Using Manipulatives - TeacherVision Using Manipulatives Using Manipulatives Manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics. They can be used to introduce, practice, or remediate a concept. Teaching Strategies: Using Manipulatives What Is It? Manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics . They can be used to introduce, practice, or remediate a concept. A manipulative may be as simple as grains of rice or as sophisticated as a model of our solar system. They may be store-bought, brought from home, or teacher- or student-made. They can be used in all areas of math instruction-teaching number and operations, algebra, geometry, measurement, and so on-and across all grade levels. The following are a few manipulatives that might be used at the elementary level. Number and Operations 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.

Place-value mats can be used to show each digit's place value when using base-10 blocks. Base-10 blocks can be used to model the variety of ways a number can be represented, and to model regrouping when adding, subtracting, multiplying, and dividing. Money, in coins or bills , can be used to count, skip-count, or model regrouping when adding or subtracting. Fraction strips can be used to show equivalent fractions, to add fractions, to subtract fractions, or to find common denominators. Algebra Pattern blocks can be used to create, find, or extend patterns. Attribute blocks can be used to sort and classify according to shape, color, size, or other attributes. Pan balances or scales can be used with objects or weights to show equivalent values. Geometry Geoboards can be used to identify simple geometric shapes and describe their properties and to develop spatial sense. Geometric-solid models can be used to teach nets or spatial reasoning. Measurement Standard and non-standard rulers and measuring cups can be used to represent length or volume. Tiles can be used to find the area or the perimeter of an object. Data Analysis and Probability Spinners can be used to find the experimental probability of landing on a designated area.

Number cubes or dice can be used to find the experimental probability of rolling a certain number or a combination of numbers. 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 a 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, level 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). Prev Patterns in Mathematics -- Number Patterns Mathematics is especially useful when it helps you predict, and number patterns are all about prediction. What will the 50th number of this pattern be? How many cupcakes would we need if we gave a party for the town instead of just our class? Working with number patterns leads directly to the concept of functions in mathematics: a formal description of the relationships among different quantities. Recognizing number patterns is also an important problem-solving skill. If you see a pattern when you look systematically at specific examples, you can use that pattern to generalize what you see into a broader solution to a problem. Here are two activities that show the variety and fun of number patterns. (Don't forget to read the activity background for more ideas on classroom use and connections to standards.) In How Many Valentines? you figure out the number of valentines sent by an entire class. In Mystery Operations you figure out what an operation does by seeing examples.

The youngest children begin simply by counting. They count by 1s, then by 2s, 5s, and 10s. These patterns give students a natural strategy to understand addition and multiplication. When considering a number pattern such as 2, 4, 6..., a young student will ask herself, By what number can I count (add) to get to the next number in the pattern and the next and the next? As the student gets older, her knowledge of patterns advances from sums to products. When asked for the 50th number in the pattern, she will know to multiply 2 times 50. High school students can start to understand functions, such as f(x) = 2x + 2, where x is the numerical sequence 0, 1, 2, 3,?. They begin with simple in-out machines and gradually adapt their understanding to the abstractions of algebra. NCTM Standard 2 (1998) sets the purpose of patterns, functions, and algebra in mathematics education at all grade levels.

Mathematics instructional programs should include attention to patterns, functions, symbols, and models so that all students understand various types of patterns and functional relationships; use symbolic forms to represent and analyze mathematical situations and structures; use mathematical models and analyze change in both real and abstract contexts. Standards for Mathematical Practice | Common Core State Standards Initiative Common Core State Standards Initiative Print this page The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

Standards in this domain: CCSS.Math.Practice.MP8 CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem.

Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. CCSS.Math.Practice.MP2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments.

They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. CCSS.Math.Practice.MP4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. CCSS.Math.Practice.MP5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.

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, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. CCSS.Math.Practice.MP6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.

They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. CCSS.Math.Practice.MP7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective.

They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3.

Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily.

Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. Kindergarten-Grade 12 Conctere-Representational-Abstract Sequence of Instruction Videos Purpose The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning.

