FOT Math
The following general sequence of activities is appropriate for helping children develop meaning for the four basic operations:
1. Concrete—modeling with materials: Use a variety of verbal problem settings and manipulative materials to act out and model the operation. 2. Semiconcrete—representing with pictures: Provide representations of objects in pictures, diagrams, and drawings to move a step away from the concrete toward symbolic representation. 3. Abstract—representing with symbols: Use symbols (especially numeric expressions and number sentences) to illustrate the operation.
What is an ordinal number?
A number that tells the position of something in a list. 1st, 2nd, 3rd, 4th, 5th etc.
What is Strategic Competence?
Ability to formulate, represent, and solve mathematical problems.
SPLITTING THE PRODUCT INTO KNOWN PARTS:
As children gain assurance with some basic facts, they can use their known facts to derive others. The strategy, known as splitting the product, is based on the distributive property of multiplication. It can be approached in terms of "one more set," "twice as much as a known fact," or "known facts of 5." The idea of one more set can be used for almost all multiplication facts. If a certain multiple of a number, N, is known, the next multiple can be determined by adding the single-digit number, N. For example, to find 3×5 if doubles are already known, you can think of 2×5 (10) and add one more 5 (to get 15). The computation is slightly more difficult if the addition requires renaming (e.g., below, where 8×7 is found by adding one more set of 7 to the known fact 7×7 = 49).
COMPATIBLE NUMBERS:
As in mental computation, using compatible numbers—numbers that go together naturally and are easy to work with mentally—is often helpful in estimation. For example, 14/31 + 2/7 are messy fractions to add, but 14/31 is almost ½, and 2/7 is about ¼. By changing these fractions to compatible numbers that are about the same makes an estimate of ¾ quick and easy.
CONCEPTUAL DEVELOPMENT OF DECIMALS:
As students develop decimal concepts, they should relate decimals to what they already know about common fractions and place value. For ease of discussion, we consider these relationships separately, but you should keep in mind that you will need to weave them together in your teaching. You should also keep in mind that it is important to relate decimals to everyday experiences familiar to the students, since contextual situations often help students understand the mathematics. The video described in Math Link 12.3 shows young children becoming familiar with the use of decimals in their lives.
COMPATIBLE-NUMBER PAIRS:
Asking children to think of "compatible-numbers pairs" for different problems in division and to tell why the numbers are compatible will help them come up with thoughtful ideas and will give you insight into their number sense.
What are attribute, or logic, blocks?
Attribute blocks, sometimes called logic blocks, provide an excellent model for classification activities and help develop logical thinking.
What is Adaptive Reasoning?
Capacity for logical thought, reflection, explanation, and justification.
What is a cardinal number?
Cardinal numbers (or cardinals) are numbers that say how many of something there are, such as one, two, three, four, five.
DEVELOPING MEANINGS FOR THE OPERATIONS:
Children encounter the four basic operations in natural ways when they work with many diverse problem situations. By representing these problem situations (e.g., acting them out, using physical models, or drawing pictures), they develop meanings for addition, subtraction, multiplication, and division. Mastery of basic facts and later computational work with multidigit examples must be based on a clear understanding of the operations.
COMBINATIONS PROBLEMS:
Combinations problems involve still another sort of multiplicative structure. Here the two factors represent the sizes of two different sets and the product indicates how many different pairs of things can be formed, with one member of each pair taken from each of the two sets.
COMMUTATIVITY:
Commutativity applies to multiplication just as it does to addition. It is, therefore, a primary strategy for helping students learn the multiplication facts.
COMPARISON PROBLEMS:
Comparison, or finding the difference, involves having two quantities, matching them one to one, and noting the quantity that is the difference between them. Problems of this type can also be solved by subtraction, even though nothing is being taken away.
What is Conceptual Understanding?
Comprehension of mathematical concepts, operations, and relations.
ESTIMATION:
Computational estimation is a process of producing answers that are close enough to allow for good decisions without performing elaborate or exact computations. The video Choosing a Method, excerpted in this chapter's Snapshot of a Lesson, nicely illustrates that there are many different real-world situations for using estimation. Computational estimation is typically done mentally.
ADDING DOUBLES AND NEAR DOUBLES:
Doubles are basic facts in which both addends are the same number, such as 4+4 or 9+9. Most children learn these facts quickly, often parroting them before they come to school. Connecting doubles facts to familiar situations often helps students remember them (e.g., two hands shows 5+5 = 10, an egg carton shows 6+6 = 12, two weeks on a calendar shows 7+7 = 14).
