One glaring weakness in many American mathematics classrooms is the nature of questions teachers ask. This manifests itself in two pedagogical realms: first, the kinds of mathematical questions or tasks teachers tend to pose are often of a closed nature which call of nothing more than a calculation (or short series of calculations) and for which the answer is a single number that requires no further interpretation or thought on the part of the student. The second area of difficulty arises in the types of questions teachers ask when they want to help students who are having difficulty performing a given task (which generally consists of something like that mentioned previously, although this problem arises in more complex problem situations).

My focus here will be on the the first situation: what kinds of tasks might comprise "good questions" for math students in elementary school?

Mathematical Tasks

In their book MAKING SENSE: Teaching and Learning Mathematics With Understanding, Hiebert, et al., propose three features for appropriate mathematical tasks: 1) the tasks make the subject problematic for students; 2) the tasks connect with where the students are; and 3) the tasks engage students in thinking about important mathematics.

What do the authors mean by the word "problematic"? It is clear from their book that it does not mean to make the mathematics puzzling or difficult (though either or both of those may be collateral results). Rather, it means to ask questions that make students have to think, to make use of what they know, to move further along the path of their mathematical knowledge and abilities. Thus, a given problem may effectively problematize mathematics for one group of students that would not have that impact on other students. Asking a simple addition question to Kindergarten students who haven't learned what addition is has the potential to problematize what has heretofore been easily addressed by counting. Given the same problem, most third grade students would see the situation as a routine exercise. But for the third graders, a question about division could be problematic.

That makes the second requirement for good problems crucial. And it also raises an important issue that is often missed in the Math Wars: teachers who have good understanding of where their students are mathematically are the ones best positioned to judge when a problem is appropriate for them. The better the teacher knows each student, the better able the teacher is to adjust tasks to individuals. A problem that is completely beyond the capabilities of all students in a class would likely not be a good one. But for far too many teachers, if a problem occurs in the textbook as part of a lesson, the teacher feels obligated to give that problem, as written, to all students. As a math coach, I've seen teachers follow blindly whatever is presented in their text, even when a given problem may take all or most students far off track with the main goal(s) of the lesson. When guest-teaching in their classrooms, I have frequently deleted or changed the sequence of the problems in the book when presenting them to the students. The teachers are generally very surprised by this, but when we discuss it afterwards, most are also unable to think of any clear reason I might choose to do this.

For example, in teaching a lesson on equivalent fractions to fourth graders from the EVERYDAY MATH book, I purposely omitted a problem that introduced mixed numbers, feeling from previous observation of the students that it would unnecessarily bog down the lesson for them in what was to my mind a secondary consideration. The notion that the text is a resource, not a bible, and that the teacher is the best judge of how to use that resource with a given student or classroom seemed radical to even the veteran teachers with whom I worked.

The third point the authors raise connects perfectly with the previous two while also raising another question about teacher content and pedagogical-content knowledge: many teachers do not seem to be able to readily identify the key mathematical idea(s) in a lesson or to distinguish between major and minor mathematical ideas in a unit or lesson. This makes successful decision making about what problems to use, omit, or revise very difficult if not impossible for them. Coupled with either relative ignorance of and/or lack of concern with where students actually are mathematically, confusion about how mathematical ideas are interrelated and what a reasonable hierarchy among them might be leaves teachers little choice but to follow what is in the text with complete passivity. While for some, this may be a desirable path of least resistance, it cannot be an acceptable teacher practice if we seriously expect to see significant improvement in mathematics teaching and learning. Thus, teachers must consider (and textbook authors must strive to make as clear as possible) what the important mathematical residue is for each lesson, chapter, and unit. By mathematical residue, I mean the key idea(s), concept(s), procedure(s), and connections that students are expected to take away from a task, lesson, or larger structure. The mathematical residue should also point clearly towards what students can actually DO with what they are expected to have learned from successfully engaging in the task.

