# Use Appropriate Tools Strategically – Part II

Teachers’ Role in Tool Building and Using
Teachers play a critical role in the development of the strategic use of tools. First, they make a diversity of tools available to students. From the beginning of the year, students should know where the math tools are in the room and how they will be used throughout the class. From the beginning, a teacher should declare: “Just like many, you play a sport of your choice and get better at it with practice. Similarly, each one of you should become an expert in a mathematics tool or strategy and its usage in mathematics concepts and procedures.” Then the teacher should present varieties of situations or problems where that tool is applicable and effective (e.g., different types of rules or protractors). Then she works on the strategic use of the tools. The first step is modeling their appropriate use.  One can begin by using phrases like “I bet a ruler would help me divide this into even pieces” or “I wonder if using a graph paper would help me organize my work” or “I think I could use a calculator to double check my accuracy on this one.” The key factor in getting students to use mathematical tools efficiently is exposure—multiple and varied exposures.

Students need to see how teachers make decisions about using tools so they know what appropriate use is. Using a calculator to solve 50 + 50 is not appropriate—but it is appropriate to check a complicated computation. A teacher should not admonish students who choose tools inappropriately. Instead, she should ask them to share their reasoning for using a particular tool in a specific way. They should also make explicit how a particular tool or approach will connect to an abstract mathematics idea.

As the year continues and new tools are introduced, students will be able to apply their current knowledge of mathematical tools to the new ones. It is important to plan tasks that will require multiple learning tools.

In order for students to be proficient they need to start using the tools independent of the teacher. They will then pick tools based on the needs of the problem and plan. They will also be able to visualize the results after using the tool.

Introductory Part of the Lesson (Teacher Directed: Didactic and Socratic Roles)
For Teachers: Decisions

• What are the goals of this lesson?
• What language, concept, procedures, and skills do I want students to develop?
• What activities and tools are best suited for this purpose?
• What tools and methods my students are already familiar with?

Middle Part of the Lesson (Socratic and Coaching Roles)

• Have the students acquired the concept/procedure using this tool, what tools can further expand, enhance, or deepen these ideas?
• Can they apply these tools in problem solving with my support and in collaboration with their colleagues?

Last Part of the Lesson (Coaching and Supporting Roles)

• What problems can I assign students that will provide opportunities of applying these and previous tools in solving them?
• Can they create/construct problems that can be solved by these tools?
• Discussion to establish the efficiency of tools and develop proficiency and competence in the use of tools and integrate these tools with earlier tools.

Teachers must recognize that tools do not produce understanding, problem solving, and solutions. These come when teachers ask questions and make connections between the tool and the concept and when students do the same. Providing students with protractors does not ensure that they will measure the angles and find angle sums of triangles with accuracy. Similarly, a graphing calculator doesn’t consider user error or misconception when graphing a linear equation. The teacher should therefore bring to students’ attention the strength and limitations of the tool and its usage.

User error (including a broken ruler) can occur with a basic ruler or a calculator. If we have the concept and understanding, we adjust. We try the tool again, maybe a little differently.

Initially it is the teacher’s questions that help students in the tools’ usage, but then the teacher needs to transfer their usage to students and the questioning process to facilitate this. The questions we want students to ask when selecting and using tools include:

• Do I need a tool in this situation?
• Which tool will work for this situation?
• Is this the right tool?
• How does it work?
• What tool is the best to use in this situation?
• Do the results align with what I was expecting?
• Do the results make sense?

Most importantly, when children select, use, make decisions, compare the usage of tools, they think and develop metacognitive processes. As a result, their cognitive ability and potential increase. They become better learners. This development is centered on teachers facilitating the process by asking questions and creating cognitive dissonance in students’ minds. Thus, we encourage the thinking behind the tool as well as the procedure for using the tool. We should require our students to predict what their findings might be prior to using the tool and then require them to reflect on the results and if the answers make sense.

For Students: Problem Solving
The standard says:
[S]tudents consider the available tools…. Proficient students…make sound decisions about when each of these tools might be helpful.
[They]… use technological tools to explore and deepen their understanding.

