Examining the use of multiple writing and discourse tasks in 5th grade mathematics

Abstract

The purpose of this study is to examine how fifth grade students were impacted by the infusion of multiple writing tasks in mathematics. In this study, writing tasks provided opportunities for students to communicate prior knowledge, share ideas to construct and justify arguments, for reflection, and assessment. In this deductive qualitative study, students’ work samples were analyzed. Findings indicated that students grew in their understanding of mathematics and ability to self-reflect and self-evaluate through multiple opportunities to write for a variety of purposes. The opportunities for constructing mathematical understanding with activities that included writing and discourse also fostered learning between peers. The findings suggest a variety of opportunities to write and engage in mathematics discourse encouraged reflection, evaluation, and learning. Implications for future research include the need to examine the impact of these activities on students’ mathematics understanding as measured by assessments or an analysis of student work samples.     

Use of comics to enhance students’ learning for the development of the twenty-first century competencies in the mathematics classroom

Abstract

This paper discusses the use of comics in teaching mathematics in the secondary mathematics classroom. We explicate how the use of comics in teaching mathematics can prepare students for the twenty-first century competencies. We developed an alternative teaching package using comics for two lower secondary mathematics topics. This alternative teaching package consists of (1) several sets of comic strips expounding all related mathematical concepts in a lively way; (2) tiered practice questions for learning reinforcement; and (3) a set of proposed lesson outlines with suggestions on how to use the comics for mathematics teaching. We also report how one of the teachers in our study used this teaching package in her mathematics lessons. Her lessons were video-recorded and 11 students were interviewed to help us understand how the mathematics comics lessons were enacted and the students’ perception of comics as instruction. We identified instances in which the teacher tweaked the provided resource to further enhance student learning and incorporated elements of the twenty-first century competencies during her lessons. Through selected student interviews, we also identified instances in which students commented on their gain from the new approach from the perspective of the twenty-first century competencies.     

Opportunities to learn: Mathematics textbooks and students’ achievements

This study explores how textbooks function in education. It asked whether opportunities provided in math textbooks to engage in tasks demanding different levels of understanding correlate with students’ achievements on tasks demanding equivalent levels of understanding on a standardized exam. The textbooks evaluated were two 8^th grade mathematics textbooks used by students in the Arab community in Israel, showing that Textbook A makes more cognitive demands than Textbook B. The study correlated textbooks’ cognitive demand with the scores of all 8^th grade students in the Arab community who completed the national math test in 2015 and studied in schools using either Textbook A or B (N = 4040), while attending to mediating variables. The findings show that if a textbook provides the opportunity to engage in tasks demanding higher levels of understanding, students using this book will have higher scores. The study shows that gender and SES play an important role in how opportunities provided in textbooks interact with students’ scores. Many factors influence variations in mathematics achievements within and between nations. The findings illuminate textbooks’ ability to provide opportunities to learn mathematics. As a result, they raise new questions about how teachers use textbooks and about the role of textbooks in promoting access and equity in mathematics education. Although the work explored specific textbooks, its findings shed light on how learning opportunities relate to achievements more generally.     

数学开放性问题和平行任务的设计:处理数学学习差异的一种途径

普及教育推行以来,处理课堂中学生的学习差异一直是一个世界性的课题。在数学科,按照数学问题或习题的难度进行分层教学是一贯的做法,期望不同能力的学生逐阶而上。然而,激发学生的数学思考,提供不同学生更多的学习机会,是处理差异化教学的关键。在面对混合能力的大班教学中,运用数学开放问题,结合平行练习任务进行分层教学或是一条有效的处理学生数学学习差异的途径。     

初中数学“章起始课”的探索与思考

在数学课程标准研制与修订中,一个重要的改进是强调整体把握课程,掌握课程的知识、方法、思想体系,促进学生核心素养、数学核心素养的发展。在数学教学中,如何上好数学“章起始课”的探索和实践是一个重要的抓手。围绕章起始课的教学要遵循六条基本原则:统领性、整体性、逻辑性、发展性、温故性、激趣性。     

Fostering the acquisition of subtraction strategies with interleaved practice: An intervention study with German third graders

