Instrumentation in Mathematics

Research studies have evaluated the effectiveness of manipulatives as a tool in mathematics instruction. One line of research has studied the Concrete-Representational-Abstract (CRA) sequence of instruction, a form of explicit instruction that moves students from concrete manipulatives topictorial representations of those manipulatives and finally to abstract concepts. Butler, Miller, Crehan, Babbitt, & Pierce  (2003) compared the effectiveness of teaching fraction concepts to students with learning disabilities using a CRA approach versus a Representational-Abstract (RA) approach (starting with pictorial representations and moving to abstract concepts, with no concrete manipulatives). Fifty middle school students with mild to moderate disabilities were assigned to the CRA or the RA group. While both groups improved their understanding of fractions, the CRA group had overall higher scores than the RA group. A study by Witzel, Mercer, & Miller (2003) also supports the effectiveness of a CRA approach for developing the basics mathematics skills of students with learning disabilities. Students were taught to solve algebraic equations using either a CRA approach or a traditional approach.

The study involved 34 matched pairs of students in grades 6 and 7 who either had been diagnosed with learning disabilities or were categorized as at risk for learning problems. After a 4-week intervention, both groups showed improvement, but those taught with the CRA group significantly outperformed those who had received traditional instruction. In another CRA study (Maccini & Hughes, 2000) six adolescents with learning disabilities used algebra tiles to represent algebra word problems during the concrete phase of instruction. The students were able to transition successfully to pictorial and ultimately symbolic representations of the problems. Other studies have focused more specifically on concrete manipulatives. Marsh and Cook (1996) studied the use of Cuisenaire rods as a support for solving word problems with three third grade students with learning disabilities.

The students were not only more successful at selecting the correct operation when using the manipulatives but continued to improve after the manipulatives were withdrawn. Cass, Cates, Smith, and Jackson (2003) used case-study methods to investigate the effectiveness of teaching perimeter and area concepts using manipulatives (geoboards). Study participants were three fourth grade students with learning disabilities, all of whom improved in their ability to solve these geometric problems. Although research on manipulatives is in its early stages, available research supports their value. Reimer and Moyer (2005) investigated the performance of 19 third grade students during a 2-week unit on fractions that used virtual manipulatives. Over half of the students improved their conceptual understanding of fractions on a teacher-designed measure. In another study of 19 second grade students, Moyer, Niezgoda, and Stanley (2005) observed that virtual base-10 blocks enabled students to demonstrate more sophisticated strategies and explanations of place value.

Bolyard and Moyer-Packenham (2006) studied the use of virtual manipulatives with 99 sixth grade students learning addition and subtraction of integers.The students showed significant gains in achievement, and the researchers concluded that virtual manipulatives can support learning these concepts. Two studies have compared the use of virtual manipulatives to more traditional materials. Suh and Moyer (2007) compared the use of concrete and virtual manipulatives in third grade students studying algebraic thinking. Both types of manipulatives were associated with higher achievement and increased flexibility in representing algebraic concepts. Steen, Brooks, and Lyon (2006) compared the academic achievement of a group of first grade students who used virtual manipulatives for practice in geometry instruction (treatment group) to another group who did not (control group). A total of 31 students were randomly assigned to either the treatment or control group. Achievement was measured by the Grade One and Grade Two assessments provided by the classroom textbook’s publisher.

The treatment group improved significantly on both the Grade One and Grade Two tests, while the control group showed significant improvement only on the Grade One test. The treatment teacher also noted that her students showed increased motivation and increased time on task. In addition to their study of base-10 blocks, Moyer, Niezgoda, and Stanley (2005) also reported on a project in a kindergarten class of 18 students. The students were engaged in a 3-day lesson on patterns. On the first day, they used wooden pattern blocks, on the second day they used Web-based virtual pattern blocks, and on the third day they drew patterns freehand on construction paper. When using the virtual blocks, most students created more complex patterns and used more total blocks. The researchers further noted that the second graders in the base-10 block study showed more sophisticated strategies after using the virtual manipulatives. In addition, they note that the English Language Learners were able to demonstrate their understanding of place-value concepts even though they could not explain them verbally.