These phrases focus on the student. The goals for students are: Use available tools recognizing the strengths and limitations of each. Use estimation and other mathematical knowledge to detect possible errors. Identify relevant external mathematical resources to pose and solve problems. Use technological tools to deepen understanding of mathematics.

Students learn through the questions teachers pose to help them think and sort through the ideas that are forming. Helping students to generate questions results in their seeking and using tools.

• What information do I have?
• What is stated in the problem?
• How will I represent the information in the problem?
• What tool(s) can help me visualize and represent this information to understand its nature?
• What do I know that is not stated in the problem?
• What approach should I consider trying first?
• What other mathematical tools could I use to visualize and represent situations and conditions in the problem?
• What is the expected range of the answer for this problem?
• Should I make an estimate for the answer?
• What estimate can I make for the solution? Should I change the numbers for that purpose?
• What will be the unit of my estimate?
• In this situation, would it be helpful to use…a graph, number line, ruler, diagram, calculator, or a manipulative?
• Why was it helpful to use…?
• What can using a ____ show us that ____ may not?

The following problem illustrates students’ reasoning:
Three-fourth of the yard was converted into a vegetable garden. Two-third of the garden is used for gardening herbs. What fraction of the garden is herbs?

Student One: I am going to assume that the garden is a rectangular shape. Let me represent the rectangle as my whole. The herb garden is going to be ⅔ of. Approach One: I can see my problem as coloring first ¾ of the garden representing the garden and then I color ⅔ of the ¾ in another color. That will represent the herb garden as: ⅔ of ¾ = ¼+¼=½.

That is a correct approach to get the answer; however, it does not help us to arrive at the procedure.

Student Two: I am going to assume that the garden is a rectangular shape. Let me represent it as a 1 by 1 rectangle. The herb garden is going to be a ¾ × ⅔ or ⅔ × ¾ rectangle. By definition ¾ × ⅔ is the area of the rectangle with the dimensions of ¾ and ⅔. Therefore, the vertical side of the rectangle is divided into four equal parts and the horizontal side into three equal parts and the ¾ × ⅔ rectangle is formed.

As the diagram suggests, this area is 2 by 3 parts out of 3 by 4 parts and that is represented as (2×3) out of (3×4) or . In other words, we have

The representational tool “area as multiplication” is more efficient and strategic compared to the “groups of” tool for multiplication. Although at the whole number level, the four models of multiplication: repeated addition, groups of, an array, and the area of rectangle are equally good as an introduction, the most efficient is the area model.

When students are sufficiently familiar with the tools, the teacher should pose problems that are tool specific and help students to sharpen their tools by practicing them daily till they are proficient, a better tool is available, or a better way of using that tool is possible.

I suggest teachers devote about 20% to 25% of time in every lesson on tool building (achieving proficiency in the use of physical tools and fluency in the use of thinking tools). For example, arriving at a particular definition relating to a mathematical idea is part of conceptual development. But, mastery in using definitions, concepts, and procedures is tool building. As students get older, they need to add more versatile tools or create new tools by combining tools to their mathematical toolbox.

The teacher starts with definitions, develops conceptual understanding and arrives at the procedure, but the children should not remain at the definition and procedural level; they should solve problems—solving problems builds tools and their efficient use builds mathematics stamina. Proficiency in the use of tools and concepts are built when we use them regularly and apply them in different settings. Some concepts become tools and then these tools are used to learn new concepts and new tools. This iterative process continues in learning mathematics. When we use tools, we learn new mathematics. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a whole number and concluded that it must be a different kind of number.

Versatile tools (exact, efficient, elegant) build mental schemas/models that last. What makes a tool like the Cuisenaire rods, Base Ten blocks, Invicta balance, Empty Number line, Bar Model or the area model truly powerful is that it is not just a special-purpose trick or temporary crutch for a particular type of problem but is faithful to the mathematics and is applicable to many domains and concepts. Because these tools help students make sense of mathematics, they last. And that is also why the CCSS and SMP mandates them.

