Mathematical learning models that depend on prior knowledge and instructional strategies

We present mathematical learning models—predictions of student’s knowledge vs amount of instruction—that are based on assumptions motivated by various theories of learning: tabula rasa, constructivist, and tutoring. These models predict the improvement (on the post-test) as a function of the pretest score due to intervening instruction and also depend on the type of instruction. We introduce a connectedness model whose connectedness parameter measures the degree to which the rate of learning is proportional to prior knowledge. Over a wide range of pretest scores on standard tests of introductory physics concepts, it fits high-quality data nearly within error. We suggest that data from MIT have low connectedness (indicating memory-based learning) because the test used the same context and representation as the instruction and that more connected data from the University of Minnesota resulted from instruction in a different representation from the test.

Figure 1

(Color) Predictions of knowledge $K\left(t\right)$ from our three “pure” models. We have assumed that $k_a={\alpha }_{\mathit{\text{memory}}}={\alpha }_{\mathit{\text{connected}}}∕2$, which matches the learning rates of simple connected and pure memory models at $t=0$ (where $K_{T0}=0.5$). The learning rate is proportional to the slope. Thus, the tutoring model educates fastest everywhere, even for $t<0$ (i.e., before the pretest). The simple connected-external model (not shown) has the same time dependence as the pure memory model (although it differently scales depending on $K_{\mathit{\text{external}}}$).

Figure 2

(Color) Fits of pure memory, connectedness, and simple connected-external models to the normalized gain of (A) the MIT 2003 class, (B) MIT 2005 class, and (C) UMn 1997-1999 classes. The tutoring model is shown in (B) to illustrate its positive curvature in a normalized gain plot. For these plots, each data set was divided into bins of equal spacing except for two inclusive bins at the extremes of the pretest range (hence the slightly wider separation of the edge points), and the average and STD of the mean of the normalized gains of the students in each bin were plotted against their average pretest score. ${\alpha }_{\mathit{\text{memory}}}{t}_{\mathrm{instr}}$ and ${\alpha }_{\mathit{connted}}{t}_{\mathit{instr}}$ were free parameters for the pure memory and the simple connected model fit. In the connectedness model fit, $\beta$ was determined by extrapolating between pure memory and the simple connected model. Results of fits are in Table .