\;{\text{for}}\quad p = 4,5, \ldots ,19 \\ \end{aligned} $$

Level 3:

$$ \begin{aligned} \beta_{00c} &= \gamma_{000} + \gamma_{001} {\text{comp}}_{c} + \zeta_{c} \\ \beta_{p0c} &= \gamma_{p00} \quad \quad \quad \quad \quad \quad \quad {\text{for}}\quad p = 1,8,9,10,11 \\ \beta_{p0c} & =\gamma_{p00} + \gamma_{p01} {\text{comp}}_{c} \quad \quad {\text{for}}\quad p = 2, \ldots ,7,12, \ldots ,19 \\ \end{aligned} $$

$wle_{tsc}$ is the WLE estimate of overall student ability for student s in classroom c at time t, $inter_{tsc}$, $post_{tsc}$, and $final_{tsc}$ are indicators for test occasion, and $integers_{tsc}$, $fractions_{tsc}$, $line_{tsc}$, and $noline_{tsc}$ are indicators for subscale. $comp_{c}$ is an indicator for classroom c being in the comparison group. $\zeta_{c}$ is a random classroom-level effect assumed to follow a normal distribution. The student-level random effects $\zeta_{sc}$ are assumed to follow a multivariate normal distribution with unrestricted covariance matrix. $\varepsilon_{tsc}$ is a time-specific error assumed to follow a normal distribution.

Table 7 presents the estimated variance components from fitting this model to the data. Tables 2, 3, and 4 (in the text) contain a subset of the estimated parameters (and their linear combinations). Figures 11 and 13 are drawn using estimates from this model.

Table 7 Estimated variance components from Model 2

1.2.3 Model 3

The third model extends the first model to include indicator variables for student pretest score group (with the low group as the reference category), as follows:

Level 1:

$$ {\text{wle}}_{tsc} = \pi_{0sc} + \pi_{1sc} {\text{inter}}_{tsc} + \pi_{2sc} {\text{post}}_{tsc} + \pi_{3sc} {\text{final}}_{tsc} + \varepsilon_{tsc} $$

Level 2:

$$ \begin{aligned} \pi_{psc} &= \beta_{p0c} + \beta_{p1c} {\text{med}}_{sc} + \beta_{p2c} {\text{high}}_{sc} + \zeta_{psc} \quad {\text{for}}\quad p = 0,2,3 \\ \pi_{1sc} &= \beta_{10c} + \beta_{p1c} {\text{med}}_{sc} + \beta_{p2c} {\text{high}}_{sc} \\ \end{aligned} $$

Level 3:

$$ \begin{aligned} \beta_{00c} &= \gamma_{000} + \gamma_{001} {\text{comp}}_{c} + \zeta_{c} \\ \beta_{0qc} &= \gamma_{0q0} + \gamma_{0q1} {\text{comp}}_{c} \quad \quad {\kern 1pt} {\text{for}}\quad q = 1,2 \\ \beta_{1qc} & = \gamma_{1q0} \quad \quad \quad \quad \quad \quad \quad \;{\text{for}}\quad q = 0,1,2 \\ \beta_{pqc} &= \gamma_{pq0} + \gamma_{pq1} {\text{comp}}_{c} \quad \quad {\text{for}}\quad p = 2,3\;and\;q = 0,1,2 \\ \end{aligned} $$

$wle_{tsc}$ is the WLE estimate of student achievement on the scale of interest for student s in classroom c at time t; this model was fit using the overall subscale and then the integers and fractions subscales as the dependent variable. $inter_{tsc}$, $post_{tsc}$, and $final_{tsc}$ are indicators for test occasion. $med_{sc}$ and $high_{sc}$ are indicator variables for a student being in the medium or high pretest score group, respectively. $comp_{c}$ is an indicator for classroom c being in the comparison group. $\zeta_{c}$ is a random classroom-level effect assumed to follow a normal distribution. The $\zeta_{sc}$ are student-level random effects assumed to follow a multivariate normal distribution with unrestricted covariance matrix.⁸ $\varepsilon_{tsc}$ is a time-specific error assumed to follow a normal distribution.

Table 8 gives the estimated variance components from fitting this model to the data. Figures 10 and 12 are drawn using estimates from this model.

Table 8 Estimated variance components from Model 3 for different scales as dependent variable