What is one-to-one correspondence?
Each object to be counted must be assigned one and only one number name. As shown in Figure 7-8, a one-to-one correspondence between each shell and the number name was established.
EQUAL-GROUPS PROBLEMS:
Equal-groups problems involve the most common type of multiplicative structure, where you are dealing with a certain number of groups, all the same size. When both the number and size of the groups are known (but the total is unknown), the problem can be solved by multiplication. The problem given earlier, about Andrew's trading card collection, is an example of an equal-groups problem. When the total in an equal-groups problem is known, but either the number of groups or the size of the group is unknown, the problem can be solved by division. Two distinct types of division situations can arise, depending on which part is unknown. These two types of division situations are known as measurement and partition.
AREA AND ARRAY PROBLEMS:
Finally, area and array problems also are typical examples of multiplicative structure. The area of any rectangle (in square units) can be found either by covering the rectangle with unit squares and counting them all individually or by multiplying the width of the rectangle (number of rows of unit squares) by the length (number of unit squares in each row). Similarly, in a rectangular array—an arrangement of discrete, countable objects (such as chairs in an auditorium)—the total number of objects can be found by multiplying the number of rows by the number of objects in each row. The array model for multiplication can be especially effective in helping children visualize multiplication. It may serve as a natural extension of children's prior work in making and naming rectangles using tiles, geoboards, or graph paper.
THINKING STRATEGIES FOR SUBTRACTION FACTS:
For each basic addition fact, there is a related subtraction fact. In some mathematics programs, the two operations are taught simultaneously. The relationship between them is then readily emphasized, and learning the basic facts for both operations proceeds as if they were in the same family.
What are unit fractions?
Fractions with numerators of 1.
CHOOSING ESTIMATION STRATEGIES:
Give your students problems that encourage and reward estimation. For example, you can expect your students to be able to compute 78 + 83 mentally and get an exact answer, but giving them 78342 + 83289 will encourage them to use estimation. Make sure the numbers are messy enough that students will want to estimate the answer, rather than compute an exact answer. Make sure students are not computing exact answers and then rounding to produce estimates. Research has documented that students frequently use this technique (McIntosh and Sparrow, 2004). Unfortunately, this often goes undetected. You should talk with and observe students to see if they are truly estimating. Ask students to tell how they made their estimates. Individual students often develop unique approaches to estimation. By sharing their different approaches, students develop an appreciation of each other's strategies. Research has shown that asking students to share and compare strategies used in producing their estimates significantly enhances their learning (Star, Kenyon, Joiner, and Rittle-Johnson 2010).
What is Productive Disposition?
Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.
MEASUREMENT (REPEATED-SUBTRACTION) PROBLEMS:
In division situations of the measurement (or repeated-subtraction) type, you have equal sized groups, you know how many objects are in each group, and you must determine the number of groups.
PARTITION (SHARING) PROBLEMS:
In division situations of the partition (or sharing) type, a collection of objects is separated into a given number of equivalent groups and you seek the number in each group. By contrast with measurement situations, here you already know how many groups you want to make, but you don't know how many objects must be put in each group.
What are nominal numbers?
Nominal numbers provide essential information for identification but do not necessarily use the ordinal or cardinal aspects of the number.
ADJUSTING:
Number sense has many dimensions, one of which is recognizing when an estimate is a little more or a little less than the exact answer would be. As students develop their estimation skills, it becomes natural for them to refine an initial estimate.
SUBTRACTING ONE AND ZERO:
Once they have learned strategies for adding 1 and adding 0, most children find it rather easy to learn the related subtraction facts involving 0 and 1. They can profit from work with materials and from observing patterns similar to those used for addition facts.
FLEXIBLE ROUNDING:
Rounding to get numbers that are easier to work with is a very useful estimation strategy. Rounding is a more sophisticated strategy than front-end estimation because rounding changes, reformulates, or recomposes the numbers in the original problem. When using rounding for estimation, students should use flexible rounding, as shown in Figure 10-12. Flexible rounding results in numbers that are close but are also compatible. Flexible rounding is appropriate for all operations with all types of numbers. It is important for students to know they have the freedom to choose different estimation strategies, but they also need to realize that different strategies produce different estimates. Accepting a range of reasonable estimates, rather than a single best estimate, fosters number sense and encourages students to choose and use their own strategies.