Students need help understanding the concepts that manipulatives represent Teachers play an important role in helping students understand the concepts that manipulatives represent. This was highlighted in a 1-year study of 10 middle school mathematics teachers and their use of manipulatives (Moyer, 2001). Teachers who were unable to represent mathematics concepts themselves were more likely to use manipulatives as a diversionary rather than instructional activity. The CRA studies described above also demonstrate the importance of structure and guidance in linking concrete materials to abstract concepts.

For example, students in Maccini and Hughes’ study (2000) were taught not only to use algebra tiles to represent word problems but also to use a structured strategy in solving them. In the Reimer and Moyer study (2005), students benefited from another important aspect of guidance: feedback. When students were interviewed regarding their impressions of the virtual manipulatives, an emergent theme was their appreciation for the immediate feedback possible with the computer-based manipulatives. Using visuals in teaching results in a greater degree of learning. The basic premise of the concept of visual literacy, defined as the ability to interpret images as well as to generate images for communicating ideas and concepts.

During a rehearsal of Debussy’s La Mer, Toscanini found himself unable to describe the effect he hoped to achieve from a particular passage. After a moment’s thought, he took a silk handkerchief from his pocket and tossed it high into the air. The orchestra, mesmerized, watched the slow, graceful descent of the silken square. Toscanini smiled with satisfaction as it finally settled on the floor. “There,” he said, “play it like that”.

The presence of visual elements in today’s teaching and learning is increasing as the integration of images and visual presentations with text in textbooks, instructional manuals, classroom presentations, and computer interfaces broadens (Benson, 1997; Branton, 1999; Dwyer as cited in Kleinman & Dwyer, 1999). Although the educational community is embracing visual enhancements in instruction, the connection of visual and verbal information is evident throughout history. According to the poet Simonides, “Words are the images of things” (as cited in Benson, p. 141); similarly, Aristotle stated that, “without image, thinking is impossible” (as cited in Benson, p. 141). Characters in alphabets began as pictures with meaning (West, 1997). These symbols portray a man-made language with no distinction between words and pictures, just as musical notes convey the language of music.

Only after the printing press was invented were illustrations and type separated, with illustrations often falling by the wayside. Recent history shows a reversal in this separation with greater reliance on visually oriented approaches to information presentation. The results are leading to a visualization movement in modern computing whereby complex computations are presented graphically, allowing for deeper insights as well as heightened abilities to communicate data and concepts. Visualization helps make sense of data that may have seemed previously unintelligible. Leonardo da Vinci, in recognizing the impossibility of recording volumes of data, translated words into drawings from different aspects. As history repeats itself, we may find that a great deal of information is better presented visually rather than verbally.

In education’s continuing mission of meeting the needs of learners, an apparent shift from the long-standing process of reading, writing, counting, and text memorization skills that may have been appropriate for the medieval clerk, are giving way to skills of analysis and innovation that are considered desirable in today’s modern cultures (West,1997). Proficiency with words and numbers is insufficient and must be supplemented with additional basic skills as new and emerging technologies permeate activities of daily living. Viewing change with fear and skepticism often accompanies shifts such as these that can revolutionize society. Even Socrates portrayed the new technology of the written word as dangerous and destructive, artificial and rigid, and unresponsive and insensitive. As more visual elements are incorporated to achieve an optimal balance between verbal and visual cues in education, interdependence between the two modes of thought will be fostered.

Kellner (1998) proposes that multiple literacies are necessary to meet the challenges of today’s society, literacies that include print literacy, visual literacy, aural literacy, media literacy, computer literacy, cultural literacy, social literacy, and ecoliteracy. Research suggests that using visual treatments in lessons enhances learning with varying degrees of success. Chanlin (1998) reports how lessons with no graphics, still graphics, or animated graphics influence students with different prior knowledge levels as they attain procedural and descriptive knowledge. When prior knowledge is low, graphics, either still or animated, are better for learning descriptive facts than lessons with text only, yet learning procedural facts does not appear to differ with the use of text or graphics.

