# 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 12^{th} 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

*, as those students who rely on these skills show less mathematical competence compared to their peers.*

**the shift away from magnitude-based calculations is vital in development of higher-order mathematical skill**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.

I have been wondering lately whether the fixation in fuzzy math curriculum of focusing on number sense for too long (still using manipulatives in 5th grade – come on) might be based on the belief that such practices could prepare one to develop breakthroughs in theoretical studies like set theory or quantum mechanics later. Might a mind not so heavily constrained by embedded algorithms be able to puzzle out solutions where none have yet been found? This study seems to indicate the answer is no. It’s not just that the student would be slower at solving the problem, the brain could get stuck in doing the basic maths. I liked Roger’s analogy to speaking foreign languages. It is easier to pick up new discrete vocabulary if you aren’t focused on translating every single word back and forth. There must be automaticity.

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I think one item that is often overlooked is that, as a species, we already have knowledge in rudimentary problem solving. This rudimentary problem solving ability can be far from efficient depending on what task we must accomplish. One might argue, then, that our time in elementary should be partially spent refining this problem solving ability. The best way to refine this ability? Domain knowledge. The more knowledge one has (facts able to recall, for example), the easier it becomes to refine problem solving search. If it is indeed problem solving ability that we want to foster, then working on expanding domain-specific knowledge should be an important priority at the elementary level.

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Good summary, Brian. With respect to the latter study, I have always held that one of the goals of elementary education ought to be the familiarization of students with abstract representations and comfort and fluency with the manipulation of symbolic expressions in the solution of problems. In my view, while the memorization of math facts is important for the reduction of load on working memory, I suspect that the greater benefit comes in the development of a precursor “confidence” in pure symbolic manipulation. In my own mathematical development I have observed a few points where a quantum-leap of both fluency and cognitive grasp at the meaning level took place, and in all cases they were associated with a sort of epiphany in which I began to “trust the math” at a different level. Looking back I have come to understand that mathematics is a sixth sense all of its own (or seventh, etc. Who’s counting?). I was taught that I’d never be fluent in french until I stopped reflexively translating everything back into English, and formulating my french statements in English first, then translating. You had to think, speak and hear “in french”, not through a mental translator. I think exactly the same thing is true in mathematics. You don’t develop that sixth sense until you loosen your grasp on the visual & tactile fall-backs. I don’t say “verbal” because in my experience the verbal faculty seems intrinsically tied to mathematics. Maybe I’m wrong there, but I find I “hear” mathematics as I do it. Perhaps it has something to do with how memorization of math facts work — I believe Ansari’s work on this associated audio memory with memorization. It may tell us something about where our abstract “organs” reside in the brain — I dunno there, only speculating as a non-expert …

Anyway, all this suggests we should be alarmed at the tendency for many modern elementary math teaching resources seem geared to teach … no TRAIN … children from an early age to reflexively always model abstract things concretely or visually. My experience and instincts tell me that while this is a natural predisposition, mathematical education is very much other than a return to the jungle, it is learning to soar above those primitive faculties and ought, rather, to direct children to develop a talent for moving in the other direction, toward comfort and mastery with the abstract domain.

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“… asked to estimate the number of sots on screen…”

Wait a minute! They were shown what, an indiscrete photo of the office New Year’s party … ?

🙂

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Haha. It was quite the party I have heard! (I have also updated this little spelling error so as not to confuse further readers about the status of said party.)

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