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Journal articles on the topic 'Worthwhile mathematical tasks'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

Blume, Glendon W., Judith S. Zawojewski, Edward A. Silver, and Patricia Ann Kenney. "Implementing the Professional Standards for Teaching Mathematics: Focusing on Worthwhile Mathematical Tasks in Professional Development: Using a Task from the National Assessment of Educational Progress." Mathematics Teacher 91, no. 2 (February 1998): 156–61. http://dx.doi.org/10.5951/mt.91.2.0156.

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Worthwhile mathematical tasks engage the problem solver in sound and significant mathematics, elicit a variety of solution methods, and require mathematical reasoning. Such problems also prompt responses that are rich enough to reveal mathematical understandings. Just as good classroom practice engages students in worthwhile mathematical tasks, sound professional development does the same with teachers. Providing teachers with opportunities to engage in worthwhile mathematical tasks and to analyze the mathematical ideas underlying those tasks promotes the vision of the Professional Standards for Teaching Mathematics (NCTM 1991).

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2

Caulfield, Richard, Shelly Sheats Harkness, and Robert Riley. "Surprise! Turn Routine Problems into Worthwhile Tasks." Mathematics Teaching in the Middle School 9, no. 4 (December 2003): 198–202. http://dx.doi.org/10.5951/mtms.9.4.0198.

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Textbook problems are often mundane, uninteresting, and allow for very little mathematical thinking or discourse. Because students in our township scored lowest in the category of ratio and proportional reasoning on our statewide assessment, we–three eighth-grade teachers–decided to work together to address this area of weakness. We modified a typical textbook problem related to this mathematical concept in an attempt to foster students' thinking, mathematical understanding, and discourse. In this article, we describe what happened in the classroom when we introduced our “modified” textbook problem.

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3

Mason, John. "Generating Worthwhile Mathematical Tasks in Order to Sustain and Develop Mathematical Thinking." Sustainability 12, no. 14 (July 16, 2020): 5727. http://dx.doi.org/10.3390/su12145727.

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Making use of a phenomenological stance which first and foremost values the lived experience of learners, six tasks are used to illustrate what it might mean for a mathematical task to be deemed worthy of being offered to learners. These take the form of encounters with, and opportunities to develop, pervasive mathematical themes, use of mathematical powers and experience of mathematical concepts and topics. Comments about how worthwhile mathematical tasks can evolve centre around developing the propensity, the habit of mind to extend, vary and generalise for oneself. Mathematical thinking is sustained by developing this disposition.

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4

Brahier, Daniel J., and William R. Speer. "Worthwhile Tasks: Exploring Mathematical Connections through Geometric Solids." Mathematics Teaching in the Middle School 3, no. 1 (September 1997): 20–28. http://dx.doi.org/10.5951/mtms.3.1.0020.

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After a full day of teaching, you take some time reflecting and planning lessons for the next several days. While paging through your course of study, you identify, among others, the following objectives that need to be developed with your middle school mathematics class:

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Chamberlin, Michelle T., and Judith Zawojewski. "A Worthwhile Mathematical Task for Students and Their Teachers." Mathematics Teaching in the Middle School 12, no. 2 (September 2006): 82–87. http://dx.doi.org/10.5951/mtms.12.2.0082.

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Worthwhile mathematical tas ks not only prompt students to learn mathematics, they also prompt teachers to learn and improve their teaching in their own mathematics classrooms. When teachers use worthwhile tasks, they have to learn “what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge” (NCTM 2000, p. 19).

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6

Breyfogle, M. Lynn, and Lauren E. Williams. "From the Classroom: Designing and Implementing Worthwhile Tasks." Teaching Children Mathematics 15, no. 5 (December 2008): 276–80. http://dx.doi.org/10.5951/tcm.15.5.0276.

