"Many students come to believe that school mathematics consists of mastering formal procedures that are completely divorced from real life, from discovery, and from problem solving.' - Alan Schoenfeld

Background

Alan Schoenfeld is currently the Elizabeth and Edward Conner Professor of Education and Affiliated Professor of Mathematics at the University of California, Berkeley. He has also been the President of the American Education Research Association. In 2008, he was awarded the Senior Scholar Award by the AERA's Special Interest Group for Research in Math Education. Much of his work has focused on problem solving.

Degrees

Professor Schoenfeld has acquired numerous degrees from both Stanford and Queens College. He first received his B.S. in Mathematics from Queens College in 1968. He then received his M.S. in mathematics from Stanford University in 1969, followed by his Ph.D. in 1973. He is currently teaching mathematics and education courses at the University of California, Berkeley.

Schoenfeld, A. H. (Ed.) (1983). Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography. Washington, DC: Mathematical Association of
America.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Schoenfeld, A. H. (Ed.) (1987). Cognitive science and mathematics education. Hillsdale, NJ: Erlbaum.
Burkhardt, H., Groves, S., Schoenfeld, A. H., and Stacey, K. (Eds.) (1988). Problem solving: A world view (Proceedings of the problem solving theme group at the V International Congress on
Mathematical Education, Adelaide, Australia). Nottingham, England: Shell Centre for Mathematical Education.
diSessa, A, Gardner, M., Greeno, J., Reif, F., & Schoenfeld, A. H. (Eds.) (1990). Toward a scientific practice of science education. Hillsdale, NJ: Erlbaum.
Schoenfeld, A. H. (Ed.) (1990). A Source Book for College Mathematics Teaching. Washington, DC: Mathematical Association of America.
Schoenfeld, A. H. (Ed.) (1992). Research methods in and for the learning sciences, a special issue of The Journal of the Learning Sciences, Volume 2, No. 2.
Schoenfeld, A. H. (Ed.) (1994). Mathematical thinking and problem solving. Hillsdale, NJ: Erlbaum.
Dubinsky, E., Schoenfeld, A. H., & Kaput, J. (Eds.) (1994). Research in Collegiate Mathematics Education. I. Washington, DC: Conference Board of the Mathematical Sciences.
Kaput, J. Schoenfeld, A. H., & Dubinsky, E., (Eds.) (1996). Research in Collegiate Mathematics Education. II. Washington, DC: Conference Board of the Mathematical Sciences.
Schoenfeld, A. H. (Ed.) (1997). Student Assessment in Calculus A report of the NSF Working Group on Assessment in Calculus. Washington, DC: Mathematical Association of America.
Schoenfeld, A. H., Kaput, J., & Dubinsky, E. (Eds.) (1998). Research in Collegiate Mathematics Education. III. Washington, DC: Conference Board of the Mathematical Sciences.
Schoenfeld, A. H. (1998) Issues in Education, Volume 4, Number 1. The issue presents and critiques Schoenfeld's theory of teaching-in-context.

Schoenfeld, Alan H. (1999) (Special Issue Editor). Examining the Complexity of Teaching. Special issue of the Journal of Mathematical Behavior, 18 (3).
Dubinsky, E., Schoenfeld, A. H., & Kaput, J. (Eds.) (2000). Research in Collegiate Mathematics Education. IV. Washington, DC: Conference Board of the Mathematical Sciences.
Ferrini-Mundy, J., Joyner, J., Reyes, B., Schoenfeld, A. H., & E. Silver, E. (Eds.) (2000) Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Holton, D., Artigue, M., Kirchgraber, U, Hillel, J., Niss, M. & Schoenfeld, H. (Eds.) (2001). The teaching and learning of mathematics at the University Level. Dordrecht: Kluwer.
Schoenfeld, A. H. (Ed.) (in press). A study of teaching: Multiple lenses, multiple views. Journal for research in Mathematics Education monograph series. Reston, VA: National Council of Teachers of Mathematics.
Schoenfeld, A. H. (Ed.) (in preparation). Assessing Mathematical proficiency. Cambridge: Cambridge University Press.

