Pablo Dartnell

The effect of analogies on learning to solve algebraic equations

A total of 236 seventh grade students who had never been taught algebraic equations before, attending 10 Chilean schools of varying socio-economic status, were randomly divided into two groups at each school. The students in one group watched a 15-minute video teaching them how to solve five different first-degree linear equations using a traditional symbolic strategy, while in the other group, the students watched a 15-minute video teaching them how to solve the same equations using four analogies for solving an equation: a two-pan balance for the equals sign, a box for a variable, candies for numbers, and guessing the number of candies inside a box. The students were then tested on 12 equation solving problems, all of them written, using only symbolic notation. The group that watched the analogies video performed significantly better. Students with a below-average mathematics GPA who watched the analogies video did as well as students with an above-average GPA who watched the symbolic strategy video. Students who watched the analogies video also reached a better conceptual understanding, were better at making generalizations, did significantly better on reasoning problems involving equations, and had a better affective reaction.

Strategies used by students on a massively multiplayer online mathematics game

We analyze the logs of an online mathematics game tournament, played simultaneously by thousands of students. Nearly 10,000 students, coming from 356 schools from all regions in Chile, registered to the fourth tournament instance. The children play in teams of 12 students from the same class, and send their personal bets to a central server every 2 minutes. Each competition lasts about one clock hour and takes place within school hours. Students are pre-registered and trained by their school teacher. The teacher is responsible for reviewing curriculum contents useful for improving performance at the game and coaches students participating in trial tournaments taking place a few weeks before the national tournament. All bets are recorded in a database that enables us to analyze later the sequence of bets made by each student. Using cluster analysis with this information, we have identified three types of players, each with a well-defined strategy.

Teaching modeling skills using a massively multiplayer online mathematics game

One important challenge in mathematics education is teaching modeling skills. We analyze the logs from a game-based learning system used in a massively multiplayer online tournament. Students had to detect an input–output pattern across 20 rounds. For each round, they received an input and had 2 minutes to predict the output by selecting a binary option (2 points if correct, −1 otherwise), or writing a model (4 points if model prediction was correct, −4 otherwise), or refraining (1 point). Thousands of 3rd to 10th grade students from hundreds of schools simultaneously played together on the web. We identified different types of players using cluster analysis. From 5th grade onwards, we found a cluster of students that wrote models with correct predictions. Half of the 7th to 10th grade students that detected patterns were able to express them with models. The analysis also shows diffusion within the teams of modeling strategies for simple patterns.

Long-distance neural synchrony correlates with processing strategies to compare fractions

Adults use different processing strategies to work with fractions. Depending on task requirements, they may analyze the fraction components separately (componential processing strategy, CPS) or consider the fraction as a whole (holistic processing strategy, HPS). It is so far unknown what is the brain coordination dynamics underlying these types of fraction processing strategies. To elucidate this issue, we analyzed oscillatory brain activity during a fraction comparison task, presenting pairs of fractions either with or without common components. Results show that CPS induces a left frontal-parietal alpha phase desynchronization after the onset of fraction pairs, while HPS induces an increase of phase synchrony on theta and gamma bands, over frontal and central-parietal sites, respectively. Additionally, the HPS evokes more negative ERPs around 400 ms over the right frontal scalp than the CPS. This ERP activity correlates with the increase of Theta phase synchrony. Our results reveal the emergence of different functional neural networks depending on the kind of cognitive strategy used for processing fractions.

Combinatorial constructions associated to the dynamics of one-sided cellular automata

In this paper we study combinatorial constructions which lead to one-sided invertible cellular automata with different dynamical behavior: equicontinuity, existence of equicontinuous points but not equicontinuity, sensitivity and expansivity. In particular, we provide a simple characterization of the class of equicontinuous invertible one-sided cellular automata and we construct families of expansive one-sided cellular automata.

Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge

Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called “flexibility” whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena.

Topic Study Group No. 27: Learning and Cognition in Mathematics

Deakin Research Online, Deakin University’s Research Repository Deakin University CRICOS Provider Code: 00113B Topic study group no. 27: learning and cognition in mathematics Citation: Williams, Gaynor, Van Dooren, Wim, Dartnell, Pablo, Lindmeier, Anke and Proulx, Jérôme 2017, Topic study group no. 27: learning and cognition in mathematics, in ICME-13 : Proceedings of the 13th International Congress on Mathematical Education, Springer, Cham, Switzerland, pp. 501505. DOI: 10.1007/978-3-319-62597-3_54

On Two Mechanisms Associated To Learning: A Mathematical Point Of View

We study two important features of the mechanisms living organisms seem to use to solve recurrent problems when able to choose strategies from a known set. The first one is forgetting and its influence on the ability of the organism to learn the chance of success of the known strategies. The other feature is selection of strategies according to their relative strengths. Specifically, we compare exponential and hyperbolic forgetting models, and we prove that when the agent has only one strategy available, the estimates for the strategy success rate using the exponential model never converge (in probabilistic terms), whereas the ones using the hyperbolic model converge almost surely. When more strategies are available and proportional selection is used, we prove several results that generalize the one strategy case.