However, students with a high level of prior knowledge of the subject responded better with the animated form of graphics in learning descriptive facts, but responded better with still graphics when learning procedural knowledge. Chanlin’s (1998) study suggests that students with different prior knowledge levels respond differently to contrasting presentation forms for achieving learning tasks, and that the effectiveness of visual design in learning is related to the prior knowledge of the students. Animated graphics are not superior to still graphics and may even be distracting to learning if the motions are inconsistent with how students process the visual information. An additional study by Chanlin (1999) suggests that providing visual control of animated graphics enhances learning, particularly in males.

Kleinman and Dwyer (1999) examined the effects of specific visual skills in facilitating learning. Their findings indicate that the use of color graphics in instructional modules as opposed to black and white graphics promotes achievement, particularly when learning concepts.

An earlier study by Myatt and Carter (as cited in Heinich et al., 1999) suggests that most learners prefer color visuals to black and white visuals, but that no significant difference in the amount of learning occurs except when color is related to the content to be learned. In addition, the study indicates that young learners prefer simple visuals and older students prefer complex visuals, yet simpler visuals are usually more effective regardless of the age group. Additionally, students do not necessarily learn best from the kinds of pictures they prefer to view.

McKay (1999) considered the learners’ cognitive styles as well as their experience in the subject matter in a study comparing the use of text-only instructional materials with text-plus graphics instructional materials. Subjects categorized as novice learners with verbal cognitive styles performed best with text plus graphics, while the novice imagery participants did better with the text only material; these findings were contrary to the expected outcomes. However, learners as a whole showed more improvement in test scores when using the text-plus-graphic format. Furthermore, novice learners from both learning style categories showed a greater improvement in scores than the experienced learners. The differences in responses by the novice and experienced learners are similar to reports from other studies (Chanlin, 1998; Mayer et al.,1996).

Many forms of graphics for instruction and enhancing understanding exist. Visual organizers that incorporate illustrations and text to depict patterns of concepts and ideas serve as organizational frameworks to promote thinking and learning (Tarquin & Walker, 1997). Frameworks assist learners in visualizing how ideas may be related to prior knowledge, subordinate ideas, and information from other sources. Story maps that can be depicted as vertical or horizontal flow maps, Venn diagrams that prove useful in analyzing similarities and differences between two or more concepts, and frameworks for webbing that encourage thought regarding the whole and its parts are examples of visual organizers. KWL frameworks link prior knowledge with what the learner wants to know and with what the learner has learned; the framework can be expanded to address what the learner still wants to learn, serving as a catalyst for further research. Feature analysis frameworks use a grid design to represent the relationships of concepts within a category.

The use of visuals in education, although consistently shown to aid in learning, must be carefully planned. The use of visuals that steer the learner to the exciting or entertaining aspects of presentation rather than encouraging thoughtful analysis of the underlying meaning may interfere with the intent of the lesson (Sherry, 1996). In addition, Dwyer (as cited in Williams & Dwyer, 1999) suggests that visuals must be properly used in the educational setting since visualization alone does not function to maximize student achievement. The study by Williams and Dwyer of the effect of metaphoric strategies in the achievement of learning objectives indicates that the use of verbal and visual metaphors to complement visualized instruction is not always an effective instructional strategy.

Teaching Aids for Mathematics

Many researchers and teachers advocate the use of manipulatives, hands-on objects like base-10 blocks, to support students’ mathematics learning. However, the evidence supporting this practice is mixed. Results from our prior research suggested a novel explanation for how manipulatives help children learn. Teachers need to use teaching aids to teach math effectively. The most effective teaching aids are manipulatives such as cubes, counters, pattern blocks, geoboards, place values blocks and plastic money. Students enjoy using manipulatives and, with proper instruction, can use the manipulatives to understand and master math concepts.