When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level. What is it? Each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, beans and bean sticks, pattern blocks). Students are provided many opportunities to practice and demonstrate mastery using concrete materials The math concept/skill is next modeled at the representational (semi-concrete) level which involves drawing pictures that represent the concrete objects previously used (e.g. tallies, dots, circles, stamps that imprint pictures for counting) Students are provided many opportunities to practice and demonstrate mastery by drawing solutions The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols) Students are provided many opportunities to practice and demonstrate mastery at the abstract level before moving to a new math concept/skill.

As a teacher moves through a concrete-to-representational-to-abstract sequence of instruction, the abstract numbers and/or symbols should be used in conjunction with the concrete materials and representational drawings (promotes association of abstract symbols with concrete & representational understanding) [ back to top ] What are the critical elements of this strategy? Use appropriate concrete objects to teach particular math concept/skill (see Concrete Level of Understanding/Understanding Manipulatives-Examples of manipulatives by math concept area). Teach concrete understanding first. Use appropriate drawing techniques or appropriate picture representations of concrete objects (see Representational Level of Understanding/Examples of drawing solutions by math concept area). Teach representational understanding second. Use appropriate strategies for assisting students to move to the abstract level of understanding for a particular math concept/skill (see Abstract Level of Understanding/Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers). When teaching at each level of understanding, use explicit teaching methods (see the instruction strategy Explicit Teacher Modeling). How do I implement the strategy? When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding). Provide students many practice opportunities using concrete objects.

When students demonstrate mastery of skill by using concrete objects, describe & model how to perform the skill by drawing or with pictures that represent concrete objects (representational level of understanding). Provide many practice opportunities where students draw their solutions or use pictures to problem-solve. When students demonstrate mastery drawing solutions, describe and model how to perform the skill using only numbers and math symbols (abstract level of understanding). Provide many opportunities for students to practice performing the skill using only numbers and symbols. After students master performing the skill at the abstract level of understanding, ensure students maintain their skill level by providing periodic practice opportunities for the math skills. How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems? Helps passive learner to make meaningful connections Teaches conceptual understanding by connecting concrete understanding to abstract math process By linking learning experiences from concrete-to-representational-to-abstract levels of understanding, the teacher provides a graduated framework for students to make meaningful connections.

Blends conceptual and procedural understanding in structured way [ back to top ] Additional Information Research Support For The Instructional Features Of This Instructional Strategy: Allsopp (1999); Baroody (1987); Butler, Miller, Crehan, Babbit, & Pierce (2003); Harris, Miller, & Mercer (1993); Kennedy and Tips (1998); Mercer, Jordan, & Miller (1996); Mercer and Mercer (2005); Miller, Butler, & Lee (1998); Miller and Mercer, 1995; Miller, Mercer, & Dillon (1992); Peterson, Mercer, & O'Shea. (1988); Van De Walle (2005); Witzel, Mercer, & Miller (2003). Create seperate pages Examples of manipulatives (concrete objects) by math concept level Suggestions for using manipulatives (concrete objects) What is it? The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills.

Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed. Understanding manipulatives (concrete objects) To use math manipulatives effectively, it is important that you understand several basic characteristics of different types of math manipulatives and how these specific characteristics impact students who have learning problems. As you read about the different types of manipulatives, click on the numbers beside each description to view pictures of these different types of manipulatives. General types of math manipulatives: Discrete - those materials that can be counted (e.g. cookies, children, counting blocks, toy cars, etc.). Continuous - materials that are not used for counting but are used for measurement (e.g. ruler, measuring cup, weight scale, trundle wheel). See example - 1 Suggestions for using Discrete & Continuous materials with students who have learning problems: Students who have learning problems need to have abundant experiences using discrete materials before they will benefit from the use of continuous materials. This is because discrete materials have defining characteristics that students can easily discriminate through sight and touch.