Theory

Dr. Schoenfeld began studying math and how students learned. He wanted to find a new way for students to approach math in order to make it more meaningful for them. Many children don't know how useful math can be in the real world, simply because they have always been taught only how to master formulas.

Alan Schoenfeld is known best for his studies of mathematics and how students should learn it. This theory, known as Mathematical Problem Solving, focuses on solving problems through a variety of ways to help students understand the subject better. He believes that teaching mathematics does not only mean educating children on the basic equations and how to plug numbers into them, but how to analyze and understand the meaning behind the formulas. Students must be taught to seek solutions rather than just memorizing procedures, expore patters rather than memorizing formulas, and formulating conjectures rather than doing exercises. Students must have a broader understanding of mathematics in order to excel in it.

According to Dr. Schoenfeld, four categories of skills are necessary in order to be successful in math:

successinmath.png

As the above image shows, students must have control, resources, heuristics, and beliefs in order to be successful in mathematics. Control includes being able to determine when certain strategies should be used. Resources are the procedural knowledge that a student has. This is basically the equations taught in a typical math class. Heuristics are the different strategies that students have to solve these math problems, including drawing figures, working backwards, identifying patterns, etc. Finally, the beliefs that a student has are the views that the student has when approaching a problem.

Another main element in Dr. Schoenfeld's mathematical problem solving theory deals with metacognition, or "thinking about thinking." Basically, metacognition means having the knowledge to know when to use particular strategies for problem solving. With the mathematical metacognition that students need, they will be able to apply mathematics to real life situations that they face every day. It can also create a community of learners outside of the classroom that use math in different contexts.

The following image show a complex view of the Mathematical Problem Solving Theory that Schoenfeld is known for:

mathematicalproblemsolving.jpg

Since this image can be difficult to read and extremely confusing to the eye, let's analyse it further to make sense of it and extract the most important points. To begin with, we see that this is a basic summary of Alan Schoenfeld's publication Learning to Think Mathematically. In this writing, he examines the goals of teaching math. Teaching needs enculturation, or using the values of the community. This theory uses sociology, anthropology, and constructivism. Constructivism is a method of teaching in the classroom in which students are engaged in activities, teachers serve as facilitators to learning, and learning is more social. Mathematical teaching also needs problem solving. Problem solving includes exercises that require thinking, and he wants students to view these problems as real life situations.

In the next portion of the chart, we see that Schoenfeld then looks at the effect of these goals on mathematical thinking. In this chart, mathematical thinking is defined as developing a mathematical point of view and developing competence with the tools of the trade. He wants these goals to empower students in their mathematical thinking. From this, he proposes a new framework for exploring mathematical thinking. The critical components of this framework are core knowledge, problem solving strategies, effective use of resources, having a mathematical perspective, and engagement in mathematical practices. Core knowledge includes perceptual knowledge, formulas, and basic math skills that are required to solve problems. Problem solving strategies are the heuristics that were discussed previously. It is critical for students to have an effective use of resources, which involves using metacognition. Having a mathematical perspective involves being able to look at problems with a mathematical mindset. Finally, engagement in mathematical practices is crucial in order for students to retain the knowledge that they have obtained in learning how to critically solve problems.

One of the best ways to apply this to the classroom is to focus on giving students word problems that are relevant to the students. Real life problems will focus students' attention on how the math concept can be applied to the world they live in. The best teaching method to use to support the Mathematical Problem Solving method is constructivism.

