programs from prekindergarten through grade 12 should enable all students
Reasoning is an integral part of doing mathematics. Students should enter the
middle grades with the view that mathematics involves examining patterns
and noting regularities, making conjectures about possible generalizations,
and evaluating the conjectures. In grades 68 students should sharpen
and extend their reasoning skills by deepening their evaluations of their
assertions and conjectures and using inductive and deductive reasoning
to formulate mathematical arguments. They should expand the audience for
their mathematical arguments beyond their teacher and their classmates.
They need to develop compelling arguments with enough evidence to convince
someone who is not part of their own learning community.
In the middle grades, students should have frequent and diverse experiences with mathematics reasoning as they
Students should discuss their reasoning on a regular basis with the teacher and with one another, explaining the basis for their conjectures and the rationale for their mathematical assertions. Through these experiences, students should become more proficient in using inductive and deductive reasoning appropriately.
Students can use inductive reasoning to search for mathematical relationships
through the study of patterns. Consider an example from a classroom in
which rising seventh-grade students were studying figurate numbers (drawn
from classroom observation and partially described in Malloy ).
The teacher began by explaining triangular numbers and then asked the students to generate representations for the first five triangular numbers. The students visualized the structure of the numbers to » draw successive dot triangles, each time adding at the bottom a row containing one more dot than the bottom row in the previous triangle (see fig. 6.31). Next the teacher asked the students to predict (without drawing) how many dots would be needed for the next triangular number. Reflecting on what they had done to generate the sequence thus far, they quickly concluded that the sixth triangular number would have six more dots than the fifth triangular number. These students were engaged in recursive reasoning about the structure of this sequence of numbers, using the just-formed number to generate the next number. This approach was repeated for several more "next" numbers in the sequence, and it worked well.
The teacher asked the students to check to see if Tamika's observation was true for other numbers in the display. After they verified the observation, the teacher asked them to use this method to find the next triangular number. Some students were unable to see how it could be done, but Curtis used Tamika's observation as follows: "Using Tamika's method, the seventh number is (7)(8)/2, which is 28." Several students checked this answer by using the recursive » method of adding 7 to the sixth triangular number to find the seventh triangular number (21 + 7 = 28). The teacher then asked the students to check Tamika's method for the next few triangular numbers to verify that it worked in those instances. She next asked if Tamika's method could be used to find the 100th triangular number. Darnell said, "If Tamika is right, the hundredth triangular number should be (100)(101)/2."
In general, the students agreed that the method of multiplying and dividing by
2 was useful because it seemed to work and because it did not require
knowing the nth term in order to find the (n +
1)th term. However, some students were not convinced that the method
was correct. It lacked the intuitive appeal of the recursive method
they used first, and it did not appear to have a mathematical basis.
The teacher decided that it was worth additional class time to develop
a mathematical argument to support Tamika's method. She began by
asking students to notice that each triangular number is the sum
of consecutive whole numbers, which they readily saw from the dot
triangles. Then the teacher demonstrated Gauss's method for finding
the sum of consecutive whole numbers, applying it to the first seven
whole numbers. She asked the students to add the numbers from 1
to 7 to those in the reversed sequence, 7 to 1, as shown in figure
6.33, to see that the seventh triangular number1 + 2 + 3 +
4 + 5 + 6 + 7could also be expressed as (7)(8)/2. After the
students completed this exercise, the teacher asked them to express
the general relationship in words. They struggled, but they came
up with this general rule: If you want to find a particular triangular
number, you multiply your number by the next number and divide by
2. The students wrote the rule this way:
The example illustrates what reasoning and proof can look like in the
middle grades. Although mathematical argument at this level lacks the
formalism and rigor often associated with mathematical proof, it shares
many of its important features, including formulating a plausible conjecture,
testing the conjecture, and displaying the associated reasoning for evaluation
by others. The teacher and students used inductive reasoning to reach
a generalization. They noted regularities in a pattern (growth of triangular
numbers), formulated a conjecture about the regularities (Tamika's rule),
and developed and discussed a convincing argument about the truth of the
Middle-grades students can develop arguments to support their conclusions in varied topics, such as number theory, properties of geometric shapes, and probability. For example, students who encounter the rules of divisibility by 2 and by 3 in number theory know that even numbers are divisible by 2 and numbers whose digits add to a number divisible by 3 are divisible by 3. A teacher might ask students » to formulate a rule for divisibility by 6 and develop arguments to support their rule.
Some students might begin by listing some multiples of 6: 12, 18, 24, and 30. They could examine the numbers and try to detect patterns resembling those in other rules they have learned. Students might observe that all the numbers are even, which allows them to infer divisibility by 2. They could also look at the sums of the digits of the multiples and notice that the sums of the digits are all divisible by 3, just as in the test for divisibility by 3. Noting that 2 3 = 6, they might conclude that if the number is divisible both by 2 and by 3, then it must be divisible by 6, which might lead them to form the following conjecture for determining whether a number is divisible by 6: Check to see if the number is even and if the sum of its digits is divisible by 3.