As students master an understanding of specific readiness concepts for specific measurement concepts/skills through the use of discrete materials (e.g. counting skills), then continuous materials can be used. Types of manipulatives used to teach the Base-10 System/place-value (Smith, 1997): Proportional - show relationships by size (e.g. ten counting blocks grouped together is ten times the size of one counting block; a beanstick with ten beans glued to a popsicle stick is ten times bigger than one bean). - Non-linked proportional - single units are independent of each other, but can be "bundled together (e.g. popsicle sticks can be "bundled together in groups of 'tens' with rubber bands; individual unifix cubes can be attached in rows of ten unifix cubes each). See examples - 1 | 2 | 3 | - Linked proportional - comes in single units as well as "already bundled" tens units, hundreds units, & thousands units (e.g. base ten cubes/blocks; beans & beansticks).

Se examples - 1 | 2 | 3 | 4 | 5 Non-proportional - use units where size is not indicative of value while other characteristics indicate value (e.g. money, where one dime is worth ten times the value of one penny; poker chips where color indicates value of chip; an abacus where location of the row indicates value). A specified number of units representing one value are exchanged for one unit of greater value (e.g. ten pennies for one dime; ten white poker chips for one blue poker chip, ten beads in the first row of an abacus for one bead in the second row). See example - 1 Suggestions for using proportional and non-proportional manipulatives with students who have learning problems: Students who have learning problems are more likely to learn place value when using proportional manipulatives because differences between ones units, tens units, & hundreds units are easy to see and feel. Due to the very nature of non-proportional manipulatives, students who have learning problems have more difficulty seeing and feeling the differences in unit values. Examples of manipulatives (concrete objects) Suggested manipulatives are listed according to math concept/skill area.

Descriptions of manipulatives are provided as appropriate. A brief description of how each set of manipulatives may be used to teach the math concept/skill is provided at the bottom of the list for each math concept area. Picture examples of some of the manipulatives for each math concept area can be accessed by clicking on the numbers found underneath the title of each math concept area. This is not meant to be an exhaustive list, but this list does include a variety of common manipulatives. The list includes examples of "teacher-made" manipulatives as well "commercially-made" ones. Counting/Basic Addition & Subtraction Popsicle sticks & rubber bands for bundling Unifix cubes (individual cubes can be combined to represent "tens") Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values) Description of use: Students are first taught to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g.

ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side. Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds." Multiplication/Division Pictures See examples - 1 | 2 | 3 | Containers & counting objects (paper dessert plates & beans, paper or plastic cups and candy pieces, playing cards & chips, cutout tag board circles & golf tees, etc.). Containers represent the "groups" and counting objects represent the number of objects in each group. (e.g. 2 x 4 = 8: two containers with four counting objects on each container) Counting objects arranged in arrays (arranged in rows and columns). Color-code the "outside" vertical column and horizontal row helps emphasize the multipliers . See examples - 1 Counting objects, one set light colored and one set dark colored (e.g. light & dark colored beans; yellow & blue counting chips; circles cut out of tag board with one side colored, etc.). Description of use: Light colored objects represent positive integers and dark colored objects represent negative integers.

When adding positive and negative integers, the student matches pairs of dark and light colored objects. The color and number of objects remaining represent the solution. Fractions Pictures See examples - 1 | 2 | 3 | Fraction pieces (circles, half-circles, quarter-circles, etc.) Fraction strips (strips of tag board one foot in length and one inch wide, divided into wholes, ½'s, 1/3's, ¼'s, etc. Fraction blocks or stacks. Blocks/cubes that represent fractional parts by proportion (e.g. a "1/2" block is twice the height as a "1/4" block). Description of use: Teacher models how to compare fractional parts using one type of manipulative. Students then compare fractional parts. As students gain understanding of fractional parts and their relationships with a variety of manipulatives, teacher models and then students begin to add, subtract, multiply, and divide using fraction pieces. Geometry Pictures See examples - 1 Geoboards (square platforms that have raised notches or rods that are formed in a array). Rubber bands or string can be used to form various shapes around the raised notches or rods.