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Teachers often need to alter mathematical tasks that they find in their district-adopted set of curriculum materials or develop new ones if none is present on a particular topic. However, how to best go about this work is not always clear. How do you make effective decisions about alterations? What should you keep in mind as you consider developing tasks to help your students with a particular idea or misconception? These and other questions were central in our minds as we developed a task to help students learn about elapsed time.

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7

Armstrong, Barbara E. "Implementing the Professional Standards for Teaching Mathematics: Teaching Patterns, Relationships, and Multiplication as Worthwhile Mathematical Tasks." Teaching Children Mathematics 1, no. 7 (March 1995): 446–50. http://dx.doi.org/10.5951/tcm.1.7.0446.

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The goal of presenting students with worthwhile tasks that enable them to make connections is to ensure the development of mathematical insights. Determining instructional activities that meet this goal, however, can be a complex task in itself.

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8

Reys, Barbara J., and Vena M. Long. "Implementing the Professional Standards for Teaching Mathematics: Teacher as Architect of Mathematical Tasks." Teaching Children Mathematics 1, no. 5 (January 1995): 296–99. http://dx.doi.org/10.5951/tcm.1.5.0296.

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The first standard presented in the Professional Standards for Teaching Mathematics (NCTM 1991) highlights the importance of choosing and using worthwhile mathematical tasks. Teachers are curriculum architects charged with ensuring the quality of the mathematical tasks in which their students engage.

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9

Russo, James, Toby Russo, and Anne Roche. "Using Rich Narratives to Engage Students in Worthwhile Mathematics: Children’s Literature, Movies and Short Films." Education Sciences 11, no. 10 (September 27, 2021): 588. http://dx.doi.org/10.3390/educsci11100588.

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Using children’s literature to support mathematics instruction has been connected to positive academic outcomes and learning dispositions; however, less is known about the use of audiovisual based narrative mediums to support student mathematical learning experiences. The current exploratory, qualitative study involved teaching three lessons based on challenging, problem solving tasks to two classes of Australian Year (Grade) 5 students (10 and 11 year olds). These tasks were developed from various narratives, each portrayed through a different medium (movie clip, short film, picture story book). Post lesson interviews were undertaken with 24 students inviting them to compare and contrast this lesson sequence with their usual mathematics instruction. Drawing on a self-determination theory lens, our analysis revealed that these lessons were experienced by students as both highly enjoyable and mathematically challenging. More specifically, it was found that presenting mathematics tasks based on rich and familiar contexts and providing meaningful choices about how to approach their mathematical work supported student autonomy. In addition, there was evidence that the narrative presentation supported student understanding of the mathematics through making the tasks clearer and more accessible, whilst the audiovisual mediums (movie clip, short film) in particular provided a dynamic representation of key mathematical ideas (e.g., transformation and scale). Students indicated an eclectic range of preferences in terms of their preferred narrative mediums for exploring mathematical ideas. Our findings support the conclusion that educators and researchers focused on the benefits of teaching mathematics through picture story books consider extending their definition of narrative to encompass other mediums, such as movie clips and short films.

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10

Henningsen, Marjorie, and Mary Kay Stein. "Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning." Journal for Research in Mathematics Education 28, no. 5 (November 1997): 524–49. http://dx.doi.org/10.5951/jresematheduc.28.5.0524.

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In order to develop students' capacities to “do mathematics,” classrooms must become environments in which students are able to engage actively in rich, worthwhile mathematical activity. This paper focuses on examining and illustrating how classroom-based factors can shape students' engagement with mathematical tasks that were set up to encourage high-level mathematical thinking and reasoning. The findings suggest that when students' engagement is successfully maintained at a high level, a large number of support factors are present. A decline in the level of students' engagement happens in different ways and for a variety of reasons. Four qualitative portraits provide concrete illustrations of the ways in which students' engagement in high-level cognitive processes was found to continue or decline during classroom work on tasks.

