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.
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.
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.