What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’     

The Development of Teacher Knowledge in Support of Student Mathematical Inquiry

Abstract

Applying the Mathematical Knowledge for Teaching framework, we discuss the components of teacher knowledge that might be useful in supporting mathematical inquiry, and examine ways in which we strive to develop this knowledge within a middle grades mathematics program for undergraduate students who are prospective teachers. Using sample activities from multiple courses in the program, we offer general principles of instruction for supporting mathematical inquiry at all grade levels. We contend that an awareness and application of the multiple facets of mathematical knowledge for teaching can be critical to supporting mathematical inquiry across the K-16 spectrum.     

Creativity and mathematical problem posing: an analysis of high school students' mathematical problem posing in China and the USA

In the literature, problem-posing abilities are reported to be an important aspect/indicator of creativity in mathematics. The importance of problem-posing activities in mathematics is emphasized in educational documents in many countries, including the USA and China. This study was aimed at exploring high school students' creativity in mathematics by analyzing their problem-posing abilities in geometric scenarios. The participants in this study were from one location in the USA and two locations in China. All participants were enrolled in advanced mathematical courses in the local high school. Differences in the problems posed by the three groups are discussed in terms of quality (novelty/elaboration) as well as quantity (fluency). The analysis of the data indicated that even mathematically advanced high school students had trouble posing good quality and/or novel mathematical problems. We discuss our findings in terms of the culture and curricula of the respective school systems and suggest implications for future directions in problem-posing research within mathematics education.     

An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge

The importance of students’ problem-posing abilities in mathematics has been emphasized in the K-12 curricula in the USA and China. There are claims that problem-posing activities are helpful in developing creative approaches to mathematics. At the same time, there are also claims that students’ mathematical content knowledge could be highly related to creativity in mathematics, too. This paper reports on a study that investigated USA and Chinese high school students’ mathematical content knowledge, their abilities in mathematical problem posing, and the relationships between students’ mathematical content knowledge and their problem-posing abilities in mathematics.     

Educating pre-service teachers: towards a critical inquiry workforce

ABSTRACT

The Australian Professional Standards for Teachers identify a range of supposedly demonstrable capabilities for graduate teachers. Elaborations privilege the realization of Standards through mentoring, and feedback from senior colleagues. As manifestations of the logic of neo-liberalism they operate as audit technologies for pre-service teachers and their learning. In this paper, we argue for the preparation of graduate teachers who can engage in critical inquiry as means for expanding professional learning, developing pedagogical practices and improving student learning. We report on the preparation of 4th year pre-service teachers to undertake critical inquiry into an aspect of pedagogic practice during their final practicum placement. We first address instrumental framings of teacher preparation. A case is then made for critical practitioner inquiry as an alternative. Empirical data is drawn from the ‘practice architectures’ of an Australian teacher education program as these relate to developing pre-service teacher inquiry designs. We present inquiry questions, abstracts and reflections developed by pre-service teachers over a seven-year period in two discipline groupings, Health and Physical Education and Mathematics and Science, as evidence of possibilities for preparing graduates for a critical inquiry workforce. We conclude in arguing that these possibilities are vital in times framed by a narrowing technical and standardized educational environment.     

Reaching the Mountaintop: Addressing the Common Core Standards in Mathematics for Students with Mathematics Difficulties

The Common Core State Standards provide teachers with a framework of necessary mathematics skills across grades K‐12, which vary considerably from previous mathematics standards. In this article, we discuss concerns about the implications of the Common Core for students with mathematics difficulties (MD), given that students with MD, by definition, struggle with mathematical skills. We suggest that instruction centered on the Common Core will be challenging and may lead to problematic outcomes for this population. We propose that working on foundational skills related to the Common Core standards is a necessary component of mathematics instruction for students with MD, and we provide teachers with a framework for working on foundational skills concurrent with the Common Core standards. We caution, however, that implementation of the Common Core is in its infancy, and the implications of the Common Core for students with MD need to be monitored carefully.     

