Worksheets are ubiquitous in early math education. Students practice a particular operation repeatedly until they acquire proficiency and fluency. Practice naturally leads to faster performance.
However, it is unclear whether this reflects a continual improvement in the same process, such as faster calculation, or if at least some of the speedup with practice is driven by qualitative changes in how problems are being solved. For example, after calculating 11 x 12 many times, you may become faster at executing the calculation, or you may just "know" that the answer is 132.
In a new paper published in the Journal of Experimental Psychology: Learning, Memory, and Cognition, Tenison and Anderson (2016) combine computational modeling, participant strategy reports, and neuroimaging to address this issue.
Participants were taught how to perform a novel mathematical operation, and then solved six blocks of problems. Three problems were frequently repeated within each block (practiced problems), while another 18 problems were only presented once per block (novel problems).
After answering a problem and receiving corrective feedback, participants were asked to select which of four strategies they used to solve the problem: retrieved the correct answer, calculated the correct answer, partially retrieved and partially calculated, or other.
Overall, problem-solving speed and accuracy increased for practiced problems, but not novel problems.
To more precisely characterize changes in performance over the course of learning, reaction times for practiced problems were modeled for each item and participant to estimate the likelihood that each trial belonged in one of three learning states.
Trials were then classified according to majority state. Although there was some speedup within each learning state, the majority of speed gains in performance reflected transitions between states.
Next, the authors compared the behavioral and neurological profile of each state. Participants primarily reported using calculation to solve problems classified as State 1, and retrieval to solve problems classified as State 2.
This self-reported shift from calculation to retrieval is consistent with the fact that item difficulty only modulated performance in State 1 — difficult items require more effortful calculation — but this should not influence how long it takes to retrieve the correct answer from memory.
Moreover, there was greater brain activity in regions supporting mathematical computation (left horizontal intraparietal sulcus) for State 1 vs. State 2 items.
Behaviorally, performance for State 2 and State 3 items was very similar: participants primarily reported using retrieval strategies, and item difficulty did not influence performance. However, the neuroimaging data showed decreased activity in a brain region associated with effortful retrieval (left lateral inferior prefrontal cortex), and increased activity in a brain region associated with response execution (left motor cortex), for State 3 vs. State 2 items.
Together, these results paint a picture of how the processes guiding math performance change over the course of learning: Performance is initially driven by effortful calculation, which can eventually be bypassed when the correct answer can be retrieved from memory.
With enough practice, this explicit memory retrieval process becomes automatized, requiring less cognitive engagement and effectively turning problem-solving into a rapid stimulus-response task.
Citation:
- Tenison, C., & Anderson, J. R. (2016). Modeling the distinct phases of skill acquisition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 42(5), 749–767. https://dx.doi.org/10.1037/xlm0000204
Note: This article is in the Basic / Experimental Psychology topic area. View more articles in the Basic / Experimental Psychology topic area.