Last time, I explored some ideas about what comprises good mathematical questions in mathematics classrooms taken from MAKING SENSE: Teaching and Learning Mathematics With Understanding, by Hiebert, et al. In this entry, I want to introduce ideas from a book by Peter Sullivan and Pat Lilburn:

**Good Questions for Math Teaching: Why Ask Them and What to Ask [K-6]**

Sullivan and Lilburn's Criteria

In the introduction to their book, Sullivan and Lilburn list three main criteria for good questions: a) They require more than remembering a fact or reproducing a skill;

b) Students can learn by answering the questions, and the teacher learns about each student from the attempt; and c) There may be several acceptable answers.

The first of these features, while not something everyone sees as important in all mathematics classrooms, is essential to non-routine questions. It takes no teacher skill to come up with questions that only ask for students to show that they have memorized a fact (e.g., "What do we call the answer in a multiplication problem?" or "What is the sum of the interior angles in any triangle?") or reproduce a skill (e.g., "Find the product of 27 x 36" or "What is the slope of the line that passes through the points (3, 2) and (0, 6)?") Teachers can easily make up such problems or find them in textbooks, ancillary materials, supplementary workbooks, on-line, or generated by software. There isn't much to debate here: no one is suggesting that there is no place for such problems, and as they are in fact ubiquitous in American classrooms, I have no interest in debating or analyzing their use. As practice, as routine exercises, as reinforcement, as "drill and skill" or "drill and kill," depending on your viewpoint, they simply are part of the landscape. What is of more interest to me, and perhaps of more controversy for some, emerges from the other two criteria.

The second property these authors want to see in good questions are those that are both somehow instructive in their own right to students and which carry added information that the teacher learns about each student as s/he attempts to solve the problem. This suggests that good problems themselves add something to student knowledge and understanding that might not be readily obtained from, say, listening to direct instruction or doing routine exercises.

At the same time, the authors value problems that tell the teacher something about the student that, I assume, helps better assess student thinking and learning, allows for better feedback and instruction, and which, once again, would not be evident from having students complete more routine exercises.

The final property they see as part of good problems is the possibility of more than one acceptable answer. Some parents and critics of reform teaching find this sort of thing highly troubling. Unfortunately, coupled with an increasing emphasis on process over product, critics and parents (and some pro-reform teachers), appear to believe that the idea here is that the "right answer" doesn't matter. From my perspective, the "process vs. product" debate is fruitless if people on either side insist on either supporting one extreme or the other, or upon claiming that people on the other side only value either process or product. For what may be a small minority of pro-reform teachers, I can only say that believing that the answer doesn't matter at all leaves out a piece of the learning puzzle that they can't afford to ignore. But the other extreme view, that all that matters is "getting the right answer" and everything else is a waste of time seems to me to be destructive for kids if taken too far (kids can do an excellent job, make a small mistake that could stem from misunderstanding, carelessness, or something as trivial as reversing digits in copying down the answer (or mis-marking a multiple choice answer sheet). It's not that it's destructive for kids to think there's a single right answer when in fact there is; it's not that it's destructive for them to know they made an error, if in fact they did; what is destructive is to denigrate excellent and clear thinking on the part of students if in fact that have done this and given evidence of it. And that is, of course, when of the problems with multiple-choice problems or any assessment that fails to credit students for demonstrating what they know (or rewarding them for a lucky guess): not only are students taught that the answer is everything, but also that it's better to be lucky than good (or at least just as good to be lucky), and that gaming the system suffices. It also teaches them that if you get a problem right (even through luck or a series of mistakes that just get you to the right place blindly), there's no point in thinking about or having to explain your answer (and I suspect this is behind SOME of the resistance on the part of kids to being asked to talk or write about their mathematical thinking: doing so can unmask their luck, their underlying confusion, or even cheating).