Mathematically proficient students are able to identify relevant external mathematical resources such as people, books, digital content on a website and use them to pose or solve problems. They use technological tools to explore and deepen their understanding of concepts and make connections between concepts and procedures.

Technology Tools
Hands-on tools are useful; however, as we prepare students for the world of work and the power of technology, students need to understand the range of use, strengths, and limitations of these tools. Today, technological computations tools (e.g., calculator) are common outside the classroom, so the classroom needs to reflect this reality. With the different technologies—calculators, smart phones, tablets, and laptops, the question of when and how to use technology becomes even more important.

Technology can allow greater opportunities to visualize, explore, predict, and compare mathematical ideas. A parallel practice for teachers, therefore, is to augment the use of appropriate technological tools for mathematics instruction at the appropriate time. The use depends on teachers deciding first the mathematical goals of instruction and then which tools may be most effective in accomplishing them.

The question of when to use technology tools is a question about the most productive use of valuable classroom time. The answer to that question may lie in the reasons we teach mathematics today:

• to understand numbers and patterns found in nature (number concept, numbersense, numeracy)
• to acquire math tools, know when and how to use those tools
• to make fast and accurate predictions and check the reasonableness of answers
• to grow and maintain mental power
• to identify unknowns in a situation, represent and deal with them
• to think logically and clearly when solving problems
• to feel comfortable with quantitative and spatial thinking demands in a technological world

Using a calculator as a tool should be a strategic decision. Calculators should be used with caution in elementary school (that means very carefully). Calculators should be available as computational tools, particularly when many or cumbersome computations are needed to solve problems. However, when the focus of the lesson is on developing computational skills or algorithms, the calculator should not be used. It should be a tool that provides access, simplifies the task, or confirms accuracy. It doesn’t make sense for a fifth grader to use a calculator for 8 + 13. However, it may make sense for a first grader to confirm the sum that way. But, for 18.1759 + 27.19427, use the calculator.

In Elementary School, the following conditions are satisfied then and only then I will allow the use of calculators: (a) student knows the arithmetic facts, (b) good in estimation, (c) understands the concept/procedure for which the calculator is being used (e.g., in how many 4 are there in .04, the student knows which one is the divisor and the dividend). Teachers should encourage students in all contexts to estimate first and then if necessary to use calculators. Students who know “about” how much an inch is can tell when they use a ruler if their answer is reasonable. Students who understand how to round and estimate when multiplying money know if what they plug into a calculator makes sense.

Students should have the opportunities to discuss when they might use tools (e.g., concrete, technology, paper/pencil, mental math), and they should know when and where tool use is appropriate. Tool use is not an ideological decision: it is neither dedication to “times tables,” “long division,” and “repetition and memorization” nor is it allegiance to “fuzzy math” reform—preaching concept over content, exposure over mastery, insight over “right.” Tool use is the judicious decision of both and is the integration of the linguistic, conceptual, and procedural components. This can happen if we:

• allow students to spend less time on tedious calculations and more time first on language development, conceptual understanding and solving problems
• help students develop better number sense (number concept, arithmetic facts, and place value)
• allow students to study mathematical concepts they could not attempt if they had to perform the related calculations themselves and use the tool to develop the concept by seeing the patterns
• allow students who would normally be turned off by math because of frustration or boredom to increase their mathematical understanding and help them to acquire fluency of arithmetic facts by seeing number relationships
• simplify tasks while helping students determine the best methods for solving problems
• make students more confident about their math abilities, once a problem is solved using a calculator, make the number manageable and help them solve without the calculator.

While few educators deny the usefulness of calculators at the high school level, we need to rethink their use in the lower grades. Inappropriate use of calculators prevents students from seeing the underlying structure and beauty in mathematics, inhibits them from seeing mathematical relationships, and gives them a false sense of confidence about their math ability. Students who do not do long division, who quickly pull out their calculator to find the answer, do not understand the underlying principle of division. For example, when I asked eleventh grade students: What is the largest remainder you could expect when you divide a six-digit number by 11? most took out their calculators and started calculating by choosing random numbers. Many gave unreasonable answers.