Description of use: Concepts such as area and perimeter can be demonstrated by counting the number of notch or rod "units" inside the shape or around the perimeter of the shape. Beginning Algebra Pictures See examples - 1 | 2 | Containers (representing the variable of "unknown") and counting objects (representing integers) -e.g. paper dessert plates & beans, small clear plastic beverage cups 7 counting chips, playing cards & candy pieces, etc. Description of use: The algebraic expression, "4x = 8," can be represented with four plates ("4x"). Eight beans can be distributed evenly among the four plates. The number of beans on one plate represent the solution ("x" = 2). Suggestions for using manipulatives (Burns, 1996) Talk with your students about how manipulatives help to learn math. Set ground rules for using manipulatives. Develop a system for storing manipulatives. Allow time for your students to explore manipulatives before beginning instruction. Encourage students to learn names of the manipulatives they use. Provide students time to describe the manipulatives they use orally or in writing. Model this as appropriate.

Introduce manipulatives to parents Examples of drawing solutions by math concept level What is it? At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. It is appropriate for students to begin drawing solutions to problems as soon as they demonstrate they have mastered a particular math concept/skill at the concrete level. While not all students need to draw solutions to problems before moving from a concrete level of understanding to an abstract level of understanding, students who have learning problems in particular typically need practice solving problems through drawing. When they learn to draw solutions, students are provided an intermediate step where they begin transferring their concrete understanding toward an abstract level of understanding. When students learn to draw solutions, they gain the ability to solve problems independently. Through multiple independent problem-solving practice opportunities, students gain confidence as they experience success. Multiple practice opportunities also assist students to begin to "internalize" the particular problem-solving process. Additionally, students' concrete understanding of the concept/skill is reinforced because of the similarity of their drawings to the manipulatives they used previously at the concrete level. Drawing is not a "crutch" for students that they will use forever.

It simply provides students an effective way to practice problem solving independently until they develop fluency at the abstract level. Examples of drawing solutions by math concept level The following drawing examples are categorized by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes"). In each category there are a variety of examples demonstrating how to use these drawings to solve different types of computation problems. Click on the numbers below to view these examples. Abstract What is it? Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers What is it? A student who problem-solves at the abstract level, does so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with numbers and math symbols only. Abstract understanding is often referred to as, "doing math in your head." Completing math problems where math problems are written and students solve these problems using paper and pencil is a common example of abstract level problem solving. Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers Students who are not successful solving problems at the abstract level may: - Not understand the concept behind the skill.

Suggestions: Re-teach the concept/skill at the concrete level using appropriate concrete objects (see Concrete Level of Understanding). Re-teach concept/skill at representational level and provide opportunities for student to practice concept/skill by drawing solutions (see Representational Level of Understanding). Provide opportunities for students to use language to explain their solutions and how they got them (see instructional strategy Structured Language Experiences). - Have difficulty with basic facts/memory problems Suggestions: Regularly provide student with a variety of practice activities focusing on basic facts. Facilitate independent practice by encouraging students to draw solutions when needed (see the student practice strategies Instructional Games, Self-correcting Materials, Structured Cooperative Learning Groups, and Structured Peer Tutoring). Conduct regular one-minute timings and chart student performance. Set goals with student and frequently review chart with student to emphasize progress. Focus on particular fact families that are most problematic first, then slowly incorporate a variety of facts as the student demonstrates competence (see evaluation strategy Continuous Monitoring & Charting of Student Performance). Teach student regular patterns that occur throughout addition, subtraction, multiplication, & division facts (e.g.

"doubles" in multiplication, 9's rule - add 10 & subtract one, etc.) Provide student a calculator or table when they are solving multiple-step problems. - Repeat procedural mistakes. Provide fewer #'s of problems per page. Provide fewer numbers of problems when assigning paper & pencil practice/homework. Provide ample space for student writing, cueing, & drawing. Provide problems that are already written on learning sheets rather than requiring students to copy problems from board or textbook. Provide structure: turn lined paper sideways to create straight columns; allow student to use dry-erase boards/lap chalkboards that allow mistakes to be wiped away cleanly; color cue symbols; for multi-step problems, draw color-cued lines that signal students where to write and what operation to use; provide boxes that represent where numerals should be placed; provide visual directional cues in a sample problem; provide a sample problem, completed step by step at top of learning sheet. Provide strategy cue cards that student can use to recall the correct procedure for solving problem. Provide a variety of practice activities that require modes of expression other than only writing *Key Idea Student learning & mastery greatly depends on the number of opportunities a student has to respond!!