In any event, the point here that must be noted carefully: the authors say that there MAY be multiple acceptable answers to good problems. So they allow that problems with only one acceptable answer could also be good. But what does it mean for a problem to have multiple acceptable answers? Before I look at one example from their book, I want to suggest that a problem that has multiple solutions that are correct within the terms of the problem are quite possible. This fact can arise in a case where there is more than one right way to accomplish a task that has a mathematical component but which is not asking for just a number. Or perhaps more than one number meets the criteria of a problem. Even in cases where there is a sole correct numerical solution, there may be more than one good way to arrive at that answer.

Unfortunately, these reasonable possibilities do not strike everyone as "okay" for kids. This believe may arise from a narrow understanding of what mathematics is and how it develops historically. Unfortunately, the history of mathematics is not much stressed even for mathematics majors at the college level. Few university mathematics departments offer a course in math history any more, not even for students who plan to teach at the secondary level and who could benefit greatly getting a more honest historical view of how the field has developed and grown. Such an understanding can help make the subject more human for students and teachers alike, and can also increase its accessibility. The fact that Newton and Leibniz independently developed the calculus, one of many such examples of parallel invention in science and mathematics, helps validate the notion that not ever powerful idea must wait for a single, unique person and set of circumstances to bring it to fruition.

Finally, the idea of open-ended problems is not radical or unique to American mathematics education reform. Jerry Becker and Shigeru Shimada edited an excellent volume entitled,

For a look at open-ended questions drawn from American elementary classrooms, along with ways to find, create, and evaluate student responses to them, see OPEN-ENDED QUESTIONS in ELEMENTARY MATHEMATICS by Mary Kay Dyer and Christine Moynihan.

In the introduction to their book, Sullivan and Lilburn list three main criteria for good questions: a) They require more than remembering a fact or reproducing a skill;

b) Students can learn by answering the questions, and the teacher learns about each student from the attempt; and c) There may be several acceptable answers.

The first of these features, while not something everyone sees as important in all mathematics classrooms, is essential to non-routine questions. It takes no teacher skill to come up with questions that only ask for students to show that they have memorized a fact (e.g., "What do we call the answer in a multiplication problem?" or "What is the sum of the interior angles in any triangle?") or reproduce a skill (e.g., "Find the product of 27 x 36" or "What is the slope of the line that passes through the points (3, 2) and (0, 6)?") Teachers can easily make up such problems or find them in textbooks, ancillary materials, supplementary workbooks, on-line, or generated by software. There isn't much to debate here: no one is suggesting that there is no place for such problems, and as they are in fact ubiquitous in American classrooms, I have no interest in debating or analyzing their use. As practice, as routine exercises, as reinforcement, as "drill and skill" or "drill and kill," depending on your viewpoint, they simply are part of the landscape. What is of more interest to me, and perhaps of more controversy for some, emerges from the other two criteria.

The second property these authors want to see in good questions are those that are both somehow instructive in their own right to students and which carry added information that the teacher learns about each student as s/he attempts to solve the problem. This suggests that good problems themselves add something to student knowledge and understanding that might not be readily obtained from, say, listening to direct instruction or doing routine exercises.

At the same time, the authors value problems that tell the teacher something about the student that, I assume, helps better assess student thinking and learning, allows for better feedback and instruction, and which, once again, would not be evident from having students complete more routine exercises.