The “physically distributed learning” (PDL), is that action with manipulatives supports learning when it provides a way for children to simultaneously and iteratively adapt and interpret their environment. Research in the area of arithmetic supports PDL, but it has not been examined in geometry. This paper examines the possibility that physical action may have different effects in arithmetic and geometry tasks in the context of two studies with kindergarten, first- and second-grade students. In the arithmetic study, students solved addition word problems using manipulatives and drawings. In the geometry study, students completed a shape identification task with manipulatives and drawings.

On addition tasks, children were more successful with manipulatives than with pictures, replicating previous arithmetic studies. On geometry tasks, children also benefited from manipulation, but in different ways. Using manipulatives and pictures led to similar overall performance. However, all children were more likely to rotate their paper or physical shape when this action could help them identify the shape. In addition, when asked to make non-triangles into triangles, children using manipulatives (pipe cleaners) reshaped non-triangles, while those who worked with pictures drew a prototypical triangle (roughly equilateral) on top of these shapes. These results suggest that while the spatial nature of geometry tasks may change the effects of manipulatives on problem solving, manipulation still helps children expand their investigations in the physical environment and thereby advance their thinking.

Manipulatives. Teachers frequently use fraction pies to teach fraction operations and meanings.

On one hand, working with manipulatives improves performance on mathematical tasks in some cases (Chao et al., 2000). For example, adults calculate the value of a group of coins more quickly and accurately when they sort them (Kirsh, 1995). Manipulating objects also helps students learn many mathematical concepts (Behr, Wachsmuth, & Post, 1988; Fuson & Briars, 1990). Manipulatives have been shown to support learning in both arithmetic and geometry contexts (Glenberg, Jaworski, Rischal, & Levin, 2006; Jaworski, 2003; Olkun, 2003; Reimer & Moyer, 2005; Sarama & Clements, 2004). For example, in arithmetic, base-10 blocks helped second-graders learn multi-digit addition (Fuson & Briars, 1990). In geometry, a curriculum involving manipulating dynamic computer-based graphs improved fourth-graders’ concepts of two-dimensional space (Sarama, Clements, Swaminathan, McMillen, & Gonzalez Gomez, 2003).

In addition, early elementary students who use manipulatives for extended periods of time perform better on achievement tests (including both arithmetic and geometry problems) than those who do not (Sowell, 1989). At the same time, other research suggests that concrete objects like manipulatives do not improve problem solving and learning. First, meta-analyses and reviews of studies comparing student achievement in mathematics classrooms that uses manipulatives in instruction to those that do not show no overall advantage for manipulative use (Fennema, 1972; Sowell, 1989).

Second, focused studies with individual students illustrate some of the pitfalls of working with concrete objects. In some cases, the more concrete an object is, the less useful it is for solving problems (Bassok, 1989; Simons & Keil, 1995; Sloutsky, Kaminski, & Heckler, 2005). For example, in a series of studies, DeLoache, Uttal and colleagues showed that playing with a scale model of a room (thereby increasing the salience of the model’s concrete properties) can impair children’s ability to use that model as a referent to find an object hidden in an identical full-size room (e.g., DeLoache, 2000; Uttal, Scudder, & DeLoache, 1997).

Learning with Manipulatives
One idea is that exposure to multiple representations leads to better understanding of underlying mathematical principles (Moreno & Mayer, 1997). This view implies that the best instructional strategy is to use multiple manipulatives to teach mathematics concepts. Another hypothesis is that manipulatives support a concrete to abstract shift in conceptual understanding. Here the best manipulatives would be those that provide an analogy for or represent abstract concepts (Hall, 1998). In other words, useful manipulatives have structures that mirror the semiotic systems they are meant to represent, such that each action on a manipulative corresponds to a semiotic action, one-to-one. Another view is that external resources primarily help problem solvers keep track of problem elements without wasting internal memory resources (Cary & Carlson, 1999).

