Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

Exploring the model development process in discrete-event simulation: insights from six expert modellers

This paper explores the model development process in discrete-event simulation (DES) by reporting on an empirical study that follows six expert modellers while building simulation models. DES is a widely used modelling approach, however little is known about the modelling processes and methodology adopted by modellers in practice. Verbal Protocol Analysis is used to collect data, where the participants are asked to speak aloud while modelling. The results show that the expert modellers spend a significant amount of time on model coding, verification and validation, and data inputs. The modellers iterate often between modelling activities. Patterns of modelling behaviour are identified, suggesting that the modellers adopt distinct modelling styles. This study is useful in that it provides an empirical view of existing DES modelling practice, which in turn can inform existing research and simulation practice as well as teaching of DES modelling to novices.     

Examining novice teacher leaders’ facilitation of mathematics professional development

This paper reports on novice teacher leaders’ efforts to enact mathematics PD through an analysis of their facilitation in workshops conducted at their schools. We consider the extent to which teacher leaders facilitated the Problem-Solving Cycle model of PD with integrity to its key characteristics. We examine the characteristics they enacted particularly well and those that were the most problematic to enact. Facilitators were generally successful with respect to workshop culture and selecting video clips for use in the PD workshops. They had more difficulty supporting discussions to foster aspects of mathematics teachers’ specialized content knowledge and pedagogical content knowledge. We suggest a number of activities that may help to better prepare novice PD leaders to hold effective workshops. Furthermore, we conjecture that leaders of mathematics PD draw from a construct we have labeled Mathematical Knowledge for Professional Development (MKPD), and we posit some domains that may comprise this construct.     

Co-facilitation of study groups around animated scenes: the discourse of a moderator and a researcher

This paper examines the methodological issues pertaining to study group facilitation assisted by representations of teaching. The study groups, which comprised high school geometry teachers, aimed at achieving two goals: enhancing professional development among teachers and studying the practical rationality of geometry teaching. Therefore, session facilitation involved playing two roles that were often contradictory. A specific co-facilitation model was developed to achieve both goals. This paper examines the characteristics of the discourse of each of the facilitators. The results of this study are used to explain how the facilitators’ discourse created a relevant environment for achieving both goals.     

Novice facilitators’ changes in practices: From launching to managing discussions about mathematics teaching

Novice facilitators of professional development (PD) programs for mathematics teachers often face challenges in leading productive discussions and achieving the goals of these programs. Although research in this area is gradually accumulating, not much is known about how novice facilitators address these challenges and change their practices accordingly. This paper presents case studies of two novice facilitators of PD programs in two different countries. The analyses look at their work over one year, to illustrate the changes in their practices while managing discussions. The results show that although the facilitators operated in different contexts, their practices and their processes of change resembled, suggesting that these processes are not idiosyncratic. We argue that novice facilitators’ changes in practices correspond to changes in their resources, orientations, goals, and identities and that PD program teams can support these changes.     

Facilitation practices in decision workshops

Decision workshops, sometimes called decision conferences, help a group of decision makers gain a shared understanding of a decision problem, analyse issues and commit to an action plan under the guidance of an experienced facilitator. This work seeks to identify best practice in the early stages of the facilitation of such workshops when the emphasis is placed on problem structuring and the main issues of a complex decision problem are identified and explored. Four decision workshops, based on the same hypothetical scenario but facilitated by a different person, were organized. Video material of the simulated workshops was analysed to compare and contrast the facilitated meetings including the problem structuring methods used. A framework for studying facilitation practices emerged. The effect of a facilitator on the structuring of the problem, the group decision process and the outcome of the workshop was studied. The results of the work indicate that a facilitator's style and approach to the workshop may have an impact on the action plan devised. Further research is required to generalize the findings of our work.     