If properly used, technology is an important means to achieve the goals of instruction. However, specific examples of technology use do not start appearing in the content standards (CCSS-M) until the middle school grades, and most appear at the high school level.

There are elements of mathematics at all levels (not just elementary) where some basic facts simply must be memorized (of course, after understanding and using them in problems). For example, to succeed in algebra, one should have mastered operations on fractions and integers. And success in mastering fractions is dependent on the mastery of:

• understanding, fluency, and applicability of multiplication tables,
• divisibility tests,
• prime factorization, and
• short division.

Similarly, at each level new skills and relationships emerge (e.g., laws of exponents, etc.), which need to be automatized (once again first understanding and then using them in problem solving situations) and without rote.

Many teachers, not just at third, fourth, and fifth grade levels, but at the high school and college levels, report that students do not know their basic facts or the concept of fractions. This is due to the ineffective use of tools. That is, students are overly dependent on concrete models, counting, number line, calculators, graphic organizers, multiplication charts, etc.

Others disagree, however, claiming that calculators help younger students grasp the underlying principles of mathematics by allowing them to spend time on those principles rather than on the rote computations necessary to solve them.

To build a beautiful piece of furniture, a carpenter does not use only one tool but recognizes the relationship between the object to be made and the tools needed. It is up to individual teachers, of course, to find the right way to achieve the many and complex goals of mathematics learning. And most teachers strive to do just that.

Examples of Tool Usage

Common Core Curriculum Standards (CCSS-M, 2010) call for specific situations and context to use specific tools. For example, the following geometric standard calls for the use of classic geometric tools:
Make formal geometric constructions with a variety of tools and methods (compass     and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle, bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

This high school standard on “Interpreting Categorical and Quantitative Data” suggests the use of several appropriate tools. Here’s an example:
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

The high school standards for number and quantity include this paragraph: Calculators, spreadsheets, and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. They can be used to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with non-integer components.

In high school algebra, the standards suggest uses for spreadsheets of computer algebra systems: A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave. Here’s an example:
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Kindergarten through Second Grade The focus of these three years is to master Additive Reasoning and its applications and identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures. To achieve these goals, the tools may include counters, Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, and concrete geometric shapes (e.g., pattern blocks, 3-d solids). Students should also have experiences with educational technologies such as virtual manipulatives and mathematical games and toys that support conceptual understanding, but calculator is not advisable. During classroom instruction, students should have access to various mathematical tools as well as paper (for applying alternative methods of adding and subtracting—concrete models, Empty Number Line, Decomposition/recomposition, transforming a problem by translating, by place value methods, and standard procedure), and determine which tools are the most appropriate to use. For example, find the difference: 93 – 46.

Here are the methods in order of efficiency.
Concrete Tools
Cuisenaire rods, BaseTen blocks

Pictorial/Representational Tools
Hundreds’ Chart, Empty Number Line (small-big, big-small, small-big-small jumps)

Abstract/Symbolic/Procedural Tools
Decomposition/ recomposition methods:

The standard procedure:

The fundamental concept in these grades is decomposition/recomposition. Decomposing and recomposing numbers should be done with manipulatives and models until it becomes something students can do mentally. Then we should go to the standard algorithms and mnemonic devices.

Third and Fourth Grade The focus of these two years is to master Multiplicative Reasoning and its applications; identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures; and introducing the concept of fractions. To achieve these goals, the tools may include Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, concrete geometric shapes (e.g., pattern blocks, 3-d solids), and factions strips. By the end of fourth grade, mathematically proficient students have mastered numeracy skills (ability to execute the four whole number operations correctly, consistently, fluently, with understanding in the standard form), use available tools (including estimation) when solving problems and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter and area. Or they may use graph paper or a number line to represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative size of units within a system and express measurements given in larger units in terms of smaller units.