A good example that Schoenfeld used to show how teachers can help students master difficult formulas is to use flashcards. Below shows some examples of the flashcards used. Students can work alone or in pairs to fill in the blanks in each card. Blank cards can be used for students to write their own.

flashcards1.png

flashcards2.png

In the first figure, students could work together in groups to first find the pattern, then fill in the blanks. While finding the answers to the blanks, students will be learning how to discover patterns at the same time. In the second figure, students can work together to write the equations for the area of each figure. These area equations will help them to expand their view of area to include variables rather than only numbers. In these examples, the role of the student is to discover patterns and equations, work together as a team, and use their metacognition to solve the puzzles. The role of the teacher would be to guide the students as needed, as well as to introduce and explain simple problems to the students and work through them to give them the idea of what is expected. The teacher then can assess the students on the completion of the puzzles, as well as the work done to show their understanding. Finally, students can be assessed on teamwork and collaboration.

When reading through Professor Schoenfeld's works, it is easy to tell what he expects the curriculum of math classes to look like. After all, he is still a practicing math professor himself. What is interesting to see is that his views of mathematics curricula have been criticized, because he wants to teach students how to think, rather than force equations onto students. To Schoenfeld, "mathematical 'content' depends on one's point of view" (http://gse.berkeley.edu/faculty/ahschoenfeld/Schoenfeld_WhatDoWeKnow.pdf). He believes that the typical math class curriculum is familiar and comfortable, because it shows exactly what a student will be exposed to and can clearly show what courses they will be prepared for. However, this is dangerous. Mathematical thinking is composed of much more than simply knowing facts, formulas, and processes. There is a big difference between what a person can mathematically do and what they know. In other words, it is about using what you know.

The curriculum of a math class taught under this theory will be dramatically different than others. One of the main differences in the curriculum will be the amount of real-life problems that are presented to the class. The curriculum of a typical direct instruction classroom contains lessons with lecture, followed by example problems, followed by rigorous practice by the students. Using Schoenfeld’s theory, a mathematics class will be based more on asking general questions about a broad topic, then gradually come to more specific conclusions and derive formulas from this. This will enable students to become more enthusiastic about the topic when they discover the equations on their own. The curriculum cannot take on a minimalistic approach to the basics, however. Students still need to learn the content in some form. Therefore, curricula can be created that contains a good mix of both basic skills and discovery work. A constructivist approach will be the main method of teaching students. Most of the lessons can consist of class discussions, in which students will contribute to the discovery of mathematical procedures.

Assessing the curriculum, according to Schoenfeld, will be based on student success. If students can successfully solve unfamiliar problems with the knowledge they have attained, then the curriculum is successful. Assessments for the students can include their contributions to in-class discussions, as well as content knowledge assessments such as tests and quizzes.

The impact that this new curriculum will have on teaching will change the way educators think about teaching math altogether. Lessons will be taught through class discussions, and teachers will become more passive. They will present the topic, and coax students when needed to discover the material on their own. They will also present real-life problems that can help
students problem solve. This is completely different from the “drill and kill” method that has been used in the past.

## Table of Contents

BackgroundDegreesPublicationsTheoryPedagogyAssessmentReferences"Many students come to believe that school mathematics consists of mastering formal procedures that are completely divorced from real life, from discovery, and from problem solving.'- Alan Schoenfeld## Background

Alan Schoenfeld is currently the Elizabeth and Edward Conner Professor of Education and Affiliated Professor of Mathematics at the University of California, Berkeley. He has also been the President of the American Education Research Association. In 2008, he was awarded the Senior Scholar Award by the AERA's Special Interest Group for Research in Math Education. Much of his work has focused on problem solving.

## Degrees

Professor Schoenfeld has acquired numerous degrees from both Stanford and Queens College. He first received his B.S. in Mathematics from Queens College in 1968. He then received his M.S. in mathematics from Stanford University in 1969, followed by his Ph.D. in 1973. He is currently teaching mathematics and education courses at the University of California, Berkeley.## Publications

The following is a list of the publications that Professor Schoenfeld has created. The list was taken from http://gse.berkeley.edu/faculty/AHSchoenfeld/AHSchoenfeld.html#Degrees, which is Professor Schoenfeld's site at the University of Berkeley.Schoenfeld, A. H. (Ed.) (1983).

Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography.Washington, DC: Mathematical Association ofAmerica.

Schoenfeld, A. H. (1985).

Mathematical problem solving. Orlando, FL: Academic Press.Schoenfeld, A. H. (Ed.) (1987).

Cognitive science and mathematics education. Hillsdale, NJ: Erlbaum.Burkhardt, H., Groves, S., Schoenfeld, A. H., and Stacey, K. (Eds.) (1988).

Problem solving: A world view(Proceedings of the problem solving theme group at the V International Congress onMathematical Education, Adelaide, Australia). Nottingham, England: Shell Centre for Mathematical Education.

diSessa, A, Gardner, M., Greeno, J., Reif, F., & Schoenfeld, A. H. (Eds.) (1990).

Toward a scientific practice of science education.Hillsdale, NJ: Erlbaum.Schoenfeld, A. H. (Ed.) (1990).

A Source Book for College Mathematics Teaching. Washington, DC: Mathematical Association of America.Schoenfeld, A. H. (Ed.) (1992).

Research methods in and for the learning sciences, a special issue ofThe Journal of the Learning Sciences, Volume 2, No. 2.Schoenfeld, A. H. (Ed.) (1994).

Mathematical thinking and problem solving. Hillsdale, NJ: Erlbaum.Dubinsky, E., Schoenfeld, A. H., & Kaput, J. (Eds.) (1994).

Research in Collegiate Mathematics Education. I.Washington, DC: Conference Board of the Mathematical Sciences.Kaput, J. Schoenfeld, A. H., & Dubinsky, E., (Eds.) (1996).

Research in Collegiate Mathematics Education. II.Washington, DC: Conference Board of the Mathematical Sciences.Schoenfeld, A. H. (Ed.) (1997).

Student Assessment in Calculus A report of the NSF Working Group on Assessment in Calculus.Washington, DC: Mathematical Association of America.Schoenfeld, A. H., Kaput, J., & Dubinsky, E. (Eds.) (1998).

Research in Collegiate Mathematics Education. III.Washington, DC: Conference Board of the Mathematical Sciences.Schoenfeld, A. H. (1998)

Issues in Education, Volume 4, Number 1.The issue presents and critiques Schoenfeld's theory of teaching-in-context.Schoenfeld, Alan H. (1999) (Special Issue Editor).

Examining the Complexity of Teaching.Special issue of theJournal of Mathematical Behavior, 18 (3).Dubinsky, E., Schoenfeld, A. H., & Kaput, J. (Eds.) (2000).

Research in Collegiate Mathematics Education. IV.Washington, DC: Conference Board of the Mathematical Sciences.Ferrini-Mundy, J., Joyner, J., Reyes, B., Schoenfeld, A. H., & E. Silver, E. (Eds.) (2000)

Principles and Standards for School Mathematics.Reston, VA: National Council of Teachers of Mathematics.Holton, D., Artigue, M., Kirchgraber, U, Hillel, J., Niss, M. & Schoenfeld, H. (Eds.) (2001).

The teaching and learning of mathematics at the University Level. Dordrecht: Kluwer.Schoenfeld, A. H. (Ed.) (in press).

A study of teaching: Multiple lenses, multiple views. Journal for research in Mathematics Education monograph series.Reston, VA: National Council of Teachers of Mathematics.Schoenfeld, A. H. (Ed.) (in preparation).

Assessing Mathematical proficiency.Cambridge: Cambridge University Press.## Theory

Dr. Schoenfeld began studying math and how students learned. He wanted to find a new way for students to approach math in order to make it more meaningful for them. Many children don't know how useful math can be in the real world, simply because they have always been taught only how to master formulas.

Alan Schoenfeld is known best for his studies of mathematics and how students should learn it. This theory, known as Mathematical Problem Solving, focuses on solving problems through a variety of ways to help students understand the subject better. He believes that teaching mathematics does not only mean educating children on the basic equations and how to plug numbers into them, but how to analyze and understand the meaning behind the formulas. Students must be taught to seek solutions rather than just memorizing procedures, expore patters rather than memorizing formulas, and formulating conjectures rather than doing exercises. Students must have a broader understanding of mathematics in order to excel in it.