The more opportunities for successful practice that you provide (i.e. practice that doesn't negatively impact student learning characteristics), the more likely it is that your student will develop mastery of that skill. The symbolic language of math Produced by Charles Wells Revised 2017-01-18 Introduction to this website website TOC website index blog THE SYMBOLIC LANGUAGE OF MATH The symbolic language of math is a distinct special-purpose language. Unlike mathematical English , it is not a variety of English. It has its own rules of grammar that are quite different from those of English. You can usually read expressions in the symbolic language in any math article written in any language. This chapter discusses aspects of the symbolic language that may cause difficulties to newcomers. It is not a systematic introduction to the symbolic language. You can find more information in the links in The languages of math. Warning: The terminology used here to talk about symbolic expressions is nonstandard. See Variations in terminology for more detail. The chapter More about the languages of math discusses topics that involve both the symbolic language and mathematical English. Symbolic expressions The symbolic language consists of symbolic expressions written in the way mathematicians traditionally write them. A symbolic expression consists of symbols arranged according to specific rules.

Every symbolic expression is one of two types: symbolic assertion and symbolic term. Every expression in the symbolic language is either a symbolic assertion or a symbolic term. Symbolic assertions A symbolic assertion is a complete statement that stands alone as a sentence. A symbolic assertion says something. Symbolic assertions play the same role in the symbolic language as assertions do in math English . A symbolic assertion may contain variables and it may be true for some values of the variables and false for others. Examples "$\pi\gt0$" is a symbolic assertion. It is true. "$\pi=3$" is a symbolic assertion. It is false, but it is nevertheless a symbolic assertion. "$x\gt0$" is true for $x=42$ and many other numbers and false for $x=-.233$ and many other numbers. The symbolic assertion "$x^2-x-2=0$" is true for the numbers $x=-1$ and $x=2$, but not for any other number. "$x^2-y^2=(x-y)(x+y)$" is a symbolic assertion with two variables. It is true for all numbers $x$ and $y$. Such a thing is commonly called an algebraic identity or identity relation.

The assertion "$x^2\geq0$" is true for all real numbers, but not for all complex numbers. So the data type of a variable matters in determining whether an assertion containing the variable is always true or not. The assertion "$x^2\lt0$" is false for all real numbers. The assertion "${{x}^{2}}-6x+4y\gt0$" is true for some values of $x$ and $y$ and false for others. For example, it is false for $x=1$ and $y = 0$ and true for $x= 1$ and $y = 2$. There is more about symbolic assertions like this one in the section on constraints Symbolic statements A symbolic statement is a symbolic assertion without variables. A symbolic statement is either true or false. A symbolic statement is regarded as a special case of a symbolic assertion. What makes it special is that it contains no variables. Examples $\pi\gt0$ and $3^2=9$ are true symbolic statements. $\pi\lt0$ and $2+3=6$ are false symbolic statements. Even though false, they are still regarded as symbolic statements.

Symbolic terms A symbolic term is a symbolic expression that refers to some mathematical object. A symbolic term names something. Terms play the same role in the symbolic language that descriptions do in math English. Examples The expression “${{3}^{2}}$” is a symbolic term. It is another name for the number $9$. "$x^2$" is a symbolic term containing a variable $x$. This means the term has variable meaning depending on which value is substituted for $x$. For example, if you set $x=-2$, you get $(-2)^2$, another name for $4$. ${{x}^{2}}-6x+4y$ (mentioned above) is symbolic term with two variables. If you substitute $2$ for $x$ and $3$ for $y$ then the expression denotes the integer $4$. Variations in terminology The names “symbolic assertion” and “symbolic term” are not standard usage in math. In mathematical logic: Symbolic assertions may be called formulas or predicates. True assertions may be called propositions or sentences.

All these words, as well as my use of “ term ”, can cause cognitive dissonance: Many people would refer to “${{\text{H}}_{\text{2}}}\text{O}$” as “the formula for water”, but it is not a formula in sense of logic because it does not make a statement. In everyday usage “proposition” may mean a statement to be debated, or a proposal for action, but in math logic the meaning is simply a statement . The name “ term ” comes from mathematical logic. The expression “${{x}^{3}}\left( {{y}^{2}}-1 \right)$” contains symbolic terms ${{x}^{3}}$ and ${{y}^{2}}-1$, which would in math English be called factors rather than "terms". This kind of conflict between different parts of math happens all the time. Neither side is right or wrong. Get used to it. Non-algebraic expressions Symbolic expressions don’t have to have algebraic form and they do not have to name numbers.

