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This paper frames children's mathematics as mathematics. Specifically, it draws upon our knowledge of children's mathematics and applies it to understanding the prime number theorem. Elementary school arithmetic emphasizes two principal operations: addition and multiplication. Through their units coordination activity, children construct two mathematical worlds: an additive world and a multiplicative world. Understanding how children might map between their additive and multiplicative worlds provides insights into the prime number theorem. It also helps us appreciate the power of children's mathematics, constructed through the coordination of their own mental actions. [For the complete proceedings, see ED630210.]

This study examined the effects of an academic intervention, associated with music, on the conceptual understanding of musical notation and arithmetic of fractions of first-year students of high school from a mixed Spanish multicultural and socioeconomic public school. The students (N = 12) had previous concepts about musical instruction, as well as operations with fractions, particularly addition. This is an observational study in which a battery of four tasks was administered before and after an instruction based on a musical environment, music being a semiotic function. The instruction included 9 sessions of 50 minutes each. The results prior to the intervention show deficiencies in a concept that was not new to the students, however, after the intervention the students were competent in addition with fractions. [For the complete proceedings, see ED657822.]

Despite curriculum expectations, many students, including a disproportionate number of girls, do not 'just know' (retrieve) single-digit addition facts by Year 3. The current study employed structured interviews to explore which strategies Year 3/4 students (n = 166) used when solving more difficult addition combinations. Results revealed that students preference the near-doubles strategy when the difference between the addends was one, the bridging-through-10 strategy when one of the addends was a nine, and the count-on-from-larger strategy when a derived strategy was more effortful. Moreover, whereas boys were more inclined to use derived strategies, girls were almost three times more likely to use the count-on-from-larger strategy.

With the growing prevalence of mobile apps for self-directed learning, educational games increasingly find their place in everyday routines, becoming accessible to a broad audience. Despite the growing ease of content creation by artificial intelligence computing, the challenge of designing effective and engaging Serious Games remains, particularly in managing cognitive resources and ensuring quality engagement, notably influenced by the game's interaction modalities. This study explores these challenges within the context of a casual mobile mental arithmetic game, investigating the differential impacts of tap and swipe interaction variants on cognitive load and reward-based engagement. The study presents the findings of an international field study on Google Play. In a between-group design, the two casual interaction paradigms were compared regarding their impact on practice performance, cognitive load and effect on classic casual game rewarding represented through points, leaderboards and badges. The findings show that tap interaction can optimise cognitive load with a better mental math practice performance than the more indirect swipe gesture. A combination of elementary tap interaction with point rank and interaction precision badges indicates to enhance practice motivation. The results are synthesised into interaction design recommendations for casual mental math mobile games. [For the full proceedings, see ED636095.]

The development of mathematical reasoning is a key proficiency for mathematics within the Australian Curriculum. However, reasoning can be difficult for teachers to assess, particularly with pen and paper tests. In this study, interview tasks were designed across three curriculum areas at three different levels to assess student reasoning through the use of examples and non-examples. Non-examples can be used to assist in building boundaries and deepening conceptual understanding. Through the interview, teacher and student dialogue can help students to demonstrate reasoning and clarify concepts through explanation and justification.

We present the results of a research project on arithmetic-algebraic thinking that was carried out jointly by a team in Mexico and another in Quebec. The project deals with the concepts of variable and covariation between variables in the sixth grade at the elementary level and the first, second, and third years of secondary school--namely, children from 11 to 14 years old. We target secondary students (first year or K7) in this article. Our objective relates to the development of a gradual generalization in arithmetic-algebraic thinking in a socio-cultural approach to the learning of mathematics. We experimented with investigative situations using a paper-and-pencil approach and technology. We analyze the emergence, in this context, of a visual abstraction, the production of institutional and non-institutional representations, a sensitivity to contradiction, and, finally, the concepts of variable and of covariation between variables. [For the complete proceedings, see ED629884.]