Artifacts as sources for problem-posing activities

The problem-posing process represents one of the forms of authentic mathematical inquiry which, if suitably implemented in classroom activities, could move well beyond the limitations of word problems, at least as they are typically utilized. The two exploratory studies presented sought to investigate the impact of problem-posing activities when they are implemented in meaningful situations involving the use of cultural artifacts, with its associated mathematics, and particular teaching methods. These situations fall under those defined by Stoyanova and Ellerton (1996) as semi-structured situations. Furthermore, these studies investigated the potential that problem-posing activities have for identifying and stimulating critical and creative thinking in mathematics.     

Summative and Formative Assessments in Mathematics Supporting the Goals of the Common Core Standards

Being proficient in mathematics involves having rich and connected mathematical knowledge, being a strategic and reflective thinker and problem solver, and having productive mathematical beliefs and dispositions. This broad set of mathematics goals is central to the Common Core State Standards for Mathematics.

High-stakes testing often drives instructional practice. In this article, I discuss test specifications and sample assessment items from the two major national testing consortia and the prospects that their assessments will be positive levers for change.

For more than 20 years, the Mathematics Assessment Project has focused on the development of assessments that emphasize productive mathematical practices, most recently creating formative assessment lessons (FALs) designed to help teachers build up student understandings through focusing on student thinking while engaging in rich mathematical tasks. This article describes our recent work.     

Developing Biology Lessons Aimed at Teaching for Understanding: A Domain-specific Heuristic for Student Teachers

Teaching for understanding requires teachers to organize thought-demanding activities which continually challenge students to apply and extend their prior knowledge. Research shows that student teachers often are unable to develop lessons in teaching for understanding. We explored how a domain-specific heuristic can assist student biology teachers in developing problem-posing lessons according to teaching for understanding. Worksheets of lesson plans were analyzed according to criteria for problem-posing lessons. Furthermore, student teachers’ perceptions of the design heuristic’s usefulness were categorized in a cyclical process. In general, the heuristic appeared helpful to most student teachers for designing problem-posing lessons satisfactory according to the criteria. Furthermore, teachers indicated that using the heuristic deepened their subject matter knowledge and their awareness of pupils’ prior knowledge.     

The Common Core State Standards: An Opportunity to Enhance Formative Assessment in History/Social Studies Classrooms

This article discusses the opportunity that the Common Core State Standards (CCSS) present for enhancing formative assessment (FA) in history and social studies classrooms. There is evidence that FA can enhance learning for students if implemented well. Unfortunately, teachers continue to be challenged in implementing FA in their classrooms. We examined reading standards for literacy in history and social studies in lesson plans created by preservice teachers in order to discuss opportunities presented by the CCSS for enhancing FA.     

“Better to be a pessimist”: A narrative inquiry into mathematics teachers' experience of the transition to the Common Core

The Common Core State Standards (CCSS) are a focus of state education policy today influencing curriculum implementation and assessment in public schools. The purpose of this narrative inquiry is to understand how high school mathematics teachers experience the transition period. Based on interviews with mathematics teachers in a high school in the Midwest, we aim at bringing teachers' voices to the forefront. Through teachers' stories, we find that: a) Teachers live in the different zones of enactment; and b) Teachers' ecological view of agency should be used as a link to a transition to the CCSS for creating a genuine dialogue. This article is significant in two ways. First, it informs the administrators and policy makers of how there will be inconsistencies across states, districts, schools and classrooms in the implementation and assessment, and second, it helps to explain the need for new professional development approaches.     

小学数学专家教师与普通教师的专业知识水平与表现的比较研究

通过对174名小学数学专家教师和307名小学数学普通教师的教师专业知识进行测查,研究者发现,总的来说,我国的小学数学教师在数学史、数学思想等方面的知识较差;小学数学教师的PCK总体水平较低,具体地说,教师缺乏“有关学习者”的知识,大部分小学数学教师缺乏探究的意识和解决开放性问题的能力;专家教师和普通教师在教师专业知识的几个大维度上有显著的差异。基于这个研究结果,研究者提出在小学数学教师培训课程中应该补充数学文化思想史方面的内容,鼓励教师参与学生数学学习方面的研究,提升教师的探究意识和问题解决能力,并形成优势互补、分层设计的教师培训课程模式。     

Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.     

