By Shigeo Katagiri

Translated of the rewritten version from Shikgeo Katagiri (2004)., Mathematical Thinking and How to Teach It. Meijitosyo Publishers, Tokyo. Copyright of English version has CRICED, University of Tsukuba. All rights reserved.

Table of Contents

Chapter 1 The Aim of Education and Mathematical Thinking

Chapter 2 The Importance of Teaching to Cultivate Mathematical Thinking

2.1 The Importance of Teaching Mathematical Thinking

2.2 Example 1: How Many Squares are There?

Chapter 3 The Meaning of Mathematical Thinking and How to Teach It

3.1 Characteristics of Mathematical Thinking

3.2 Substance of Mathematical Thinking

List of Types of Mathematical Thinking

I. Mathematical Attitudes

II. Mathematical Thinking Related to Mathematical Methods

III. Mathematical Thinking Related to Mathematical Contents

Chapter 4 Detailed Discussion of Mathematical Thinking Related to Mathematical Methods

Chapter 5 Detailed Discussion of Mathematical Thinking Related to Mathematical Substance

Chapter 6 Detailed Discussion of Mathematical Attitudes Chapter 7 Questions for Eliciting Mathematical Thinking

Chapter 1

The Aim of Education and Mathematical Thinking

1. The Aim of Education: Scholastic Ability From the Perspective of “Cultivating Independent Persons”

School-based education must be provided to achieve educational goals. “Scholastic ability” becomes clear when one views the aim of school-based education.

The Aim of School Education

The aim of school education is described as follows in a report by the Curriculum Council: “To cultivate qualifications and competencies among each individual school child, including the ability to find issues by oneself, to learn by oneself, to think by oneself, to make judgments independently and to act, so that each child or student can solve problems more skillfully, regardless of how society might change in the future.”

This guideline is a straightforward expression of the preferred aim of education.

The most important ability that children need to gain at present and in the future, as society, science, and technology advance dramatically, is not the ability to correctly and quickly execute predetermined tasks and commands, but rather the ability to determine for themselves what they should do, or what they should charge themselves with doing.

Of course, the ability to correctly and quickly execute necessary tasks is also necessary, but from now on, rather than adeptly imitating the skilled methods or knowledge of others, the ability to come up with one’s own ideas, no matter how small, and to execute one’s own independent, preferable actions (ability full of creative ingenuity) will be most important. This is why the aim of education from now on is to instill the ability (scholastic ability) to take these kinds of actions. Furthermore, this is something that must be instilled in every individual child or student. From now, it will be of particular importance for each individual school child to be able to act independently (rather than the entire class acting independently as a unit). Of course, not every child will be able to act independently at the same level, but each school child must be able to act independently according to his or her own capabilities. To this end, teaching methods that focus on the individual are important.

2. The Scholastic Ability to Think and Make Judgments Independently Is Mathematical Thinking

– Looking at Examples –

The most important ability that arithmetic and mathematics courses need to cultivate in order to instill in students this ability to think and make judgments independently is mathematical thinking. This is why cultivation of this “mathematical thinking” has been an objective of arithmetic and mathematics courses in Japan since the year 1950. Unfortunately, however, the teaching of mathematical thinking has been far from adequate in reality.

One sign of this is the assertion by some that “if students can do calculations, that is enough.”

The following example illustrates just how wrong this assertion is:

Example: “Bus fare for a trip is 4,500 yen per person. If a bus that can seat 60 people is rented out, however, this fare is reduced by 20% per person. How many people would need to ride for it to be a better deal to rent out an entire bus?”

This problem is solved in the following manner:

Solving Method When a bus is rented – One person’s fee: 4,500×0.8=3,600

In the case of 60 people: 3,600×60=216,000

With individual tickets, the number of people that can ride is

216,000÷4,500=48 (people).

Therefore, it would be cheaper to rent the bus if more than 48 people ride.