Fifth and Sixth Grades The focus of these two years is on Proportional Reasoning, introduction to integers and equations, and their applications to quantitative and geometrical situations. Mastery of the concept of and operations on fractions in different forms—parts to whole, comparison of quantities (ratio, rate, etc.), comparison of a quantity with a standard (decimals and percents), comparison of comparisons (proportions), concept of integers and its applications; identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures and their relationships. To achieve these goals, the tools may include Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, fraction and decimal strips, geoboard, concrete geometric shapes (e.g., pattern blocks, 3-d solids), and factions strips. Technological tools—calculators, Apps, Geometric Sketch pad, etc. Mathematically proficient students consider the available tools (including estimation) when solving problems and decide when certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to accurately create graphs and solve problems or make predictions from real world data.

At any level, what is critical is that students are given opportunities to use each tool and to learn when its use is appropriate. For example, is it better to use a tape measure or a ruler to measure the length of a room? Why? In what situation would you use a protractor? Why would pattern blocks be a tool for helping students understand the need for a common denominator when adding or subtracting fractions? Is this the only tool students should experience with this specific content? Questions such as these will help teachers determine how tools foster mathematics learning most effectively.

# Use Appropriate Tools Strategically: Right Tools for the Right Job – Part I

One of the Standard of Mathematics Practice (SMP 5, CCSSI 2010, p. 7) calls for selecting appropriate tools and using them strategically. The two words “appropriate” and “strategically” apply to students as well as teachers. What does appropriate and strategic mean in the use of a tool? The answer depends on our interpretation of tools, our expectations for using them, and their role in gaining mathematical maturity for our students.

Simply, a tool is anything that aids in accomplishing a task—learning a concept/procedure. It is appropriate if it makes the concept transparent and provides the learner access to the concept. A tool is an appropriate tool in the context of what it is for and who is using it and for what purpose. Appropriateness of a tool, thus, is a function of the concept, the user, and the standard of mastery expected. A tool is appropriate if it helps the student learn the concept at the expected level.

Without strategic use, any tool, including an appropriate one may be ineffective and may not produce optimal results. However, we need to have a common definition of “using a tool strategically.” If the tool produces optimal results—develops language, concepts, and procedures with rigor and efficiently, the tool is being used strategically.

The number line is sometimes regarded just as a visual aid for children—as a physical tool. It is, in fact, a sophisticated image used even by mathematicians; it is a thinking tool. For young children, it helps develop early mental images of addition and subtraction that connect arithmetic with measurement, mental arithmetic, and standard algorithms. Rulers are just number lines built to specifications. In Kindergarten and first grade, it is the starting of solving a problem like 9 – 5 = ?

This number line image shows “the distance from 5 to 9.” It gives visual and conceptual richness to the problem and in extension the flexibility of thought. For example, if children are given the problem:

My team scored 9 points on Monday and 5 points on Tuesday. Then,

• How many more points did they score on Monday than Tuesday?
• How many fewer points did they score on Tuesday than Monday?
• What was the difference between the scores on Monday and Tuesday?
• How many more points should they have scored on Tuesday so that their score was the same as on Monday?
• How many less points should they have scored so that their score would have been as on Tuesday?
• How many extra points did they score on Monday if the goal of the game was to score only five points?

The number line can answer all of the questions raised in diverse contexts. Children who see subtraction that way can use this model to see the problems with larger quantities and different numbers. For example, let us consider the problem: 63 –27 as “the distance between 28 and 63.” To do so without crossing out digits and borrowing and following a rule, they may only barely understand.

But this number line easily explains the procedure and extends to mental calculations and applications in real life situations. In fact, it leads them to forming mental models of subtraction and helps achieve fluency in problem solving—both addition and subtraction. The number line model also extends naturally to decimals, fractions, integers, and elapsed time. For example, when students are asked to solve the problem:
If the temperature in the morning was -20 and reached 50 at noon time, what was the change in the temperature?

Many students answer it as 30, showing that they do not have the conceptual understanding and visual image in their mind for the problem. However, with the use of the number line, they can see that the distance from -2 to 5 is the number we must add to -2 to get 5: From -2 to 0 is 2 units and from 0 to 5 as 5 units, therefore, the total distance from -2 to 5 as 7, and they can generalize to solve a problem like: 42 – (-36), which can also be seen as distance from -36 to 42, using the Empty Number Line as the sum of distances from -36 to 0 (=36) and then from 0 to 42 as (=42) or 36 + 42.