Examples Texts on group theory use the expression " ${{\text{S}}_{n}}$ " to denote the group of all permutations of an $n$-element set, with composition as operation. The group ${{\text{S}}_{3}}$ contains six elements. You can write them out and make a table of the group multiplication (described here ). All true statements about ${{\text{S}}_{3}}$ are implied by the symbol. Operations can be performed on groups, just as they can for numbers. One operation is the product of two groups . In particular one can form the product ${{\text{S}}_{n}}\times\mathbb{Z}/3$. ( $\mathbb{Z}/3$ is another group.) This is a symbolic term just like "$a\times b$", but in group theory rather than in elementary algebra. Each branch of mathematics is concerned with certain particular kinds of mathematical objects, and every one of them studies many different kinds of operations on the objects, expressed (usually) in symbolic notation.

Reading symbolic expressions Distinguish between assertions and terms A fundamental difficulty many people new to algebra have is that they don't pay attention to the difference betweeen assertions and terms. Examples An expression such as “$x\alpha y$, where $\alpha$ is any old symbol, may be an assertion (saying something) or a term (naming something). “$x \lt y$” is an assertion – a complete statement. If $x$ and $y$ have specific real number values, then the statement is either true or false. “$x+y$” is a term – an expression to be evaluated. For this expression, if you plug in $x=2$ and $y=3$ then it names the number $5$. See encapsulated computation below. To write "If $x\lt y$, then $x\lt y+1$" is the same same as saying, "If $x$ is less than $y$, then $x$ is less than $y+1$". Not only is it OK to say it, it's true. To say "If $x+y$ then $x\lt x+y$" is nonsense.

The expression "$x+y$" is not a sentence and so can't be a clause after "if". In the same way, it is nonsense to say "If my house, then it is white". Division and fractions Two symbols used in the study of integers are notorious for their confusing similarity. The expression "$m/n$" is a term denoting the number obtained by dividing $m$ by $n$. Thus "$12/3$" denotes $4$ and "$12/5$" denotes the number $2.4$. The expression "$m|n$" is the assertion that "$m$ divides $n$ with no remainder". So for example "$3|12$", read "$3$ divides $12$" or "$12$ is a multiple of $3$", is a true statement and "$5|12$" is a false statement. Notice that $m/n$ is an integer if and only if $n|m$. Not only is $m/n$ a number and $n|m$ a statement, but the statement "one is an integer if and only if the other is true" is correct only after the $m$ and $n$ are switched!

It is wise to be a bit paranoid about whether you really understand a particular kind of math notation. Be patient When you see a complicated assertion or term you have to be patient. You must stop and unwind it. Read the tiresomely long example of unwinding an expression in Zooming and Chunking. Giving names to symbolic expressions Turning symbolic terms into functions The expression "${{x}^{2}}-1$" is a symbolic term. You may define a function $f$ whose value at $x$ is given by the expression ${{x}^{2}}-1$. After we say that, "$f$" is a name for the function. You can also give names to symbolic assertions. Example Let $P(x)$ be the expression “$x\gt1$”. In this case, you could write statements such as “$P(3)$ is true” and “$P(1/2)$ is false", as well as more complicated statements such as "For any number $x$, if $P(x)$ then $x\gt0$." Don’t let this notation mislead you into thinking “$P(3)$” is a number.

