Arithmetical fluency project

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Arithmetical Fluency Project

PI: Julie Fiez

1. Abstract

This project examines the neural mechanisms that support the use of drill-based training to yield robust mathematical learning. We believe that such training will be most effective when the controlled retrieval and manipulation of symbolic information is made relatively inaccessible during practice. Under such conditions, we predict that basic representations of numerical magnitude in inferior parietal cortex will become more refined and accessible, through a reinforcement-based learning process that is mediated by the basal ganglia. Because basic representation of number magnitude is a component of both simple and complex mathematical tasks, significant and long-lasting benefits are expected on tasks that require far transfer of the trained skill. This work should lay the groundwork for using known properties of basal ganglia function to optimize drill-based training regimes. More generally, it has the potential to illustrate how effective bridges can be built between neuroscience and education.

2. Glossary

basal ganglia

inferior parietal cortex

refinement of representations

distance effect


3. Research question

Does drill-based training on multi-operand addition and subtraction problems lead to the refinement, via mediation of basal ganglia reinforcement learning mechanisms, of “number sense” representations in inferior parietal cortex?

4. Background and Significance

Nine years ago, John Bruer authored a critical and widely-cited commentary on the relationship between neuroscience and education (Bruer, 1997). This article performed a tremendous service by highlighting the dangers in naïvely and prematurely using neuroscience data to justify curricular design (e..g, “brain-based education”). Bruer ended his commentary by stating, “Looking to the future, we should attempt to develop an interactive, recursive relationship among research programs in education, cognitive psychology, and systems neuroscience. Such interaction would allow us to extend and apply our knowledge of how mind and brain support learning.” The aim of this project is to lay the groundwork for such a research program in the domain of mathematics. Our overarching hypothesis is that most drill-based techniques engage a reinforcement learning system that is supported by the basal ganglia. If this is the case, then properties of basal ganglia function that have been discovered through systems neuroscience and cognitive neuroscience research should provide a novel perspective on: 1) a mechanistic account of why drill-based approaches are effective in developing fluency in basic skills, 2) why improvement at the basic level can enhance higher-level abilities, and 3) how drill-based approaches– used in isolation, and as embedded components of tutoring software -- might be optimized to best take advantage of this learning mechanism. To us, this latter objective would truly demonstrate the type of recursive relationship between neuroscience and education that was envisioned by Bruer.

5. Dependent measures

a. Behavioral measures to assess robustness of learning. We will acquire pre vs. post training behavioral data on a number of tasks designed to measure immediate, intermediate, and far transfer. They are described below:

Assessment of immediate transfer: Subjects will be given a page containing multi-operand addition and subtraction problems. We consider this a measure of immediate transfer because it provides a pen-and-paper equivalent to the training task. We will measure overall solution time and accuracy.

Assessment of intermediate transfer: Subjects will be given a set of complex math problems to solve. We consider this to be a form of intermediate transfer because addition and subtraction could be considered a component skill of complex problem solving. We will measure overall solution time and accuracy.

Assessment of far transfer: We will perform a set of problems in which they have to determine which of two Arabic numerals, point of line bisection, or display of dots is larger. We consider this to be an assessment of far transfer because it does not involve addition or subtraction. Our primary dependent measure will be reaction time.

Assessment of accelerated learning: We are investigating the feasibility of using the Algebra Tutor to assess whether future learning is accelerated.

b. Neuroimaging measures to assess mechanisms of learning

We will conduct a 1-hour imaging study the day before training and the day after training.

Assessment of training task: Subjects will complete 60 two-operand/single-operand addition problems. They will have 3 s to respond on each trial, then feedback will be delivered and a 15-s fixation interval would be used to allow the blood-flow response to decay back to baseline. Our dependent measure will be the BOLD signal evoked on each trial.

Assessment of adaptation effects: Subjects will be shown sequences of dot displays. For 12 blocks the sequences will contain dots with an identical quantity of items, for 12 blocks the quantity will vary by one above or below a central magnitude, and for 12 blocks the quantity will vary by three above or below a central magnitude. Our dependent measure will be the BOLD signal evoked on each trial.

6. Independent variables

During training we will manipulate problem difficulty by varying the number of operands in addition and subtraction problems. Across a variety of learning assessments, our critical variable will be pre vs. post-test assessment. Within our measures of magnitude estimation, a critical parameter will be the distance between two items of comparison (e.g., two Arabic numbers); this parameter provides a measure of “distance effects,” which we hypothesize will be reduced as basic number representations become more refined and accessible.

7. Hypotheses

a. Behavioral evidence for robust learning. We predict that we will replicate prior demonstrations of immediate and near transfer from drill-based training. We hypothesize that we will also see long-lasting transfer to our magnitude comparison tasks.

b. Neural mechanisms. We predict that the striatum and the inferior parietal cortex will be active during our training task and that the response in each of these regions will be different for correct versus incorrect trials. Following training, we expect to see reduced activation in both the striatum and parietal cortex.

We hypothesize that prior to training subjects should show reduced parietal activation for sequences in which the numerical quantity of presented items is identical or only slighly varying, relative to sequences in which there is a large amount of numerical variation. Following training, we reason that the distributed representations of two quantities that are similar in magnitude become more precisely tuned – that is, there is less overlap between the representations. If this is the case, we should see similar amounts of adaptation for sequences with identical quantities of items, but less adaptation for sequences with slight variations in item quantities.

8. Findings

We aim to commence this work in October 2006. Preliminary behavioral investigations provide strong support that our behavioral hypotheses will be confirmed.

9. Explanation

We believe that we will see evidence for far transfer behaviorally, because training will refine the underlying neural representations that support performance on nearly any task that involves evaluation or computation of magnitude. This is accord with neuroimaging findings that have shown activation of inferior parietal cortex across a wide range of mathematical tasks. Additionally, we believe that the use of speeded responding and monetary reinforcement will maximize basal ganglia mediated learning and minimize the accessibility of information from power, but slower and more effortful symbolic manipulation systems.

The predicted patterns of activation during our training task would reflect central involvement of the striatum in many different forms of feedback-based learning, coupled with cortical targets of feedback that are dependent upon the locus of representations that underlie accurate task performance. The predicted changes on our magnitude comparison task would be consistent with our idea that fluent and nearly automic performance arises from representations that have become more refined and accessible through practice.

10. Descendents

11. Further information

Bruer JT (1997). Education and the brain: A bridge too far. Educational Researcher, November, 4-16.

Haverty LA (1999). The Importance of Basic Number Knowledge to Advanced Mathematical Problem Solving. Doctoral Dissertation published in Doctoral Abstracts International.

Dehaene S, Molko N, Cohen L, Wilson AJ (2004). Arithmetic and the brain. Current Opinions in Neurobiology, 14:218-224.