The teacher should also challenge students to consider possible limitations of
their reasoning. For example, she could ask them to use 12 as an example
to consider whether it is always true that the product of two factors
of a number is itself a factor of that number. The students should note
that although 6 and 4 are both factors of 12, 6 4 is
not. In this way, the teacher can help students become appropriately cautious
in making inferences about divisibility on the basis of factors. Such
an exploration should lead to the correct generalization that combining
criteria for divisibility, which worked with divisibility by 6, works
only when the two factors are relatively prime.
Teachers in the middle grades can help students appreciate and use the power of mathematical reasoning by regularly engaging students in thinking and reasoning in the classroom. Fostering a mathematically thoughtful environment is vital to supporting the development of students' facility with mathematical reasoning.
The teacher plays an important role by creating or selecting tasks that are appropriate
to the ages and interests of middle-grades students and that call for
reasoning to investigate mathematical relationships. Tasks that require
the generation and organization of data to make, validate, or refute a
conjecture are often appropriate. For example, the examination of patterns
associated with figurate numbers discussed above shows how a teacher can
use the task both to stimulate students' investigation and to develop
facility with mathematical reasoning and argumentation. Suitable tasks
can arise in everyday life, although many will arise within mathematics
Teachers also serve as monitors of students' developing facility with reasoning. In order to use inductive reasoning appropriately, students need to know its limitations as well as its possibilities. Because many elementary and middle-grades tasks rely on inductive reasoning, teachers should be aware that students might develop an incorrect expectation that patterns always generalize in ways that would be expected on the basis of the regularities found in the first few terms. The following hypothetical example shows how a » teacher could help students develop a healthy appreciation for the power and limits of inductive reasoning.
A teacher asks students to determine how many segments of different lengths can be made by connecting pegs on a square geoboard that is 5 units on each side (a 55 square geoboard). Because the number of segments is large and some students will have difficulty being systematic in representing the segments on their geoboards, the teacher encourages the students to examine simpler cases to develop a systematic way to generate the different segments. The students approach this task by examining the number of segments on various subsquares on a 55 geoboard, looking at the growth from a 11 square to a 44 square, as shown in figure 6.34.
teacher orchestrates a class discussion about the numbers in the table.
Most students quickly detect a pattern of growth and are prepared
to predict the answer for a 55 geoboard20
different segmentsbecause (2 + 3 + 4 + 5) + 6 = 14 + 6 = 20.
In fact, many students are prepared to state a more general conjecture:
The number of segments for an NN square
geoboard is the sum 2 + 3 + ... + (N + 1).
After the students have made
their prediction, the teacher asks them to check its accuracy by
actually making all the possible segments of different lengths on
a 55 geoboard, as in
figure 6.36. Because of their prior experience in systematically
generating all possible segments, most of the students are able
to find all the possibilities. In fact, most recognize that they
need only to count the "new" segments and check to be sure that
the segments are of different lengths from one another and from
the segments already counted in the previous cases. The students
note that there are twenty segments, as predicted, and most are
content with the observation that all the new segments are of different
lengths. But » some students discover
that two segmentsAB and CD in figure 6.36are
both five units long. Thus, there are only nineteen different lengths,
rather than twenty as predicted. Most of the students are surprised
at this result, although they recognize that it is correct.
A teacher can use an example such as this as a powerful reminder that students
should be cautious when generalizing inductively from a small number of
cases, because not all patterns generalize in ways that we might wish or
expect from early observations. This important lesson allows students to
develop a healthy skepticism in their work with patterns and generalization.
Teachers need to monitor students' developing facility not only with inductive reasoning but also with deductive reasoning. In the middle grades, students begin to consider assertions such as the following: The diagonals of any given rectangle are equal in length. (See the "Geometry" section of this chapter for more discussion of how this assertion might be generated and verified by students.) An assertion such as this is tricky, at least in part because it is an implicitly conditional statement: If a shape is a rectangle, then its diagonals are equal in length. Thus, it is probably not surprising that some students will misapply this idea by inferring that any quadrilateral with diagonals of equal length must be a rectangle. Doing so reflects the erroneous view that if a statement is true then its converse is true. In this instance, the converse is not true because nonrectangular isosceles trapezoids also have diagonals of equal length, as do many other quadrilaterals.
Teachers in the middle grades need to be mindful of complexities in logical thinking and be alert in order to help students reason correctly. In this example, a teacher might have students use dynamic geometry software to investigate which types of quadrilaterals have diagonals of equal length. The software could allow students to see changes in the lengths of the diagonals instantly as they change the shape of the quadrilateral. A teacher might have students investigate quadrilaterals in general and particular types of quadrilaterals, including rectangles, squares, parallelograms, rhombuses, and trapezoids. The teacher might ask students to note which shapes have diagonals of equal length. If no one found such a shape, the teacher could ask them to construct an isosceles trapezoid with a given set of vertices, and the students would then see that this trapezoid has diagonals of equal length. This type of investigation can lead students to understand that even when a statement is true, its converse may be false.
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