Understand math cognition.

Improve math education.

We aim to understand and improve fundamental abilities of symbolic mathematical thinking: reading and writing numbers, calculation, algorithmic thinking, and more.

Our Goals

For each mathematical ability, we ask about…

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    Cognitive mechanisms

    Characterize the cognitive mechanisms underlying the mathematical ability

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    Learning disorders

    Identify specific learning disorders that disrupt the performance in this aspect of math

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    Individual differences

    Discover why some people are better than others in this aspect of math

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    Personalized learning

    Improve the mathematical ability by creating learning methods that can be adapted to each person’s cognitive profile

About the Mathematical Thinking Lab

Many people say that “mathematics is a different kind of language”. What they often mean is that similar to language, mathematics allows using a basic set of symbols to express complex meanings by combining these symbols according to certain rules. For example, the base-10 system allows combining digits or words into numbers; operators such as + and ÷ allow combining these numbers to perform basic arithmetic; and mathematical procedures combine simple calculations to attain more complex goals, e.g., solving an equation.

Our lab examines how, when children and adults do mathematics, their mind handles these systems of symbols and rules via cognitive mechanisms such as language and executive functions. We investigate this question for ‘the average person’, but we also examine individual differences, i.e., how and why some people are better than others in particular aspects of math, and which precise learning disorders disrupt specific mathematical abilities. To investigate these issues we use a variety of methods from Cognitive Psychology and Neuropsychology. For more details, visit the Research page.

We believe that cognitive research should not remain in the lab, but rather be used to improve education. Thus, we create methods and tools to diagnose mathematical learning disorders in individual children and adults, and we develop research-based educational methods that should make math learning easier and more efficient.

The Mathematical Thinking Lab is affiliated with Tel Aviv University‘s School of Education and Sagol School of Neuroscience.

Selected publications

  • Elementary math in elementary school: The effect of interference on learning the multiplication table. Cognitive Research: Principles and Implications, 7, 101. doi:10.1186/s41235-022-00451-0 [Abstract]

    Elementary math in elementary school: The effect of interference on learning the multiplication table.

    Memorizing the multiplication table is a major challenge for elementary school students: there are many facts to memorize, and they are often similar to each other. This similarity creates interference in memory. Here, we examined whether learning would improve if the degree of interference is reduced, and which memory process are responsible for this improvement. In a series of 16 short training sessions over 4 weeks, first-grade children learned 16 multiplication facts – 4 facts per week. In 2 weeks the facts were dissimilar from each other (low interference), and in 2 control weeks the facts were similar (high interference). Learning in the low-similarity, low-interference weeks was better than in the high-similarity weeks. Critically, this similarity effect originated in the specific learning context, i.e., the grouping of facts to weeks, and could not be explained as an intrinsic advantage of certain facts over others. Moreover, the interference arose from the similarity between facts in a given week, not from similarity to the previously-learned facts. Similarity affected long-term memory – its effect persisted 7 weeks after training has ended; and it operated on long-term memory directly, not via mediation of working memory. Pedagogically, the effectiveness of the low-interference training method, which is dramatically different from currently-used pedagogical methods, seems to call for reconsideration of the way we teach the multiplication table in school.
  • Serial and syntactic processing in the visual analysis of multi-digit numbers. Cortex, 134, 162-180. doi:10.1016/j.cortex.2020.10.012 [Abstract]

    Serial and syntactic processing in the visual analysis of multi-digit numbers.

    The visual analysis of letter strings is a separate cognitive process from the analysis of digit strings. Recent studies have hypothesized that these processes are not only separate but also qualitatively different, in that they may encode information specific to numbers or to words. To examine this hypothesis and to shed further light on the visual analysis of numbers, we asked adults to read aloud multi-digit strings presented to them for brief durations. Their performance was better in digits on the number’s left side than in digits farther to the right, with better performance in the two outer digits than their neighbors. This indicates the digits were processed serially, from left to right. Visual similarity of digits increased the likelihood of errors, and when a digit migrated to an incorrect position, it was most often to an adjacent location. Interestingly, the positions of 0 and 1 were encoded better than the positions of 2-9, and 2-9 were identified better when they were next to 0 or 1. To accommodate these findings, we propose a detailed model for the visual analysis of digit strings. The model assumes imperfect digit detectors in which a digit’s visual information leaks to adjacent locations, and a compensation mechanism that inhibits this leakage. Crucially, the compensating inhibition is stronger for 0 and 1 than for the digits 2-9, presumably because of the importance of 0 and 1 in the number system. This sensitivity to 0 and 1 makes the visual analyzer specifically adapted to numbers, not words, and may be one of the brain’s reasons to implement the visual analysis of numbers and words in two separate cognitive processes.
  • Parallel and serial processes in number-to-quantity conversion. Cognition, 204, 104387. doi:10.1016/j.cognition.2020.104387 [Abstract]

    Parallel and serial processes in number-to-quantity conversion.

