A REVIEW OF ENCOURAGING MATHEMATICAL THINKING: DISCOURSE AROUND A RICH PROBLEM
by: Rita M. Paton
Canutillo Middle School
February 26, 2000
In reviewing the Math Forum’s Bridging Research and Practice Group’s (BRAP) paper entitled Encouraging Mathematical Thinking: Discourse around a Rich Problem, I have gained insight into ways to improve what we teachers do for our students, and ourselves. I was impressed to find relevant, practical, student centered suggestions, which allow for individual discovery and mathematical growth.
The main thrust of this project is the use of discourse to allow students to reevaluate their own knowledge, incorporate new information, and make sense of mathematical thought. Discourse, as the BRAP group defines it, is "The use of questioning, listening, writing, and reflection as a means of encouraging reciprocal conservation—the kind of teaching that allows every person to have a voice in creating mathematical understanding."
It is through rich discourse that students turn the frequent disharmony between experience and new discoveries into an opportunity to grow in understanding, confidence, and mathematical abilities. This discourse with peers, in a safe and supportive environment, allows students to speak in common vocabulary, share ideas and uncertainties, hypothesize on outcomes, analyze results and data, and reflect on how new information can be incorporated into their own mathematical repertoire. Video clips from both Susan Stein "Repeating a Student's Question" and Susan Boone’s classes "Four Students Work Together" were examples of effective student discourse from
answering questions, predicting outcomes, reacting to the cylinder demonstration, organizing and calculating numerical values, and analysis of the results. The students led this dialogue and the teachers merely summarized information, and directed transitions in the lesson.
Many of us use leading questions, non-leading questions, paraphrasing, summarizing and listening discussed in the BRAP paper, but not always to the extent that would truly promote student success. I was most impressed with Jon Bosden's "Paraphrasing, Using Math Vocabulary" and Art Mabbott's "Paraphrasing Using Math Vocabulary", use of paraphrasing student ideas and comments in order to transition ordinary student language, albeit accurate in its content, into technical mathematical language. Both teachers acknowledge the student’s accurate contribution, but incorporate precise
terms such as diameter, radius, and greater circumference. Too often we expect students to be fluent in "our language" of mathematics just because a
term or concept was introduced in prior studies. The goals of healthy
discourse are to allow students to express predictions and concepts in their
own language, and to shape this vocabulary.
However, there comes a time to insist on the use of formal vocabulary essential to the more abstract formulas or algorithms. This was observed in many of the video clips when project teachers ask leading questions to build on student comments. Specific ways to accomplish this could be: The statement is true, but if you dig deeper what would you find? Why is the radius squared more important than the height? How does radius relate, in mathematical terms to the circumference or to the volume? Can you relate "bigger something" to one of these terms? If students can answer these questions, then some level of deeper understanding has been reached.
I perceived three benefits of the students expressing their thoughts. First, teachers can get a realistic picture of student thoughts, no matter how accurate or erroneous. This tells the teacher where to go next, whether to redirect the discourse, reteach a skill or concept, or proceed as planned.
Second, my observations mirror Susan Stein's experiences captured in the video clip "One Student Corrects Another". Students will speak up to build onto, clarify or correct another student's statements once they are confident in their own understanding. Furthermore, each student must be allowed to correct his earlier predictions or comments based on new information generated during the discourse. Thus, the students become the voice in the classroom, teaching their classmates in a common language, and at a mutual level of understanding.
Finally, the accountability for learning concepts, incorporating vocabulary, and problem solving belongs to the individual student. Cooperative learning groups and curving grades may allow an individual to pass, but do not ensure that he obtained the necessary level of mathematical understanding. Through discourse the individual must evaluate his observations and explain how they have affected his understanding. He cannot luck into correct answers without thinking.
The result of healthy discourse observed by the BRAP group was that students realize that they can do it. They become their own resource. Students can identify gaps in their own understanding and vocabulary, and ask pertinent questions to fill them. I see how students would gain confidence as they confirm their intuition, and on many occasions tackle their misconceptions. Students realize that being less than correct in their predictions or "not getting it' at first, are merely challenges and not insurmountable obstacles. They are now open to learning why it happened, as observed in
Susan Stein's student's reaction to the initial cylinder demonstration of
the volume disparities.