The final property they see as part of good problems is the possibility of more than one acceptable answer. Some parents and critics of reform teaching find this sort of thing highly troubling. Unfortunately, coupled with an increasing emphasis on process over product, critics and parents (and some pro-reform teachers), appear to believe that the idea here is that the "right answer" doesn't matter. From my perspective, the "process vs. product" debate is fruitless if people on either side insist on either supporting one extreme or the other, or upon claiming that people on the other side only value either process or product. For what may be a small minority of pro-reform teachers, I can only say that believing that the answer doesn't matter at all leaves out a piece of the learning puzzle that they can't afford to ignore. But the other extreme view, that all that matters is "getting the right answer" and everything else is a waste of time seems to me to be destructive for kids if taken too far (kids can do an excellent job, make a small mistake that could stem from misunderstanding, carelessness, or something as trivial as reversing digits in copying down the answer (or mis-marking a multiple choice answer sheet). It's not that it's destructive for kids to think there's a single right answer when in fact there is; it's not that it's destructive for them to know they made an error, if in fact they did; what is destructive is to denigrate excellent and clear thinking on the part of students if in fact that have done this and given evidence of it. And that is, of course, when of the problems with multiple-choice problems or any assessment that fails to credit students for demonstrating what they know (or rewarding them for a lucky guess): not only are students taught that the answer is everything, but also that it's better to be lucky than good (or at least just as good to be lucky), and that gaming the system suffices. It also teaches them that if you get a problem right (even through luck or a series of mistakes that just get you to the right place blindly), there's no point in thinking about or having to explain your answer (and I suspect this is behind SOME of the resistance on the part of kids to being asked to talk or write about their mathematical thinking: doing so can unmask their luck, their underlying confusion, or even cheating).

In any event, the point here that must be noted carefully: the authors say that there MAY be multiple acceptable answers to good problems. So they allow that problems with only one acceptable answer could also be good. But what does it mean for a problem to have multiple acceptable answers? Before I look at one example from their book, I want to suggest that a problem that has multiple solutions that are correct within the terms of the problem are quite possible. This fact can arise in a case where there is more than one right way to accomplish a task that has a mathematical component but which is not asking for just a number. Or perhaps more than one number meets the criteria of a problem. Even in cases where there is a sole correct numerical solution, there may be more than one good way to arrive at that answer.

Unfortunately, these reasonable possibilities do not strike everyone as "okay" for kids. This believe may arise from a narrow understanding of what mathematics is and how it develops historically. Unfortunately, the history of mathematics is not much stressed even for mathematics majors at the college level. Few university mathematics departments offer a course in math history any more, not even for students who plan to teach at the secondary level and who could benefit greatly getting a more honest historical view of how the field has developed and grown. Such an understanding can help make the subject more human for students and teachers alike, and can also increase its accessibility. The fact that Newton and Leibniz independently developed the calculus, one of many such examples of parallel invention in science and mathematics, helps validate the notion that not ever powerful idea must wait for a single, unique person and set of circumstances to bring it to fruition.

Finally, the idea of open-ended problems is not radical or unique to American mathematics education reform. Jerry Becker and Shigeru Shimada edited an excellent volume entitled,

**The Open-Ended Approach: A New Proposal for Teaching Mathematics.**It explores many examples of quite challenging and engaging problems from Japanese K-12 math classrooms that illustrate what is meant by problems with more than one reasonable or valid solution. And there is clear evidence from the TIMSS videos taken in Japan that such problems are used regularly in ordinary middle school lessons. It seems almost obvious that a non-routine problem, even one drawn from quite typical mathematics content, can fit this criterion.

For a look at open-ended questions drawn from American elementary classrooms, along with ways to find, create, and evaluate student responses to them, see OPEN-ENDED QUESTIONS in ELEMENTARY MATHEMATICS by Mary Kay Dyer and Christine Moynihan.

An Example from Elementary Mathematics

Sullivan and Lilburn offer a host of examples from different content bands and grade levels. The following is in their chapter on number in a section of fractions problems for grades 3-4. On page 27, they pose:

Before going further, try to decide how you would sort these fractions into exactly two groups. Can you think of more than one way to do so? What did you focus on? Do you believe that there is a single best way to sort them? What alternatives might students you teach come up with? Try not to read further until you've come up with at least three ways to sort the fractions that make mathematical sense to you or that reflect things you think students would think of.

Okay, I'll assume you've tried to sort the fractions a few different ways. Here are some possibilities:

A) 3/4 and 6/10 B) 2/5, 1/3, 1/10.