“$P(3)$” is a statement, namely the statement “$3\gt1$”. Of course, $P$ may be thought of as a function $f:\mathbb{R}\to \{\text{true, false}\}$. Using notation such as “$P(x)$” for statements occurs mostly but not entirely in texts on logic. (This claim needs lexicographical research.) An overview of its use in first-order logic is given in Mathematical reasoning. See also the Wikipedia articles on various kinds of logic: A symbolic expression in algebra is both of these things: $\bullet$ The name of a mathematical object $\bullet$ An encapsulated computation of the mathematical object it names If you are fairly proficient in algebra, you already know this subconsciously about algebraic expressions. Examples The expression “$2\cdot 2+3$” is both a name for the number $7$ and a description of a particular calculation that gives $7$. In the math ed literature, we say that the expression “$2\cdot 2+3$” encapsulates the process of multiplying $2$ by itself and adding $3$. See the article What is the object of the encapsulation of a process? .

The expression “$63/9$” is also a name for the number $7$ and encapsulates a different calculation that results in $7$. The expression “$7$” is a name for the number $7$. It encapsulates a proper-name calculation, because in our culture “$7$” is the default symbolic name of the number. A proper-name calculation is like referring to "Henry" in a conversation where those present know which Henry you are talking about. In the case of $7$, the context is that we are talking about math, where everyone is supposed to know what the symbol "$7$" means. The expression “$371$” is our default name for $371$. It is in decimal notation and encapsulates the calculation “$3\cdot 100+7\cdot 10+1$”. The expression “The largest positive root of ${{x}^{3}}-9{{x}^{2}}+15x-7$” is a name for $7$, but that fact requires a more difficult calculation that $2\cdot 2+3$ or $63/9$.

Indeed, you don’t even know that the expression is a correctly formed name of a number until you work out that ${{x}^{3}}-9{{x}^{2}}+15x-7$ has a positive root. The expression \[3{{x}^{2}}-2x-5\] names a variable number . Like most variable mathematical objects, some statements about it must be said to be neither true nor false. For example, "$3{{x}^{2}}-2x-5\leq12$" is neither true nor false. On the other hand, it is a nice calculus exercise to show that “$3{{x}^{2}}-2x-5\le -6$” is definitely false for any $x$. Non-algebraic expressions Most math objects can be combined into new constructions, making expressions like algebraic expressions except that the variables represent structures or objects instead of numbers. Groups, various kinds of spaces, and lots of math objects you never heard of can be combined into "products" and "coproducts", and many of them have "quotients", "function spaces" and other constructions.

Most Wikipedia articles about important kinds of math objects describe some of these constructions. The expressions representing such things can still be thought of as both an encapsulated computation and as the name of another math object. Symbolic expressions as trees Symbolic expressions such as "$4(x-2)=3$" and the very similar looking "$4x-2=3$" have different abstract structures. The difference results in different solutions: $x=11/4$ and $x=5/4$ respectively. The abstract structures are largely invisible, with the only hint about the difference being the presence or absence of parentheses. There are other ways to exhibit symbolic expressions that make the abstract structure much more obvious. One way is to use trees. Examples of the tree representation of expressions are given in the following posts in Gyre&Gimble: I expect to include examples like these in a future revision of this article. Grammar of the symbolic language Arrangement of symbols is meaningful In symbolic expressions, the symbols and the arrangement of the symbols both communicate meaning. Examples “${{\sin }^{2}}x$” , “$\sin 2x$” and “$2\sin x$” all mean different things.

“$x{{2}^{\sin }}$” is meaningless. “${{\sin }^{2}}x$” and “${{\left( \sin x \right)}^{2}}$” mean the same thing. If you took a class in precalculus or calculus, you may have had this fact expressed explicitly or you may have learned it by osmosis (see “osmosis theory” in the Handbook .). Subexpressions An expression may contain several subexpression s. The rules for forming expressions and the use of delimiters let you determine the subexpressions. Examples The subexpressions in “${{x}^{2}}$” are “x” and “$2$”. The subexpressions in “${{(2x+5)}^{3}}$” are "$2$", "$x$", “$2x$”, "$5$", “$2x+5$” and "$3$". Math English subexpressions A phrase in math English can be a subexpression of a symbolic expression.