Physically Distributed Learning
In a series of studies, we investigated children’s problem solving and learning in arithmetic tasks (fractions and division) (Martin, Lukong, & Campbell, April, 2006; Martin & Schwartz, 2005; Schwartz & Martin, 2006). Our results suggested a novel explanation for how manipulatives help children learn. We suggest that action with manipulatives supports learning when it provides a way for children to simultaneously and iteratively adapt and interpret their environment. We call this process “physically distributed learning” (PDL). When children first attempt problems with manipulatives, they start with some basic ideas and actions that can help them. For example, for the problem “Sarah has three apples and Maria has four apples. How many apples do they have together?” a child might think, “The problem asked about how many they both have, I need to figure out how many altogether.” She might also try out some different actions like moving or grouping pieces. These actions and ideas can develop each other (or coevolve) (Martin, 2004).

As children work on these problems, they move the materials in different ways. For example, they might try making groups of different numbers of pieces for the problem above. As they think about the problem, they may decide that certain actions and configurations are more helpful than others. For this problem, they might be thinking about combining Sarah’s three apples and Maria’s four apples. Consequently, they might first make a group of three or a group of four pieces. Then they may realize that they can combine a group of three pieces and a group of four pieces to model the total number of apples the girls have together.

As they solve multiple problems using manipulatives, children’s actions and ideas could coevolve so that their concepts about the broader mathematics topic the problems address develop into more coherent structures (Martin et al., April, 2006). For example, they may come to understand addition and subtraction as related operations that involve a part-part-whole model whose parts can be combined and separated to compose and decompose the whole. This type of structure can support the development of mental problem solving strategies. In turn, practicing these mental strategies can lead to memorizing addition and subtraction facts (Carpenter, Fennema, Franke, Levi, & Empson, 1999).

The Role Of Manipulatives
One reason is geometry tasks often require operations on different objects and have different problem solving outcomes than arithmetic tasks. In geometry, the objects operated on are often visual, spatial, or concrete, whereas in arithmetic, they are primarily numbers. Many answers for geometry tasks involve outcomes like identifications or measurements rather than the numerical answers arithmetic tasks frequently involve. Manipulatives could be more helpful in geometry than arithmetic because they are similar to the objects operated on and resemble the outcomes of problem solving. On the other hand, this similarity may interfere with problem solvers’ formation of abstractions and therefore impede their development of more advanced concepts in the area (Goldstone & Sakamoto, 2003; Sloutsky et al., 2005). Another reason is that children work with spatial representations in many elementary geometry contexts (e.g., shapes or computer-based coordinate grids).

In this case, using manipulatives would not require the same translation between a verbal or numeric representation and a spatial representation as in the case of an arithmetic problem. This difference could make hands-on materials more useful for geometry than arithmetic, as there is less cognitive load involved in working with single rather than multiple representations (Ainsworth, 2006). Conversely, translating between representations can help children learn, particularly in the long term (Ainsworth & VanLabeke, 2004; Moreno & Mayer, 1997). Therefore, children may benefit more from working with manipulatives on arithmetic tasks. In terms of the PDL theory, a major difference is that children’s early ideas and actions may be more similar in geometry than in arithmetic. In geometry, physical shapes and children’s concepts of shapes may be nearly identical, particularly in early stages of learning. For example, children initially identify shapes by their appearance (e.g., it’s a circle because it looks like a frisbee) (Clements, 2003; van Hiele, 1986).

In contrast, on addition problems, children’s ideas could be about parts and wholes while their actions could involve moving tile pieces around. Because ideas and actions are more similar in geometry, coevolution and distributing work to the environment (the processes of PDL) may operate more easily. Children may develop their ideas and actions faster if these are more similar. On the other hand, this very similarity may impede development. For example, in fraction problem solving, children transferred better when they learned about adding fractions with manipulatives that were less like fractions (tile pieces) than with manipulatives that were more like fractions (pie pieces) (Martin & Schwartz, 2005). Children may not have had to construct fraction structures mentally with the pie pieces because they could leave that distributed to the environment.