A) 1/3 and 1/10 B) 3/4, 2/5, 6/10

A) 3/4, 2/5, 1/3 B) 6/10, 1/10

A) 2/5 and 1/3 B) 3/4, 6/10, 1/10

A) 6/10 B) 3/4, 2/5, 1/3, 1/10

Can you see at least one reasonable mathematical basis for each sorting above? Are the multiple valid mathematical reasons for any of the sortings? Are there sortings you found not listed above? Do they involve mathematical ideas not reflected in any of those I've offered? Can you think of any that students might come up with that would indicate confusion? How would you know?

I hope it is clear that tasks like the one above are most effective for learning when: 1) students are required to explain their thinking; 2) students share their explanations with the teacher and each other; 3) students offer and accept feedback constructively and freely from peers and the teacher; 4) a classroom culture has been established to facilitate all of the above, students have had opportunities to model or see modeled all relevant aspects of the expected behaviors, and students feel safe to share all their ideas and to offer and ask for feedback; and 5) this classroom culture establishes that it is important to understand other people's thinking and to make one's own thinking understandable.

"A friend of mine put these fraction into two groups: 3/4, 2/5, 1/3, 6/10, 1/10. What might the two groups be?"Looking at this in terms of the authors' three criteria for good problems, we can note that it fits the first one: the problem calls for something other than the recall of a fact or demonstrating a procedure. Second, since students may not have been asked to do anything like this problem, they may learn from the process of trying to solve it. Further, the teacher can gain information about what students know about fractions, possible areas of confusion, and willingness to engage in and communicate about mathematics through how each chooses to answer the question. Finally, it should be obvious that while there is no clear-cut right answer to this problem, there are many acceptable answers.

Before going further, try to decide how you would sort these fractions into exactly two groups. Can you think of more than one way to do so? What did you focus on? Do you believe that there is a single best way to sort them? What alternatives might students you teach come up with? Try not to read further until you've come up with at least three ways to sort the fractions that make mathematical sense to you or that reflect things you think students would think of.

Okay, I'll assume you've tried to sort the fractions a few different ways. Here are some possibilities:

A) 3/4 and 6/10 B) 2/5, 1/3, 1/10.

A) 1/3 and 1/10 B) 3/4, 2/5, 6/10

A) 3/4, 2/5, 1/3 B) 6/10, 1/10

A) 2/5 and 1/3 B) 3/4, 6/10, 1/10

A) 6/10 B) 3/4, 2/5, 1/3, 1/10

Can you see at least one reasonable mathematical basis for each sorting above? Are the multiple valid mathematical reasons for any of the sortings? Are there sortings you found not listed above? Do they involve mathematical ideas not reflected in any of those I've offered? Can you think of any that students might come up with that would indicate confusion? How would you know?

I hope it is clear that tasks like the one above are most effective for learning when: 1) students are required to explain their thinking; 2) students share their explanations with the teacher and each other; 3) students offer and accept feedback constructively and freely from peers and the teacher; 4) a classroom culture has been established to facilitate all of the above, students have had opportunities to model or see modeled all relevant aspects of the expected behaviors, and students feel safe to share all their ideas and to offer and ask for feedback; and 5) this classroom culture establishes that it is important to understand other people's thinking and to make one's own thinking understandable.

What's Next?

In Part 3, I will look at a companion volume to the Sullivan and Lilburn book: GOOD QUESTIONS FOR MATH TEACHING: WHY ASK THEM AND WHAT TO ASK Grades 5-8 by Lainie Schuster and Nancy Canavan Anderson.

In part 4, I will investigate the other issue I raised at the beginning of this entry: what kinds of questions can teachers ask to help get students "unstuck." What kind of classroom discourse practices do teachers need to promote to effectively make students more independent, able problem solvers and learners of mathematics.

In part 4, I will investigate the other issue I raised at the beginning of this entry: what kinds of questions can teachers ask to help get students "unstuck." What kind of classroom discourse practices do teachers need to promote to effectively make students more independent, able problem solvers and learners of mathematics.

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