Engaging Students in Problem Solving Through Questioning

1. Engaging Students in Problem Solving Through Questioning Jim Rahn LL Teach, Inc. www.jamesrahn.com James.rahn@verizon.net

2. NCTM Process Standards • Problem solving • Reasoning and Proof • Communication • Connections • Representation

3. Problem Solving  • Instructional programs from pre-K through grade 12 should enable all students to— • Build new mathematical knowledge through problem solving • Solve problems that arise in mathematics and in other contexts • Apply and adapt a variety of appropriate strategies to solve problems • Monitor and reflect on the process of mathematical problem solving

4. Goals for Engaging Students in Problem Solving • Improve pupils' willingness to try problems and improve their perseverance when solving problems. • Improve pupils' self-concepts with respect to the abilities to solve problems. • Make pupils aware of the problem-solving strategies. • Make pupils aware of the value of approaching problems in a systematic manner. • Make pupils aware that many problems can be solved in more than one way. • Improve pupils' abilities to select appropriate solution strategies. • Improve pupils' abilities to implement solution strategies accurately. • Improve pupils' abilities to get more correct answers to problems.

5. In today’s workshop • We will focus on • questioning that can take place during both the investigations and the follow up activities in both DG and DAA • The type of questions you can ask • Specific examples of questions for sample lessons • Brainstorming additional questions that will help develop conceptual understand for upcoming lessons

6. Michael T. Battista of Kent State University writes • that algebra teaching should focus on the basic skills of today, not those of 40 years ago. Problem solving, reasoning, justifying ideas, making sense of complex situations, and learning new ideas independently—not paper-and-pencil computation—are now critical skills for all Americans. In the Information Age and the Web era, obtaining the facts is not the problem; analyzing and making sense of them is. “The Mathematical Miseducation of America’s Youth,” The Phi Delta Kappan. February, 1999

7. Classroom Situation 1

8. Assign Roles • Teacher • Susie • Arthoro • Tommy • Joshua • Sam • Jacob • Travis • Chorus – Everyone but the teacher

9. Laura, the teacher, had finished teaching a section on solving systems of linear equations using the elimination method. After reminding her students of the method of substitution that they had learned the previous day, she used two examples to illustrate how each of the methods could be employed to find the values of two unknowns. Before assigning exercises for in-class practice, she asked whether students had any questions about either of the methods she had discussed.

10. Name some good characteristics about the teacher’s interaction with the students • tried to assure that the students knew the steps for solving linear equations in one or two variables • reviewed procedures when students requested additional examples • insisted that the students ask questions • called on specific students to ask or answer questions • tried to assess whether the students remembered the algorithm and methods from previous lessons

11. Describe some weaknesses in the questioning • Not sure what students can or can’t do at the end of the session • Limited responses from students • Little evidence of student work or thinking • Questions controlled the student responses or answers • Little reveal about student understanding, misunderstandings, and competence in either the computational or conceptual domain • Questions did not inquire whether the students understood why or when certain procedures were used • Most of the questions were process questions or only solicited dichotomous responses

12. Name some specific ways the student responses were not addressed • Not sure what difficulty that Susie was experiencing or what she hoped to learn from the new example • Questions did not try to did not try to detect the problem area • Susie's response, "It does not matter"' remained unexplored • The teacher made an assumption about Susie’s need

13. Joshua's remark, "How come we are substituting?“ highlighted his confusion on the most fundamental part of the lesson or possibly other students • Is there more than one approachthat could be used? • How many of students fully understood the processes?

14. General Characteristics of Teacher Questions

15. Teachers ask lots of questions • Most questions control students' learning • focus students' attention on specific features of the concepts the class in exploring • Central to the type of learning that takes place in the classroom • Some are aimed at recall of information • Others provoke problem solving or concept development. • Establish/validate students' perceptions about what is important to know • Procedural Questions

16. Better questions: • Are directed toward evaluating students' thinking • Help students make connections between previously learned ideas and new ideas • Help students practice reasoning from the data and learn to argue a point of view, and • Help students examine mathematics from more than one perspective.

17. Wagner and Parker (1993) said: • It has been shown that some students can successfully give correct answers to problems; however, the depth of their misunderstandings or the nature of their misconceptions can become obvious only when they were asked to explain their thinking. • This suggests that unless students are asked to explain their thinking, a teacher may not know which concepts the students understand.