I regularly experience numerous “hurrahs” in my regular and gifted classes with student centered activities that focus around concrete, tactile manipulatives as tools for solving rich problems, like the cylinder problem discussed here. By affirming their beliefs, discourse between students aided those who felt confident in their knowledge, by affirming their beliefs. All those still assimilating their ideas benefited by hearing each discussion resulting from another’s struggle with the mathematics.
I can see all students benefiting from discourse, but I know from experience, and collaboration with peers, that some student populations present a challenge to even the most well-intentioned interventions. In my very low socio-economic district, we face a larger than normal remedial population of students overwhelmed by the curriculum. These students are served under Special Education, Section 504 services, and English as a Second Language. By middle school level almost 15% of our students are struggling with concepts that are 2 to 5 years below grade level. Although we work with inclusion, dual language, and intervention programs beginning with Head Start and preschool, we continue to see students struggling with basics, the language, and a lack of self-esteem. It has been a challenge to present activities and maintain discourse with those lacking a concrete foundation of both mathematical concepts and English vocabulary.
The BRAP Group touched on this when discussing how it is critical for effectiveness that the cylinder lesson was introduced at the appropriate point of the curriculum. They felt that if students lacked sufficient prior knowledge, problem solving attempts might fail or not yield results that would support further discourse. How can we address students who lack the foundation, but yet would benefit greatly from exposure to a concrete problem, like the cylinder problem? These populations are the most in need of the benefits of discourse, yet they are the hardest to reach. The benefits would include vocabulary development, building confidence in their understanding of mathematics, and developing a solid mathematical foundation. I would like to see these students be the focus in the ongoing research and dialogue of the BRAP group.
I often perceive this problem: "How can skills observed in a concrete context, such as with manipulatives, be transferred and applied to more abstract problem solving?" The BRAP Group’s observations and conclusions suggest that discourse can help bridge the concrete and the abstract. I would also like to propose that discourse be the key in the seamless vertical alignment of mathematics, Pre-K through Calculus. It is not enough to merely get an answer, but rather to be able to justify the underlying mathematics. If students learn to verbalize throughout their school years, just imagine the power behind their solutions. I am excited by the myriad benefits of student discourse, combined with concrete mathematics and rich problems throughout their education.
The cylinder problem is rich indeed, loaded with geometric vocabulary and algebraic relationships between dimensions, circumference, and volume. Yet it is easy to model with paper serving as a net of the surface area. It has many levels of complexity suitable for all grades. Elementary students will create paper models and compare the resulting volumes. Discourse will focus around their predictions, results, and applications to other models. Important vocabulary and concepts such as radius, circumference, volume, area, more, less, and equivalent are introduced.
The middle school lesson reinforces the concepts of the elementary lesson, expanding the explanations to include constant surface area, dimensions, square units, and calculating the volume in cubic units. By comparing the resulting volumes, students can see patterns emerge. They compare predictions to results, and pose questions such as: Why did this happen? How many other cylinders could we make with the paper? Would other cylinders also show these results? All of these leading questions prompt the students to make testable models, chart data, generate algorithms, do calculations, and analyze the data.
I would first have students plot the data on a grid, connecting the points as Judy Koenig directed her student to do in "Several Leading Questions and Paraphrasing; Adding an Example". By presenting the graph to the class and interpreting relationships between the variables, students increase vocabulary and can predict measurements for other cylinders they did not build. They can then test those predictions using the algorithms or a spreadsheet program.
Technology used in this manner can be a great asset in mathematics, especially when the calculations are secondary to the focus of the lesson. These tools can aid in extrapolating data beyond the four cylinders to an infinite number of cylinders, as witnessed in Jon Basden’s class discussion "Teacher as Confidence Builder" and "Not Just One Right Way". The radius and volume can be extrapolated conveniently from the dimensions created from the constant area. The graphing functions can be employed to visually relate the radius to the volume of the cylinders.
It is important that technology be incorporated into mathematics classes regularly. Graphing calculators, spreadsheets and software are important tools that allow us to give more attention to analysis of data, graphs, and patterns. Discourse is an essential part of the appropriate use of technology. Discourse helps us see "the big picture", and allows the students to focus on transferring skills and concepts to similar problems.