Number line, on one hand, unifies arithmetic, making sense of what is otherwise often seen as a collection of independent and hard-to-remember rules and, on the other hand, it is generalizable and one can leap into algebra. The number line remains useful as students study data, graphing, and algebra: two number lines, at right angles to each other, label the addresses of points on the coordinate plane.

To find the difference 231 – 197 by counting on the number line by tens or ones is an inefficient use of the number line. But treating the problems as an addition problem and using the number line as Empty Number Line is effective and efficient as it improves numbersense and mental math.

A tool by itself is neither appropriate nor strategic. It is its use that determines whether it is appropriate and strategic.

Tools are meant to help teachers and students make sense of mathematics and its role in the world around us. They are to make teaching efficient and to support accurate, rigorous, and proficient learning. It is, therefore, our responsibility to know how to select, understand, and use the tools strategically to develop our students’ proficiencies in learning and their competence in mathematics. Ultimately, the strategic use of tools is when teachers are able to transfer the control of the use of tools to students and they use them strategically.

For students it means that they acquire the facility to use appropriate tools strategically in learning and solving problems in mathematics. It is one of the important skills of mathematically proficient students.

Teachers’ Role in Using Tools Strategically
An important element of the strategic use of tools depends on the goals of instruction. A teacher first considers mathematical goals of her instruction and then decides which tools may be most effective in accomplishing them. It means to select tools to get the concept across the students and then to use them with optimal results in learning and achievement.

Appropriateness of a tool means that the concept becomes transparent to the students and they can see the congruence of representations through the tools and the abstract/symbolic form. To achieve this, the teacher asks:

• Does it show the concept exactly—is the representation transparent?
• Is it efficient to demonstrate the concept or procedure?
• Is it easy to work with, to manipulate?
• Does the student see it efficiently and clearly?
• Can this tool be used to extrapolate, generalize, and abstract the concept from the current manifestation?
• Is it available to them?
• Will the student be able to use this tool easily and effectively?

Effective, appropriate, and strategic use of tools is important at all grade levels, but the types of tools and how they are used can differ. Golf players know when to use which “iron.” They constantly practice their usage of the tools. The same is true for a “budding” mathematician and a real mathematician alike. In developing students’ capacity to “use appropriate tools strategically,” teachers make clear to students why the use of tools will aid their problem solving processes.

Proficient students are sufficiently familiar with tools appropriate for their grade to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

Using Appropriate Tools Strategically
Mathematically proficient students consider the available tools when solving a problem and they use them strategically. The framers of CCSS-M seem to refer to two kinds of tools: physical and thinking tools. In the case of physical tools, one is looking for proficiency, and in the case of thinking tools, one wants fluency.

The physical tools (commercially prepared or constructed by teachers and students) might include pencil and paper, concrete manipulative models (sundry counting objects, fingers, TenFrames, Cuisenaire rods, Algebra tiles, Base-Ten blocks, Invicta Balance, fraction strips, games and toys, straight edge, rulers, diagrams, two-way tables, graphs, graphic organizers, protractor, compass, calculator, spreadsheet, computer algebra systems, statistical package, or dynamic geometry software, geometry sketch pad, geogebra, iPad apps, Smart Phone Apps, Graphing Calculator, Algebra Computer System, Statistics Package, Spread Sheet, etc.).

Manipulatives are objects that appeal to several senses and that can be touched, moved about, rearranged, and otherwise handled by children. Using manipulatives in the early grades is one way of making mathematics learning more meaningful to students as they are used to make abstract ideas more concrete and transparent. Modeling with manipulatives is the first step in creating an environment where students can begin to understand abstract mathematical concepts in a variety of contexts and ways. For example, an elementary teacher might have students select different color tiles to show repetition in a patterning task. A middle or high school teacher might have established norms for accessing tools during the students’ group learning and problem solving processes to make things and see geometrical relationships from them.