Sixth-graders must be able to solve a problem of this level. Is it sufficient, however, to solve this problem just by being able to do formal calculation (calculation on paper or mental calculation, or the use of an abacus or calculator)? Regardless of how skilled a student is at calculation on paper, and regardless of whether or not a student is allowed to use a calculator at will, these skills alone are not enough to solve the problem. The reason is that before one calculates on paper or with a calculator, one must be able to make the judgment “what numbers need to be calculated, what are the operations that need to be performed on those numbers, and in what order should these operations be performed?” If a student is not able to make these judgments, then there’s not much point in calculating on paper or with a calculator. Formal calculation is a skill that is only useful for carrying out commands such as “calculate this and this” (a formula for calculation) once these commands are actually specified. Carrying out these commands is known as “deciding the operation.” Therefore, “deciding the operation” for oneself in order to determine which command is necessary to “calculate this and this” is an important skill that is indispensable for solving problems.

Deciding the operation clearly determines the meaning of each computation, and decides what must be done based on that meaning. This is why “the ability to clarify the meanings of addition, subtraction, multiplication, and division and determine operations based on these meanings” is the most important ability required for computation.

Actually, there is something more important – in order to correctly decide which operations to use in this way, one must be able to think in the following manner “I would like to determine the correct operations, and to do so, I need to recall the meanings of each operation, and think based on these meanings.” This thought process is one kind of mathematical thinking.

Even if a student solves the group discount problem as described above, this might not be sufficient to conclude that he truly understood the problem. This is why it is important to “change the conditions of the problem a little” and “consider whether or not it is still possible solve the problem in the same way.” These types of thinking are neither knowledge nor a skill. They are “functional thinking” and “analogical thinking.”

For instance, let’s try changing one of the conditions by “changing the bus fare from 4,500 yen to 4,000 yen.” Calculating again as described above results in an answer of 48 people (actually, a better way of thinking is to replace 4,500 above with 4,000 – this is analogical thinking). In this way, one should gain confidence in one’s method of solving the problem, as one realizes that the result is the same: 48 people.

The above formulas are expressed in a way that is insufficient for students in fourth grade or higher. It is necessary to express problems using a single formula whenever possible. When these formulas are converted into a single formula based on this thinking, this is the result:4,500×(1-0.2) ×60÷4,500

When viewed in this form, it becomes apparent that the formula is simply 60×(1-0.2)

What is important here is the idea of “reading the meaning of this formula.” This is important “mathematical thinking regarding formulas.” Reading the meaning of this formula gives us: full capacity × ratio

For this reason, even if the bus fare changes to 4,000 yen, the formula 60×0.8=48 is not affected. Furthermore, if the full capacity is 50 persons and the group discount is 30%, then regardless of what the bus fare may be, the problem can always be solved as “50×0.7=35; the group rate (bus rental) is a better deal with 35 or more people.” This greatly simplifies the result, and is an indication of the appreciation of mathematical thinking, namely “conserving cogitative energy” and “seeking a more beautiful solution.”

Students should have the ability to reach the type of solution shown above independently. This is a desirable scholastic ability that includes the following aims:

• Clearly grasp the meaning of operations, and decide which operations to use based on this understanding

• Functional thinking

• Analogical thinking

• Expressing the problem with a better formula

• Reading the meaning of a formula

• Economizing thought and effort (seeking a better solution)

Although this is only a single example, this type of thinking is generally applicable. In other words, in order to be able to independently solve problems and expand upon problems and solving methods, the ability to use “mathematical thinking” is even more important than knowledge and skill, because it enables to drive the necessary knowledge and skill.

Mathematical thinking is the “scholastic ability” we must work hardest to cultivate in arithmetic and mathematics courses.

3. The Hierarchy of Scholastic Abilities and Mathematical Thinking

As the previous discussion makes clear, there is a hierarchy of scholastic abilities. When related to the above discussion, and limited to the area of computation (this is the same as in other areas, and can be generalized), these scholastic abilities enable the following (from lower to higher levels):

1. The ability to memorize methods of formal calculation and to carry out these calculation

2. The ability to understand the rules of calculation and how to carry out formal calculation

3. The ability to understand the meaning of each operation, to decide which operations to use based on this understanding, and to solve simple problems

4. The ability to form problems by changing conditions or abstracting situations

5. The ability to creatively make problems and solve them

The higher the level, the more important it is to cultivate independent thinking in individuals. To this end, mathematical thinking is becoming even more and more necessary.

