“The art of problem-solving is the heart of mathematics.” -- Wilson, Fernandez, & Hadaway, 1993
Recommended Readings
Mathematical Problem Solving
This is an intriguing paper written by three professors (James W. Wilson, Maria L. Fernandez, and Nelda Hadaway) in the Department of Mathematics Education at the University of Georgia. In it, they discuss research about how secondary students gain problem-solving skills and how instruction can be geared to best develop these skills. The article aims to determine if instruction should emphasize particular problem-solving techniques or strategies that are useful for a variety of tasks. The authors highlight the importance of using constructivist methods to inform teaching at the secondary level as well as a more cyclic framework for problem-solving that is more reflective of the genuine problem-solving process as depicted by Polya. The remainder of the article discusses problem-solving as an instructional goal, problem-solving as an instructional method, beliefs about mathematics problem-solving, and technology & problem-solving. In this discussion, the authors argue strongly for incorporating problem-solving into the secondary mathematics curriculum, using mathematics problem-solving examples and prior research to illustrate the benefits of these activities and to support their arguments.
This is an intriguing paper written by three professors (James W. Wilson, Maria L. Fernandez, and Nelda Hadaway) in the Department of Mathematics Education at the University of Georgia. In it, they discuss research about how secondary students gain problem-solving skills and how instruction can be geared to best develop these skills. The article aims to determine if instruction should emphasize particular problem-solving techniques or strategies that are useful for a variety of tasks. The authors highlight the importance of using constructivist methods to inform teaching at the secondary level as well as a more cyclic framework for problem-solving that is more reflective of the genuine problem-solving process as depicted by Polya. The remainder of the article discusses problem-solving as an instructional goal, problem-solving as an instructional method, beliefs about mathematics problem-solving, and technology & problem-solving. In this discussion, the authors argue strongly for incorporating problem-solving into the secondary mathematics curriculum, using mathematics problem-solving examples and prior research to illustrate the benefits of these activities and to support their arguments.
Teacher Help for Conceptual Level Raising in Mathematics
The aim of this article, written by Pijls, Dekker, and Van Hout-Wolters, is to determine whether high school mathematics students attain more conceptual level raising from teachers who provide process help (by stimulating students' interactions) or from teachers who provide product help (with mathematical content) during collaborative learning. Simply stated, the issue is, “should a teacher leave the learning content to the students and focus on the collaborative learning process, or take part in the collaboration, by focusing on the content of the learning” (pg. 223-224). In order to explore this question, the authors first provide theoretical background on the topic, including the results of previous studies, and then describe, in depth, their own field study, which also addresses this issue.
Click here if you are interested in reading a critical analysis that I wrote of this article.
Click below to download the full article in pdf format if you cannot access the article from the above link.
The aim of this article, written by Pijls, Dekker, and Van Hout-Wolters, is to determine whether high school mathematics students attain more conceptual level raising from teachers who provide process help (by stimulating students' interactions) or from teachers who provide product help (with mathematical content) during collaborative learning. Simply stated, the issue is, “should a teacher leave the learning content to the students and focus on the collaborative learning process, or take part in the collaboration, by focusing on the content of the learning” (pg. 223-224). In order to explore this question, the authors first provide theoretical background on the topic, including the results of previous studies, and then describe, in depth, their own field study, which also addresses this issue.
Click here if you are interested in reading a critical analysis that I wrote of this article.
Click below to download the full article in pdf format if you cannot access the article from the above link.
teacher_help_for_conceptal_level_raising.pdf | |
File Size: | 288 kb |
File Type: |
Outside-In-Inside-Out: Seventh Grade Students' Mathematical Thought Processes
In this paper, Dr. Lillie Albert, Associate Professor of Mathematics Education at Boston College, investigates the relationship between students' oral thought processes and written thought processes in mathematics. Expanding upon Vygotsky's research involving the role of social exchanges and the zone of proximal development (ZPD) in student learning, Albert argues that the practice of writing in mathematics can be used to aid students in reaching a new zone of learning: the zone of proximal practice (ZPP) in which students are able to self-direct and self-regulate their own thought processes regarding mathematics concepts and ideas. To support this argument, the researchers conducted a case study of seventh grade students engaged in collaborative mathematics problem-solving activities. Questioning was used to facilitate learning processes and to co-construct problem-solving strategies, which was effective in improving students’ understanding of the process as well as the mathematical concepts involved. Interview protocols and writing samples were used to describe students' strategies for solving problems and to record students' oral and written thought processes. Results indicate that the writing process can be an effective tool for deepening conceptual understanding and advancing the learner from the ZPD to the ZPP.
Click below to download the full article in pdf format.