18. The Form • A closed form question seeks a particular answer • These questions usually are stated to solicit right or wrong or true or false answers. • An open form (how or why) solicit additional information from the students The Content • Determines the type of information that teacher obtains about students' thinking

19. The Purpose • To engage inactive students in the activity • To test students' on specific skills • To encourage students to explore mathematical relationships • To help build connections • To create a sense of community • To build group relationships among students. • To establish individual accountability and to detect individual progress

20. Reasons for asking Questions • To develop problem-solving skills • Ask questions that foster an exploration • To ascertain students’ procedural knowledge or basic skills • Not adequate in determining what students can do beyond solving exercises. • To acquire data on the students’ skills and the depth of their understanding • Ask “How do you know?” questions • To develop a coherent profile of what learners know and whether they have fully grasped the concepts • To bring a lesson to immediate closure • We often rely on closed-form questions

21. Effective Questioning • Think about long- and short-term instructional goals • Identify the big idea of the lesson(s) and the mathematical outcomes that students can achieve

22. 10 Things to Think about: • What do I want the students to know at the end of this lesson or unit? • How do I know whether they really know it? • How does this new concept relate to the ones that the class has discussed? • How do I assess whether the students realize the connections? • What are common misconceptions and how can I determine whether my students have these misconceptions?

23. 10 Things to Think about: • What question can I ask to measure the different layers of student understanding? • How can I help students focus on similarities and differences within a lesson? • What questions can I ask that will allow me to determine whether students can use the procedure in context? • How do I determine whether they can use a procedure in a new setting? • How should I phrase a question to meet the needs of students with special needs?

24. Comparing Closed and Open Ended Questions From Classroom Situation 1

25. Example 1 "Does everyone understand the method of elimination?" "When is using the elimination method in solving systems of linear equations more advantageous than using other methods? Why does elimination yield a solution?" • Stated in a closed form. • Has the potential to determine the number of people who claim understanding of a piece of mathematics. • Asks students to analyze and evaluate various methods. To answer these questions, the students need to pay attention to both the content and the context of the methods discussed in class and to assess specific features of each method.

26. Example 2 "What should you consider when deciding which method to use in solving instances of systems of linear equations? How do you decide which method is more efficient?" "Are you clear on the difference between the methods of elimination and substitution?" • Stated in a closed form. • Has the potential to determine the number of people who claim it has the potential to determine the number of people who claim understanding of a piece of mathematics. • Asks students to analyze and evaluate various methods. To answer these questions, the students need to pay attention to both the content and the context of the methods discussed in class and to assess specific features of each method.

27. Example 3 "What could you do next and why? How could you proceed from here? How do you know that the solution you find from elimination is a solution to both equations?" "What is the next step?" • students are expected to remember specific procedures • These questions allow them to identify other methods that appear meaningful to them. • The teacher has a greater chance of determining whether the students understand why specific procedures are emphasized in the process. • Determining when and where to use each type of question depends on the purpose of each question and on the mathematical and instructional goals of the teacher,

28. Example 5 "Does anyone have any questions about what we did?" "What are some good questions to ask about what we discussed today?" • Can give the teacher a greater knowledge base about his or her students' thinking, their conceptual development, and their level of comfort with the targeted concepts and algorithms. • The teacher can then organize instruction to meet the specific needs of the group.

29. Example 6 "Is this clear to everyone?" "Identify three features of this process that are most dear to you." • Can give the teacher a greater knowledge base about his or her students' thinking, their conceptual development, and their level of comfort with the targeted concepts and algorithms. • The teacher can then organize instruction to meet the specific needs of the group.

30. Assign Readers • Teacher • Students: Everyone

31. Classroom Situation 2 The teacher has directed the students to find the equation of graph B. A student has identified (0,0) and (4,1) as two points on the line represented by graph B.

32. The teacher walks through a series of steps with the students until they find the correct equation for the line. • The students' attention is focused on completing various steps – not a connection of all the parts. • The student needs to only know how to respond to the individual questions with a correct answer. • Only the teacher's thinking process is explicit; little is known about what the students were actually thinking. • The end result is that the teacher is making all the needed connections.

33. Scaffolding or not Scaffolding • Is the teacher scaffolding the students' thinking by modeling the questions one needs to ask when finding a linear equation? • For scaffolding to be successful two important aspects need to occur in future interactions: • the teacher should discuss these particular questions and the purpose for using them • the questions need to be diminished and eventually removed. • Some students don’t immediately understand the significance of the questions because they view asking questions to be the teacher’s role – not their role. • Students need to realize why these type of questions are being asked. • The set of questions will eventually need to be removed and be able to be asked by the student.