Constructing an Inquiry Orientation from a Learning Theory Perspective: Democratizing Access through Task Design

Abstract

It is widely accepted in the mathematics education community that pedagogies oriented toward inquiry are aligned with a constructivist theory of learning, and that these pedagogies effectively support students’ learning of mathematics. In order to promote such an orientation, we first separate the idea of inquiry from its conception as a collection of methods. Then, by grounding those methods in a generally accepted theory of learning, we construct an inquiry-oriented pedagogy from a constructivist perspective. We then discuss the implications of this pedagogy for the design of mathematical tasks that democratize student access to inquiry. This work has implications for educators who wish to enact an inquiry-oriented pedagogy in their classroom in order to support their students’ problem solving and problem posing.     

The curricular importance of mathematics: a comparison of English and Hungarian teachers’ espoused beliefs

This paper reports an interview study of 45 English and 10 Hungarian teachers of mathematics. The semi‐structured interviews focused on the teachers’ professional life‐histories and invited them to discuss their beliefs about the necessary subject content for the teaching and learning of mathematics. Substantial differences emerged between the two cohorts, which accord with well‐defined national perspectives on education in general and mathematics education in particular. They reflect, at national rather than individual levels, the expectations of the curricular frameworks within which teachers operate. English teachers tended to view mathematics as applicable number and the means by which learners are prepared for a world beyond school. Hungarian teachers privileged mathematics as problem‐solving and logical thinking.     

Zen and the art of neriage: Facilitating consensus building in mathematics inquiry lessons through lesson study

One danger of integrating inquiry-based problem-solving activities into mathematics lessons is that different strategies could be accepted without in-depth discussions on the cogency and efficiency of the strategies. To overcome this issue, Japanese teachers typically go through a series of lesson-study-based teacher learning sessions and learn how to help students build consensus on the best mathematical strategy and think deeply about problem solving (neriage in Japanese). Assuming that this can also be an effective model in other cultural contexts, a video-based lesson study was conducted for a group of US teachers to effectively incorporate consensus building discussions in their mathematical inquiry lessons. Through the lesson study, the teachers learned to release control of class discussions to their students and help them discuss and examine different strategies. This article concludes with various aspects that the teachers learned for effectively implementing neriage in their lessons.     

电大数学专业问题提出能力培养模式构建

根据电大数学师范生数学问题提出能力的现状,以及对教与学理论的研究,构建了电大数学专业问题提出能力培养模式。该模式分为六个模块：创设问题情境、探索提出问题的方法、提出数学问题、评价数学问题、开展问题提出的教学实践、撰写与发布总结报告。     

数学文化教育中教师的缄默知识探讨

数学文化素养的提高,教师素质是关键.数学文化知识是一种缄默知识.探讨数学教师的缄默知识的内涵、特点、教育意义及基于缄默知识的数学文化教育能力具有非常重要的意义.     

论数学课题探究教学

数学课题探究教学是指,在教师指导下,围绕某一课题,运用探究的方法主动获取数学知识,独立、自主地解决数学问题,培养科学精神和创造性思维与能力的一种实践活动。它具有内容的开放性和方法的多样性等特点。其内容主要有数学基本概念和规律、数学中的综合问题、现实中的数学问题以及科学前沿中的数学问题等。数学课题探究教学包括教师示范指导和诱导、学生探究、师生共探三个基本环节。     

巧借问题之“水”浇灌学生思维之“花”——谈谈初中数学课堂教学中的问题设计

思维能力的培养是数学教学的核心任务,而问题的设计正是学生思维的导火索。在初中数学教学中,如果教师精于"问题设计",不但会影响学生的数学思维方式,而且还会激发学生的数学探究热情。笔者从事初中数学教学工作多年,在问题设计方面做了有益的尝试与探索。