Some Manipulatives
A) Place-value mats can be used to show each digit’s place value when using base-10 blocks. B) Counters can be used to teach one-to-one correspondence, ordinal numbers, and basic addition and subtraction. Two-sided counters can be used to model one-to-one correspondence, addition and subtraction, or skip-count. C) Money, coins or bills can be used to count, skip-count, or model regrouping when adding or subtracting. D) Fraction strips can be used to show equivalent fractions, to add fractions, to subtract fractions, or to find common denominators. E) Pattern blocks can be used to create, find, or extend patterns. F) Attribute blocks can be used to sort and classify according to shape, color, size, or other attributes. G) Geoboards can be used to identify simple geometric shapes and describe their properties and to develop spatial sense. H) Geometric-solid models can be used to teach nets or spatial reasoning. I) Standard and non-standard rulers and measuring cups can be used to represent length or volume. J) Spinners can be used to find the experimental probability of landing on a designated area. K) Number cubes or dice can be used to find the experimental probability of rolling a certain number or a combination of numbers.

MANIPULATIVES

(Counters) (Money, coins or bills (Fraction Strips) (Pattern Blocks)

(Attribute Blocks) (Geoboard

(Geometric-solid models) (Rulers and measuring cup)

(Spinners) (Dice)

How Can You Make It Happen?
The amount of time students need to progress from concrete to abstract understanding varies by concept and by student. It may occur within a single lesson, but more often it occurs over days, weeks, months, or even years, as the concept is revisited. A concept (for example, multiplication) may be introduced and practiced at the concrete and pictorial levels in one grade, then reviewed at these levels and practiced at the abstract level in the next grade. Students in primary grades need to see many examples of the same concept as well as a variety of non-examples of the concept. For example, these are equivalent fractions: 1/2 = 1/4

These are not equivalent fractions: 1/2, 5/8

When students can use different manipulatives to represent the same concept, their ability to understand subsequent math concepts is enhanced. When a new manipulative is going to be used in a lesson, you should allow students time to examine it and explore its use before giving them concrete directions. You can do this by allowing time at the beginning of the lesson for this unstructured use, or by making the manipulative available to students in an area of the classroom they have access to during the day, such as in a learning center. Then, before you give students the task they are to work on using the manipulative, demonstrate how to use it. For example, when using base-10 blocks, introduce the ones-blocks, model how to count them, and show students where they are placed on the place value chart.

Then, show students the ten-sticks. Have them count each block in the ten-stick to discover that there are ten blocks, and then show students where they are placed on the place-value chart. Have student pairs regroup ten-sticks into ones-blocks and model numbers such as 11, 16, or 23. In upper-level classes, lessons are more productive when teachers structure activities and allow some systematic exploration of the manipulatives, when teachers do not immediately correct deviations from the prescribed “procedure,” and when they guide rather than lead exploration of mathematical ideas (Stein 2001). Whole-Number Operations

You can model conventional ways of representing mathematical situations, but it is important for students to use representations that are meaningful to them. For example, to find the sum of 43 + 29, some students prefer to add the numbers in the tens place, 40 + 20, add the numbers in the ones place, 9 + 3, and combine them, 60 + 12 = 72, to find the sum. Introduce students to manipulatives, and have students use them to solve a problem using conventional algorithms or their own, and record their work. Students in grades 3-5 should use manipulatives as tools for solving problems, such as blocks to represent place value, counters to represent multiplication, and graphs and charts to model problems. Place Value

Have students use base-10 blocks to represent numbers in as many ways as possible. For example, the number 38 can be made with 3 ten-sticks and 8 one-blocks, 2 ten-sticks and 18 one-blocks, 38 one-blocks, and so on. Students can use the place value charts to write numbers using expanded notation: 38 is 3 tens and 8 ones

30 + 8.
Multiplication
Have students use counters to make arrays representing multiplication facts. Make an array of counters for 3 x 8, and have students record the multiplication sentence and product. Have them also make an array of 8 x 3 to reinforce their understanding of the commutative property when solving multiplication problems.

Arrays can also help students understand the distributive property. Students can find 12 x 6 by multiplying 10 by 6, and then 2 by 6, and then adding the products together to find the answer.