What Matters Most: A Comparison of Expert and Novice Teachers' Noticing of Mathematics Classroom Events

In this study, we examined 10 expert and 10 novice teachers' noticing of classroom events in China. It was found that both expert and novice teachers, who were selected from two cities in China, highly attended to developing students' mathematics knowledge coherently and developing students' mathematical thinking and ability; they also paid attention to students' self‐exploratory learning, students' participation, and teachers' instructional skills. Furthermore, compared with novice teachers, expert teachers paid greater attention to developing mathematical and high‐order thinking, and developing mathematics knowledge coherently, but paid less attention to teachers' guidance. Moreover, we further illustrated the qualitative differences and similarities in their noticing of classroom events. Finally, we discussed the findings and relevant implications.     

Conceptual modelling and the project process in real simulation projects: a survey of simulation modellers

A survey was used to obtain information on the processes and methods used by simulation experts in real projects. The 102 survey respondents answered questions about their most recent simulation project. This paper presents some of the survey results, focussing mainly on conceptual modelling and the pattern of time allocation to different topics. There are a wide range of findings that include the modellers making changes to the initial conceptual model during subsequent tasks in most of the projects usually by adding complexity, model coding taking on average about twice the time of other topics, and the topics generally occurring in single blocks of time (at the resolution of the survey data collection) but with considerable overlaps. The results give an insight into the way experts approach simulation projects and their problem solving strategies. A potential application is in training novice modellers, particularly in developing ‘craft skills’. The results also provide an empirical basis for further research, especially in conceptual modelling.     

A systematic comparison of professional exchange rate forecasts with the judgemental forecasts of novices

Professional forecasters in foreign exchange markets are not able to beat naive forecasts. In order to find reasons for this phenomenon we compare the empirical forecasts of experts with the experimentally generated forecasts of novices for the EUR/USD exchange rate in three different forecast horizons. Although the subjects are only provided with the realizations of the exchange rate and are not supported by any statistical procedures they outperform experts in accuracy. Professionals consistently expect a reversal of forgoing exchange rate changes whereas novices extrapolate trends. The judgemental forecasts appear to be unbiased and professionals appear to be biased. We demonstrate that professionals are influenced by the fundamental value—an irrelevant anchor in speculative exchange markets. The poor performance of the experts is not a common failure of human decision-making in market environments but caused by misleading information.     

Psychological aspects of equation-based modelling

Psychological aspects of equation-based modelling languages like Modelica are under-represented in literature. This does not reflect the growth of the corresponding userbase. In this paper we try to close this gap by tackling the problem from three sides: we conduct expert interviews, we conduct an experiment based on self-reported timings to analyse the effects of inheritance on understandability, and we conduct an online experiment to analyse the effects of model representations on the performance at modelling tasks. The expert interviews suggest that experienced modelling experts tend to develop their models from the top-down, while novices do the opposite. Results from the second experiment indicate that the effect of inheritance on the time to understand a model is both significant and large. The results of the last experiment imply that graphical representations outperform block-diagrams for several metrics. These results open a broad research field on the theory of good modelling practice.     

The long term impact of a coherence based model for mathematics intervention

This article presents a large-scale longitudinal study of hundreds of students across the state of Kentucky that participated in a dual-focus mathematics intervention initiative when they were in the third grade. Rather than an exclusive focus on intervention, this initiative focused on both (i) high quality pull-out intervention and (ii) coherence between pull-out intervention and classroom instruction. The study found that over half of the third grade intervention students that participated in this initiative were classified as “novice” (the lowest possible performance category) on state standardized mathematics assessments at the end of the third grade. However, over the course of the following four years, the novice reduction rate of these students was significantly (p < .01) greater than other novices in Kentucky that did not participate in the initiative. These findings indicate that when implementing intervention initiatives to help students that are struggling with mathematics, it may be important to establish coherence between pull-out intervention and classroom instruction. The long term impact of this approach among traditionally underrepresented minorities suggest that this publication may provide insight into important equity issues where long-term analyses may sometimes be needed to capture the full impact of intervention initiatives.     