Example The set $\left\{ {{n}^{3}}|n\in \mathrm{}\text{, }n\gt0 \right\}$ could also be written as$\left\{ {{n}^{3}}|n\text{ is a positive integer} \right\}$. Embedded symbolic expressions in math English Symbolic expressions in texts are usually embedded in sentences in math English, although they may stand independently. Examples "If $x\lt y$, then $x\lt y+1$." This math English sentence occurred earlier in this chapter. "The indefinite integral of the function $x^2+1$ is $\frac{x^3}{3}+x+C$, where $C$ is an arbitrary real number." The statement "$\int (x^2+1)dx=\frac{x^3}{3}+x+C$" could occur in a text by itself as a sentence, but that is uncommon except perhaps in lists. Embedded symbolic expressions in math English involves a remarkable number of subtleties. Teachers almost never tell you about these subtleties. The abstractmath article Embedding reveals a few of these secrets. Generally, students learn these facts unconsciously.

Some don't, and those generally don't become math majors. Precedence The expression $xy+z$ means $(xy)+z$, not $x(y+z)$. This is an illustration of the principle that in an algebraic expression, multiplication is performed first, then addition. We say multiplication has a higher precedence that addition. PEMDAS When two operations have the same precedence, the operations should be done from left to right. The mnemonic “Please Excuse My Dear Aunt Sally” (PEMDAS) describes the order of the common operations: Parentheses (calculate what is inside the parentheses before you do anything alse.) Exponentiation Addition and Subtraction. One more rule The names of functions of one variable generally have the highest precedence, except for unary minus , which has lowest precedent. Examples "$2\cdot 3+5$" means do the multiplication first, then add the five, getting 11, whereas "$2\cdot (3+5)$" means do the addition first, then multiply the result by $2$, getting $16$. "$4+3^2$" means first calculate $3^2=9$, getting $4+9$, then calculate $4+9$, getting $13$.

But $(4+3)^2$ means $7^2$. The expression "$\sin x+y$" means calculate $\sin x$ and add $y$ to the result. The expression "$\sin(x+y)$" means calculate $x+y$, then take the sine of the result. $-{{3}^{2}}$ requires you to calculate ${{3}^{2}}$ first, then apply the minus sign, yielding $-9$. On the other hand, ${{(-3)}^{2}}$ yields $9$. Because so many people new to math misread some of these expressions, I have acquired the habit of putting in theoretically unnecessary parentheses for clarity. So for example I would write $(\sin x)+y$ instead of $\sin x+y$ and $-({{3}^{2}})$ instead of $-{{3}^{2}}$.

There is supposed to be a rule that says that ${{2}^{{{x}^{\,y}}}}$ denotes $2{{\,}^{\left( {{x}^{\,y}} \right)}}$, but this is even more widely unknown, so I always write $2{{\,}^{\left( {{x}^{\,y}} \right)}}$. But note: ${{({{2}^{x}})}^{\,y}}={{2}^{x\,y}}$, which is not usually equal to $2{{\,}^{\left( {{x}^{\,y}} \right)}}$. Wikipedia has an excellent detailed description of the precedence rules of algebra. Irregular syntax in the symbolic language The symbolic language of math has developed over the centuries the way natural languages do. In particular, the symbolic language, like English, has definite rules and it has irregularities. Rules In English, the plural of a noun is normally formed by adding “s” or “es” according to fairly precise rules. (The plural of car is cars, the plural of loss is losses.) But English rules have exceptions.

Think mouse/mice (instead of mouses) and hold/held (instead of holded for the past tense). . The symbolic language of math has a lot of rules too. In the symbolic language, the symbol for a function is usually put to the left of the input ( argument ) and the input is put in parentheses. For example if $f$ is the function defined by $f(x)=x+1$, then the value of $f$ at $3$ is denoted by $f(3)$ (which of course evaluates to $4$.) Irregularities Just as English has irregular plurals and past tenses, the symbolic language has irregular syntax for certain expressions. Here are two of many examples of irregularities. The symbol “!” for the factorial ( MW , Wi ) function is put after the argument. For example, $6! = 720$. The parentheses around the argument of a function are omitted for the trig and log ( MW , Wi ) functions, so we typically write $\sin x$ instead of $\sin(x)$ and $\log x$ instead of $\log(x)$. More about that here. There are many other examples of irregularities in symbolic notation in these places:

Question: A way of looking at mathematical relationships and operations using abstract symbols instead of specific numbers is called what?
Answer: Algebra