Both the middle school and high school lessons use the four models of cylinders made from 8.5 X 11 inch sheets of paper. I was thrilled to see the incorporation of algebra to manipulate the algorithms to find the radius and volume. Too often my students only see an algorithm like circumference= 2*pi*r as good only for computing circumference. It is not apparent to them that with the circumference, one can also find the radius or diameter without measuring the circle directly. This ability to manipulate algorithms is essential not only for algebra and math, but also for chemistry, physics, and statistics. However, it is a skill not well developed, nor practiced nearly enough.
I would also like to see included in this lesson the development of both circumference, and area of a circle algorithms. The authors are assuming that students have an understanding of these terms and formulas, just because they may have calculated them previously. I have found that seventh graders have learned to put in the given diameter value, and multiply by "3.14", but they do not understand what it represents, especially the pi. There are numerous concrete activities that one could incorporate into this lesson or a previous one.
In order to explore the components of circumference wrap string around a can and cut to length. Then cut four strings equal to the diameter of the can. When the diameter strings are placed end to end next to the circumference string, it will take three diameters plus a little of the fourth to equal the length of the circumference string. Once students try this with many cans or circles, they see that circumference is always three diameters plus a little bit. Hence the students discover the origin of our constant "pi". If they measure the strings, and divide the circumference by the diameter, the average quotient will be about 3.14, depending on how accurately they measure.
Students can manipulate the variables to form circumference equal to diameter times the constant known as pi, or C=d*pi. Diameter can be replaced by two radii, resulting in C=2*r*pi. I have found that my students understand this term pi and why it is used only with circles, much better after this activity.
Susan Stein, "Struggling to Find Explanation" must have done some groundwork on algorithms for area of a circle. When her students were discussing area of a circle, one student stated radius times pi, and another countered that it was radius squared times pi, because they had used squares to fill the circle. They used three squares with the dimensions equal to the radius, plus a little bit of room left over to fill the circle. A diagram of this activity is below.
This reinforces the concept of pi and the relationships of the algorithms to the dimensions and to each other. Without grounding in a concrete context, algorithms are meaningless to students. I use the term 'math-speak" when information becomes like another language, without meaning to the student. Students cannot take concepts and skills to the next level if they do not understand them or their purpose.
Reflecting on the lesson should focus on interpreting graphs, applying insights, and comparing the outcomes to predictions. Such discourse may have the most impact if first shared within the small groups. Each group can then summarize the highlights to the class.
I also recommend writing in a math journal. Besides the need to practice writing skills, many students will reflect more on their own growth if only the teacher will see it. This is also a place for uncertainties to be expressed and I can follow up with that individual. I may have them do similar problems in their journal to assess specific details of their individual understanding. This is another way to create individual accountability in a group project. A fellow teacher has each group member reflect on a different aspect of the problem and binds them into a group report with their pictures, models, data tables, and graphs.
The participants’ reflections paralleled what we want to see in our own students. They went into this project predicting that they might learn more about what to do for students. However, discourse with peers gave them more confidence, colleagues with whom to learn, tools to enhance their teaching, and a new appreciation for the field of mathematics education. By focusing on the process, and not just getting an answer, they learned interventions to address many strengths and needs of the students. They also believe that we are all still learners of mathematics, and our power as teachers comes from our diverse approaches to this field.
Jon Basden said it well in his personal reflection on this project. "If collaboration and personal reflection and interaction with peers is so helpful to adults, than why not foster the same environment with students." We get the most from people we can relate to, for whom there is mutual respect, and who give us something useful. The same is true for our students.
Project teachers and the authors also agreed on the need for ongoing discourse among teachers and researchers. Awareness of new practices, tools, and ways to interact more effectively with our students and our content, will make us better teachers. Researchers need to realize that understanding and improving what happens in schools is the reason they exist. Educating ourselves, and partaking in professional discussion groups are things we need to make time for, but are unfortunately the first to go during the busy school year. I feel that I have been more successful with my students since applying visual math methodologies and creating a more student centered classroom. Both of these philosophies are the results of research on how today’s students learn.
Overall, my opinions of this videopaper, the cylinder problem, and the issue of using discourse to help foster student learning, are very positive. I believe that what makes something valuable to education is if it can assist me in meeting this challenge stated by Robert Maynard Hutchins. "The object of education is to prepare the young to educate themselves for life." Discourse to develop a deeper understanding of mathematics and its application is a skill for a lifetime.