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11

Sherrill, Carl M. "Math Riddles: Helping Children Connect Words and Numbers." Teaching Children Mathematics 11, no. 7 (March 2005): 368–75. http://dx.doi.org/10.5951/tcm.11.7.0368.

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Student-written math riddles provide worthwhile mathematical tasks that support the learning of important concepts as they engage and challenge children. Procedures for classroom teachers to use, guiding students as they create math riddles for their classmates to solve are presented.

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12

Henningsen, Marjorie A. "Reader Reflections: Triumph through Adversity: Supporting High-Level Thinking." Mathematics Teaching in the Middle School 6, no. 4 (December 2000): 244–48. http://dx.doi.org/10.5951/mtms.6.4.0244.

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Worthwhile mathematical tasks are central to students' learning because “tasks convey messages about what mathematics is and what doing mathematics entails” (NCTM 1991, p. 24). Such tasks engage students in looking for patterns, explaining ideas, justifying conclusions, testing conjectures, framing and posing problems, making decisions, comparing and contrasting problem structures and multiple solution strategies, and reasoning across multiple representations. Frequent experience with such tasks increases students' opportunities to do, think, and reason about mathematics and, therefore, increases their opportunities to learn.

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13

Calleja, James. "A spiral pattern investigation: making mathematical connections." Mathematical Gazette 104, no. 560 (June 18, 2020): 262–70. http://dx.doi.org/10.1017/mag.2020.49.

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Nowadays there is considerable agreement among educators that learning mathematics fundamentally involves making mathematics [1]. Students learn mathematics while working on tasks that they consider meaningful and worthwhile, and their interest is aroused when they can see the point of what they are being asked to do. Given that learning mathematics involves a process of meaning-making - the use of mathematical language, symbols and representations as learners negotiate ideas – activities should provide students with a variety of challenging experiences through which they can actively construct mathematical meanings for themselves.

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14

Zhang, Xilin, Yuejin Tan, and Zhiwei Yang. "Rework Quantification and Influence of Rework on Duration and Cost of Equipment Development Task." Sustainability 10, no. 10 (October 9, 2018): 3590. http://dx.doi.org/10.3390/su10103590.

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Rework is a sub-task within equipment development tasks that is revised after initial completion to meet task requirements. Some sub-tasks require multiple rework iterations due to their uncertainty and complexity, or the technology and process needs of the overall task, resulting in inefficient task implementation and resource wastage. Therefore, studying the impact of rework iterations on the duration and cost of development tasks is worthwhile. This study divides rework into foreseeable and hidden types and uses several methods to express and quantify their parameters. The main influencing factors in rework iterations—the uncertainty and complexity of the development task—are quantitatively analyzed. Then, mathematical and mapping models of the dependence between sub-tasks, uncertainty, complexity, and rework parameters are established. The impacts of rework type and rework parameters on the duration and cost of equipment development tasks are analyzed via simulation based on the design structure matrix (DSM). Finally, an example is used to illustrate the influence of different rework types and rework parameters on development tasks’ duration and cost. The results show that the duration and cost of development tasks are greater, their volatility range is wider, and the distribution is more dispersed when both foreseeable and hidden rework are considered.

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15

Anderson, Katie L. "Pattern-block frenzy." Teaching Children Mathematics 19, no. 2 (September 2012): 116–21. http://dx.doi.org/10.5951/teacchilmath.19.2.0116.

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Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K–grade 6 classrooms. This article describes a set of lessons where sixth graders use virtual pattern blocks to develop proportional reasoning. Students' work with the virtual manipulatives reveals a variety of creative solutions and promotes active engagement. The author suggests that technology is most effective when coupled with worthwhile mathematical tasks and rich classroom discussions.

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16

Bu, Lingguo. "Spinning the Cube with Technologies." Mathematics Teacher 112, no. 7 (May 2019): 551–54. http://dx.doi.org/10.5951/mathteacher.112.7.0551.

