In this chapter, the focus is on arithmetic which for the Netherlands as a trading nation is a crucial part of the mathematics curriculum.The chapter goes back to the roots of arithmetic education in the sixteenth century and compares it with the current approach to teaching arithmetic. In the sixteenth century, in the Netherlands, the traditional arithmetic method using coins on a counting board was replaced by written arithmetic with Hindu–Arabic numbers. Many manuscripts and books written in the vernacular teach this new method to future merchants, money changers, bankers, bookkeepers, etcetera. These students wanted to learn recipes to solve the arithmetical problems of their future profession. The books offer standard algorithms and many practical exercises. Much attention was paid to memorising rules and recipes, tables of multiplication and other number relations. It seems likely that the sixteenth century craftsmen became skilful reckoners within their profession and that was sufficient. They did not need mathematical insight to solve new problems. Five centuries later we want to teach our students mathematical skills to survive in a computerised and globalised society. They also need knowledge about number relations and arithmetical rules, but they have to learn to apply this knowledge flexibly and meaningfully to solve new problems, to mathematise situations, and to evaluate, interpret and check output of computers and calculators. The twenty-first century needs problem solvers, but to acquire the skills of a good problem solver a firm knowledge base—comparable with that of the sixteenth century reckoner—is still necessary.
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Many students persistently misinterpret histograms. This calls for closer inspection of students’ strategies when interpreting histograms and case-value plots (which look similar but are diferent). Using students’ gaze data, we ask: How and how well do upper secondary pre-university school students estimate and compare arithmetic means of histograms and case-value plots? We designed four item types: two requiring mean estimation and two requiring means comparison. Analysis of gaze data of 50 students (15–19 years old) solving these items was triangulated with data from cued recall. We found five strategies. Two hypothesized most common strategies for estimating means were confirmed: a strategy associated with horizontal gazes and a strategy associated with vertical gazes. A third, new, count-and-compute strategy was found. Two more strategies emerged for comparing means that take specific features of the distribution into account. In about half of the histogram tasks, students used correct strategies. Surprisingly, when comparing two case-value plots, some students used distribution features that are only relevant for histograms, such as symmetry. As several incorrect strategies related to how and where the data and the distribution of these data are depicted in histograms, future interventions should aim at supporting students in understanding these concepts in histograms. A methodological advantage of eye-tracking data collection is that it reveals more details about students’ problem-solving processes than thinking-aloud protocols. We speculate that spatial gaze data can be re-used to substantiate ideas about the sensorimotor origin of learning mathematics.
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Conducting large calculations manually with pen and paper following prescribed procedures or algorithms has been diminishing in significance for some time. In most cultures, and for many years already, individuals employ digital instruments for such computational tasks, when confronted with them in daily life. Yet, a closer examination of prevalent practices in the teaching of basic numeracy skills in adult education reveals a persistent emphasis on mastering standardized manual calculation techniques, especially with abstract and decontextualized numbers. This emphasis predominantly stems from the belief that mastering these manual procedures forms the cornerstone of all numeracy abilities. Contrastingly, our research indicates that the numeracy skills most frequently utilized and required in contemporary professions and daily activities encompass higher-order capabilities (Hoogland and Stoker, 2021; Boels et al., 2022; Hoogland and Díez-Palomar, 2022). These include interpretation, reasoning, mathematizing, estimation, critical reflection on quantitative data, and the application of digital instruments for computation. It is imperative, therefore, that numeracy education for adults prioritizes these competencies to achieve efficacy.
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