Frequently Asked Questions
Q: Can this project be used to help students in Grade 3 or earlier to learn facts?
A: This project is designed for fourth through sixth graders who are below grade level.
However, it has been successfully used with middle school students who are two or more years below grade level.
Q: Can Knowing Mathematics be used on alternate days?
A: This project is designed with the assumption that students are engaged with the program every school day.
Q: Can single units of this project be used?
A: Yes.
Q: Can my students use calculators?
A: Not in the class or in their homework for this program.
Calculators are used to check student work throughout Unit 3.
Q: Do I need manipulatives in order to teach the lessons?
A: No.
Q: Why are manipulatives not used?
A: Many students in this program will have already used manipulatives and may well already understand how operations can be modeled with manipulatives.
But, more importantly, in this program, for these students who are far below grade level, we don't want manipulatives to be a major focus.
hese students, like their peers, need to be able to operate fluently with numbers, without stopping to model the operation with manipulatives.
Instead, our focus is on operating with the symbolic representations of the numbers and operations themselves and understanding those representations well;
for example, what a digit placed in a particular column represents, how the columns are related, and how to operate with digits in the columns.
We want students to operate fluently enough that they don't need to stop and think of the meanings of the operations each time they perform an operation.
The mathematician Alfred North Whitehead said, "Civilization advances by extending the number of important operations that we can perform without thinking about them. . ."^{1}
Research in cognitive science suggests that only a small amount of information can be held at any one time in shortterm memory.^{2}
The performance of experts (including gradeschool student experts) in various fields has been explained in terms of "chunking."
Experts, when working in their fields, see strings of symbols or collections of objects or configurations of circuit boards in "chunks," rather than as several individual objects.^{3}
We want students eventually to be able to see mathematical expressions as chunks — whole sentences, rather than separate unrelated symbols.
What we know about mental processes suggests this will allow students to learn more mathematics.
Q: Why are most computations done horizontally with expressions rather than vertically in columns?
A: Expressions allow many relationships to be shown that computations with columns do not allow or show in a rather unclear way.
Also, number patterns may be easier to see (e.g., Lesson 17).
But, perhaps the most important reason is that the relationship of arithmetic with algebra appears more clearly with expressions rather than computations in columns.
However, columns are very useful for computation and students use them for this purpose in Knowing Mathematics.
Q: Why are no activities labeled "problem solving"?
A: "Problem" has acquired two meanings in mathematics education research.
Some researchers use "problem solving" to mean "accomplishing a task for which you already know a solution method."
Some researchers use "problem solving" to mean "accomplishing a task for which you don't already know a solution method."^{4}
We could specify our meaning, but there's a second difficulty — we can't always tell whether or not a particular task is a problem for every student.
Whatever you mean by "problem solving," it is very likely that your Knowing Mathematics students will be solving problems sooner or later!
Q: Why is no part of the lesson labeled "assessment"?
A: In this program, assessment is not confined to a single part of the lesson.^{5}
You will be monitoring your students' understanding in different ways throughout each lesson, from their responses to the mathematical conversation, during guided practice, during individual exercises, and when they respond to the reflection and discussion questions at the end of each lesson.
At the end of each week, students take quizzes or unit assessments.
Bibliography
^{1} Alfred North Whitehead, An Introduction to Mathematics, 1948 (first published 1911), London: Oxford University Press, p. 42.
^{2} George Miller, "The magic number seven, plus or minus two: some limits on our capacity for processing information," 1956, Psychological Review, 63, 81Ð97.
^{3} See Alan Schoenfeld, Mathematical Problem Solving, 1985, Orlando, FL: Academic Press, pp. 48Ð50; John Bransford, Ann Brown, and Rodney Cocking (Eds.), How People Learn: Brain, Mind, Experience, and School, 1999, Washington, DC: National Academy Press, pp. 20Ð24, 84Ð85.
^{4} For a discussion of the distinction between a problem and an exercise, see Alan Schoenfeld, Mathematical Problem Solving, 1985, Orlando, FL: Academic Press, p. 74. For examples of both uses of "problem," see Alan Schoenfeld, "Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics," 1992, in D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334Ð370). New York: Macmillan.
^{5} As How People Learn puts it, "Opportunities for feedback should occur continuously, but not intrusively, as a part of instruction" (p. 128).