In this paper, Dr. Lillie Albert, Associate Professor of Mathematics Education at Boston College, investigates the relationship between students' oral thought processes and written thought processes in mathematics. Expanding upon Vygotsky's research involving the role of social exchanges and the zone of proximal development (ZPD) in student learning, Albert argues that the practice of writing in mathematics can be used to aid students in reaching a new zone of learning: the zone of proximal practice (ZPP) in which students are able to self-direct and self-regulate their own thought processes regarding mathematics concepts and ideas. To support this argument, the researchers conducted a case study of seventh grade students engaged in collaborative mathematics problem-solving activities. Questioning was used to facilitate learning processes and to co-construct problem-solving strategies, which was effective in improving students’ understanding of the process as well as the mathematical concepts involved. Interview protocols and writing samples were used to describe students' strategies for solving problems and to record students' oral and written thought processes. Results indicate that the writing process can be an effective tool for deepening conceptual understanding and advancing the learner from the ZPD to the ZPP.
Click below to download the full article in pdf format.
albert_outside-in_inside-out.pdf | |
File Size: | 3646 kb |
File Type: |
Recommended Problem-Solving Websites
United States of America Mathematical Olympiad (USAMO)
This website is for the United States Mathematics Olympiad, an annual competition which entails 9 open-ended problems that challenge students' higher level thinking skills. The competition is designed to identify, recognize, and encourage the most creative and critically thinking secondary mathematics students in the country. "The USAMO is part of a worldwide system of national mathematics competitions, a movement in which both educators and research mathematicians are engaged in recognizing and celebrating the imagination and resourcefulness of our youth." On the website, parents and students can read about the history, qualifications, and details of many different American Mathematics Contests. There is also a page designed specifically for students, where they can find sample problems, practice exams, mathematics enrichment and preparation
programs, tips for success in the Olympiad, and links to other useful mathematics and problem-solving websites.
This website is for the United States Mathematics Olympiad, an annual competition which entails 9 open-ended problems that challenge students' higher level thinking skills. The competition is designed to identify, recognize, and encourage the most creative and critically thinking secondary mathematics students in the country. "The USAMO is part of a worldwide system of national mathematics competitions, a movement in which both educators and research mathematicians are engaged in recognizing and celebrating the imagination and resourcefulness of our youth." On the website, parents and students can read about the history, qualifications, and details of many different American Mathematics Contests. There is also a page designed specifically for students, where they can find sample problems, practice exams, mathematics enrichment and preparation
programs, tips for success in the Olympiad, and links to other useful mathematics and problem-solving websites.
The Art of Problem Solving
This website is designed for teachers and students to explore the Art of Problem Solving (AoPS) curriculum and online courses, designed specifically for high-performing math students in grades 6 - 12 who are interested in a challenging, problem-based program. Under the Bookstore tab, teachers can read more about the AoPS course materials and curriculum, which presents a much broader and deeper exploration of challenging mathematics concepts than a typical math curriculum. "Our texts not only introduce key results of mathematics, but also teach students how to develop new ideas themselves and how to apply these ideas to challenging problems. In so doing, we help students learn the key problem solving skills needed for success in top universities and in the most competitive careers, as well as in major national programs like MATHCOUNTS and the American Mathematics Competitions." Under the community tab, students can access online forums for discussing and collaborating with peers on particularly challenging problems. Students can also join the free Alcumus online learning system, which offers students a customized learning experience with access to over 3500 problems and solutions from the major national math competitions. It provides the opportunity for students to track their performance and for teachers to monitor their students' progress online.
This website is designed for teachers and students to explore the Art of Problem Solving (AoPS) curriculum and online courses, designed specifically for high-performing math students in grades 6 - 12 who are interested in a challenging, problem-based program. Under the Bookstore tab, teachers can read more about the AoPS course materials and curriculum, which presents a much broader and deeper exploration of challenging mathematics concepts than a typical math curriculum. "Our texts not only introduce key results of mathematics, but also teach students how to develop new ideas themselves and how to apply these ideas to challenging problems. In so doing, we help students learn the key problem solving skills needed for success in top universities and in the most competitive careers, as well as in major national programs like MATHCOUNTS and the American Mathematics Competitions." Under the community tab, students can access online forums for discussing and collaborating with peers on particularly challenging problems. Students can also join the free Alcumus online learning system, which offers students a customized learning experience with access to over 3500 problems and solutions from the major national math competitions. It provides the opportunity for students to track their performance and for teachers to monitor their students' progress online.
Cut-the-Knot: Interactive Mathematics Miscellany and Puzzles
This website provides students with a variety of interactive mathematics problems, puzzles, games, and activities from the elementary to high school level. The site covers a wide range of mathematics content, including Geometry, Algebra, Probability, Number Theory, Logic Problems, and Word Problems. Many of the problem-solving activities use Geometer's Sketchpad Applets to enable interaction between students and the computer program. With every activity, there is a detailed explanation available that students can read to understand the solutions to each problem and to enrich their knowledge of the underlying mathematics concepts involved in the puzzles and games. Some examples of my favorite activities are Knights and Knaves, Euclid's Game, A Broken Calculator, and Five Siblings. This site also provides a forum, the CTK Exchange, on which students can post problems as well as share and discuss solutions. In addition, there are links for parents and teachers to access additional websites, recommended books, and other resources for mathematics practice, problem-solving, and enrichment activities.
