Manipulatives: The Missing Link in High School Math

by Marilyn Curtain- Phillips, M. Ed.

The thrill of achievement comes from overcoming adversity in the accomplishment of an important goal ~ UnknownAlgebra, Geometry and Trigonometry can be challenging and complex subjects for many high school students in the United States. These subjects are usually taught using textbooks, workbooks and examinations. While these resources are essential in developing learning skills in math, it doesn’t encourage problem-solving skills and retention. It is not surprising to learn many students view math as boring, difficult and irrelevant, rather than fun and interesting. The scope of what it means to be successful in mathematics expands beyond procedural knowledge and skill-acquisition to include sense-making and conceptual understanding (Buckley 2004). An integrated approach is needed to motivate students to learn math in a more relaxed environment, thus eliminating persistent math anxiety, the lack of math application skills and poor standardized test performance. The Third International Mathematics and Science Study (TIMSS) tested the students of 41 nations. Children in the United States were among the leaders in the fourth grade assessments, but by high school graduation, they were almost last. High school math should be taught in a way in which students can grasp and understand skills. Manipulatives are visual objects that help illustrate mathematical relationships and applications. These resources are used primarily in elementary schools and somewhat in middle schools. Manipulatives are valuable resources for accelerating and deepening students understanding of math, yet its use is almost non-existent in high school. Marilyn Burns, Creator of Math Solutions has "used manipulative materials at all levels for 30 years." In every decade since 1940, the National Council of Teachers of Mathematics (NCTM) has encouraged the use of manipulatives at all grade levels (Bellonio), yet many high school teachers are reluctant to use this type of resource.
Almost all mathematics-teaching activities take place at the abstract level (Sharma, 1997). According to Sharma, students have a tendency to forget when taught only at the abstract stage. Thus students become frustrated because mastery was never fully attained. Students will begin to have difficulty in learning mathematics. The results in failure will cause many students to develop a fear of mathematics (Sharma, 1997). Students’ attitudes towards any curriculum area can be related to their achievement in ways that reinforce higher or lower performance (timss.org). Full 88% of Bill and Melinda Gates Foundations’ survey respondents said they had passing grades in high school. Asked to name the reasons they had left school, more respondents named boredom than struggles with coursework. Over 1 million students drop out of school each year. That includes nearly half of all African Americans, Hispanics, and Native Americans who fail to graduate from public high school with their class. Leaving many of them with a host of poor outcomes to follow, from low lifetime earnings to high incarceration rates to a high likelihood that their children will drop out of high school thus eliminating the cycle (Thornburg, 2006).
Sharma (1997) feels there are six levels of mastery of mathematical concepts: intuitive, concrete, representation (pictorial), abstract, applications and communication. According to Sharma, ideally, each mathematics concept should be introduced beginning at the communication level. Manipulatives will teach concrete understanding to the abstract math process, especially when the student may not understand the concept behind the skill. When students develop a concrete understanding of math skills, then they are more likely to perform that math skill and understand math concepts at the abstract level. Manipulatives can make math concepts come alive. According to Spikell (1993), most learners whether adults or children, will master mathematical concepts and skills more readily, if they are presented first in concrete, pictorial and symbols. By using manipulatives, pictures and symbols to model or represent abstract ideas, the stage is set for learners to understand the abstractions they represent (Spikell, 1993). Students will be able to practice and demonstrate mastery using concrete objects. Manipulatives appeal to the learning style of kinesthetic learners because they actually touch the objects. Pictures appeal visually for visual/spatial learners. "Visualization is the natural way one begins to think. Before words, images emerge" (Sharma, 1987). "Almost every mathematics idea, except simple arithmetic facts, consists of three components: linguistic, conceptual and skill/procedural" (Sharma, 1987). Therefore manipulatives offer benefits to a variety of learning styles. They also provide a change from the textbook for mathematical/logical learners.
Games with manipulatives are also valuable with helping students to apply what they learned to the real world, as well as provide a means to improve their math skills interactively. Using board games and card games along with cooperative learning are ways that students can become involved in a positive mathematical environment. Games are highly motivational to students and can be used effectively to practice specific skills. "Using games in the classroom and at home will maximize students' problem-solving competence, ability to communicate and reason mathematically, perception of the value of mathematics, and self-confidence in their ability to apply mathematical knowledge to new situations." Cooperative groups provide students a chance to exchange ideas, to ask questions freely, to explain to one another, to clarify ideas in meaningful ways and to express feelings about their learning. These skills acquired at an early age will be greatly beneficial throughout their adult working life.
Many teachers feel as though they do not know how to teach using manipulatives and therefore, hesitate to use them in the classroom. Many math teachers, who attended college more than ten years ago, were usually taught on the abstract level, textbook, pencil and paper, all through their school days. There are classes and workshops for teachers to learn how to teach using manipulatives. The companies that make the manipulatives also provide books and pamphlets on ways the material can be used. There are articles on using manipulatives in mathematics teaching journal such as the National Council of Teachers of Mathematics’ Arithmetic Teacher magazine. Manipulatives help relieve boredom in children allowing them to explore and use their imagination.
Many manipulatives are inexpensive and can be everyday objects. Money, 2 –color counters, calculators, rulers, dominoes, playing cards, button and number cubes are a few of the commonly available manipulatives that can successfully be used in the classroom. These manipulatives can be used to teach such concepts such as angles, area, decimals, factoring, estimation, fractions, measurement, counting, percent, prime numbers, probability, geometry and whole numbers.
There are companies, which specialize in manipulatives that can be ordered from a catalog or online. No matter where a school is located, materials can be made available through the mail. These companies also make available to teachers such manipulatives as tangrams, pattern blocks, fraction towers, geoboards, algebra tiles, Cuisenaire rods, miras and polyhedral models.
When I initially introduced pattern blocks to my Geometry students, I was met with opposition and disbelief. It was a consensus that these students had not used pattern blocks since elementary school. However, many students actually began to explore and build figures with these blocks. I gave them this little bit of play time before taking on the task for that particular lesson, which was angle measurement of polygons. We must remember, everyone likes to play games even adults, look at our various hobbies, such as bowling, tennis, golf, art, race cars, chess, playing cards, etc. This holds true for high school students, they still enjoy playing games. Manipulatives require a great deal of prior planning and organization. But considering the many benefits that manipulatives offer, it is well worth the effort. Students will find math more attainable, fun and exciting, hopefully returning them to the prior successes of their early math experiences.

