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Instructional
programs from prekindergarten through grade 12 should enable all students
to—
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During grades 3–5, students should be involved in an important transition in their mathematical reasoning. Many students begin this grade band believing that something is true because it has occurred before, because they have seen several examples of it, or because their experience to date seems to confirm it. During these grades, formulating conjectures and assessing them on the basis of evidence should become the norm. Students should learn that several examples are not sufficient to establish the truth of a conjecture and that counterexamples can be used to disprove a conjecture. They should learn that by considering a range of examples, they can reason about the general properties and relationships they find.
Much of the work in these grades should be focused on reasoning about mathematical relationships, such as the structure of a pattern, the similarities and differences between two classes of shapes, or the overall shape of the data represented on a line plot. Students should move from considering individual mathematical objects—this triangle, this number, this data point—to thinking about classes of objects—all triangles, all numbers that are multiples of 4, a whole set of data. Further, they should be developing descriptions and mathematical statements about relationships between these classes of objects, and they can begin to understand the role of definition in mathematics.
Mathematical reasoning develops in classrooms where students are encouraged to
put forth their own ideas for examination. Teachers and students should
be open to questions, reactions, and elaborations from others in the classroom.
Students need to explain and justify their thinking and learn how to detect
fallacies and critique others' thinking. They need to have ample opportunity
to apply their reasoning skills and justify their thinking in mathematics
discussions. They will need time, many varied and rich experiences, and
guidance to develop the ability to construct valid arguments and to evaluate
the arguments of others. There is clear evidence that in classrooms where
reasoning is emphasized, students do engage in reasoning and, in the process,
learn what constitutes acceptable mathematical explanation (Lampert 1990;
Yackel and Cobb 1994, 1996).
In grades 3–5, students should reason about the relationships that apply to the numbers, shapes, or operations they are studying. They » need to define the relationship, analyze why it is true, and determine to what group of mathematical objects (numbers, shapes, and operations) it can be applied. Consider the following episode drawn from unpublished classroom observation notes:
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In Ms. Taylor's third-grade class, students were having a discussion
of how to compute 4 |
This example shows a teacher taking advantage of an opportunity to engage students in mathematical reasoning. By asking the question "Do you think that always works?" she moved the discussion from the specific problem to a consideration of a general characteristic of multiplication problems—that a factor in a multiplication expression can itself be factored and then the new factors can be multiplied in any order.
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After students
had worked on several problems and had discussed with a partner
why "doubling then doubling again" was a strategy for multiplying
by 4, the teacher reconvened the class for further discussion. Student
responses to whether Matt's strategy would always work showed a
wide range of thinking:
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These students' explanations are tied to the specific example, but there
is evidence that some students are constructing arguments that may lead
to more-general conclusions. Carol is satisfied that "it works every time"
but does not have an argument that is based on the structure of multiplication.
Malia refers to breaking up one of the factors in the problem into two
parts, multiplying the other number by both parts, and then adding the
results—the distributive property of multiplication over addition.
Steven's explanation is based on modeling multiplication as skip-counting,
and Matt takes his original idea further by testing whether multiplying
by 6 is the same as multiplying by 2 then by 3. Although none of these
third graders' arguments is stated in a way that is complete or general,
they are beginning to see what it means to develop and test conjectures
about mathematical relationships. »
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Following this
discussion, the teacher sent students off to work on a set of multiplication
problems. Their work on the problems gave evidence that some of
them were applying aspects of the reasoning discussed in the class
session. For example, Katherine computed 74
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During grades 3–5, students should move toward reasoning that depends on relationships and properties. Students need to be challenged with questions such as, What if I gave you twenty more problems like this to do—would they all work the same way? How do you know? Through comparing solutions and questioning one another's reasoning, they can begin to learn to describe relationships that hold across many instances and to develop and defend arguments about why those relationships can be generalized and to what cases they apply (Maher and Martino 1996).
At these grades, students need experiences in learning about what constitutes a convincing argument (Hanna and Yackel forthcoming). For example, in this episode drawn from unpublished classroom observation notes, a third-grade class explored the following problem (adapted from Tierney and Berle-Carman [1995, p. 22]).
Initially, the students tried to solve the problem by just looking at the figure. For example, they reasoned:
"The triangle is bigger because it goes way up."
"I think they're the same because the triangle's taller, but the rectangle's longer."