Interleaved practice is a promising approach to foster the adaptive use of subtraction strategies. By intermixing strategies, comparison processes are evoked, which prompt more task-based strategy use. However, the effectiveness of interleaved practice in primary school mathematics has not been investigated yet. In the current study, 236 German third graders were randomly assigned to either an interleaved or a blocked condition. Both groups were instructed in using number-based strategies and the standard written algorithm for solving subtraction problems over 14 lessons. The students in the interleaved condition were prompted to compare strategies (between-comparison), while the students in the blocked condition compared the adaptivity of one strategy for different tasks (within-comparison). Our findings show that the students in the interleaved condition solved subtraction tasks with greater adaptiveness and accuracy. The effect on correctness was mediated by greater adaptive strategy use in the interleaved condition.     

One speaker,two languages: Learning opportunities in the mathematics classroom

The data reported in this article come from a large project whose goal was to explore how Latin American students in Catalonia, Spain use their two languages—Spanish and Catalan—to support their learning of mathematics in small groups with other students who are Spanish- or Catalan-dominant. For 5 years, lessons from bilingual mathematics classrooms in three public secondary schools were video-recorded and transcribed. In the presentation of findings, I discuss three language practices that emerged from the analyses of several classroom instances as follows: (1) caution with mathematical vocabulary, (2) invention of terms, and (3) word-for-word translation. One example is chosen to represent each practice and some of its situated effects. The first two examples support the view that the experience of language difficulties, either real or presumed, contributes to generating opportunities that may be beneficial to mathematics learning. The third example, where the focus on the mathematics is hindered, points to a contrasting finding. Unlike other studies in the field, which have reported the difficulties and obstacles that arise in learning and teaching due to bilingualism in the classroom, I propose a change of focus through the conceptualization of language as a potential for thinking and doing, and particularly for learning and teaching mathematics.     

The nature of student predictions and learning opportunities in middle school algebra

In this article, we describe how using prediction during instruction can create learning opportunities to enhance the understanding and doing of mathematics. In doing so, we characterize the nature of the predictions students made and the levels of sophistication in students’ reasoning within a middle school algebra context. In this study, when linear and exponential functions were taught, prediction questions were posed at the launch of the lessons to reflect the mathematical ideas of each lesson. Students responded in writing along with supportive reasoning individually and then discussed their predictions and rationale. A total of 395 prediction responses were coded using a dual system: sophistication of reasoning, and the mechanism students appeared to utilize to formulate their prediction response. The results indicate that using prediction provoked students to connect among mathematical ideas that they learned. It was apparent that students also visualized mathematical ideas in the problem or the possible results of the problem. These results suggest that using prediction in fact provides learning opportunities for students to engage in mathematical sense making and reasoning, which promotes students’ understanding of the mathematics that they learn.     

Self-Study of a Teacher's Journey toward Standards-Based Mathematics Teaching

The authors describe a journey of self-study during which one author shifted from traditional, teacher-driven approaches to a more problem-based inquiry approach to teaching mathematics. He videotaped a series of lessons taught to sixth-grade students over a semester and analyzed his teaching during discussions with his mentor at the university. The shared analysis helped him learn to involve his students more directly in their own learning. A major lesson learned was that understanding the potential value of a problem-solving approach to teaching mathematics does not guarantee corresponding changes in the classroom. Two vital elements of lesson development emerged as focal points for self-study. After first learning how to prepare worthwhile mathematical tasks, the teacher also learned the importance of implementing effective questioning strategies to help students think more deeply about the mathematics they were learning.     

M in CoMputational thinking: How long does it take to read a book?

Even at the primary level, computational thinking (CT) can support young students to prepare for participating in futures that are immersed in data. In mathematics classrooms, there are few explanations of the ways CT can support students in formulating and solving complex problems. This paper presents an example of a primary classroom investigation (8-9 year olds) over seven lessons of the problem “How long does it take to read a book?” The aim is to illustrate ways a statistical investigation can provide context for CT and demonstrate how the two complement each other to solve problems involving mathematics. The findings highlight opportunities and challenges that students face across the elements of CT—decomposition, abstraction, pattern recognition and modelling, and generalization and algorithmic thinking, including recommendations for teaching.     