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REFERENCES:

Buckley, L. A. (2004, Oct) Course Taking and Equity: The Efforts of One High School Mathematics Department Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF> Retrieved 2008-01-12 from <http://www.allacademic.com/meta/p117666_index.html>
Burns, M. How to make the most of math manipulatives – A fresh look at getting students’ heads — and hands! — around math concepts. Retrieved 2008-01-1-10 from <http://teacher.scholastic.com/lessonrepro/lessonplans/instructor/burns.htm>
Sharma, M. (1987). How to take a child from concrete to abstract. Math Note book, 5, 8-10
Sharma, M. (1997, July). Improving mathematics instruction for all. Fourth Lecture in Colloquium "Improving Schools from Within: Your Role. Pp. 2-12
Spikell, M. (1993). Teaching mathematics with manipulatives: A resource of activities for the K-12 teacher. New York: Allyn and Bacon.
Third International Mathematics and Science Study (TIMSS), Institute of Educational Sciences, United States Department of Education
Thornburg, N. (2006), April 7), Dropout Nation, Time Magazine
TIMSS 1999 International Mathematics Report. Findings from IEA's Repeat of the Third International Mathematics and Science Study at the Eighth Grade. Ina V.S. Mullis, Michael O. Martin, Eugenio J. Gonzalez, Kelvin D. Gregory, Robert A. Garden, Kathleen M. O'Connor, Steven J. Chrostowski, Teresa A. Smith. Retrieved 2008-01-12 from <http://timss.bc.edu/timss1999i/pdf/T99i_Math_TOC.pdf>
Yale-New Haven Teachers Institute Home Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete by Judith L. Bellonio Contents of Curriculum Unit 01.06.12: Narrative Math Anxiety Types of ... Retrieved 2008-01-12 from <http://www.yale.edu/ynhti/curriculum/units/2001/6/01.06.12.x.html>