However, a manipulative does not by itself carry the intended meaning and does not guarantee that mathematical understanding will result from use. It is the expertise of the teacher in the use of manipulatives and the amount of time and experiences students are given to interact with the manipulatives that lead to increased achievement.

Counters of many kinds, Base-10 blocks, Cuisenaire rods, Pattern Blocks, measuring tapes, spoons or cups, and other physical devices are all, if used strategically, of great potential value in the elementary school classroom. They are the “obvious” tools. But, physical tools should satisfy the following properties: they should (a) be exact and transparent, (b) be efficient, and (c) be elegant. The physical tools serve three purposes:

(a) generate the language of that mathematical idea,
(b) help develop the conceptual schema of the idea, and
(c) derive the procedure related to the idea.

Concepts must be developed and reinforced by the tool. The use of the tool itself should support reasoning rather than mere procedure. Reasoning develops understanding. And understanding develops mental math and strategies. The idea of understanding holds true for other tools and transfers to paper/pencil as well. With understanding, physical tools develop into thinking tools. For example, the practice of making ten by the help of Cuisenaire rods develops the mental math strategies suing making ten, for example, 8 + 6 = 8 + 2 + 6 or 4 + 4 + 6; 17 – 9 = 7 + 10 – 9 = 7 + 1, etc.

Understanding helps students realize accuracy and proficiency. Consider 9.1888 + 11.1020. If I use a calculator, I should know that my sum will be in the neighborhood of 20. I need to reconsider if my calculator result is dramatically different. This transcends grade level. For example, if I determine that my slope is negative and my line rises from left to right, then something is not right.

In other words, students must derive and understand outcomes of operations with and without a calculator but also reinforce this understanding while using the calculator. If the Sin of an angle comes out to be more than 1, then, there is something wrong. Similarly, using a protractor in measuring an angle, it is more than just “lining it up the right way.” Understanding enables them to be proficient in diverse situations and even with diverse protractors.

When we have developed the language, concept, and procedure, using physical tools, students should convert them into thinking tools and then practice the procedure and the skills related to that idea. The physical tool should always be converted into thinking tools.

The thinking tools refer to vocabulary, written or mental strategies (decomposition/recomposition, properties of operations, etc.), conceptual schemas (e.g., area model of multiplication), approaches (e.g., prime factorization for LCM, etc.), skills (e.g., facts, translation from native language to math symbols, etc.), and procedures (standard or alternative). The mathematical thinking tools deal with intellectual and cognitive skills.

Cognitive/Learning Skills
A major outcome of using concrete materials as tools for mathematics is the development of prerequisite skills to anchor mathematics ideas.

• Following sequential directions: every procedure and task analysis is dependent on this skill,
• Pattern analysis: mathematics is the study of patterns in quantity and space; recognizing, identifying, extending, creating, and applying are integral part of tool usage,
• Spatial orientation/space organization: observing and identifying spatial orientation, organization, and relationships is essential in tool usage,
• Visualization: holding and manipulating information are essential for mathematics, particularly for mental math and planning problem solving and selecting tools. Tools that have patterns, color, shape, and size (e.g., visual cluster cards, Cuisenaire rods, etc.) develop visualization and therefore enhance working memory.
• Estimating: along with number concept, numbersense, the key skill implicated in dyscalculia is estimation; using appropriate concrete tools (non-counting materials) help develop estimation,
• Deductive and inductive reasoning: The development of formal/ abstract/logical reasoning begins when children use concrete tools effectively,
• Collecting/classifying/organizing: These are developmental concepts; children begin at concrete level and then are transitioned to abstract/formal levels (e.g., collecting data—look up information on Internet, in a book, in one’s notes, and read teacher comments on home work and tests, etc.),
• Metacognition: Learning about one’s learning—what works and does not work.