Chapter 2

The Importance of Teaching to Cultivate Mathematical Thinking

2.1 The Importance of Teaching Mathematical Thinking

As we found in the previous chapter, the method of thinking is the center of scholastic ability. In arithmetic and mathematics courses, mathematical thinking is the center of scholastic ability. However, in Japan, in spite of the fact that the improvement of mathematical thinking was established as a goal more than 45 years ago, the teaching of mathematical thinking is by no means sufficient.

One of the reasons that teaching to cultivate mathematical thinking does not tend to happen is, teachers are of the opinion that students can still learn enough arithmetic even if they don’t teach in a way to cultivate the students’ mathematical thinking. In other words, teachers do not understand the importance of mathematical thinking.

The second reason is that, in spite of the fact that mathematical thinking was established as a goal, teachers do not understand what it really is. It goes without saying that teachers cannot teach what they themselves do not understand.

Therefore, we shall start out by explaining how important the teaching of mathematical thinking is.

A simple summary follows.

Mathematical thinking allows for:

(1) An understanding of the necessity of using knowledge and skills

(2) Learning how to learn by oneself, and the attainment of the abilities required for independent learning

(1) The Driving Forces to Pursue Knowledge and Skills

Mathematics involves the teaching of many different areas of knowledge, and of many skills. If children are simply taught to “use some knowledge or skill” to solve problems, they will use that knowledge or skill. In this case, however, children will not realize why they are being told to use such a knowledge or skill. Also, when new knowledge or skills are required for problem solving and students are taught what skill to use, they will be able to use that skill to solve the problem, but they will not know why this skill must be used. Students will therefore fail to understand why the new skill is good.

What is important is “how to realize” which previously learned knowledge and skills should be used. It is also important to “sense the necessity of” and “perceive the need or desirability of using” new knowledge and skills.

Therefore, it is necessary for something to act as a drive towards the required knowledge and skills. Children first understand the benefits of using knowledge and skills when they possess and utilize such a drive. This leads them to fully acquire the knowledge and skills they have used.

Mathematical thinking acts as this drive.

(2) Achieving Independent Thinking and the Ability to Learn Independently

Possession of this driving force gives students an understanding of how to learn by themselves.

Cultivating the power to think independently will be the most important goal in education from now on, and in the case of arithmetic and mathematics courses, mathematical thinking will be the most central ability required for independent thinking. By mastering this skill even further, students will attain the ability to learn independently.

The following specific example serves to clarify this point further.

2.2 Example: How Many Squares are There? This instructional material is appropriate for fourth-grade students.

1. The Usual Lesson Process

This is usually taught in the following way (T refers to the teacher, and C the children):

T: There are both big and small squares here. Let’s count how many squares there are in total.

T: (When the children start counting) First, how many small squares are there? C: 25.

T: Which squares are the second smallest?

C: (Indicate the squares using two by two segments)

T: Count the number of those squares.

T: Which squares are the next biggest size, and how many are there?

The questions continue in this manner in order of size. In each case, the teacher asks one child the number, and then asks another child if this number is correct. Alternatively, the teacher might recognize the correctness of the number, and comment “yes, that’s the right number.”

The teacher has the children count squares in order of size, and then has the children add the numbers together to derive the grand total.

2. Problems with This Method

a) When the teacher instructs children to count squares based on size, the children do not realize for themselves that they should sort the squares into groups. As a result, the children do not understand the need to sort, or the thinking behind sorting.

b) The number of squares of each size is determined either by the majority of the children’s answers, or based on the teacher’s approval. These methods are not the right way of determining the correct answer. Correctness must be determined based on solid rationales.

c) Also, if instruction regarding this problem ends this way, children will only know the answer to this particular problem. The important things they must grasp, however, are what to focus upon in general, and how to think about problems of this nature.

Teachers should, therefore, follow the following teaching method:

3. Preferred Method

(1) Clarification of the Problem – 1

The teacher gives the children the previous diagram. T: How many squares are there in this diagram?

C: 25 (many children will probably answer this easily).

Some children will probably respond with a larger number.

The children come up with the answer 25 after counting just the smallest squares. Those who think the number is higher are also considering squares with more than one segment per side. This is the source of the issue, which is not about the correct answer, but the vagueness of the mathematical problem.