Money
Have students use coins and dollars to understand the real-world use of regrouping, skip-counting, and other mathematical concepts. Have students find all the ways to show $1.00, how to regroup a dollar in order to give 4 friends 25 cents, how to make change, and how to skip-count to find the value of a group of nickels or dimes. Geometry/Spatial Reasoning

You should encourage students to explore shapes and their attributes and provide them with appropriate manipulatives and a structured environment to make this happen. Students can explore shapes by decomposing them, creating new shapes, and comparing and sorting them. Students can create shapes on geoboards, dot paper, or graph paper, and represent them in drawings or blocks. Sort by Attributes

For primary students, use different shapes in different colors and sort them by attribute. Have students determine how to classify the pieces, by color, shape, size, and so on. Pattern Blocks/Tangrams

Have students explore how shapes can be combined to make other shapes. For example, how many triangles are in a rectangle? Fractions and Decimals
Students need to be able to build on a solid conceptual understanding of place value and whole numbers when learning aboutfractions and decimals. Without this background, students may not understand how 1.0 is greater than .900, or that 1/3 is more than 5/6. In grades 3-5, students learn about, compare, and find equivalent forms of fractions. Students can use manipulatives to understand how two fractions can have the same value, as well as how they relate to whole numbers. Using a variety of manipulatives can help students compare fractions to decimal values. Fraction Strips

Have students compare otherfraction stripsto the benchmarks of 1 whole, ½ and ¼, to discover which is a greater or lesser value. Decimal Models Have students usedecimal place-value chartsand base 10 blocks to model decimal numbers. They can also use a hundreds chart and shade the blocks to represent 10 of 100, 50 of 100, and so on.

Why Is It Important?
According to learning theory based on psychologist Jean Piaget’s research, children are active learners who master concepts by progressing through three levels of knowledge–concrete, pictorial, and abstract. The use of manipulatives enables students to explore concepts at the first, or concrete, level of understanding. When students manipulate objects, they are taking the necessary first steps toward building understanding and internalizing math processes and procedures. For example, when learning to add fractions, students can use fraction strips to represent each addend. They then add the fractional parts to find the sum. After practicing with these, they can progress to finding sums for problems on paper, represented by pictures with corresponding fraction numerals (pictorial level). Over time, they will devise strategies and apply algorithms so they can find sums when given only the addition expression (abstract level).

For a learner to understand and handle a concept at the abstract or symbolic level successfully, he or she must first understand the concept at the other two levels, in the order given-concrete then pictorial. Using concrete manipulatives is the first step to using mental images and models. When students demonstrate understanding with the concept at this physical, or concrete, levels then they are ready to move to the next level, where they can apply their knowledge using representations of the objects in place of the objects themselves. Young children often solve addition and subtraction problems by counting concrete objects, beginning with their fingers. They go on to use concrete objects such as base-ten blocks and counters to develop more sophisticated problem-solving strategies based on what they know about counting (Siegler 1996). Concrete models can help students represent numbers and develop number sense; they can also help bring meaning to written symbols and can be useful in building place-value concepts.

You should try to “get inside students’ heads” as they work with concrete materials by asking questions that elicit their thinking and reasoning. In this way, you can get a better sense of what students know and don’t know, along with identifying misconceptions, thereby developing a basis for intervention strategies. You should also choose interesting tasks using manipulatives (such as an abacus) that engage students in mathematical thinking and reasoning, which builds their understanding of numbers as well as relationships among numbers (NCTM 2000). In fact, research shows that using manipulatives can contribute to the development of well-grounded, interconnected understandings of mathematical ideas. Students can more easily remember what they did and explain what they were thinking when they used manipulatives to solve a problem (Stein 2001). The following are some examples how to use manipulatives:

* Have students show three blue counters, and then match each blue counter with a red counter. * Create a sentence such as, “__ students were playing kickball and __ students were on the swings. There were __ students all together.” Have students take turns changing the numbers, and model each sentence using manipulatives. * Show five red counters and four blue counters. Then ask, “Are there more red or blue counters?” * Have students use 2-sided counters to show 3 (white) + 6 (black). Then ask, “How many more black counters are there? * Have students sort money into pennies, nickels, dimes, and quarters. How many pennies are heads up? How many nickels? * Have students show these math problems using manipulatives. Jack collected 66 baseball cards this month. If he collects the same amount each month, how many will he have in 6 months? 66 x 6 = 396 * Sue has 10 markers. She has 2 friends.