According to Dr. Schoenfeld, four categories of skills are necessary in order to be successful in math:

As the above image shows, students must have control, resources, heuristics, and beliefs in order to be successful in mathematics. Control includes being able to determine when certain strategies should be used. Resources are the procedural knowledge that a student has. This is basically the equations taught in a typical math class. Heuristics are the different strategies that students have to solve these math problems, including drawing figures, working backwards, identifying patterns, etc. Finally, the beliefs that a student has are the views that the student has when approaching a problem.

Another main element in Dr. Schoenfeld's mathematical problem solving theory deals with metacognition, or "thinking about thinking." Basically, metacognition means having the knowledge to know when to use particular strategies for problem solving. With the mathematical metacognition that students need, they will be able to apply mathematics to real life situations that they face every day. It can also create a community of learners outside of the classroom that use math in different contexts.

The following image show a complex view of the Mathematical Problem Solving Theory that Schoenfeld is known for:

Since this image can be difficult to read and extremely confusing to the eye, let's analyse it further to make sense of it and extract the most important points. To begin with, we see that this is a basic summary of Alan Schoenfeld's publication

Learning to Think Mathematically.In this writing, he examines the goals of teaching math. Teaching needs enculturation, or using the values of the community. This theory uses sociology, anthropology, and constructivism. Constructivism is a method of teaching in the classroom in which students are engaged in activities, teachers serve as facilitators to learning, and learning is more social. Mathematical teaching also needs problem solving. Problem solving includes exercises that require thinking, and he wants students to view these problems as real life situations.In the next portion of the chart, we see that Schoenfeld then looks at the effect of these goals on mathematical thinking. In this chart, mathematical thinking is defined as developing a mathematical point of view and developing competence with the tools of the trade. He wants these goals to empower students in their mathematical thinking. From this, he proposes a new framework for exploring mathematical thinking. The critical components of this framework are core knowledge, problem solving strategies, effective use of resources, having a mathematical perspective, and engagement in mathematical practices. Core knowledge includes perceptual knowledge, formulas, and basic math skills that are required to solve problems. Problem solving strategies are the heuristics that were discussed previously. It is critical for students to have an effective use of resources, which involves using metacognition. Having a mathematical perspective involves being able to look at problems with a mathematical mindset. Finally, engagement in mathematical practices is crucial in order for students to retain the knowledge that they have obtained in learning how to critically solve problems.

This information was taken from:

http://mathforum.org/~sarah/Discussion.Sessions/Schoenfeld.html

http://tip.psychology.org/schoen.html

http://gse.berkeley.edu/faculty/AHSchoenfeld/AHSchoenfeld.html

http://en.wikipedia.org/wiki/Metacognition

http://skat.ihmc.us/rid=1183793087964_1258534287_1115/Schoenfeld%201991.cmap

http://educationaldesigner.org/ed/volume1/issue2/article5/

## Pedagogy

One of the best ways to apply this to the classroom is to focus on giving students word problems that are relevant to the students. Real life problems will focus students' attention on how the math concept can be applied to the world they live in. The best teaching method to use to support the Mathematical Problem Solving method is

constructivism.A good example that Schoenfeld used to show how teachers can help students master difficult formulas is to use flashcards. Below shows some examples of the flashcards used. Students can work alone or in pairs to fill in the blanks in each card. Blank cards can be used for students to write their own.

In the first figure, students could work together in groups to first find the pattern, then fill in the blanks. While finding the answers to the blanks, students will be learning how to discover patterns at the same time. In the second figure, students can work together to write the equations for the area of each figure. These area equations will help them to expand their view of area to include variables rather than only numbers. In these examples, the role of the student is to discover patterns and equations, work together as a team, and use their metacognition to solve the puzzles. The role of the teacher would be to guide the students as needed, as well as to introduce and explain simple problems to the students and work through them to give them the idea of what is expected. The teacher then can assess the students on the completion of the puzzles, as well as the work done to show their understanding. Finally, students can be assessed on teamwork and collaboration.