As the students worked on this problem, some were convinced that they could decide if the areas were equal (or not) by whether or not they could cut the triangle into a set of haphazard pieces and fit them on the rectangle so that they cover the space (see fig. 5.31a). Others thought about how to organize the cutting and pasting by, for example, cutting the triangle into two pieces to make it into a rectangle that matches the other rectangles (see fig. 5.31b).
Still others developed ways to reason about the relationships in the figure without cutting and pasting. For example: "We folded each paper in half and each paper was the same size to begin with, so the half that's a rectangle is the same as the half that's a triangle."
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At this grade level, many students are just beginning to develop an idea about what constitutes a convincing argument. The first solution—cutting and pasting in a disorganized way—does not make use of the properties of the two shapes and therefore may not convince the student doing the cutting and pasting or other students that the areas are equal. The second solution takes into consideration geometric relationships between the particular triangle and rectangle and therefore may be more » convincing. The third solution is the beginning of a logical analysis of the relationships among the shapes—that halves of equal areas must be equal to each other.
Students in grades 3–5 should frequently make conjectures about mathematical relationships, investigate those conjectures, and make mathematical arguments that are based on their work. They need to know that posing conjectures and trying to justify them is an expected part of students' mathematical activity. Justification will have a range of meanings for students in grades 3–5, but as they progress through these grades and have more experiences with making mathematical arguments, they should increasingly base their arguments on an analysis of properties, structures, and relationships.
Sometimes students' conjectures about mathematical properties and relationships
will turn out to be wrong. Part of mathematical reasoning is examining
and trying to understand why something that looks and seems as if it might
be true is not and to begin to use counterexamples in this context. Coming
up with ideas that turn out not to be true is part of the endeavor.
These "wrong" ideas often are opportunities for important mathematical
discussions and discoveries. For example, a student might propose that
if both the numerator and denominator of a fraction are larger than the
numerator and denominator, respectively, of another fraction, then the
first fraction must be larger. This rule works in comparing 3/4 with 1/2
or 6/4 with 2/3. However, when thinking about this conjecture more carefully,
students will find counterexamples—for example, 3/4 is not larger
than 2/2 and 2/6 is smaller than 1/2.
In order for mathematical experiences such as those described in this section to happen frequently, the teacher must establish the expectation that the class as a mathematical community is continually developing, testing, and applying conjectures about mathematical relationships. In the episode in Ms. Taylor's third-grade classroom, where students explored the effects of multiplying by 2 and by 2 again, the teacher looked for an opportunity to go beyond finding the solution to an individual problem to focus on more-general mathematical structures and relationships. In this way, she helped her students recognize reasoning as a central part of mathematical activity.
Part of the teacher's role in making reasoning central is to make all students responsible both for articulating their own reasoning and for working hard to understand the reasoning of others, as shown in the following episode, drawn from unpublished classroom observation notes.
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In a fourth-grade classroom, the students were ordering fractions.
To begin this activity, the teacher had asked them to identify fractions
that are more than 1/2 and less than 1. After the students talked
in pairs, the teacher asked how they were choosing their fractions:
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By routinely questioning students in this way, the teacher is establishing the expectation that students listen carefully to one another's ideas and try to understand them.
The teacher should continually remind students of conjectures and mathematical arguments that they have developed as part of the shared classroom experience and that can be applied to further work. Teachers should look for opportunities for students to revise, expand, and update generalizations they have made as they develop new mathematical skills and knowledge. Matt's idea about tripling in Ms. Taylor's third-grade class could provide the basis for students to reason about a larger class of problems. Even students who seem to have developed a clear argument about a mathematical relationship need to be questioned and challenged when they are ready to encounter new aspects of the relationship. For example, a class of third graders had spent a great deal of time working with arrays in their study of multiplication. As a group, they were very sure that multiplication was commutative, and they could demonstrate this property using an area model. In the fourth grade, they began encountering larger numbers; when the teacher noticed that some students were using commutativity, she asked the class what they knew about it. At first they seemed certain that multiplication is commutative in all cases, but when she pressed, "But would it work for any numbers? How about 43 279 times 6 892?" they lost their confidence. They could no longer use physical models to show commutativity with such large numbers, and they needed further work to develop mental images and mathematical arguments based on what they had learned from the physical models. It is likely that these students will also need to revisit commutativity when they study computation with fractions and decimals.
The teacher will also have to make decisions about which conjectures are mathematically significant for students to pursue. To do this, the teacher must take into account the skills, needs, and understandings of the students and the mathematical goals for the class.
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8. One student, Matt,
explained, "I thought of 2