Literature typically describes mathematization, the process of transforming a real-world situation into a mathematical model, in terms of desirable actions and behaviors students exhibit. We attended to STEM undergraduate students' quantitative reasoning as they derived equations. Analysis of the meanings they held for arithmetic operations (+, -, ·, ÷) provided insight into how participants expressed real-world relationships among entities with arithmetic relationships among values. We extend the findings from K-12 literature (e.g., using multiplication to instantiate a rate) to STEM undergraduates and found evidence of new ways of justifying the usage of arithmetic operations (e.g., using multiplication to instantiate an amount). [For the complete proceedings, see ED630210.]

Students exhibiting mature number sense make sense of numbers and operations, use reasoning to notice patterns, and flexibly select the most effective and efficient problem-solving strategies (McIntosh et al., 1997; Reys et al., 1999; Yang, 2005). Despite being highlighted in national standards and policy documents (CCSS, 2010; NCTM, 2000, 2014), students' mature number sense and its nomological network are not yet well specified. For example, how does students' mature number sense relate to their knowledge of fractions and their grade-level mathematics achievement? We analyzed 129 middle school students' scores on measures of mature number sense, fraction and decimal computation, and grade-level mathematics achievement. We found mature number sense to be measurably distinct from their fraction and decimal knowledge and uniquely associated with students' grade-level mathematics achievement. [For the complete proceedings, see ED630210.]

Knowledge tracing algorithms are embedded in Intelligent Tutoring Systems (ITS) to keep track of students' learning process. While knowledge tracing models have been extensively studied in offline settings, very little work has explored their use in online settings. This is primarily because conducting experiments to evaluate and select knowledge tracing models in classroom settings is expensive. To fill this gap, we introduce a novel way of using machinelearning models to generate simulated students. We conduct experiments using agents generated by the Apprentice Learner Architecture to investigate the online use of different knowledge tracing models (Bayesian Knowledge Tracing, the Streak model, and Deep Knowledge Tracing). An analysis of our simulation results revealed an error in the initial implementation of our Bayesian knowledge tracing model that was not identified in our previous work. Our simulations also revealed a more fundamental limitation of Deep Knowledge Tracing that prevents the model from supporting mastery learning on multi-step problems. Together, these two findings suggest that Apprentice agents provide a practical means of evaluating knowledge tracing models prior to more costly classroom testing. Lastly, our analysis identifies a positive correlation between the Bayesian knowledge tracing parameters estimated from human data and the parameters estimated from simulated learners. This suggests that model parameters might be initialized using simulated data when no human-student data is yet available. [For the full proceedings, see ED615472.]

Measuring computational fluency, an aspect of procedural fluency, is complex. Many attempts to measure this construct have emphasised accuracy and efficiency at the expense of flexibility and appropriate strategy choice. Efforts to account for these latter constructs through assessing children's computational reasoning using structured interviews (e.g., MAI), are necessarily time-intensive. In this paper, we introduce a novel measure of Mental Computational Fluency with Addition (MCF-A) that attempts to incorporate these aspects by requiring children to reason from the perspective of another child. We describe results of a pilot study using the MCF-A with 169 Year 3 and 4 students.

Language is often depicted as the sine qua non of mathematical thinking, a view buttressed by findings of language-of-training effects among bilinguals. These findings, however, have been limited to studies of arithmetic. Nothing is known about the potential influence of language on the ability to learn rules about the relations among variables (e.g., algebra). To test whether arithmetic and algebraic thinking differ, Chinese-English bilinguals were trained to solve arithmetic and algebra problems in either Chinese or English and then tested on new and old problems in both languages. For arithmetic problems, solution times were always longer for English than Chinese; in both languages, solution times dropped during training; after training, solution times continued to drop for old problems, but returned to pre-training levels for new problems. In contrast, for algebra problems, solution times did not differ across language; solution times dropped during training; after training, gains in speed were preserved for both old and new problems. These findings suggest that the contribution of language to mathematical thinking may be limited to the areas of mathematics that are learned by rote and not by rule. [This paper was published in: "Proceedings of the 40th Annual Conference of the Cognitive Science Society," edited by T. T. Rogers et al., Cognitive Science Society, 2018, pp. 918-923.]

