Research on mathematics teachers as partners in task design

Mathematical tasks and tools, including tasks in the form of digital tools, are key resources in mathematics teaching and in mathematics teacher education. Even so, the ‘design’ of mathematical tasks is perceived in different ways: sometimes seen as something distinct from the teaching and learning process, and sometimes as integral to it. Whilst task design has often been carried out by designers or mathematicians (perhaps as textbook authors), the focus for this review article is research that has involved mathematics teachers as partners in the design of tasks. The article provides a state-of-the-art review of relevant literature and is presented under three headings that consider, in turn, the role of mathematical ‘tasks’; the nature of ‘task design’; and the notion of ‘partnerships for task design’ in mathematics education. Subsequently, we present current research that is providing new insights into tasks, task design, and task design partnership. Based on this, we argue that ‘task design’ needs to pay particular attention to what to design, which tools are necessary or beneficial, and under what conditions; digital tools and task resources offer particular affordances that traditional resources cannot provide; and not only do teachers benefit from being partners in task design (in terms of their professional learning) but without their involvement some aspects of task design would most likely be neglected.     

Effects of variations in task design on mathematics teachers’ learning experiences: a case of a sorting task

The goal of the study presented in this article was to examine how variations in task design may affect mathematics teachers’ learning experiences. The study focuses on sorting tasks, i.e., learning tasks that require grouping a given set of mathematical items, in as many ways as possible, according to different criteria suggested by the learners. We present an example of a sorting task for which the items to be grouped are related to basic concepts of analytical geometry that are connected to the notion of loci of points. Based on a design experiment of three iterations with practicing secondary school mathematics teachers, we report on intended and enacted objects of learning inherent in three versions of the task. Empirically based suggestions are drawn about design of sorting tasks that potentially evoke desirable learning experiences.     

Analyzing Mathematical Tasks: A Catalyst for Change?

In this study we investigate a strategy for engaging high school mathematics teachers in an initial examination of their teaching in a way that is non-threatening and at the same time effectively supports the development of teachers’ pedagogical content knowledge [Shulman (1986). Educational Researcher, 15(2), 4–14]. Based on the work undertaken by the QUASAR project with middle school mathematics teachers, we engaged a group of seven high school mathematics teachers in learning about the Levels of Cognitive Demand, a set of criteria that can be used to examine mathematical tasks critically. Using qualitative methods of data collection and analysis, we sought to understand how focusing the teachers on critically examining mathematical tasks influenced their thinking about the nature of mathematical tasks as well as their choice of tasks to use in their classrooms. Our research indicates that the teachers showed growth in the ways that they consider tasks, and that some of the teachers changed their patterns of task choice. Further, this study provides a new research instrument for measuring teachers’ growth in pedagogical content knowledge. An earlier version of this paper was presented at the American Educational Research Association Annual Meeting, New Orleans, LA, April 2002.     

Teaching primary mathematics with challenging tasks: How should lessons be structured?

Engaging students in a challenging (cognitively demanding) task and launching a mathematics lesson with a task before instruction are two characteristics of a reform-oriented approach to mathematics instruction often considered together. The authors systematically contrasted teaching with challenging tasks using a task-first lesson structure with that of a discussion-first lesson structure to three composite classes of first- and second-grade students (n = 73). Subsequent assessments of mathematical performance revealed that the discussion-first lesson structure was somewhat more efficacious in improving fluency performance but both structures similarly improved problem-solving performance. The findings suggest there is more than one way of incorporating challenging tasks into mathematics lessons to produce sizeable learning gains.     

Toward a Synthesis of Cognitive Load Theory, Four-Component Instructional Design, and Self-Directed Learning

This article explores the opportunities to apply cognitive load theory and four-component instructional design to self-directed learning. Learning tasks are defined as containing three elements: learners must (a) perform the tasks, (b) assess their task performance, and (c) select future tasks for improving their performance. Principles to manage intrinsic and extraneous load for performing learning tasks, such as simple-to-complex ordering and fading-guidance strategies, are also applicable to assessing performance and selecting tasks. Moreover, principles to increase germane load, such as high variability and self-explanation prompts, are also applicable to assessment and selection. It is concluded that cognitive load theory and four-component instructional design provide a solid basis for a research program on self-directed learning.     