The teacher should then have the children discuss “which squares they are counting when they arrive at the number 25,” and inform them that “this problem is vague and does not clearly state which squares need to be counted.” The teacher concludes by clarifying the meaning of the problem, saying “let’s count all the squares, of every different size.”

(2) Clarification of the Problem – 2

First, the teacher lets all the children count the squares independently. Various answers will be given when the teacher asks for totals, or the children may become confused while counting. The children will realize that most of them (or all of them) have failed to count correctly. It is then time to think of a way of counting that is a little better and easier (this becomes a problem for the children to solve).

(3) Realizing the Benefit of Sorting The children will realize that the squares should be sorted and counted based on size. The teacher has the children count the squares again, this time sorting according to size.

(4) Knowing the Benefit of Encoding

Once the children are finished counting, the teacher asks them to give their results. At this point, when the teacher asks “how many squares are there of this size, and how many squares are there of that size...” he/she will run into the problem of not being able to clearly indicate “which size.”

At this point, naming (encoding) each square size should be considered. It is important to make sure that the children realize that calling the squares “large, medium, and small” is not preferable because this naming system is limited. However, the children learn that naming the squares in the following way is a good system, as they state each number.

Squares with 1 segment 25

Squares with 2 segment 16

Squares with 3 segment 9

Squares with 4 segment 4

Squares with 5 segment 1

Total 55

(5) Judging the Correctness of Results More Clearly, Based on Solid Rationale The correctness or incorrectness of these numbers must be elucidated, so have one child count the squares again in front of the entire class. The student will probably count the squares while tracing each one, as shown to the right. This will result in a messy diagram, and make it hard to tell which squares are being counted. Tracing each square is inconvenient, and will make the students feel their counting has become sloppy.

(6) Coming up with a More Accurate and Convenient Counting Method

There is a counting method that does not involve tracing squares. Have the students discover that they can count the upper left vertex (corner) of each square instead of tracing, in the following manner: place the pencil on the

upper left vertex and start to trace each square in one’s head, without moving the pencil from the vertex.

By using this system, it is possible to count two-segment squares as shown in the diagram to the right, by simply counting the upper left vertices of each square. This counting method is easier and clearer.

This method takes advantage of the fact that “squares and upper left vertices are in a one-to-one relationship.” In other words, in the case of two-segment squares, once a square is selected, only one vertex will correspond to that square’s upper left corner. The flip side of this principle is that once a point is selected, if that point corresponds to the upper left corner of a square, then it will only correspond to a single square of that size. Therefore, while sorting based on size, instead of counting squares, one can also count the upper left corners.

Instead of counting squares, this method “uses a functional thinking by counting the easy-to-count upper left vertices, which are functionally equivalent to the squares (in a one-to-one relationship).”

(7) Expressing the Number of Squares as a Formula

When viewed in this fashion, the two-segment squares shown in the diagram have the same number as a matrix of four rows by four columns of dots. When one realizes that this is the same as 4×4, it becomes apparent that the total number of squares is as follows: 5×5+4×4+3×3+2×2+1×1 (A)

Students will understand that it’s a good idea to think of ways to devise different expression methods, and to express problems as formulas.

(8) Generalizing

This makes generalization simple. In other words, consider what happens when “the segment length of the original diagram is increased by 1 to a total of 6.” All one needs to do is to add 6×6 to formula (A) above. Thus, the thought process of trying to generalize, and the attempt to read formulas is important.

(9) Further Generalization

For instance (for students in fifth grade or higher), when this system is applied to other diagrams, such as a diagram constructed entirely of rhombus, how will this change the formula? (Answer: It will not change the above formula at all.)

By generalizing to see the case of parallelograms (as long as the counting involves only parallelograms that are similar to the smallest parallelogram, the diagram can be seen in the same way), the true nature of the problem becomes clear.