Reasoning about fraction magnitude is an important topic in elementary mathematics because it lays the foundations for meaningful reasoning about fraction operations. Much of the research literature has reported deficits in preservice elementary teachers' (PSTs) knowledge of fractions and has given little attention to the productive resources that PSTs bring to teacher education. We surveyed 26 PSTs using a set of 9 fraction-comparison tasks. We report the frequency of complete strategy-arguments and the perspectives (ways of reasoning) used for each item. We further examine incomplete strategy-arguments, noting substantial evidence for productive seeds of reasoning. Using data from interviews with 10 of these PSTs, we identify evidence suggesting these seeds are, in fact, productive in that they provide foundations for further development. We argue that this type of research is needed in order to further mathematics teacher education. [For the complete proceedings, see ED629884.]

Researchers, who have studied the understanding of the density property in the set of decimal numbers, have shown that the student uses the property of the discrete of natural numbers to solve tasks related to density. So, a restructuring of concepts is necessary, that is, a conceptual change from "the discrete" to "the dense". This report presents evidence from ten pre-service teachers from Mexico City that this change can initiate through the implementation of a didactic sequence. The pre-service teachers managed to visualize that a decimal number can be found in an interval. Consequently, they were able to conceive an infinity. However, several of them persisted with the idea of the existence of a successor in the set of decimal numbers. [For the complete proceedings, see ED629884.]

A challenge in digital learning games is assessing students' learning behaviors, which are often intertwined with game behaviors. How do we know whether students have learned enough or needed more practice at the end of their game play? To answer this question, we performed post hoc analyses on a prior study of the game "Decimal Point," which teaches decimal numbers and decimal operations to middle school students. Using Bayesian Knowledge Tracing, we found that students had the most difficulty with mastering the number line and sorting skills, but also tended to over-practice the skills they had previously mastered. In addition, using students' survey responses and in-game measurements, we identified the best feature sets to predict test scores and self-reported enjoyment. Analyzing these features and their connections with learning outcomes and enjoyment yielded useful insights into areas of improvement for the game. We conclude by highlighting the need for combining traditional test measures with rigorous learning analytics to critically evaluate the effectiveness of learning games. [For the full proceedings, see ED607784.]

Mathematics education scholars have generally classified students' conception of the equal sign as either operational or relational. Adding to these conceptions, Jones (2008) introduced the idea of substitutional conception. Building off these scholars, I introduce a form of understanding the equal sign that includes a transformative equivalence component and extends the conceptions of the equal sign to conceptions of equations. [For the complete proceedings, see ED629884.]

There is an ongoing debate over the psychophysical functions that best fit human data from numerical estimation tasks. To test whether one psychophysical function could account for data across diverse tasks, we examined 40 kindergartners, 38 first graders, 40 second graders and 40 adults' estimates using two fully crossed 2 × 2 designs, crossing symbol (symbolic, non-symbolic) and boundedness (bounded, unbounded) on free number-line tasks (Experiment 1) and crossing the same factors on anchored number-line tasks (Experiment 2). This strategy yielded 4 novel tasks to assess the generalizability of the models. Across all 8 tasks, 90% of participants provided estimates better fit by a mixed log-linear model than other cognitive models, and the weight of the logarithmic component ([lambda]) decreased with age. After controlling for age, the weight of the logarithmic component ([lambda]) significantly predicted arithmetic skills, whereas parameters of other models failed to do so. Results suggest that the logarithmic-to-linear shift theory provides a unified account of numerical magnitude estimation and provides uniquely accurate predictions for mathematical proficiency. [This paper was published in: "Proceedings of the 39th Annual Meeting of the Cognitive Science Society," edited by G. Gunzelmann et al., Cognitive Science Society, 2017, pp. 2943-2948.]

This paper provides insight about the development of addition and subtraction strategies for nearly 22,000 Australian primary school children in 2016. The children were each assessed by their teacher using a task-based assessment interview that identified the strategies they used to mentally perform addition and subtraction, and matched these to a growth point framework. The findings highlight the broad distribution of strategies used by children in each grade level and suggest that few children, including those in Grade 6, reach Growth Point 6 that involves the mental calculation of two-digit and three-digit numbers. These findings have important implications for classroom teaching and professional learning.