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Marilyn Curtain-Phillips teaches high school mathematics and is the author of Math Attack: How to Reduce Math Anxiety in the Classroom, at Work and in Everyday Personal Use,(1999), $15.00. You can purchase her book through our Online Bookstore.

Algebra

Introduction to Algebra

What is algebra?

Algebra is a branch of mathematics that deals with generalized arithmetical operations. It is a collection of rules for translating words into the symbolic notation of mathematics, in a manner that leaves their truth unchanged. It can otherwise be defined as a branch of mathematics concerning the study of Structure, Relation and Quantity.

Roots of Algebra

The name "Algebra" is derived from the book written by a Persian Mathematician, Muhammad bin M?s? al-Khw?rizm?. In 9th century he wrote Al-Kitab al-Jabr wa-l-Muqabala, a book on symbolic operations and systematic solutions for Linear and Quadratic equations. The word “Algebra” is derived from al-Jabr.

Symbols

Symbols are the specialty of Algebra. They make our work more convenient. For example, instead of listing the natural numbers as 1, 2, 3… we just say ‘N’. This symbolic representation helps us to generalize the characteristics of natural numbers. Algebra involves a lot of symbols. The following are the other few notations: C – Complex Numbers R – Real Numbers Q – Rational Numbers. Z – Integers. W – Whole Numbers.

Variables in Algebra

Variables are symbols that stand for unknown quantities. It must not necessarily be an alphabet. It can be any symbol. But, generally, we use x, y, z, etc. as variables. In the first algebra book, Khw?rizm? did not use any variable, even though it was all about solving equations. It is only in the 16th century that François Viète, a French mathematician, started using variables. To have a clear idea about the importance of variables, we can have a look at the following example: Mom gave Isaac a bag of apples. He did not count the apples in the bag but started eating them. After eating 3 apples, he found 4 apples were remaining in the bag. How would he find how many apples were there in the bag initially? Here, we go for variables. Let the number of apples be "x" initially. Isaac ate 3 apples. So, x – 3 will be remaining. But there are 4 apples remaining. So, x – 3 = 4. We need to get the value of x. We should get rid of that – 3. We add 3 on both the sides, which will not affect the equality.

Thus we have x = 7. So, there were 7 apples initially

Classification of Algebra

Algebra can roughly have four specialized branches:
• Elementary Algebra
• Abstract Algebra
• Linear Algebra
• Universal Algebra
  
  In Elementary Algebra, the number system and generalization of rules are studied. Constants,
  variables, expressions, and equations are the key concepts of elementary algebra.
  Abstract Algebra represents a further generalization of elementary algebra, by defining such
  constructs as groups, based on a set of initial assumptions, called axioms.
  Linear Algebra involves the extension of techniques from elementary algebra
  to the solution of systems of linear equations.
  In Universal Algebra the properties common to all algebras are studied.
  

Applications of Algebra

TAlgebra has very wide applications. Even in our everyday life we need some algebra knowledge. The principles of algebra are applied in all branches of mathematics, like calculus, geometry, topology, etc. It has its application in all sciences though.

Additional Links for Algebra

• Click here for samples
• Click here for Mathematics Dictionary

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Algebra, Present and Past

What is Algebra? Algebra is one of the many branches of mathematics. This branch of math utilizes mathematical equations to describe the relationships between any two things that may change over a period of time. Whenever a math equation has to represent something that fluctuates and does not stay the same, letters or symbols are often used to represent the varying quantity. This letter or symbol is called a variable because it varies.