4. Mathematical Thinking is the Key Ability

Here What kind of ability is required to think in the manner described above? First, what knowledge and skills are required? The requirements are actually extremely simple: Understanding the meaning of “square,” “vertex,” “segment,” and so on The ability to count to around 100

The ability to write the problem as a formula, using multiplication and addition Possession of this understanding and skills, however, is not enough to solve the problem. An additional, more powerful ability is necessary. This ability is represented by the underlined parts above, from (1) to (9):

Clarification of the Meaning of the Problem

Coming up with an Convenient Counting Method

Sorting and Counting

Coming up with a Method for Simply and Clearly Expressing How the Objects Are Sorted Encoding

Replacing to Easy-to-Count Things in a Relationship of Functional Equivalence

Expressing the Counting Method as a Formula

Reading the Formula Generalizing

This is mathematical thinking, which differs from simple knowledge or skills.

It is evident that mathematical thinking serves an important purpose in providing the ability to solve problems on one’s own as described above, and that this is not limited to this specific problem. Therefore, the cultivation of a number of these types of mathematical thinking must be the aim of this class.

Chapter 3

The Meaning of Mathematical Thinking and How to Teach It Characteristics of Mathematical Thinking

Although we have examined a specific example of the importance of teaching that cultivates mathematical thinking during each hour of instruction, for a teacher to be able to teach in this way, he must first have a solid grasp of “what kinds of mathematical thinking there are.” After all, there is no way a person could teach in such a way as to cultivate mathematical thinking without first understanding the kinds of mathematical thinking that exist. Let us consider the characteristics of mathematical thinking.

1. Focus on Sets

Mathematical thinking is like an attitude, as in it can be expressed as a state of “attempting to do” or “working to do” something. It is not limited to results represented by actions, as in “the ability to do,” or “could do” or “couldn’t do” something.

For instance, the states of “working to establish a perspective” and “attempting to analogize, and working to create an analogy” are ways of thinking. If, on the other hand, one has no intention whatsoever of creating an analogy, and is told to “create an analogy,” he/she might succeed in doing so due to having the ability to do so, but this does not mean that he/she consciously thought in an analogical manner.

In other words, mathematical thinking means that when one encounters a problem, one decides which set, or psychological set, to use to solve that problem.

2. Thinking Depends on Three Variables

In this case, the type of thinking to use is not determined by the problem or situation. Rather, the type of thinking to use is determined by the problem (situation), the person, and the approach (strategy) used. In other words, the way of thinking depends on three variables: the problem (situation), the person involved, and the strategy.

Two of these involve the connotative understanding of mathematical thinking. There is also denotative understanding of the same.

3. Denotative Understanding

Concepts are made up of both connotative and denotative components. One method which clarifies the “mathematical thinking” concept is a method of clearly expressing connotative “meaning.” Even if the concept of mathematical thinking is expressed with words, as in “mathematical thinking is this kind of thing,” this will be almost useless when it comes to teaching, because even if one understands the sentences that express this meaning, this does not mean that they will be able to think mathematically.

Instead of describing mathematical thinking this way, it should be shown with concrete examples. At a minimum, doing this allows for the teaching of the type of thinking shown. In other words, mathematical thinking should be captured denotatively.

4. Mathematical Thinking is the Driving Force Behind Knowledge and Skills Mathematical thinking acts as a guiding force that elicits knowledge and skills, by helping one realize the necessary knowledge or skills for solving each problem. It should also be seen as the driving force behind such knowledge and skills.There is another type of mathematical thinking that acts as a driving force for eliciting other types of even more necessary mathematical thinking. This is referred to as the “mathematical attitude.”

3.2 Substance of Mathematical Thinking

It is important to achieve a concrete (denotative) grasp of mathematical thinking, based on the fundamental thinking described in section 3.1. Let us list the various types of mathematical thinking.

First of all, mathematical thinking can be divided into the following three categories:

II. Mathematical Thinking Related to Mathematical Methods

III. Mathematical Thinking Related to Mathematical Contents

Furthermore, the following acts as a driving force behind the above categories: I. Mathematical Attitudes

Although the necessity of category I was mentioned above, further consideration as described below reveals the fact that it is appropriate to divide mathematical thinking into II and III. Mathematical thinking is used during mathematical activities, and is therefore intimately related to the contents and methods of arithmetic and mathematics. Put more precisely, a variety of different methods is applied when arithmetic or mathematics is used to perform mathematical activities, along with various types of mathematical contents. It would be accurate to say that all of these methods and types of contents are types of mathematical thinking. It is because of the ways of thinking that the existence of these methods and types of contents has meaning. Let us focus upon these types of contents and methods as we examine mathematical thinking from these two angles.

