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The approximate number system (ANS) is thought to support non-symbolic representations of numerical magnitudes in humans. Recently much debate has focused on the causal direction for an observed relation between ANS acuity and arithmetic fluency. Here we investigate if arithmetic training can improve ANS acuity. We show with an experimental training study consisting of six 45-min training sessions that although feedback during arithmetic training improves arithmetic performance substantially, it does not influence ANS acuity. Hence, we find no support for a causal link where symbolic arithmetic training influences ANS acuity. Further, although short-term number memory is likely involved in arithmetic tasks we did not find that short-term memory capacity for numbers, measured by a digit-span test, was effected by arithmetic training. This suggests that the improvement in arithmetic fluency may have occurred independent of short-term memory efficiency, but rather due to long-term memory processes and/or mental calculation strategy development. The theoretical implications of these findings are discussed.

The approximate number system (ANS) is a mechanism believed to mediate our ability to make fast approximate judgments of non-symbolic number in tasks such as deciding without counting which apple tree contain more apples. Large inter-individual differences have been found in ANS acuity (efficiency) and several studies have documented a relation between individual ANS acuity and achievement in symbolic arithmetic (

It has been proposed that the ANS lays the foundation for the development of symbolic math (e.g.,

There is also a possibility of a causal link in the reversed direction, that math education or familiarization with symbolic numbers modifies and sharpens the ANS. It is well-known that ANS acuity improves with age (e.g.,

To summarize, previous research has indicated a relation between non-symbolic magnitude discrimination (ANS acuity) and math ability. This relation has been hypothesized to indicate a causal link. The direction of this link has, however, been debated, and it has even been suggested that a bi-directional causal association exits. By and large the research has been correlational in nature, except for a few studies training people on non-symbolic number tasks, and does therefore not allow causal conclusions. To our knowledge, no study has tried to experimentally, with random assignment and control group, investigate the effect of manipulating people’s familiarity with mental manipulation of numbers in terms of basic arithmetic calculation on ANS acuity. Accordingly, the aim of this study was to investigate the possible effect on ANS acuity of sustained exposure to mental arithmetic training with feedback. A transfer effect on arithmetic fluency following an improvement in ANS acuity would have significant theoretical implications for the interpretation of the association between math performance and non-symbolic magnitude processing efficiency.

Approximate number system acuity is, of course, not the only factor related to arithmetic performance. Research shows that short-term memory performance is also such a factor, although the relation is complex and likely depends on several auxiliary factors (

Forty-six participants (10 males, 36 females, _{age}

All participants carried out two tasks during pre- and post-test; a task measuring ANS acuity with an adaptive testing method and a task measuring short-term memory for numbers. Both tasks are described in detail below. Between pre- and post-test participants in the experimental condition carried out six 45-min sessions of targeted arithmetic fluency training. The training sessions were conducted on separate days and are described in detail below. Previous research has shown that it is possible that merely engaging in a non-symbolic numeric discrimination task is sufficient to induce an improvement in arithmetic fluency (

An adaptive test based on the ZEST algorithm was used to measure ANS acuity (

Approximate number system acuity thresholds were measured using non-symbolic representations of numerosity in the form of clusters of spatially intermixed yellow and blue dots presented on an intermediate gray background in each trial (

The response was indicated by a press on a yellow or blue marked key on the computer keyboard to indicate what dots were in majority. In both pre- and post-tests participants carried out 240 trials, which has been shown to provide a reliability of 0.80 using this adaptive procedure (

Approximate number system threshold for obtaining 80% correct discriminations was estimated by the ZEST algorithm (a modification of the Bayesian QUEST algorithm,

Short-term memory was measured with a digit span test. On each trial, a sequence of digits appeared on a computer screen for 10 s before disappearing. Participants were instructed to memorize and recalled the sequence in the presented order. Starting sequence length was four digits. Sequence length increased with one digit provided that the participant gave the correct answer in three out of four sequences of a particular length. When this criterion was not met, the test was interrupted. The maximum number of digits successfully recalled was used as a measure of the participant’s digit span/working memory capacity.

Participants in the experimental condition received six 45-min sessions of targeted arithmetic fluency training. The first session was conducted directly after the pre-test and the last session was conducted directly prior to the post-test. The other four sessions were conducted on separate days between pre- and post-test. The approximate time between pre- and post-test was 6 days. In each training session participants solved as many arithmetic problems (whole number addition, subtraction, multiplication, and division) as possible. All problems were generated randomly with the constraints described below. The algorithm generating the problems also made sure that an approximately equal number of problems for addition, subtraction, multiplication, and division were generated. Addition and subtraction problems consisted of addends, minuends, and subtrahends of 1–3 digits. For subtraction problems, all differences were positive. Multiplication and division consisted of problems with one single digit factor/divisor and one factor/dividend with 1–3 digits. For division problems, the quotient was always an integer. Directly after answering a problem, participants were shown their own answer together with the correct solution and color-coded feedback (Green = Correct, Red = Wrong).

