What Neuroscience can tell us about Arithmetic Success
“Then, little by little, one may introduce [students] to the power of symbolic mathematical notation and the shortcuts it provides — but at this stage, great care should be taken never to divorce such symbolic knowledge from the child’s quantitative intuitions.” -Stanislas Dehaene
In The Number Sense, Dehaene explains that many animals exhibit an ability to approximate numerical quantities. This ability is also present in human infants. It is quite likely that the ability to approximate numerical quantities is biologically primary knowledge and is part of our evolutionary growth as a species. Of interest in mathematics education is how students begin to “fine-tune” this approximate system and develop symbolic understanding of the Arabic numeral system.
Dr. Daniel Ansari has been a part of several studies exploring these ideas as they pertain to neurocognitive architecture and arithmetic fluency. He describes arithmetic fluency as “the speed and efficiency with which correct solutions to numerical computations are generated.” The first study I wish to discuss looked at numerical predictors of arithmetic success. The study used the tasks described below.
- Numeral Ordering: students saw three numbers (either all one-digit or all two-digit) equidistant from the centre number and had to determine if the numbers were in increasing order.
- Numeral Comparison: students saw two numbers and had to determine which numeral was larger.
- Dot Comparison: students saw two arrays of dots and had to determine which array contained more dots. Since strategies such as using density or area to determine relative size are not as predictive of mathematics achievement, the researchers ensured number of dots and area/average size of dots were incongruent so that utilizing a non-numerical strategy would be the more difficult strategy.
- Object Matching: students were shown an array of 1-6 objects and were asked to match the cardinality to two other arrays (only one of these other arrays contained the same number of objects).
- Counting: students saw 1-9 dots and had to count them quickly and correctly.
- Number Line Estimation: students were shown a horizontal line with endpoints 0 and 100 and had to estimate where a given number (given both verbally and on screen) would be on the number line.
- Dot Quantity Estimation: students were shown an array of dots quickly (too fast to count) and were asked to estimate the number of dots on screen.
The study analyzed results taken from 1391 Dutch school children between grades 1 to 6 (inclusive). In general, performance significantly improved for each task as grade level increased. Of interest are the predictive values of the tasks on arithmetic ability over the grade levels. It was found that the number line estimation task was a strong predictor of arithmetic ability in grades 1 and 2, and tapered off for the later grades. This result was shared with the numeral comparison task – those students who were better able to compare two numbers tended to have higher arithmetic ability in grades 1 and 2. Numeral ordering was the poorest predictor for grade 1, yet increased gradually so that it was the strongest predictor of arithmetic ability by grade 6. A figure summarizing all tasks and their relative predictive values is given below.
This study has interesting implications for teaching, suggesting that encoding relative magnitudes of Arabic numerals and mapping them onto the number line are important for arithmetic success in primary mathematics, while numeric ordering is most important for later-middle years mathematics. There were no distinct predictors for grades 3, 4 or 5; however, one might argue that numerical ordering becomes more important and relative magnitudes become less important as students progress through these grades.
Tying nicely into this study is a second one in which the authors observed areas of the brain while participants answered single digit arithmetic. A correlation was found between activated areas and arithmetic fluency on the PSAT mathematics test. It is known that activation in regions of the left inferior parietal lobe is associated with fact retrieval, while activation in regions of the right bilateral intraparietal sulcus is associated with procedural calculations such as processing or representing relative magnitudes.
In this analysis, thirty-three 12th grade participants were presented with a series of single-digit addition and subtraction statements that were either correct or incorrect (for example 3+2=5 or 3+2=6). Students were tasked with determining the validity of the statement. Incorrect solutions differed from the actual solution by +1 or -1. It was determined that individuals who scored higher on the PSAT mathematics test showed greater activation in regions of the left inferior parietal lobe, or the region associated with fact retrieval. Those with lower scores on the PSAT test showed more activation in the regions associated with processing relative magnitudes.
Again, we have some significant applications to teaching. First and foremost, it must be noted that reliance on procedural or magnitude-based mechanisms to solve basic arithmetic problems is associated with deficient mathematics competence as students move from elementary grades to the middle school and high school grade levels. This might be in part due to the higher cognitive demand of mathematical problems as one moves up through the grade levels. A reliance on more procedural-based mechanics rather than fact recall may strain the working memory and prevent the construction of appropriate neurological structures required to support higher learning in mathematics. Another important aspect to take away from this study is the importance of teachers understanding that the shift away from magnitude-based calculations is vital in development of higher-order mathematical skill, as those students who rely on these skills show less mathematical competence compared to their peers.
So we see that there is a period of growth between the primary years and middle years in which students ideally transition from a quantitative approach to a symbolic approach to numbers. During the primary years we often see students using quantitative mechanics to solve mathematical problems, such as counting on their fingers. It is important to help develop this cardinal approach to numbers in the primary grades, as it is predictive of arithmetic fluency at this level. However, we have also seen that those students who fail to transition to a more symbolic approach with fact recall show signs of mathematical deficiency in the middle and high school years. Thus, it is imperative for teachers to help develop an ordinal approach to numerals as students progress through grades 3 to 6. In addition to this, utilizing strategies, such as low stakes single-digit arithmetic fluency tests, that increase fact recall in students is equally as important. It very well may be that development of fact recall aids in developing the neurological pathways required for success in higher-level mathematics.