For this reason, three logical categories can be derived.

Specific details are provided below.

List of Types of Mathematical Thinking

I. Mathematical Attitudes

1. Attempting to grasp one’s own problems or objectives or substance clearly, by oneself

(1) Attempting to have questions

(2) Attempting to maintain a problem consciousness

(3) Attempting to discover mathematical problems in phenomena

2. Attempting to take logical actions

(1) Attempting to take actions that match the objectives

(2) Attempting to establish a perspective

(3) Attempting to think based on the data that can be used, previously learned items, and assumptions

3. Attempting to express matters clearly and succinctly

(1) Attempting to record and communicate problems and results clearly and succinctly

(2) Attempting to sort and organize objects when expressing them

4. Attempting to seek better things

(1) Attempting to raise thinking from the concrete level to the abstract level

(2) Attempting to evaluate thinking both objectively and subjectively, and to refine thinking

(3) Attempting to economize thought and effort

II. Mathematical Thinking Related to Mathematical Methods

1. Inductive thinking

2. Analogical thinking

3. Deductive thinking

4. Integrative thinking (including expansive thinking)

5. Developmental thinking

6. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)

7. Thinking that simplifies

8. Thinking that generalizes

8. Thinking that specializes

9. Thinking that symbolize

10. Thinking that express with numbers, quantifies, and figures

III. Mathematical Thinking Related to Mathematical Contents

1. Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)

2. Focusing on constituent elements (units) and their sizes and relationships (Idea of units)

3. Attempting to think based on the fundamental principles of expressions (Idea of expression)

4. Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)

5. Attempting to formalize operation methods (Idea of algorithm)

6. Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)

7. Focusing on basic rules and properties (Idea of fundamental properties)

8. Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)

9. Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)

Once the student has written part of the number table, he/she can induce that “it is possible to move from one multiple of 8 to another by going down one row, and then left two columns.” When stated the same way for multiples of 4, it is also possible to induce that “it is possible to move from one multiple of 4 to another by going down one row, and then left two columns.”

Then considering “why it is possible to make this simple statement” and “whether or not it is still possible to state this for numbers over 99, and why this is the case” is deductive thinking. Next, consider what to base an explanation of this on. One will realize at this point that it is possible to base this on how the number table is created. This is also deductive thinking, and is based upon the following.

Since this number table has 10 numbers in each row, “going one position to the right increases the number by one, and going one position down increases the number by ten.” Based upon this, it is evident that going down one position always adds 10, and going left two positions always subtracts 2. Combining both of these moves always results in an increase of 8 (10-2=8). Therefore, if one adds 8 to a multiple of 4 (or a multiple of 8), the result will always be a multiple of 4 (8). This explains what is happening.

By achieving results with one’s own abilities in this way, it is possible to gain confidence in the correctness of one’s conclusion, and to powerfully assert this conclusion. Always try to explain the truth of what you have induced, and you will feel this way. Also, think about general explanations based on clear evidence (the creation of the number table). This is deductive thinking.

Example 2: Deductive thinking is not just used in upper grades, but is used in lower grades as well.

Assume that at the start of single-digit multiplication in 3rd grade, the problem “how many sheets of paper would you need to hand out 16 sheets each to 8 children” is presented. When the children respond with “16×8,” the teacher could run with this response and say, “all right, let’s consider how to find the answer to this.”

This is not adequate, however. The students must be made to thoroughly understand the fundamental reasoning behind the solution. It is important that students independently consider “why this is the way the problem is solved.”

The child will probably explain the problem by saying that “in this problem, eight 16s are added: 16+16+16+16+16+16+16+16.” This is based on the meaning of multiplication

Chapter 7

Questions for Eliciting Mathematical Thinking

Teaching should focus on mathematical thinking. Teachers need to first think of how they can help children appreciate and gain the ability to use mathematical thinking. When children get stuck, rather than helping them directly with useful knowledge and skills, teachers must prepare a way to teach the mathematical thinking required to elicit the knowledge and skill and moreover to teach the attitude that leads to this thinking methods. Also, this assistance must be of a general nature, and must be applicable to many different situations. Assistance should take a form that is frequently helpful when one focuses upon it. This is because this kind of assistance is useful in many different situations. By repeatedly providing it, a student can grow accustomed to this type of mathematical thinking. This kind of assistance is not something taught directly, but something that should be used by children themselves to overcome problems. Therefore, this assistance should take the form of questions.