It is possible that merely engaging in the arithmetic fluency training is sufficient to induce an improvement in arithmetic fluency. We designed the protocol for the control condition to control for this possibility and thereby separate a possible effect of targeted arithmetic fluency training on ANS acuity from an effect introduced by engaging in a task including numerical content. Accordingly, in the control condition participants carried out the same pre- and post-test as in the experimental condition. In addition, participants in the control condition carried out two 45-min session of solving arithmetic problems, one directly after the pre-test and one directly prior to the post-test. These sessions were the same as the corresponding sessions in the experimental condition with the exception that no feedback was given when participants had answered a problem.

Weber fractions for two participants in the experimental condition (both pre- and post-test) and for two participants in the control condition (post-test only) were lost due to apparatus failure. Data were scanned for outliers (|z| > 3), which resulted in one weber fraction in the post session of the control condition being excluded. The data is avilable at

Each of the two ANS tests was composed of 240 trials. In a study with the aim of assessing psychometric properties, we have previously estimated a reliability approaching 0.80, using this adaptive procedure and amount of trials (

Comparing performance on the first and last session of arithmetic fluency training demonstrated a large improvement by training from an average of 185 to 303 correctly solved tasks, a difference of 118 tasks (

The time spent on each problem decreased from 13.6 to 8.3 s/problem, a 40% increase in speed (

To verify that these performance increases were due to training effects three two-way ANOVAs were calculated with session (first/last) and experimental condition (experimental group/control group) as independent variables and each of the above performance indices as dependent variables. For the number of problems solved correctly there was no main effect of experimental condition,

For the time spent at each problem, there was no main effect of experimental condition,

For the proportion of attempted problems solved correctly, there was no main effect of experimental condition,

To investigate effects of arithmetic fluency training on ANS acuity, a two-way ANOVA with experimental condition and pre-/post-test as independent variables, and weber-fraction as dependent variable was calculated. As can be seen in

To investigate effects of arithmetic training on short-term memory (Digit span) a two-way ANOVA with experimental condition and pre-/post-test as independent variables, and digit span as dependent variable was calculated. As can be seen in

Recent studies have indicated that non-symbolic arithmetic training transfers to an improvement in math performance (

We thus investigated if ANS acuity would improve with symbolic arithmetic training. Our results indicated that arithmetic training over six 45 min sessions lead to a large improvement in the number of correctly solved tasks and operation speed. Further, the proportion correctly solved tasks also improved showing that the training effect is not just due to a speed increase, but results in a genuine better precision in arithmetic operations. Also, the lack of improvement in the control condition makes it unlikely that the training results in better performance due to an increased task familiarity. This improvement in arithmetic fluency is noteworthy by itself. Because mental arithmetic ability is highly useful in a society increasingly dependent on numeric information, more exposure to direct mental arithmetic and immediate feedback during math education could contribute to a large performance gain that would be worthwhile with respect to people’s everyday undertakings.

In contrast to previous research showing that non-symbolic arithmetic training transfers to enhancement in math performance (e.g., ^{1}, to math performance. It is thus still possible that the type of training used in the present study might have an effect on non-symbolic arithmetic. Future research should investigate this possibility. As with any training study it could be the case that more extensive training would have resulted in training effects. Our participants, however, received a similar amount of training as participants in previous studies that have indicated training effects in the opposite causal direction (e.g.,

A possible limitation to our study is that we did not use a measure of ANS acuity with control for visual cues. Indeed, recent research using a procedure to create stimuli developed by ^{2}. It will be an important topic for future research to address to what extent visual cues influences training paradigms of the type used here.

There was no effect of arithmetic training on short-term memory. This is in line with results showing that training on short-term memory resulting in improvement is often tied to strategy and is highly context dependent. What is learnt, then, when arithmetic fluency is improved? A possibility is that learning is a mere strengthening of long-term memory math facts (e. g., 12 + 9 = 21) that are retrieved from memory more rapidly and with less error. Another possibility is that more general “rules” for efficient mental calculation are learned. An interesting venue for future research is to develop ways to measure the exact nature of how human number manipulation is improved by training, which may also have important implications for educational settings.

ML and AW developed the study concept. All authors contributed to the study design. Testing and data collection were performed under the supervision of ML and AW. All authors contributed in the data analysis and interpretation. ML, AW, and LP wrote the paper. All authors approved the final version of the paper for submission.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Swedish Research Council sponsored this research. We are indebted to Towa Swärd, Mikael Andersén, and Philip Millroth for help with the data collection.

With a “non-symbolic comparison” task participants decide which of two sets of objects that is the more numerous. With a “non-symbolic arithmetic” task, participants view two sets of objects, and decide whether a third set of objects is more or less numerous than either the sum or difference between these two sets.

Note however, that there is an ongoing debate about choice of methods to control for visual cues, and that more elaborate methods that control over several dimensions including convex hull result in measures that are apparently not related to measures obtained with more basic controls (see