Comparing the efficacy of early arithmetic instruction based on a learning trajectory and teaching-to-a-target.

It is common for learning advisors to receive a request such as conversation practice or simple linguistic support in our institution simply because learners find them accessible and friendly. Although this may not be the usual role of a learning advisor, it can be the beginning of a long-lasting reflective dialogue. Learning advisors are aware that the learners’ initial interest is not necessarily an opportunity for reflection on their learning process. Nevertheless, they also acknowledge the fact that learner readiness for reflection varies depending on the learner. Thus, while advisors appear to be “conversation partners” at first, they endeavor to create reflective dialogues by incorporating advising strategies in each session. As the learning trajectory that the learner follows is not always straightforward, keeping an open mind and engaging in each advising session is crucial. The following stories depict the importance of openness to learners’ needs and maintaining continuous advising sessions in order to generate transformational learning.

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Designing Tasks for a Dynamic Online Environment: Applying Research into Students’ Difficulties with Linear Equations

Mathematics ◽

10.3390/math9050557 ◽

2021 ◽

Vol 9 (5) ◽

pp. 557

Author(s):

Morten Elkjær ◽

Uffe Thomas Jankvist

Keyword(s):

Design Theory ◽

Teaching And Learning ◽

Linear Equations ◽

Task Design ◽

Tree Structure ◽

Learning Trajectory ◽

Variation Theory ◽

Online Environment ◽

School Students ◽

Personalised Learning

Despite almost half a century of research into students’ difficulties with solving linear equations, these difficulties persist in everyday mathematics classes around the world. Furthermore, the difficulties reported decades ago are the same ones that persist today. With the immense number of dynamic online environments for mathematics teaching and learning that are emerging today, we are presented with a perhaps unique opportunity to do something about this. This study sets out to apply the research on lower secondary school students’ difficulties with equation solving, in order to eventually inform students’ personalised learning through a specific task design in a particular dynamic online environment (matematikfessor.dk). In doing so, task design theory is applied, particularly variation theory. The final design we present consists of eleven general equation types—ten types of arithmetical equations and one type of algebraic equation—and a broad range of variations of these, embedded in a potential learning-trajectory-tree structure. Besides establishing this tree structure, the main theoretical contribution of the study and the task design we present is the detailed treatment of the category of arithmetical equations, which also involves a new distinction between simplified and non-simplified arithmetical equations.

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A Proposed Instructional Theory for Integer Addition and Subtraction

Journal for Research in Mathematics Education ◽

10.5951/jresematheduc.43.4.0428 ◽

2012 ◽

Vol 43 (4) ◽

pp. 428-464 ◽

Cited By ~ 49

Author(s):

Michelle Stephan ◽

Didem Akyuz

Keyword(s):

Number Line ◽

Teaching Experiment ◽

Learning Trajectory ◽

Realistic Mathematics Education ◽

Instructional Theory ◽

Education Theory ◽

Addition And Subtraction ◽

Support Students ◽

Grade Classroom ◽

Viable Model

This article presents the results of a 7th-grade classroom teaching experiment that supported students' understanding of integer addition and subtraction. The experiment was conducted to test and revise a hypothetical learning trajectory so as to propose a potential instructional theory for integer addition and subtraction. The instructional sequence, which was based on a financial context, was designed using the Realistic Mathematics Education theory. Additionally, an empty, vertical number line (VNL) is posited as a potentially viable model to support students' organizing their addition and subtraction strategies. Particular emphasis is placed on the mathematical practices that were established in this setting. These practices indicate that students can successfully draw on their experiences with assets, debts, and net worths to create meaning for integer addition and subtraction.

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