    Converting a multi-digit number to quantity requires processing not only the digits but also the number’s decimal structure, thus raising several issues. First, are all the digits processed in parallel, or serially from left to right? Second, given that the same digit, at different places, can represent different quantities (e.g., “2” can mean 2, 20, etc.), how is each digit assigned to its correct decimal role? We presented participants with two-digit numbers and asked them to point at the corresponding locations on a number line, while we recorded their pointing trajectory. Crucially, on some trials, the decade and unit digits did not appear simultaneously. When the decade digit was delayed, the decade effect on finger movement was delayed by the same amount. However, a lag in presenting the unit digit delayed the unit effect by 35 ms less than the lag duration, a pattern reminiscent of the psychological refractory period, indicating an idle time window of 35 ms in the units processing pathway. When a lag transiently caused a display of just one digit on screen, the unit effect increased and the decade effect decreased, suggesting errors in binding digits to decimal roles. We propose that a serial bottleneck is imposed by the creation of a syntactic frame for the multidigit number, a process launched by the leftmost digit. All other stages, including the binding of digits to decimal roles, quantification, and merging them into a whole-number quantity, appear to operate in parallel across digits, suggesting a remarkable degree of parallelism in expert readers.
  • Track it to crack it: Dissecting processing stages with trajectory tracking. Trends in Cognitive Sciences, 23(12), 1058-1070. doi:10.1016/j.tics.2019.10.002 [Abstract]

    Track it to crack it: Dissecting processing stages with trajectory tracking.

    A central goal in cognitive science is to parse the series of processing stages underlying a cognitive task. A powerful yet simple behavioral method that can resolve this problem is finger trajectory tracking: by continuously tracking the finger position and speed as a participant chooses a response, and by analyzing which stimulus features affect the trajectory at each time point during the trial, we can estimate the absolute timing and order of each processing stage, and detect transient effects, changes of mind, serial versus parallel processing, and real-time fluctuations in subjective confidence. We suggest that trajectory tracking, which provides considerably more information than mere response times, may provide a comprehensive understanding of the fast temporal dynamics of cognitive operations.
  • A cognitive model for multi-digit number reading: Inferences from individuals with selective impairments. Cortex, 101, 249-281. doi:10.1016/j.cortex.2017.10.025 [Abstract]

    A cognitive model for multi-digit number reading: Inferences from individuals with selective impairments.

    We propose a detailed cognitive model of number reading. The model postulates separate processes for visual analysis of the digit string and for oral production of the verbal number. Within visual analysis, separate sub-processes encode the digit identities and the digit order, and additional sub-processes encode the number’s decimal structure: its length, the positions of 0, and the way it is parsed into triplets (e.g., 314987 → 314,987). Verbal production consists of a process that generates the verbal structure of the number, and another process that retrieves the phonological forms of each number word. The verbal number structure is first encoded in a tree- like structure, similarly to syntactic trees of sentences, and then linearized to a sequence of number-word specifiers. This model is based on an investigation of the number processing abilities of seven individuals with different selective deficits in number reading. We report participants with impairment in specific sub-processes of the visual analysis of digit strings – in encoding the digit order, in encoding the number length, or in parsing the digit string to triplets. Other participants were impaired in verbal production, making errors in the number structure (shifts of digits to another decimal position, e.g., 3,040 → 30,004). Their selective deficits yielded several dissociations: first, we found a double dissociation between visual analysis deficits and verbal production deficits. Second, several dissociations were found within visual analysis: a double dissociation between errors in digit order and errors in the number length; a dissociation between order/length errors and errors in parsing the digit string into triplets; and a dissociation between the processing of different digits – impaired order encoding of the digits 2- 9, without errors in the 0 position. Third, within verbal production, a dissociation was found between digit shifts and substitutions of number words. A selective deficit in any of the processes described by the model would cause difficulties in number reading, which we propose to term “dysnumeria”.