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The rise of dynamic modeling and 3-D design technologies provides appealing opportunities for mathematics teachers to reconsider a host of pedagogical issues in mathematics education, ranging from motivation to application and from visualization to physical manipulation. This article reports on a classroom teaching experiment about cube spinning, integrating traditional tools, GeoGebra (www.geogebra.org), and 3-D design and printing technologies. It highlights the rich interplay between worthwhile mathematical tasks and the strategic use of diverse technologies in sustaining sense making and problem solving with a group of prospective teachers.

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17

Greiner-Petter, André, Abdou Youssef, Terry Ruas, Bruce R. Miller, Moritz Schubotz, Akiko Aizawa, and Bela Gipp. "Math-word embedding in math search and semantic extraction." Scientometrics 125, no. 3 (June 9, 2020): 3017–46. http://dx.doi.org/10.1007/s11192-020-03502-9.

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AbstractWord embedding, which represents individual words with semantically fixed-length vectors, has made it possible to successfully apply deep learning to natural language processing tasks such as semantic role-modeling, question answering, and machine translation. As math text consists of natural text, as well as math expressions that similarly exhibit linear correlation and contextual characteristics, word embedding techniques can also be applied to math documents. However, while mathematics is a precise and accurate science, it is usually expressed through imprecise and less accurate descriptions, contributing to the relative dearth of machine learning applications for information retrieval in this domain. Generally, mathematical documents communicate their knowledge with an ambiguous, context-dependent, and non-formal language. Given recent advances in word embedding, it is worthwhile to explore their use and effectiveness in math information retrieval tasks, such as math language processing and semantic knowledge extraction. In this paper, we explore math embedding by testing it on several different scenarios, namely, (1) math-term similarity, (2) analogy, (3) numerical concept-modeling based on the centroid of the keywords that characterize a concept, (4) math search using query expansions, and (5) semantic extraction, i.e., extracting descriptive phrases for math expressions. Due to the lack of benchmarks, our investigations were performed using the arXiv collection of STEM documents and carefully selected illustrations on the Digital Library of Mathematical Functions (DLMF: NIST digital library of mathematical functions. Release 1.0.20 of 2018-09-1, 2018). Our results show that math embedding holds much promise for similarity, analogy, and search tasks. However, we also observed the need for more robust math embedding approaches. Moreover, we explore and discuss fundamental issues that we believe thwart the progress in mathematical information retrieval in the direction of machine learning.

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18

Schulman, Steven M. "Squares on a Checkerboard." Teaching Children Mathematics 21, no. 2 (September 2014): 84–90. http://dx.doi.org/10.5951/teacchilmath.21.2.0084.

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19

Panasuk, Regina M., and Yvonne Greenleaf. "Using ROOTine Problems for Group Work in Geometry." Mathematics Teacher 91, no. 9 (December 1998): 794–98. http://dx.doi.org/10.5951/mt.91.9.0794.

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How can mathematics teachers structure classroom activities so that students will be intellectually challenged? How can they create a learning environment that encourages students to communicate and reason mathematically, make decisions collaboratively, and acquire mathematics skills and concepts that they thoroughly understand? The Professional Standards for Teaching Mathematics (NCTM 1991) suggests that mathematics teachers need to focus on the major components of teaching: worthwhile tasks, discourse, and students' active participation and involvement. Cooperative-learning approaches offer practical classroom techniques that teachers can use to motivate all their students to learn and appreciate mathematics (Davidson 1990).

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20

Piñeiro, Juan Luis, Olive Chapman, Elena Castro-Rodríguez, and Enrique Castro. "Prospective Elementary Teachers’ Pedagogical Knowledge for Mathematical Problem Solving." Mathematics 9, no. 15 (July 30, 2021): 1811. http://dx.doi.org/10.3390/math9151811.