TI图形计算器让数学实验进入课堂教学

数学实验对于培养学生创新性思维、创新意识和实践能力具有特殊的作用.在以TI图形计算器为平台的实验环境中进行富有创造性的个性化学习,可以实现数学CAI从演示、练习型向探索型的发展,从而使数学实验真正进入课堂.在教学中应注意:(1)应让学生有足够的时间参与到数学实验活动中;(2)努力提高教师应用TI图形计算器的技术能力和科研能力;(3)现行评价方式应予以改变.     

Using student notebooks to measure opportunity to learn in Botswana and South African classrooms

This article describes how a detailed notebook analysis was used to assess and compare the opportunity to learn of a sample of grade 6 students from 126 classes in South East Botswana and North West Province, South Africa. Students’ mathematics notebooks provided the main data source for estimating how much time is spent on the subject during the 6th grade year, and what mathematics is covered relative to what the study’s student test asked the sample of students. The notebook analysis shows that in sample classes in both countries, most teachers were giving an average number of mathematics lessons that was considerably lower than the number officially programmed under each country’s ministry guidelines. It also reveals that, on average, teachers in both samples of classes covered just over a third of the possible topics related to the test items between the pre-test and post-test. Nevertheless, substantial differences in students’ opportunities to learn in the two countries are evident.     

Pre-service mathematics teachers’ learning and teaching of activity-based lessons supported with spreadsheets

In this study, 12 pre-service mathematics teachers worked in teams to develop their knowledge and skills in using teacher-led spreadsheet demonstrations to help students explore mathematics concepts, stimulate discussions and perform authentic tasks through activity-based lessons. Pre-service teachers’ lesson plans, their instruction of the lessons designed, experiences and lesson enactment outcomes were examined. The pre-service teachers in the study were able to develop and demonstrate their knowledge and skill adequately in designing and enacting activity-based mathematical lessons supported with spreadsheets. The results also showed that the pre-service teachers’ use of the spreadsheet as an instructional tool promoted student in-depth mathematical concept formation and an activity-based learning approach to make lessons less teacher centred and more interactive.     

Teacher Actions to Maximize Mathematics Learning Opportunities in Heterogeneous Classrooms

The basic unit of school based mathematics teaching is the lesson. This article is a contribution to understanding teacher actions that facilitate successful lessons, defined as those that engage all students, especially those who may sometimes feel alienated from mathematics and schooling, in productive and successful mathematical thinking and learning. An underlying assumption is that lessons can seek to build a sense in the students that their experience has elements in common with the rest of the class and that this can be done through attention to particular aspects of the mathematical and socio-mathematical goals. We examine three teacher actions that address the mathematical goals: using open-ended tasks, preparing prompts to support students experiencing difficulty, and posing extension tasks to students who finish the set tasks quickly; as well as actions that address the socio-mathematical goals by making classroom processes explicit. To illustrate and elaborate these actions, we describe a particular lesson taught to a heterogeneous upper primary (age 11–12) class.     

Teaching mathematics after hours

The quantum of instruction time allocated to curriculum subjects such as mathematics facilitates greater exposure to knowledge and skill development, leading to higher levels of achievement. There are a number of manifestations of time to consider when investigating the quantum of time in mathematics education. The OECD have investigated the time allocated to mathematics by schools internationally, while also reporting on the prevalence of private tuition and time spent on homework. However, to date, no attention has been afforded to the provision of ‘hidden curriculum time’ for mathematics. This study seeks to advance the work of the OECD and describes a case study that sought to ascertain if teachers in Ireland provide additional mathematics lessons outside of school hours. The authors examined how prevalent this practice is, how many additional minutes some students receive as a result of this practice and ascertained teachers’ reasons for providing/not providing these additional mathematics classes. The results from this case study show that the majority of teachers at Senior Cycle provide these classes while a large proportion of Junior Cycle teachers also do. In extreme cases, these additional classes expose students to an additional 88.3 h of mathematics over the two-year Senior Cycle programme.