Fostering construction of a meaning for multiplication that subsumes whole-number and fraction multiplication: A study of the Learning Through Activity research program

We report on a teaching experiment intended to foster a concept of multiplication that would both subsume students’ multiple-groups concept of whole number multiplication and provide a conceptual basis for understanding multiplication of fractions. The teaching experiment, which used a rigorous single-subject methodology, began with an attempt to build on students’ multiple-groups concept by promoting generalizing assimilation. This was not totally successful and led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to several significant conclusions, which in turn led to a revised hypothetical learning trajectory. The revised trajectory aims to foster a concept of multiplication as a change in units.     

Empirically-based hypothetical learning trajectories for fraction concepts: Products of the Learning Through Activity research program

The Measurement Approach to Rational Number (MARN) Project, a project of the ongoing Learning Through Activity (LTA) research program, produced eleven hypothetical learning trajectories (HLTs) for promoting fraction concepts. Four of these HLTs are the subject of research reports. In this article, we present the other seven HLTs We judged that the data and analyses of these seven would not separately make sufficient contributions to merit individual research reports. However, presenting these seven HLTs together was intended to meet the following goals:1. To give a broad set of examples of HLTs developed based on the LTA theoretical framework.2. To complete a set of HLTs that provide a comprehensive example of HLTs built on prior HLTs.3. To make available for future research and development the full set of HLTs generated by the MARN Project.LTA researchers have focused on how learners abstract a concept through their mathematical activity and how the abstractions can be promoted. The MARN Project continued this inquiry with rigorous single-subject teaching experiments.     

An empirically-based trajectory for fostering abstraction of equivalent-fraction concepts: A study of the Learning Through Activity research program

Promoting deep understanding of equivalent-fractions has proved problematic. Using a one-on-one teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.     

Promoting reinvention of a multiplication-of-fractions algorithm: A study of the Learning Through Activity research program

Whereas proficiency in performing the canonic multiplication-of-fractions algorithm is common, understanding of the algorithm is much less so. We conducted a teaching experiment with a fifth-grade student, based on an initial hypothetical learning trajectory (HLT), to promote reinvention of the multiplication-of-fractions algorithm. The instructional intervention built on two concepts, recursive partitioning and distributive partitioning. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the student through which she could abstract the necessary concepts. The results of the teaching experiment were analyzed and, based on conclusions from the research, a revised HLT was generated. Recursive partitioning and distributive partitioning proved to be a strong foundation for construction of the algorithm.     

Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework

We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.     

Further elaboration of the Learning Through Activity research program

This article was prompted by the thoughtful commentaries of Norton, Tzur, and Dreyfus. Their commentaries pointed out important ideas that were left implicit or only partially explained. The clarifications made here include how the Learning Through Activity (LTA) research program differs from related research programs, the relation of the LTA theoretical framework to scheme theory, our choice to not employ the construct of perturbation in explaining learning, and the structure of hypothetical learning trajectories. In addition, I discuss a type of mathematical concept that we have not discussed previously.     

Learning trajectory of a student with learning disabilities for the concept of length: A teaching experiment

This study aims to map the learning trajectory (LT) of a student with learning disabilities (LDs) regarding the unit concept in length measurement and the usage of rulers. The article draws on data from a teaching experiment with a 10-year-old student with LDs in Turkey. Data were analyzed in two stages, including microanalysis, where each successive teaching session was separately analyzed, and macroanalysis, where the teaching sessions regarding interrelated instructional goals were analyzed to construct the LT. The main findings of the study illustrate that this student with LDs eliminated her misconceptions about the unit concept and using a ruler, accomplished the determined instructional goals to a large extent, and reached a higher level of thinking with a 4-month teaching experiment designed based on her specific developmental capacity.     

A cycle for validating a learning progression illustrated with an example from the concept of function

A learning progression, or learning trajectory, describes the evolution of student thinking from early conceptions to the target understanding within a particular domain. As a complex theory of development, it requires conceptual and empirical support. In earlier work, we proposed a cycle for the validation of a learning progression with four steps: 1) Theory Development, 2) Examination of Empirical Recovery, 3) Comparison to Competing Models, and 4) Evaluation of Instructional Efficacy. A group of experts met to discuss the application of learning sciences to the design, use, and validation of classroom assessment. Learning progressions, learning trajectories, and how they can support classroom assessment were the main focuses. Revisions to the cycle were suggested. We describe the adapted cycle and illustrate how the first third of it has been applied towards the validation of a learning progression for the concept of function.     

Issues in theorizing mathematics learning and teaching: A contrast between learning through activity and DNR research programs

By continuing a contrast with the DNR research program, begun in Harel and Koichu (2010), I discuss several important issues with respect to teaching and learning mathematics that have emerged from our research program which studies learning that occurs through students’ mathematical activity and indicate issues of complementarity between DNR and our research program. I make distinctions about what we mean by inquiring into the mechanisms of conceptual learning and how it differs from work that elucidates steps in the development of a mathematical concept. I argue that the construct of disequilibrium is neither necessary nor sufficient to explain mathematics conceptual learning. I describe an emerging approach to instruction aimed at particular mathematical understandings that fosters reinvention of mathematical concepts without depending on students’ success solving novel problems.     