SEPARATION PROBLEMS:
Separation, or take away, involves having one quantity, removing a specified quantity from it, and noting what is left. Research indicates that this subtraction situation is the easiest for children to learn; however, persistent use of the words take away results in many children assuming that this is the only subtraction situation and leads to misunderstanding of the other two situations. This is why it is important to read a subtraction sentence such as 8−3 = 5 as "8 minus 3 equals 5" rather than "8 take away 3 equals 5." Take away is just one of the three types of subtraction situations.
What is Procedural Fluency?
Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Distributive Property?
The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property.
CLUSTERING:
The clustering, or averaging, strategy uses an average for estimating a sum. Whenever a group of numbers clusters around some value, students can use this two-step process to estimate the sum of all the numbers in the group.
COMBINATIONS TO 10:
The combinations-to-10 facts are those nine pairs of numbers that together make 10: 1+9, 2+8, 3+7, 4+6, 5+5, 6+4, 7+3, 8+2, and 9+1. Note that recognition of the commutative property can reduce this list to just five facts (since each of the nine facts except 5+5 has a commutative partner-pair—for example, 8+2 and 2+8 are pairs, etc.). Of the nine combinations to 10, five have already been mentioned in previous fact strategies (either counting on or doubles).
PART-WHOLE PROBLEMS:
The final type of subtraction situation is known as part-whole. In this type of problem, a set of objects can logically be separated into two parts. You know how many are in the entire set and you know how many are in one of the parts. You need to find out how many must be in the remaining part. Nothing is being added or taken away—you simply have a static situation involving parts and a whole.
FRONT-END ESTIMATION:
The front-end strategy for estimation is a basic yet powerful approach that can be used in a variety of situations. Front-end estimation involves checking (1) the leading, or front-end, digit in a number and (2) the place value of that digit. Figure 10-10 shows how front-end digits are used to obtain an initial estimate. The unique advantage of the front-end strategy is that the original problem shows all the numbers that students have to work with. This enables students to reach an estimate quickly and easily. The front-end strategy also encourages students to use number sense as they think about the computations.
What are the four basic functions of an instructor?
The instructor role involves four basic functions: planning, teaching, assessing, and analyzing.
What is the cardinality rule?
The last number name used gives the number of objects. This principle is a statement of the cardinality rule, which connects counting with how many. Regardless of which block is counted first or the order in which they are counted, the last block named always tells the cardinal number of the objects being counted.
What is the stable-order rule?
The number-name list must be used in a fixed order every time a group of objects is counted. The child in the figure started with "one" and counted "two, three, ..., seven," in a specific order. This is known as the stable-order rule.
What is the order-irrelevance rule?
The order in which the objects are counted doesn't matter. This is known as the The order in which the objects are counted doesn't matter. This is known as the order-irrelevance rule. Thus, the child can start with any object and count them in any order.. Thus, the child can start with any object and count them in any order.
What is subitizing?
The skill to "instantly see how many" in a group is called subitizing, from a Latin word meaning "suddenly." It is an important skill to develop. In fact, one instructional goal for first-grade students is to develop immediate recognition of small groups.
DOUBLES:
The strategy for doubles may need to be taught more explicitly for subtraction facts than for addition facts. It rests on the assumption that children know the doubles for addition.
COUNTING BACK:
The strategy of counting back is related to counting on in addition. It is most efficient when the number to be subtracted is 1 or 2.
COUNTING ON:
The strategy of counting on can be used for any addition facts but is most easily used when one of the addends is 1 or 2. To be efficient, it is important to count on from the larger addend.
REPEATED ADDITION:
The strategy of repeated addition can be used most efficiently when one of the factors is less than 5. The child thinks of the multiplication as repeated addition.
Associative Property?
The word "associative" comes from "associate" or "group";the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "a(bc) = (ab)c"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.
Commutative Property?
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.
ADDING TO 10 AND BEYOND:
With the strategy of adding to 10 and beyond, one addend is broken apart so that one part of it can be used with the other addend to make 10. Then the remaining part of the first addend is added to the 10 to go beyond to the final teen sum. This strategy is used most easily when one of the addends is 8 or 9. However, if children are fluent with combinations to 10, then adding to 10 and beyond can be useful in many other situations too.
What three distinct meanings of fractions are found in most elementary mathematics programs?
—part-whole, quotient, and ratio—the focus is usually on the part-whole meaning, with little development of the other two meanings. Ignoring these other meanings may be one source of students' difficulty with fractions.