In Part 2, I will explore ideas from a book by Peter Sullivan and Pat Lilburn, :

I will also look at some particular problems and how they might be good examples of what teachers can do in their classroom regardless of what curriculum they teach from.

In part 3, I will look at an expanded list of ideas about good math questions from a related book 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.

Problematizing mathematics

What do the authors mean by the word "problematic"? It is clear from their book that it does not mean to make the mathematics puzzling or difficult (though either or both of those may be collateral results). Rather, it means to ask questions that make students have to think, to make use of what they know, to move further along the path of their mathematical knowledge and abilities. Thus, a given problem may effectively problematize mathematics for one group of students that would not have that impact on other students. Asking a simple addition question to Kindergarten students who haven't learned what addition is has the potential to problematize what has heretofore been easily addressed by counting. Given the same problem, most third grade students would see the situation as a routine exercise. But for the third graders, a question about division could be problematic.

Meeting students where they are mathematically

That makes the second requirement for good problems crucial. And it also raises an important issue that is often missed in the Math Wars: teachers who have good understanding of where their students are mathematically are the ones best positioned to judge when a problem is appropriate for them. The better the teacher knows each student, the better able the teacher is to adjust tasks to individuals. A problem that is completely beyond the capabilities of all students in a class would likely not be a good one. But for far too many teachers, if a problem occurs in the textbook as part of a lesson, the teacher feels obligated to give that problem, as written, to all students. As a math coach, I've seen teachers follow blindly whatever is presented in their text, even when a given problem may take all or most students far off track with the main goal(s) of the lesson. When guest-teaching in their classrooms, I have frequently deleted or changed the sequence of the problems in the book when presenting them to the students. The teachers are generally very surprised by this, but when we discuss it afterwards, most are also unable to think of any clear reason I might choose to do this.

For example, in teaching a lesson on equivalent fractions to fourth graders from the EVERYDAY MATH book, I purposely omitted a problem that introduced mixed numbers, feeling from previous observation of the students that it would unnecessarily bog down the lesson for them in what was to my mind a secondary consideration. The notion that the text is a resource, not a bible, and that the teacher is the best judge of how to use that resource with a given student or classroom seemed radical to even the veteran teachers with whom I worked.

Mathematical residue

The third point the authors raise connects perfectly with the previous two while also raising another question about teacher content and pedagogical-content knowledge: many teachers do not seem to be able to readily identify the key mathematical idea(s) in a lesson or to distinguish between major and minor mathematical ideas in a unit or lesson. This makes successful decision making about what problems to use, omit, or revise very difficult if not impossible for them. Coupled with either relative ignorance of and/or lack of concern with where students actually are mathematically, confusion about how mathematical ideas are interrelated and what a reasonable hierarchy among them might be leaves teachers little choice but to follow what is in the text with complete passivity. While for some, this may be a desirable path of least resistance, it cannot be an acceptable teacher practice if we seriously expect to see significant improvement in mathematics teaching and learning. Thus, teachers must consider (and textbook authors must strive to make as clear as possible) what the important mathematical residue is for each lesson, chapter, and unit. By mathematical residue, I mean the key idea(s), concept(s), procedure(s), and connections that students are expected to take away from a task, lesson, or larger structure. The mathematical residue should also point clearly towards what students can actually DO with what they are expected to have learned from successfully engaging in the task.

What Next?

In Part 2, I will explore ideas from a book by Peter Sullivan and Pat Lilburn, :

**Good Questions for Math Teaching: Why Ask Them and What to Ask [K-6].**I will also look at some particular problems and how they might be good examples of what teachers can do in their classroom regardless of what curriculum they teach from.

In part 3, I will look at an expanded list of ideas about good math questions from a related book by Lainie Schuster and Nancy Canavan Anderson :

**Good Questions for Math Teaching: Why Ask Them And What to Ask, Grades 5-8**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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