A learning trajectory for university students regarding the concept of vector

This paper presents and evaluates a hypothetical learning trajectory by which students bridge the transition from elementary to university-level instruction regarding the concept of vector. The trajectory consists of an instructional sequence of five tasks and begins with a problem in context. Each task is carried out with the support of a Virtual Interactive Didactic Scenario, accompanied by exploration and guided learning sheets, in which the problem is introduced through the simulation of the movement of a robotic arm. This proposal was implemented at the beginning of the SARS-CoV-2 pandemic using various digital media. Two teaching experiments were carried out with engineering students at a Mexican public university. We present the hypothetical learning trajectory that should be followed toward solving the task, and contrast it in each case with the students’ actual learning trajectory. The results show that more than 70 % of the students successfully transitioned from the geometrical vector representation of elementary physics to the algebraic one.     

Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

The finite-to-finite strand of a learning progression for the concept of function: A research synthesis and cognitive analysis

In this paper we report validation efforts around the finite-to-finite strand of a provisional learning progression (LP) for the concept of function. We regard an LP as an empirically-verified account of how student understandings form over time and in response to instruction. The finite-to-finite strand of the LP was informed by literature on students’ thinking and learning related to functions as well as the Algebra Project’s curricular approach, which is designed for students who are traditionally underserved by mathematics education. Developing and validating an LP is a multi-step, cyclic process. Here we report on one step in this process, an item and response analysis. Data sources include 680 students’ responses to 13 multipart computer-delivered tasks. Results suggest that revisions to the items, associated scoring rubrics, and in some instances the LP are warranted. We illustrate this task, rubric, and LP revision process through an item analysis for a selected task.     

Peer interactions in a computer lab: reflections on results of a case study involving web-based dynamic geometry sketches

A case study, originally set up to identify and describe some benefits and limitations of using dynamic web-based geometry sketches, provided an opportunity to examine peer interactions in a lab. Since classes were held in a computer lab, teachers and pairs faced the challenges of working and communicating in a lab environment.Research has shown that particular teacher interventions provide motivation for the consideration of new ideas, and help uncover misunderstandings that may interfere with student progress [Towers, J. (1999). In what ways do teachers interventions interact with and occasion the growth of students’ mathematical understanding. Doctoral Dissertation, University of British Columbia, Unpublished]. Examples of student discourse presented here suggest that certain peer interactions act in similar ways—helping propel students towards new understanding. On the other hand, they also show that some peer interactions, although superficially similar to teacher interventions, may hamper student progress.     

Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.     

How students structure their investigations and learn mathematics: insights from a long-term study

This paper reports on the mathematical thinking of participants of a long-term study, now in its 17th year, who did mathematics together through their public school and early university years. In particular, it describes how fundamental ideas and images of a cohort group of students are elaborated and presented in symbolic expressions of generalized mathematical ideas while exploring problems in grades 10 and 11. From high school and university interview data, we learn from participants how they viewed their mathematical activity in structuring their investigations and justifying their solutions.     

A local instructional theory for the guided reinvention of the quotient group concept

In this paper we describe a local instructional theory for supporting the guided reinvention of the quotient group concept. This local instructional theory takes the form of a sequence of key steps in the process of reinventing the quotient group concept. We describe these steps and frame them in terms of the theory of Realistic Mathematics Education. Each step of the local instructional theory is illustrated using example instructional tasks and either samples of students’ written work or excerpts of discussions.     

A connectionist model of the organizational learning curve

Organizational learning can be understood as a spontaneous development of routines. Mathematically, this process can be described as a search for better paths on a graph whose nodes are humans and machines. Since the rules for connecting nodes depend on their ability to process goods, the slope of the learning curve may be connected to physical and psychological properties. Two suggestive examples are discussed. Guido Fioretti, born 1964, graduated in Electronic Engineering and obtained a PhD in Economics from the University of Rome “La Sapienza”. He is currently an assistant professor at the University of Bologna, Italy.His research interests span from decision theory to economics and organization science. In particular, he is interested in linking structural development to cognitive processes. The present article has been conceived as a theoretical underpinning of agent-based simulations of organizations. In particular, future applications of the Java Enterprise Simulator () may test the usefulness of the results derived herein.     

Application of a Near-Optimal Reinforcement Learning Controller to a Robotics Problem in Manufacturing: A Hybrid Approach

Optimization theory provides a framework for determining the best decisions or actions with respect to some mathematical model of a process. This paper focuses on learning to act in a near-optimal manner through reinforcement learning for problems that either have no model or the model is too complex. One approach to solving this class of problems is via approximate dynamic programming. The application of these methods are established primarily for the case of discrete state and action spaces. In this paper we develop efficient methods of learning which act in complex systems with continuous state and action spaces. Monte-Carlo approaches are employed to estimate function values in an iterative, incremental procedure. Derivative-free line search methods are used to obtain a near-optimal action in the continuous action space for a discrete subset of the state space. This near-optimal control policy is then extended to the entire continuous state space via a fuzzy additive model. To compensate for approximation errors, a modified procedure for perturbing the generated control policy is developed. Convergence results under moderate assumptions and stopping criteria are established.     

Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and span. The objective is to evaluate this didactic proposal by determining the level of match between the hypothetical learning trajectory (HLT) designed in this study with the actual learning trajectory in the second experimental cycle of an investigation design-based research more extensive. The results show a high level of match between the trajectories in more than half of the conjectures, which gives evidence that the HLT has supported, in many cases, the achievement of the learning objective, and that additionally mathematical modelling contributes to the construction of these linear algebra concepts.     

A study of the similarities between topics

This paper presents a new methodology whereby the relationships among points of interest appearing in Internet Web pages may be graphically represented. This is done by quantifying the various points and representing them graphically using an intuitive interpretation of the similarity of sets. The similarity or interrelationships existing among various lines of research may be studied based on information taken from the Web pages using function taken from Learning Theory. A visual analysis of those interrelationships has been applied to the field of machine learning during the time interval May 2003–May 2004.