Developing flexible procedural knowledge in undergraduate calculus

Mathematics experts often choose appropriate procedures to produce an efficient or elegant solution to a mathematical task. This flexible procedural knowledge distinguishes novice and expert procedural performances. This article reports on an intervention intended to aid the development of undergraduate calculus students’ flexible use of procedures. Two sections of the same course were randomly assigned to treatment and control conditions. Treatment students completed an assignment on which they resolved derivative-finding problems with alternative methods and compared the two resulting solutions. Control students were assigned a list of functions to differentiate. On the post-intervention test, treatment students were more likely to use a variety of solution methods without prompting than the control. Moreover, the set of treatment section solutions were closer to those of a group of mathematics experts. This study presents evidence that not only is flexible procedural knowledge a key skill in tertiary mathematics, it can be taught.     

Exploring Images of Parental Participation in Mathematics Education: Challenges and Possibilities

In this article, we draw on research within a large project on parental involvement in mathematics education in working-class Latino communities. Our research is situated within a sociocultural framework and, in particular, the concept of funds of knowledge. We also draw on research on parental involvement in education, particularly that which critically examines issues of power and perceptions of parents. We build on the concept of dialogic learning and on the characterization of parents as intellectual resources and present a model for parental involvement in mathematics in which parents engage as (a) parents, (b) learners, (c) facilitators, and (d) leaders. In particular, in this article, we focus on the third component—parents as facilitators of mathematics workshops for the community at large—centering on some of the challenges as parents and teachers engage in this type of collaboration. We also look at the possibilities afforded by a model for parental involvement that views parents as intellectual resources. By looking at examples of interactions among parents and teachers, and among parents and children, in mathematics workshops, we challenge conventional notions about parental involvement—in particular, as they apply to working-class, language/ethnic “minority” parents.     

Exploring Images of Parental Participation in Mathematics Education: Challenges and Possibilities

In this article, we draw on research within a large project on parental involvement in mathematics education in working-class Latino communities. Our research is situated within a sociocultural framework and, in particular, the concept of funds of knowledge. We also draw on research on parental involvement in education, particularly that which critically examines issues of power and perceptions of parents. We build on the concept of dialogic learning and on the characterization of parents as intellectual resources and present a model for parental involvement in mathematics in which parents engage as (a) parents, (b) learners, (c) facilitators, and (d) leaders. In particular, in this article, we focus on the third component—parents as facilitators of mathematics workshops for the community at large—centering on some of the challenges as parents and teachers engage in this type of collaboration. We also look at the possibilities afforded by a model for parental involvement that views parents as intellectual resources. By looking at examples of interactions among parents and teachers, and among parents and children, in mathematics workshops, we challenge conventional notions about parental involvement—in particular, as they apply to working-class, language/ethnic "minority" parents.     

Reducing the graphical user interface of a dynamic geometry system

Graphical user interfaces (GUIs) of dynamic or interactive geometry software (DGS) allow users to interact with the DGS by using a computer mouse. Clicking on a GUI icon performs an action like choosing a construction tool or manipulating an object. For novices, it may be difficult to recognize and recall the icons needed for a task. Learning mathematics and learning the use of a dynamic geometry system at the same time could lead to cognitive overload. Several DGS systems try to solve this problem by offering different GUIs: expert users can choose between a wide range of icons, while for novice users only the most basic icons are presented. By preselecting a specific set of icons, a teacher can adapt a DGS to create a tool, which meets specific pedagogical demands. Two experiments were conducted to investigate the effects of reducing GUIs of a DGS. In experiment 1, which was carried out with full and reduced interfaces of the DGS Cinderella, the eye movements and gaze points of the users were recorded by an eye tracker. The time taken by users to find given icons in different types of interfaces was measured. In experiment 2, students measured the angle sums of polygons using the DGS Cinderella with a full or a reduced interface. No significant effects of GUI reduction were found in both experiments. The results are discussed and ideas for future research are presented.     

Teacher Mentoring as a Community Effort

This article presents the findings from a study of a mentoring program for novice mathematics and science teachers, which was provided by their teacher education program. This study reports the findings of interviews with novice math and science teachers, their mentors, and the mentoring program administrators to explore stakeholder perceptions of mentoring support. Findings suggest the importance of using multiple mentoring strategies to develop, support, and retain high‐quality math and science teachers in the teaching profession. This study contributes to what is known about the role that teacher education programs may play in mentoring novice math and science teachers who have graduated from their programs.     

The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.