Show how many markers each friend will get. * Sue has 10 markers. She wants each friend to get 2 markers. Show how many friends can get markers. * There are 245 students in the second grade. If they are divided into two groups to watch a play, how many students will be in each group? 245 divided by 2 = 122 R1. There will be 122 students in one group and 123 students in the other group. * Four t-shirts are on sale for $6.00. How much does each one cost? 600 divided by 4 = 150. Each shirt is $1.50. * Find fractions that are equivalent to 1. Equivalent fractions: 4/4, 2/2, 8/8 * Add 1/4 + 3/6. Find the common denominator and then add. The common denominator for 4 and 6 is 12. Change 1/4 to equivalent fraction 3/12. Change 3/6 to equivalent fraction 6/12. The answer is 9/12. Change to equivalent fraction 3/4. Alternate: Change 3/6 to equivalent fraction 1/2. The common denominator for 4 and 2 is 4. Change 1/2 to 2/4, and add to find 3/4.

Graphing Calculator
A graphing calculator (also graphics / graphic calculator) typically refers to a class of handheld calculators that are capable of plotting graphs, solving simultaneous equations, and performing numerous other tasks with variables. Most popular graphing calculators are also programmable, allowing the user to create customized programs, typically for scientific/engineering and education applications. Due to their large displays intended for graphing, they can also accommodate several lines of text and calculations at a time. Some of the more recent graphing calculators are capable of color output, and also feature animated and interactive drawing of math plots (2D and 3D), other figures such as animated Geometry theorems, preparation of documents which can include these plots and drawings, etc. This is giving the new graphing calculators a presence even in high school courses where they were formerly disallowed.

Some calculator manufacturers also offer computer software for emulating and working with handheld graphing calculators. Many graphing calculators can be attached to devices like electronic thermometers, pH gauges, weather instruments, decibel and light meters, accelerometers, and other sensors and therefore function as data loggers, as well as WiFi or other communication modules for monitoring, polling and interaction with the teacher. Student laboratory exercises with data from such devices enhances learning of math, especially statistics and mechanics. Since graphing calculators are usually readily user-programmable, such calculators are also widely used for gaming purposes, with a sizable body of user-created game software on most popular platforms.

Even though handheld gaming devices fall in a similar price range, graphing calculators offer superior math programming capability for math based games. However, for developers and advanced users like researchers, analysts and gamers, 3rd party software development involving firmware mods, whether for powerful gaming or exploiting capabilities beyond the published data sheet and programming language, is a contentious issue with manufacturers and education authorities as it might incite unfair calculator use during standardized high school and college tests where these devices are targeted. Nowadays graduate (Masters) students and researchers have turned to advanced Computer Aided Math software for learning as well as experimenting. Types

Graphing calculators can be sub-divided into two categories (both need the graphical display): 1. Numerical (or just graphing) Calculators – non-CAS and producing numerical results, at most represented as a fraction. In some countries, graphing calculators are not permitted in high school science tests or certain basic math tests. See examples below. 2. CAS (or symbolic) Calculators – most advanced calculators capable of producing a symbolic result (in expression or equation form), usually utilizing a Computer Algebra System (CAS). Symbolic/CAS calculators are posing a challenge to high school and undergraduate educators.

They can make math easier to learn for high schoolers, provided school curriculum evolves towards this advantage. Testing based on tedious hand calculation is also being forced to evolve towards more creative testing. Such tests are often more challenging and expensive to design and can’t be recycled as much, but encourage a genuine deeper appreciation of the art of mathematics and critiqueing a fallacy. CAS calculators are therefore usually permitted only in select advanced math or calculus tests, thereby being more of a classroom learning tool for many users who then switch to a permitted and speedy numerical non-CAS graphing or scientific calculator for tests and exams. ( 1 ) ( 2 )

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