34. Employing a scaffolding-interaction pattern limits what students are able to contribute because it directs their thinking in a predetermined path based only on how the teacher would have solved the problem. • Students need more opportunities to articulate their thinking so that they can build on prior knowledge and make their ideas clear to the teacher and their classmates.

35. Assign Readers • Teacher • Becky • Mark • Sam • Keith

36. Classroom Situation 3 Students were given a point on a line (5, 9.5) and the slope of the line as 1.5. They were asked to find the y-intercept so they could write the equation of the line. One student, Becky offered a unique way to use the graphing calculator to find the equation of the line. The teacher asks questions or replied in ways that helped Becky articulate her thinking. This would help both the teacher and the other students.

37. Focusing • A focusing-interaction pattern requires the teacher to listen to students' responses and guide them based on what the students are thinking rather than how the teacher would solve the problem. • This pattern of interaction serves many purposes: • allowing the teacher to see more clearly what the students were thinking • requiring the students to make their thinking clear and articulate so that others can understand what they are saying • values student thinking and • encourages students to contribute in the classroom

38. Things to note about this example • Becky has used a unique approach with the graphing calculator to find the equation of the line. • The teacher recognizes that Becky's method is unique. • Becky is asked to explain what she did to everyone in the class. • To assist Becky in articulating her strategy and to aid everyone else's sense-making, the teacher suggests that Becky go back through the process while everyone else follows along on their graphing calculators. • The teacher interrupts with addition questions when she thinks Becky's strategy might be confusing

39. This situation does not allow students' attention "to fade or change or be interrupted" • The teacher "tries to anticipate what the other students might not understand and asks clarifying questions to keep all students’ attention focused. • The classroom discussion then turned to figuring out why Becky's strategy worked and pursuing how changes in the slope and y-intercept in the equation effect the shifts in the line on the graph. • The long-term benefits of focusing make it imperative that mathematics teachers "focus" more often.

40. Classroom Situation 2 Revised . The teacher has directed the students to find the equation of graph B. A student has identified (0,0) and (4,1) as two points on the line represented by graph B.

41. How has the lesson been revised? • The teacher interprets the pause differently – more clarification is needed • The teacher asks the students to articulate what the slope is by referring back to a previous problem they solved. • Students are asked what they mean and to decide if they agree or disagree with others' ideas. • The teacher repeats important information and keeps students focused on the components of slope • The teacher values student language (steep), but also subtly offers the more mathematically appropriate language (slope).

42. The teacher does not do the thinking for the students • Students are helped to make connections and articulate their thinking. • These actions Not only does this action value and draw out student thinking but it also supports two of the teacher's goals: • (1) help students make connections and • (2) encourage multiple representations

43. Thinking about Questions in DM

44. Discovering Geometry – Chapter 4 Triangle Sum • Page 201 • If some groups finish early, ask “What conjectures did you use as the basis of your proof?” • Notice the questions within the proof. • Page 202 • Example • Page 202 • Sharing Ideas • Page 203 • What questions might you ask students who are finding difficulty in solving problems 6-7? 8-9?

45. Discovering Advanced Algebra – Chapter 4 – Solving Equations • Page 182 • Lesson Example A • What questions might you add? • Page 184 • Discussing the Lesson • What other questions might you ask? • Page 185 • Lesson Example • What question might you add? • Page 186 • Discussing the Investigation • What additional questions might you ask?

46. DG - DAA • Select a section and study questions suggested throughout the text. • What additional questions might you ask? • What are your reasons for the questions? • Select a section and study questions suggested throughout the text. • What additional questions might you ask? • What are your reasons for the questions? Discovering Geometry Discovering Advanced Algebra

47. Some Suggestions on Ways to Change your Questioning Patterns

48. Record your lesson • Listen to the questions you ask during an upcoming lesson. • Were the questions effective? • How can you tell? • Did the questions result in single answers or explanations from the students? • Were you able to tell if the students had true understanding of the mathematical topics? • What kind of questions would you suggest to the teacher?

49. Think about the BIG IDEA Remind yourself of the big ideas and think if your questions are helping the students achieve understanding that BIG IDEA. Make sure that you prepare questions that address these multiple approaches and misconceptions, prompting a discussion about when particular approaches are better than others and how to explain why each misconception is faulty. Close each lesson with a summarizing question that reiterates the BIG IDEA.

50. Additional Thoughts When you use questions not mentioned in the textbook, write them in your teacher’s edition. Talk with other teachers to see what questions they have added.