This website provides students with a variety of interactive mathematics problems, puzzles, games, and activities from the elementary to high school level. The site covers a wide range of mathematics content, including Geometry, Algebra, Probability, Number Theory, Logic Problems, and Word Problems. Many of the problem-solving activities use Geometer's Sketchpad Applets to enable interaction between students and the computer program. With every activity, there is a detailed explanation available that students can read to understand the solutions to each problem and to enrich their knowledge of the underlying mathematics concepts involved in the puzzles and games. Some examples of my favorite activities are Knights and Knaves, Euclid's Game, A Broken Calculator, and Five Siblings. This site also provides a forum, the CTK Exchange, on which students can post problems as well as share and discuss solutions. In addition, there are links for parents and teachers to access additional websites, recommended books, and other resources for mathematics practice, problem-solving, and enrichment activities.
Other Recommended Websites
For Students
Algebra Lab: This site provides interactive study materials for students to explore with practice examples covering a range algebra topics, from basic operations to introductory calculus.
Free Math Help: Students can use this site as a forum for posting problems and questions about a variety of different topics in mathematics, including Algebra, Geometry, Trigonometry, Statistics, Calculus, and much more.
Math.com: Students can access homework help,practice problems, and lessons from subjects in basic math to calculus. Students can also use the online calculator and tools from this website if they do not have a graphing calculator at home.
Free Math Help: Students can use this site as a forum for posting problems and questions about a variety of different topics in mathematics, including Algebra, Geometry, Trigonometry, Statistics, Calculus, and much more.
Math.com: Students can access homework help,practice problems, and lessons from subjects in basic math to calculus. Students can also use the online calculator and tools from this website if they do not have a graphing calculator at home.
For Teachers
National Council of Teachers of Mathematics: "From planning to assessment, NCTM books have it all"The NCTM website provides leadership and professional development to support teachers in ensuring the best mathematics education for all students.
Massachusetts Curriculum Frameworks: From the Department of Elementary and Secondary Education, this site provides the curriculum frameworks and standards for the state of Massachusetts. Every Massachusetts teacher should be familiar with the information provided here.
The MacTutor History of Mathematics Archive: This is a great source of information on the history of mathematics, famous mathematicians, mathematical societies, chronologies of discoveries, mathematics quotations, and many other interesting topics. Teachers can use this site to incorporate historical aspects and background information into their mathematics courses.
Biographies of Women Mathematicians: This site was created as part of a an on-going project at Agnes Scott College to display the many achievements of women in the field of mathematics. It provides a series of biographical essays on the contributions of women mathematicians as well as additional resources about women in mathematics.
Math for America: The Math for America (MfA) community of teachers is a nonprofit organizationwith a mission to improve secondary mathematics education for public schools all over the US. This site provides teaching resources such as activities and lesson plans, virtual manipulatives, and technology-specific sites.
Massachusetts Curriculum Frameworks: From the Department of Elementary and Secondary Education, this site provides the curriculum frameworks and standards for the state of Massachusetts. Every Massachusetts teacher should be familiar with the information provided here.
The MacTutor History of Mathematics Archive: This is a great source of information on the history of mathematics, famous mathematicians, mathematical societies, chronologies of discoveries, mathematics quotations, and many other interesting topics. Teachers can use this site to incorporate historical aspects and background information into their mathematics courses.
Biographies of Women Mathematicians: This site was created as part of a an on-going project at Agnes Scott College to display the many achievements of women in the field of mathematics. It provides a series of biographical essays on the contributions of women mathematicians as well as additional resources about women in mathematics.
Math for America: The Math for America (MfA) community of teachers is a nonprofit organizationwith a mission to improve secondary mathematics education for public schools all over the US. This site provides teaching resources such as activities and lesson plans, virtual manipulatives, and technology-specific sites.
For Everyone
Math Forum: This is an online resource for improving mathematics learning, teaching, and communication, designed by teachers, mathematicians, researchers, students, and parents. It provides a wide range of problems and puzzles, online mentoring, research, team problem solving, collaborations, and professional development to improve mathematics education. The following are some of the useful links included: Teacher2Teacher is a resource for teachers and parents who have questions about teaching mathematics,
Ask Dr. Math is a forum for students to ask questions and review difficult concepts, and Math Tools offers technology tools, lessons, activities, and support materials for teaching and learning mathematics.
Ask Dr. Math is a forum for students to ask questions and review difficult concepts, and Math Tools offers technology tools, lessons, activities, and support materials for teaching and learning mathematics.
Boston College: It's never too early to start thinking about college! Check out the academic departments, admissions programs, faculty, organizations, and other resources that are available at Boston College.
For all those who are thinking about getting involved in the field of academia, click here to discover all that the Lynch School of Education at Boston College has to offer.
For all those who are thinking about getting involved in the field of academia, click here to discover all that the Lynch School of Education at Boston College has to offer.