The differential effects of task complexity on domain-specific and peer assessment skills

In this study the relationship between domain-specific skills and peer assessment skills as a function of task complexity is investigated. We hypothesised that peer assessment skills were superposed on domain-specific skills and will therefore suffer more when higher cognitive load is induced by increased task complexity. In a mixed factorial design with the between-subjects factor task complexity (simple, n?=?51; complex, n?=?59) and within-subjects factor task type (domain-specific, peer assessment), secondary school students studied four integrated study tasks, requiring them to learn a domain-specific skill (i.e. identifying the six steps of scientific research) and to learn how to assess a fictitious peer performing the same skill. Additionally, the students performed two domain-specific test tasks and two peer assessment test tasks. The interaction effect found on test performance supports our hypothesis. Implications for the teaching and learning of peer assessment skills are discussed.     

Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.

Connecting engagement and focus in pedagogic task design

Teachers of mathematics face a particular tension, which the authors call the planning paradox If teachers plan from objectives, the tasks they set are likely to be unrewarding for the pupils and mathematically impoverished. Planning from tasks may increase pupils' engagement but their activity is likely to be unfocused and learning difficult to assess. By seeking inspiration from research in the areas of curriculum design, the nature of authenticity in the classroom and the use of tools, and by looking retrospectively at the design of computer‐based tasks that have underpinned their research for many years, the authors recognise a theme of purposeful activity, leading to a planned appreciation of utilities for certain mathematical concepts. The authors propose utility as a third dimension of understanding, which can be linked to purpose in the effective design of tasks. The article concludes with a set of heuristics to guide such planning.     

Selecting learning tasks: Effects of adaptation and shared control on learning efficiency and task involvement

Complex skill acquisition by performing authentic learning tasks is constrained by limited working memory capacity [Baddeley, A. D. (1992). Working memory. Science, 255, 556–559]. To prevent cognitive overload, task difficulty and support of each newly selected learning task can be adapted to the learner’s competence level and perceived task load, either by some external agent, the learner herself, or both. Health sciences students (N = 55) participated in a study using a 2 × 2 factorial design with the factors adaptation (present or absent) and control over task-selection (program control or shared control). As hypothesized, adaptation led to more efficient learning; that is, higher learning outcomes combined with less effort invested in performing the learning tasks. Shared control over task-selection led to higher task involvement, that is, higher learning outcomes combined with more effort directly invested in learning. Adaptation also produced greater task involvement.     

Contrasting executive function development among primary school children from Hong Kong and Germany

Previous research findings indicate that young children from East Asia outperform their counterparts from Europe and North America on executive function (EF) tasks. However, very few cross-national studies have focused on EF development during middle childhood. The current study assessed the EF performance of 170 children in grades 2 and 4 from Hong Kong (n = 80) and Germany (n = 90) in a cross-sectional design. Children completed tasks assessing the main components of EF, namely inhibition (child-friendly Stroop task), updating (Object Span task), and shifting (Contingency Naming task). Results of multilevel models showed that all three EF measures differentiated well between younger and older children across the full sample. However, contrary to our hypothesis and previous research, we did not find any significant differences in EF performance between children from Hong Kong and Germany at primary school age. Our findings highlight the possibility that issues related to the measurement of EF and features specific to Hong Kong and Germany underlie our results.

     

Associations of Number Line Estimation With Mathematical Competence: A Meta‐analysis

The number line estimation task is widely used to investigate mathematical learning and development. The present meta‐analysis statistically synthesized the extensive evidence on the correlation between number line estimation and broader mathematical competence. Averaged over 263 effect sizes with 10,576 participants with sample mean ages from 4 to 14 years, this correlation was r = .443. The correlation increased with age, mainly because it was higher for fractions than for whole numbers. The correlation remained stable across a wide range of task variants and mathematical competence measures (i.e., counting, arithmetic, school achievement). These findings demonstrate that the task is a robust tool for diagnosing and predicting broader mathematical competence and should be further investigated in developmental and experimental training studies.     

