Polya’s four-step approach to problem solving:
Preparation: Understand the problem (Identify mathematical goals [Objectives])
Thinking Time: Devise a plan (Create and find the problem)
Insight: Carry out the plan (Keep trying until something works [Anticipate student solutions]).
Verification: Look back (Implement and reflect on the problem)
Classroom Discourse:
The interactions that occur throughout a lesson.
Goal of Discourse:
To keep the cognitive demand high while students are learning and formalizing mathematical concepts
When Discourse happens:
Discourse can occur before, during, or after solving a problem, but the after phase is particularly important as it is this discussion that is supposed to help students connect their problem to more general or formal mathematics, and to make connections to other ideas..
Mathematics Vocabulary:
English language learners, other students with more limited language skills, and students with learning disabilities—need to use mathematical vocabulary and articulate mathematics concepts in order to learn both the language and the concepts of mathematics.
Strategic Competence:
Students begin to take ownership of ideas
Productive Disposition:
Develop a sense of power in making sense of mathematics.
General questions for students to build Understanding::
• What did you do that helped you understand the problem?
• Was there something in this problem that reminded you of another problem we’ve done?
• Did you find any numbers or information you didn’t need? How did you know that the information was not important?
• How did you decide what to do?
• How did you decide whether your answer was right?
• Did you try something that didn’t work?
How did you figure out it was not going to work out?
• Can something you did in this problem help you solve other problems?
Balanced Discussions:
Helps students learn how to do mathematics.
Five “Talk Moves”:
- Revoicing (What It Means and Why): This move involves restating the statement as a question in order to clarify, apply appropriate language. Ex: “So, first you recorded your measurements in a table?”
- Rephrasing (What It Means and Why): Asking students to restate someone else’s ideas in their own words.
Ex: “Who can share what Ricardo said, but in your own words?” - Reasoning (What It Means and Why): This asks the student what they think of the idea proposed by another student. Ex: “Do you agree or disagree with Johanna”?
- Elaborating (What It Means and Why): This is a request for students to challenge, add on, elaborate, or give an example. Ex: “Can you give an example?”
- Waiting (What It Means and Why): Ironically, one “talk move” is to not talk. Ex: “This question is important. Let’s take some time to think about it.”
Questioning Considerations:
- The “level” of the question: They are leveled in various models.
- Type of knowledge that is targeted: Both procedural/ conceptual knowledge are important, questions must target both. Procedural questions: “How did you solve this, what steps did you use?” Conceptual knowledge include, “Will this rule always work?”
- Pattern of questioning: One common pattern of questioning: Teacher asks question, student answers question, teacher confirms or challenges answer. This is “initiation-response-feedback” or “IRF” pattern . Another pattern is “funneling,” when a teacher probes students in order to get them to a particular answer.
- Who is thinking of the answer. Focus on both procedures/concepts, and think about your questioning patterns. Be sure that such efforts engage all students, use strategies so everyone must think of the answer.
- How you respond to an answer: Don’t confirm a correct solution, because you lose engaging students in strong discussions about mathematics.
Metacognition:
Conscious monitoring (being aware of how and why you are doing something) and regulation (choosing to do something or deciding to make changes) of your own thought process.
Metacognitive Skills:
Talk about the problem.
How can it be solved?
Identify a strategy to solve the problem.
Notice how your strategy helped you solve the problem. Keep thinking about the problem.
Does it make sense?
Is there another way to solve it?
How Much to Tell and not to Tell:
Information can and should be shared as long as it does not solve the problem [and] does not take away the need for students to reflect on the situation and develop solution methods they understand.
Three things teachers need to tell students:
Mathematical conventions. The symbols used in representing “three and five equals eight” as “3 + 5 = 8” are conventions (+ and =). Terminology and labels are also conventions.
• Alternative methods. When an important strategy does not come naturally from students, the teacher introduces the strategy, being careful to introduce it as “another” way, not the only or the best way.
• Clarification/formalization of students’ methods: Help students clarify or interpret their ideas, point out related ideas. A student may add 38 and 5 by noting that 38 and 2 more is 40 with 3 more making 43. This strategy can be related to the Make 10 strategy used to add 8 + 5.
Writing to Learn:
- The act of writing is a reflective process. As students make an effort to explain their thinking and defend their answers, they will spend more focused time thinking about the ideas involved.
- A written report is a rehearsal for the discussion period. It is difficult for students to explain how they solved a problem 15 minutes after they have done so. Students can always refer to a written report when asked to share. Even a kindergarten child can show a picture and talk about it.
- A written report is also a written record that remains when the lesson is finished. The reports can be collected and looked at later. The information can be used for planning, for finding out who needs help or opportunities to extend their knowledge, and for evaluation and parent conferences.
Graphic organizers used for writing:
They help students know what to write.