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Research on mathematics teachers’ knowledge has generally focused more on mathematics concepts than mathematical processes. This paper addresses the latter with a focus on mathematical problem solving (PS). It reports on a study that investigated the pedagogical knowledge for PS of prospective elementary school teachers of mathematics (PTs). Participants were 149 PTs at a university in Spain. They were at the end of their teacher education program. Data sources consisted of a questionnaire on knowledge of learning PS and a questionnaire on knowledge of teaching PS. Findings indicated that the PTs held combination of different levels of knowledge of PS learning and teaching. Many of them demonstrated appropriate knowledge of many characteristics for (1) PS learning consisting of student as a problem-solver, PS as a worthwhile task, non-cognitive factor related to PS, and (2) PS teaching consisting of PS teaching approaches, discourse in PS, intervention during stuck state in PS, PS assessment, and PS resources. However, there were also contradictions and limitations to their knowledge with implications for teacher education. These combination of appropriate and inappropriate knowledge resulted in some conflicts that are related to teaching actions and would limit student’ learning of PS.

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21

Montoro, Ana B., Carmen Gloria Aguayo-Arriagada, and Pablo Flores. "Measurement in Primary School Mathematics and Science Textbooks." Mathematics 9, no. 17 (September 2, 2021): 2127. http://dx.doi.org/10.3390/math9172127.

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The STEM (science, technology, engineering and mathematics) approach to education has acquired considerable prominence among teachers in recent years. Putting forward integrated proposals is nonetheless complex and many educators opt to implement the ones set out in textbooks. We consequently deemed it worthwhile to analyse how content common to mathematics and science is addressed in primary school textbooks with a view to determining whether the approaches adopted complement one another and are compatible with STEM education. More specifically, in light of the importance of measurement in both areas of learning and in everyday life, we describe the meaning of mass and volume found, in two publishers’ textbooks. Based on the components of the meaning of measurement and deploying content analysis techniques, we analysed the explanations and tasks set out in these mathematics and science books to identify the similarities and differences in the handling of those magnitudes in the two subjects. Our findings showed the proposals for teaching mass to pursue similar objectives in the earliest grades, addressing matters that could be included in STEM proposals. On the contrary, inconsistencies were detected in the distribution of volume measurement-related content, as well as in the strategies, units and tools used in the two areas.

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22

Zwick, Analia, Gonzalo A. Alvarez, Joachim Stolze, and Omar Osenda. "Quantum state transfer in disordered spin chains: How much engineering is reasonable?" Quantum Information and Computation 15, no. 7&8 (May 2015): 582–600. http://dx.doi.org/10.26421/qic15.7-8-4.

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The transmission of quantum states through spin chains is an important element in the implementation of quantum information technologies. Speed and fidelity of transfer are the main objectives which have to be achieved by the devices even in the presence of imperfections which are unavoidable in any manufacturing process. To reach these goals, several kinds of spin chains have been suggested, which differ in the degree of fine-tuning, or engineering, of the system parameters. In this work we present a systematic study of two important classes of such chains. In one class only the spin couplings at the ends of the chain have to be adjusted to a value different from the bulk coupling constant, while in the other class every coupling has to have a specific value. We demonstrate that configurations from the two different classes may perform similarly when subjected to the same kind of disorder in spite of the large difference in the engineering effort necessary to prepare the system. We identify the system features responsible for these similarities and we perform a detailed study of the transfer fidelity as a function of chain length and disorder strength, yielding empirical scaling laws for the fidelity which are similar for all kinds of chain and all disorder models. These results are helpful in identifying the optimal spin chain for a given quantum information transfer task. In particular, they help in judging whether it is worthwhile to engineer all couplings in the chain as compared to adjusting only the boundary couplings.

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23

BONE, PAUL, ZOLTAN SOMOGYI, and PETER SCHACHTE. "Estimating the overlap between dependent computations for automatic parallelization." Theory and Practice of Logic Programming 11, no. 4-5 (July 2011): 575–91. http://dx.doi.org/10.1017/s1471068411000184.