It goes without saying that the goal of teaching based on these kinds of questions is for children to gain the ability to ask these questions of themselves, and to learn how to think for themselves.

Questions related to mathematical thinking and attitudes must be posed based on a perspective of what kinds of questions must be asked. This must be considered in advance. Questions must be created so that the problem solving process elicits mathematical thinking and attitudes. The following offer a list of question analyses designed to cultivate mathematical thinking, based on a consideration of these kinds of questions. In other words, this question analysis list is comprised of questions derived from the main types of mathematical thinking used at each stage of the problem solving process.

The A questions on this list deal with mathematical attitudes, with the stage indicated as “A11” and so on. Questions related to mathematical thinking related to mathematical methods are marked with M, and questions related to mathematical ideas are marked with I. Types of thinking corresponding to the question are given in parentheses ( ).

[List of Questions Regarding Mathematical Thinking]

Questions Regarding Mathematical Attitudes

A11 What kinds of things (to what extent) are understood and usable? (Clarifying the problem)

A12 What is needed to understand, and can this be stated clearly? (Clarifying the problem)

A13 What kinds of things (from what point) are not understood? What does one want to find? (Clarifying the problem)

A14 Does anything seem strange? (A questioning attitude)

Questions Regarding Thinking Related to Methods

M11 What is the same? What is shared? (Abstraction)

M12 Clarify the meaning of the words and use them by oneself. (Abstraction)

M13 What (conditions) are important? (Abstraction)

M14 What types of situations are being considered? What types of situations are being proposed? (Idealization)

M15 Use figures (numbers) for expression. (Diagramming, quantification)

M16 Replace numbers with simpler numbers. (Simplification)

M17 Simplify the conditions. (Simplification)

M18 Give an example. (Concretization)

Questions Regarding Thinking Related to Contents

I11 What must be decided? (Functional)

I12 What kinds of conditions are not needed, and what kinds of conditions are not included? (Functional)

Questions Regarding Mathematical Attitudes

A21 What kind of method seems likely to work? (Perspective)

A22 What kind of result seems to be possible? (Perspective)

Questions Regarding Thinking Related to Methods

M21 Is it possible to do this in the same way as something already known? (Analogy)

M22 Will this turn out the same thing as something already known? (Analogy)

M23 Consider special cases. (Specialization)

Questions Regarding Thinking Related to Contents

I21 What should one consider this based on (what unit)? (Units, sets)

I22 What seems to be the approximate result? (Approximation)

I23 Is there something else with a similar meaning (properties)? (Expressions, operations, properties)

Questions Regarding Mathematical Attitudes

A31 Try using what is known (what will be known). (Logic)

A32 Are you approaching what you seek? (Logic)

A33 Can this be said clearly? (Clarity)

Questions Regarding Thinking Related to Methods

M31 What kinds of rules seem to be involved? Try collecting data. (Induction)

M32 Think based on what is known (what will be known). (Deduction)

M33 What must be known before this can be said? (Deduction)

M34 Consider a simple situation (using simple numbers or figures). (Simplification)

M35 Hold the conditions constant. Consider the case with special conditions. (Specialization)

M36 Can this be expressed as a figure? (Diagramming)

M37 Can this be expressed with numbers? (Quantification)

Questions Regarding Thinking Related to Ideas

I31 Think based on units (points, etc.). (Units)

I32 What unit (what scope) should be used for thinking? (Units, sets)

I33 Think based on the meaning of words (words used to express methods, or methods themselves). (Expressions, operations, properties)

I34 Try following a predetermined procedure (calculations). (Algorithms)

I35 What is this (formula or symbol) expressing? (Formulas, expressions)

I36 Can I express this in a formula? (Formulas)

Questions Regarding Mathematical Attitudes

A41 Why is this (always) correct? (Logical)

A42 Can this be said more accurately? (Accuracy)

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