Design and Reflection Help Students Develop Scientific Abilities: Learning in Introductory Physics Laboratories

Design activities, when embedded in an inquiry cycle and appropriately scaffolded and supplemented with reflection, can promote the development of the habits of mind (scientific abilities) that are an important part of scientific practice. Through the Investigative Science Learning Environment (ISLE), students construct physics knowledge by engaging in inquiry cycles that replicate the approach used by physicists to construct knowledge. A significant portion of student learning occurs in ISLE instructional labs where students design their own experiments. The labs provide an environment for cognitive apprenticeship enhanced by formative assessment. As a result, students develop interpretive knowing that helps them approach new problems as scientists. This article describes a classroom study in which the students in the ISLE design lab performed equally well on traditional exams as ISLE students who did not engage in design activities. However, the design group significantly outperformed the non-design group while working on novel experimental tasks (in physics and biology), demonstrating the application of scientific abilities to an inquiry task in a novel content domain. This research shows that a learning environment that integrates cognitive apprenticeship and formative assessment in a series of conceptual design tasks provides a rich context for helping students build scientific habits of mind.     

Long‐Term Episodic Memory in Children with Attention‐Deficit/Hyperactivity Disorder

Twenty‐nine grade‐matched 4th–8th‐grade males, 12 with attention‐deficit/hyperactivity disorder (ADHD) (age M= 12.2 years, SD= 1.48), and 17 without (age M= 11.5, SD= 1.59), completed two working memory tasks (digit span and the Simon game) and three long‐term episodic memory tasks (a personal event memory task, story memory task, and picture recognition task). In line with clinical observations, children with ADHD performed worse than peers on all working memory tasks, but performed as well as or better than peers on long‐term episodic tasks, demonstrating particularly detailed memory for personally experienced past events. Participants' parents also completed questionnaires about their children's memories in daily life. Parents rated children with ADHD lower than children without ADHD on working and semantic memory (e.g., remembering names, spelling, and math), but rated them as high or higher on memory for events. Implications for theory and educational practice are discussed.     

Authentic Tasks Online: A synergy among learner,task, and technology

Fostering synergies amongst learner, task, and technology to create innovative and immersive distance learning environments runs counter to the widespread practice of incorporating traditional classroom pedagogical strategies into Web‐based delivery of courses. The most widely accepted model of online higher education appears to be one of reductionism, whereby learning management systems facilitate the design of easily digested packets of information, usually assessed by discrete stand‐alone tests and academic assignments. This article describes a model for the development of authentic tasks that can assist in designing environments of increased, rather than reduced, complexity. It provides a robust framework for the design of online courses, based on the work of theorists and researchers in situated learning and authentic learning. It describes the characteristics of a task's design that facilitates the requirements of an entire course of study being readily satisfied by its completion, where the students make the important decisions about why, how, and in what order they investigate a problem. The article describes several learning environments that were investigated in depth in the study, and explores the synergies that exist between the learners, tasks, and technology engaged in authentic learning settings. The article leads readers to a conceptual understanding of the role of authentic tasks in supporting knowledge construction and meaningful learning, and illustrates the principles of authentic task design for online learning environments.     

Individual differences in relative metacomprehension accuracy: variation within and across task manipulations

In recent decades, increasing numbers of studies have focused on metacomprehension accuracy, or readers’ ability to distinguish between texts comprehended more vs. less well. Following early findings that suggested readers are fairly poor at doing so, a number of studies have identified specific tasks to supplement a single reading of text that have resulted in greater metacomprehension accuracy. One assumption underlying these studies is that, in the absence of such tasks, metacomprehension accuracy is uniformly poor, and given their implementation, readers uniformly improve. Here we describe the individual variation that occurs both in the absence (e.g., within a single text reading manipulation) and presence (e.g., within a rereading or selective rereading task manipulation) of these supplementary tasks (N = 214), in order to make a case for greater attention to individual differences in metacomprehension accuracy. We also introduce a new manipulation in metacomprehension research, selective rereading, and argue that certain types of tasks may be more likely to reveal individual differences in metacomprehension accuracy due to the nature of the task being more or less demanding on working memory capacity.     