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AbstractResearchers working on the automatic parallelization of programs have long known that too much parallelism can be even worse for performance than too little, because spawning a task to be run on another CPU incurs overheads. Autoparallelizing compilers have therefore long tried to use granularity analysis to ensure that they only spawn off computations whose cost will probably exceed the spawn-off cost by a comfortable margin. However, this is not enough to yield good results, because data dependencies may also limit the usefulness of running computations in parallel. If one computation blocks almost immediately and can resume only after another has completed its work, then the cost of parallelization again exceeds the benefit. We present a set of algorithms for recognizing places in a program where it is worthwhile to execute two or more computations in parallel that pay attention to the second of these issues as well as the first. Our system uses profiling information to compute the times at which a procedure call consumes the values of its input arguments and the times at which it produces the values of its output arguments. Given two calls that may be executed in parallel, our system uses the times of production and consumption of the variables they share to determine how much their executions would overlap if they were run in parallel, and therefore whether executing them in parallel is a good idea or not. We have implemented this technique for Mercury in the form of a tool that uses profiling data to generate recommendations about what to parallelize, for the Mercury compiler to apply on the next compilation of the program. We present preliminary results that show that this technique can yield useful parallelization speedups, while requiring nothing more from the programmer than representative input data for the profiling run.

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24

"Call for Manuscripts: Discourse – August 2011." Mathematics Teaching in the Middle School 17, no. 1 (August 2011): 33. http://dx.doi.org/10.5951/mathteacmiddscho.17.1.0033.

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Discourse is the mathematical communication that occurs in a classroom. Effective discourse happens when students articulate their own ideas and seriously consider their peers' mathematical perspectives as a way to construct mathematical understandings. Encouraging students to construct their own mathematical understanding through discourse is an effective way to teach mathematics, especially since the role of the teacher has transformed from being a transmitter of knowledge to one who presents worthwhile and engaging mathematical tasks.

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25

"Call for Manuscripts: Discourse." Mathematics Teaching in the Middle School 15, no. 9 (May 2010): 513. http://dx.doi.org/10.5951/mtms.15.9.0513.

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Discourse is the mathematical communication that occurs in a classroom. Effective discourse happens when students articulate their own ideas and seriously consider their peers' mathematical perspectives as a way to construct mathematical understandings. Encouraging students to construct their own mathematical understanding through discourse is an effective way to teach mathematics, especially since the role of the teacher has transformed from being a transmitter of knowledge to one who presents worthwhile and engaging mathematical tasks.

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26

"Call for Manuscripts: Discourse: September 2010." Mathematics Teaching in the Middle School 16, no. 2 (September 2010): 109. http://dx.doi.org/10.5951/mtms.16.2.0109.

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Discourse is the mathematical communication that occurs in a classroom. Effective discourse happens when students articulate their own ideas and seriously consider their peers' mathematical perspectives as a way to construct mathematical understandings. Encouraging students to construct their own mathematical understanding through discourse is an effective way to teach mathematics, especially since the role of the teacher has transformed from being a transmitter of knowledge to one who presents worthwhile and engaging mathematical tasks.

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27

"Readers Write: March 2009." Mathematics Teaching in the Middle School 14, no. 7 (March 2009): 387. http://dx.doi.org/10.5951/mtms.14.7.0387.

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Beauty is in the eyes of the beholder. I have been a long-time user of the Connected Mathematics Project materials and have literally worked every problem offered for all three grade levels so as to anticipate how the problem might play out in the classroom. My knowledge of mathematics and teaching has grown by leaps and bounds from solving each problem; from studying the teacher editions, in themselves a wealth of support; and by working with colleagues and students. I enter my classroom believing anything can happen if I let it. I have seen students at work on worthwhile mathematical tasks in my own classroom, as well as the classrooms of many others. My stories are many, my enthusiasm abounds, and my belief that every child can learn is steadfast.

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