Information Taken From:

http://gse.berkeley.edu/faculty/AHSchoenfeld/Schoenfeld_MathTeachingAndLearning.pdf

## Assessment

When reading through Professor Schoenfeld's works, it is easy to tell what he expects the curriculum of math classes to look like. After all, he is still a practicing math professor himself. What is interesting to see is that his views of mathematics curricula have been criticized, because he wants to teach students how to think, rather than force equations onto students. To Schoenfeld, "mathematical 'content' depends on one's point of view" (http://gse.berkeley.edu/faculty/ahschoenfeld/Schoenfeld_WhatDoWeKnow.pdf). He believes that the typical math class curriculum is familiar and comfortable, because it shows exactly what a student will be exposed to and can clearly show what courses they will be prepared for. However, this is dangerous. Mathematical thinking is composed of much more than simply knowing facts, formulas, and processes. There is a big difference between what a person can mathematically do and what they know. In other words, it is about using what you know.

The curriculum of a math class taught under this theory will be dramatically different than others. One of the main differences in the curriculum will be the amount of real-life problems that are presented to the class. The curriculum of a typical direct instruction classroom contains lessons with lecture, followed by example problems, followed by rigorous practice by the students. Using Schoenfeld’s theory, a mathematics class will be based more on asking general questions about a broad topic, then gradually come to more specific conclusions and derive formulas from this. This will enable students to become more enthusiastic about the topic when they discover the equations on their own. The curriculum cannot take on a minimalistic approach to the basics, however. Students still need to learn the content in some form. Therefore, curricula can be created that contains a good mix of both basic skills and discovery work. A constructivist approach will be the main method of teaching students. Most of the lessons can consist of class discussions, in which students will contribute to the discovery of mathematical procedures.

Assessing the curriculum, according to Schoenfeld, will be based on student success. If students can successfully solve unfamiliar problems with the knowledge they have attained, then the curriculum is successful. Assessments for the students can include their contributions to in-class discussions, as well as content knowledge assessments such as tests and quizzes.

The impact that this new curriculum will have on teaching will change the way educators think about teaching math altogether. Lessons will be taught through class discussions, and teachers will become more passive. They will present the topic, and coax students when needed to discover the material on their own. They will also present real-life problems that can help

students problem solve. This is completely different from the “drill and kill” method that has been used in the past.

Information taken from:

http://gse.berkeley.edu/faculty/AHSchoenfeld/Schoenfeld_WhatDoWeKnow.pdf

## References

For more information on Professor Alan Schoenfeld, follow these links:

Math Forum: Learning and Mathematics: Metacognition- http://mathforum.org/~sarah/Discussion.Sessions/Schoenfeld.html

Mathematical Problem Solving- http://tip.psychology.org/schoen.html

GSE Profiles: Alan Schoenfeld- http://gse.berkeley.edu/faculty/AHSchoenfeld/AHSchoenfeld.html

Metacognition- http://en.wikipedia.org/wiki/Metacognition

Schoenfeld 1991- http://skat.ihmc.us/rid=1183793087964_1258534287_1115/Schoenfeld%201991.cmap

Bridging the Cultures of Educational Research and Design- http://educationaldesigner.org/ed/volume1/issue2/article5/

What Do We Know About Mathematics Curricula?- http://gse.berkeley.edu/faculty/AHSchoenfeld/Schoenfeld_WhatDoWeKnow.pdf

Mathematics Teaching and Learning- http://gse.berkeley.edu/faculty/AHSchoenfeld/Schoenfeld_MathTeachingAndLearning.pdf

The Math Wars- http://gse.berkeley.edu/faculty/AHSchoenfeld/Schoenfeld_MathWars.pdf