Mathematical Skills
The purpose of many physical tools is to acquire abstract/formal tools to prepare students for college and careers. This is achieved when they have these tools:

• linguistic: read the problem (e.g. focus on instructions), know the vocabulary, rewrite the problem in one’s own words, underline and understand the key words, recall and define the key terms, translate terms from English language to mathematical language and symbols, ask questions, etc.;
• conceptual: describe what the problem means, identify what mathematical concept is involved, what the unknowns are, what the knowns are, draw diagrams/figure/curve, make tables, create relationships between knowns and unknowns, write mathematical expressions, equations/inequalities, see patterns, solve a special case, recall an analogous situation or problem, consult a related solved problem, generalize, etc.;
• arithmetic: know decomposition/recomposition of numbers, master arithmetic facts, understand place value, describe the relationship between the quantities, estimate the outcome, create an empty number line, make a bar model, make a concrete model, draw a picture, create or use a graphic organizer, etc.;
• algebraic (identify the variables, write a formula, equation, or inequality, construct a table, chart, graph, or diagram, sketch the function, identify the parent function, create a prime factor tree or successive prime division chart, use a graphic organizer, etc.),
• geometric tools (draw a figure or diagram, classify data or information, look for spatial relationships, etc.);
• probabilistic and statistical (draw a Venn diagram, make a graph, create a tree-diagram, make lists, make a model, consult result tables, guess and check, etc.)

Mathematically proficient students gain entry to the problem situation and the solution process by using appropriate physical tools, manipulative materials—such as Cuisenaire and BaseTen blocks, for example, at the elementary school level, fraction strips, fraction bars, algebra tiles at higher grades or thinking tools (writing relationships between knowns and unknowns) to model a problem. For example, mathematically proficient high school students analyze graphs of functions and solutions and their behaviors with a graphing calculator and realize that technology can enable them to visualize the results of different assumptions on the conditions of the problem, explore their consequences, compare predictions with data, and the role of assumptions and constraints on the solution process.

Thinking tools also develop the ability to make sound decisions about when each of these tools might be helpful and gain the insight from their optimal use and also their limitations. This certainly requires that students gain sufficient competence with the tools to recognize the differential power and efficiency they offer.

It also requires that their learning include opportunities to decide for themselves which tool serves them best and why. In order for students not to become dependent on a particular tool and strategy and to develop flexibility of thought, it is important that the curriculum and teaching include the kinds of problems that involve the use of different tools. Students use tools efficiently and deepen their understanding by using different tools to solve the same problem. For example, from time to time, a particular tool is used until students develop a competency that would allow them to make sound decisions about which tool to use. The proficiency in the use of a tool is developed when we use it frequently, discuss its use from different perspectives, and apply it in several problems. In many situations, paper and pencil are inefficient and using them is not strategic. We must therefore develop the notion that mental computations are possible, reliable, and often more efficient. However, students should have skills to detect possible errors by strategically using estimation and other mathematical knowledge.

Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful (e.g., the flat piece in the BaseTen blocks kit represents 10×10 =102 at the third grade level, 1×1 = 12 at fifth grade level, and a×a = a2 at seventh grade level).

As we explore the connections between different types of concepts (e.g. numbers relationships) to use them flexibly, we need to explore the similar interactions between different types of tools to be able to use them flexibly and strategically.  For example, using BaseTen blocks for place value or for addition and subtraction operations encourages children to count, but combining BaseTen blocks and Cuisenaire rods precludes that possibility. Similarly, learning how to solve linear equations can follow the sequence for the strategic use of several tools:

• Invicta balance to derive and learn the properties of equality,
• Cuisenaire rods, BaseTen blocks, and Algebra tiles to learn arithmetic and algebraic manipulations and then to arrive at the procedures, and properties of operations,
• Paper and pencil to record these activities and procedures, then practice these operations formally,
• Graphing tool to see the behavior of the equations, functions, and solutions,
• Using computer algebra system (CAS) to take more complex equations and see their relationships and behaviors. To have proficiency in the strategic use of tools, the role of questions and classroom discussions is critical. The teacher can ask questions to help students to identify, select and use tools effectively.

To have proficiency in the strategic use of tools, the role of questions and classroom discussions is critical. The teacher can ask questions to help students to identify, select and use tools effectively.