How Concrete is Concrete?

If we want to make something concrete in mathematics education, we are inclined introduce, what we call, ‘manipulatives\u27, in the form of tactile objects or visual representations. If we want to make something concrete in a everyday-life conversation, we look for an example. In the former, we try to make a concrete model of our own, abstract, knowledge; in the latter, we try to find an example that the others will be familiar with. This article first looks at the tension between these two different ways of making things concrete. Next another role of manipulatives, will be discussed, namely that of means for scaffolding and communication. In this role, manipulatives may function as means of support in a process that aims at helping students to build on their own thinking while constructing more sophisticated mathematics

Design research in mathematics education : the case of an ict-rich learning arrangement for the concept of function

The concept of function is a central but difficult topic in secondary school mathematics curricula, which encompasses a transition from an operational to a structural view. The question in this paper is how to design and evaluate a technology-rich learning arrangement that may foster this transition. With domain-specific pedagogical knowledge on the learning of function as a starting point, and the notions of emergent modeling and instrumentation as design heuristics, such a learning arrangement was designed for grade 8 students and field tested. The results suggest that these design heuristics provide fruitful guidelines for the design of both a hypothetical learning trajectory and concrete tasks, and can be generalized to other design processes

Word problems versus image-rich problems: an analysis of effects of task characteristics on students’ performance on contextual mathematics problems

This article reports on a post hoc study using a randomised controlled trial with 31,842 students in the Netherlands and an instrument consisting of 21 paired problems. The trial showed a variability in the differences of students’ results in solving contextual mathematical problems with either a descriptive or a depictive representation of the problem situation. In this study the relation between this variability and two task characteristics is investigated: (1) complexity of the task representation; and (2) the content domain of the task. We found indications that differences in performance on descriptive and depictive representations of the problem situation are related to the content domain of the problems. One of the tentative conclusions is that for depicted problems in the domain of measurement and geometry the inferential step from representation of the problem situation to the mathematical problem to be solved is smaller than for word problems

Changing representation in contextual mathematical problems from descriptive to depictive: The effect on students’ performance

Research on solving mathematical word problems suggests that students may perform better on problems with a close to real-life representation of the problem situation than on word problems. In this study we pursued real-life representation by a mainly depictive representation of the problem situation, mostly by photographs. The prediction that students perform better on problems with a depictive representation of the problem situation than on comparable word problems was tested in a randomised controlled trial with 31,842 students, aged 10–20 years, from primary and secondary education. The conclusion was that students scored significantly higher on problems with a depictive representation of the problem situation, but with a very small effect size of Cohen's d = 0.09. The results of this research are likely to be relevant for evaluations of mathematics education where word problems are used to evaluate the mathematical capacity of students

Investigating students’ learning trajectory: a case on triangle

Learning trajectory becomes the main issues in mathematics education research. However, there have been limited studies of students’ learning trajectory for triangle construction. Therefore, this design research was conducted to investigate students’ learning trajectory for the topic of the triangle. The study involved 22 students from 7th grade in Malang, Indonesia. Data were collected through a videotaped, a student’s worksheet, and a classroom observation. The results showed that students discovered the requirement of forming a triangle given three side lengths. In this condition, the starting point of students’ learning trajectory was the drawing of a line segment from the given three side lengths. Students examined two side lengths whether these side lengths can be joined to the line segment as a triangle or not. Students used rulers for determining those three side lengths that could form a triangle. They made a statement that the sum of any two sides of a triangle must be greater than the third side. Furthermore, teachers should consider students’ learning trajectory for achieving successfully the learning goal

Connie Myers v. Albertsons, Inc. : Brief of Appellee

Appeal of the Judgment of Michael Glasmann Based upon a Jury Verdict Second Judicial District Court Weber County, State of Uta

The influence of technology on the mathematical modelling of physical phenomena

A study is presented in which students are asked to model two physical phenomena using applications on electronic tablets : a bounce of a ball and the extension of a spring. The analysis focusses on (a) the influence of characteristics of the applications on the tablets on the decisions that groups of 16-year-old students made during the modelling phases in which reality and mathematics are related, (b) mathematisation of the phenomena and (c) interpretation of the models. The phenomena were recorded using an app that requests users establish a set of references during the mathematisation process, which makes students focus on the way the references have been set to interpret the model properly. Our findings indicated inconsistences between student decisions made during mathematisation and their considerations during interpretation of the model. To conclude we suggest reasons students experience problems in working without a pre-defined reference syste

Towards a knowledge-rich learning environment in preparatory secondary education

In this case study a novel educational programme for students in preparatory vocational education was studied. The research questions were: (1) Which teaching/learning processes occur in a simulated workplace using the concept of a knowledge-rich workplace? (2) What is the role of models and modelling in the teaching/learning processes? The curriculum project consisted of design and construction tasks. The students were collaboratively involved in the process of designing a tricycle for a real customer. This real-life activity creates opportunities for students to develop and use models, which can be used in more than in one context. The case study explored how the teachers deal with the students' explicit and implicit need for knowledge and skills. The main findings are that teachers more often provide this knowledge, rather than guide the students in reconstructing it, and towards the end of the project, knowledge tended to remain situated

Reflexões sobre Relações entre Currículo, Avaliação e Formação de Professores na Área de Educação Matemática

ResumoNo presente texto, o objetivo é trazer algumas reflexões voltadas para a articulação das pesquisas sobre currículo, avaliação e formação de professores na área de Educação Matemática, particularmente voltadas para o modo como se articulam os projetos/ações que constituem políticas públicas no Brasil, considerando-se essa tríade. Essas reflexões baseiam-se em experiências vivenciadas em nosso grupo de pesquisa e em projetos/ações governamentais de reorganização curricular e formação de professores. Destacamos a necessidade de desenvolver de forma articulada pesquisas sobre currículo, avaliação e formação de professores e que seus resultados sejam mais bem divulgados, com vistas a contribuir efetivamente na formulação de políticas públicas. Ainda no âmbito de projetos/ações que constituem políticas públicas no Brasil, observa-se que as propostas de discussão curricular, de avaliação e de formação de professores conversam pouco entre si e são implementadas como se fossem autossuficientes, com desarticulação visível