The Relations Between Children’s Reading Comprehension, Working Memory, Language Skills and Components of Reading Decoding in a Normal Sample

The primary aim of the current study was to identify the strongest independent predictors of reading comprehension using word reading, language and memory variables in a normal sample of 180 children in grades 3–5, with a range of word reading skills. It was hypothesized that orthographic processing, receptive vocabulary and verbal working memory would all make independent contributions to reading comprehension. The contributions of reading speed, receptive grammatical skills, exposure to print, visuospatial working memory and verbal learning and retrieval (a measure of longer-term retention) were also investigated. Working memory tasks that required the processing and storage of numerical and spatial material were used. One of the numerical working memory tasks was based on the number span task developed by Yuill, Oakhill, and Parkin British Journal of Psychology, 1989, 80, 351–361. A visuospatial equivalent of that task was developed from the forward Corsi block task [Corsi, Abstracts International, 1973, 34, 891]. The results revealed that, after controlling for age and general intellectual ability, the word reading and the language variables had a much stronger relation with reading comprehension than the memory variables. The strongest independent predictor of reading comprehension was orthographic processing since it captured variance in both word reading, language skills and verbal working memory. The forward Corsi task and performance on a measure of verbal learning and retrieval each made small independent contributions to reading comprehension but the contribution of verbal working memory was not significant. It was concluded that tasks measuring the interplay between short-term and long-term memory, in which new information is combined with information already stored in long-term memory, may better predict reading comprehension measured with the text available than working memory tasks which only have a short-term memory component.     

A Comparison of Two Alternate Scaling Approaches Employed for Task Analyses in Credentialing Examination Development

Credentialing examination developers rely on task (job) analyses for establishing inventories of task and knowledge areas in which competency is required for safe and successful practice in target occupations. There are many ways in which task‐related information may be gathered from practitioner ratings, each with its own advantage and limitation. Two of the myriad alternative task analysis rating approaches are compared in situ: one establishing relative task saliency through a single scale of rated importance and another employing a composite of several independent scales. Outcomes regarding tasks ranked by two practitioner groups are compared. A relatively high degree of association is observed between tasks ranked through each approach, yielding comparable, though not identical examination blueprints.     

Self-Regulation and Task Engagement as Predictors of Emergent Language and Literacy Skills

Research Findings: A growing emphasis in the literature on children’s self-regulation signals the need for increased understanding of the ways in which young children become active players in the acquisition of knowledge. In particular, self-regulation may be linked to subsequent academic achievement through greater engagement with the learning tasks and activities made available in the preschool classroom. This study tested preschoolers’ (N = 603) observed task engagement in the classroom as mediating the relations between directly assessed self-regulation and changes in their language and literacy outcomes during the preschool year. Findings indicate that self-regulation is directly related to observed task engagement as well as changes in a host of language and literacy skills. Engagement with tasks and activities in the classroom also partially mediates the association between self-regulation and changes in expressive vocabulary. Mediation through task engagement was not found for receptive language or early literacy skills. Practice or Policy: Findings suggest that the development and evaluation of clearly articulated preschool curricula designed to promote academic achievement by fostering self-regulation is an important direction for future research.     

Listening comprehension and reading comprehension in poor decoders: Evidence for the importance of syntactic and semantic skills as well as phonological skills

The relation between decoding and comprehensionin the oral and written modalities was studiedin a randomly selected group of nine-year-olds,subdivided into good, average and poordecoders. Performances on two types ofcomprehension tasks (story retelling andcloze tasks) were compared and related tophonological, syntactic and semantic abilities.(Story retelling demanded the ability to retellthe gist of a story, while the cloze tasksdemanded precise skills in drawing anaphoricreference across sentence boundaries.) Atwo-way analysis of variance using IQ ascovariate showed that poor decoders scoredlower than average and good decoders on allcomprehension tasks. This suggests a highdegree of interdependence between listeningcomprehension, reading comprehension anddecoding. The associated pattern of oralcorrelates furthermore varied with task demandsand to some extent, independent of modality.Vocabulary was related to the ability toretell a story. Syntax and, in particularphonemic awareness, were on the otherhand more strongly related to the ability todraw anaphoric reference. The results wereinterpreted in favor of ``the phonologicaldeficit hypothesis', but the interactionbetween linguistic sub-skills and task demandswas also underscored.