The very start of modern algebra began with the ancient Babylonians. They were the very first people who used an advanced math system for their time. This math used an algebraic system to do calculations, and it enabled the Babylonians to utilize formulas and calculate the answer using unknown values for a problem class that would be solved in the modern day by quadratic equations, linear equations, and indeterminate linear equations. During this same time period in the first millennium BC, most of the mathematicians in India, China, Greece, and Egypt were using geometric methods to solve such equations. The word algebra comes from the Arabic word al-jabar, which means reunion of the broken parts. A Persian Muslim
mathematician named Muhammad ibn Masa al-khwarizmi wrote a book in 820 whose title is translated to mean The book of Summary Concerning Calculating by Transposition and Reduction. There is debate in math circles on whether the mathematician Diophantes, who has always been known as the father of Algebra, or Al-Khwarizmi should actually be named the father of Algebra. There is support for Al-Khwarizmi instead of Diophantes because a lot of his work on reduction is still currently in use today. He also explained how to solve a quadratic very extensively.

In the fourteenth century the word algeber referred to bone setting, which is quite similar to the original meaning of the word. The mathematical form of the word that is spelled algebra had made its debut by the sixteenth century. A man by the name of Robert Recorde was the very first person to use the term algebra in its mathematical meaning. He was also the inventor of the equality symbol(=). The original meaning of the word Algebra would refer today to the sub class of elementary algebra. There are numerous different fields of algebra in the modern definition of the word.

Some of these other mathematic fields include elementary algebra, s-algebra, abstract algebra, geometric algebra, boolean algebra, linear algebra, and many more. The mastery of elementary algebra must be accomplished before calculus or any of the other advanced mathematics can be learned. Many careers today require that at least elementary algebra has been mastered. The symbolism that is first introduced in elementary algebra extends throughout all the fields of mathematics. This is because the symbolism is the letters in the language of math. Modern fields of algebra are vastly more advanced than the first equations and formulas that were developed by the Babylonians. Numerous jobs and careers in the modern work environment require that algebra is mastered as a legitimate requirement for employment.

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Boolean Algebra

One of the primary requirements when dealing with digital circuits is to find ways to make them as simple as possible. This constantly requires that complex logical expressions be reduced to simpler expressions that nevertheless produce the same results under all possible conditions. The simpler expression can then be implemented with a smaller, simpler circuit, which in turn saves the price of the unnecessary gates, reduces the number of gates needed, and reduces the power and the amount of space required by those gates.

One tool to reduce logical expressions is the mathematics of logical expressions, introduced by George Boole in 1854 and known today as Boolean Algebra. The rules of Boolean Algebra are simple and straight-forward, and can be applied to any logical expression. The resulting reduced expression can then be readily tested with a Truth Table, to verify that the reduction was valid.

This has a very great role to play in abstract algebra. In fact, in abstract algebra, Boolean algebra is an algebraic that captures essential properties of both set operation and logic operations. Specifically, it deals with the set operations of intersection, union, complement and the logic operations of AND, or, NOT.

Like any lattice, a Boolean algebra (A, land, lor) gives rise to a partially ordered set (A, ≤) by defining

a ≤ b precisely when a = a land b

(which is also equivalent to b = a lor b).

In fact one can also define a Boolean algebra to be a distributive lattice with least element 0 and greatest element 1, within which every element x has a complement x such that

x land x = 0 and x lor x = 1

Here land and lor are used to denote the infimum (meet) and supremum (join) of two elements. Again, if complements in the above sense exist, then they are uniquely determined.

The algebraic and the order theoretic perspective can usually be used interchangeably and both are of great use to import results and concepts from both universal algebra and order theory. In many practical examples an ordering relation, conjunction, disjunction, and negation are all naturally available, so that it is straightforward to exploit this relationship. One can also apply general insights from duality in order theory to Boolean algebras. Especially, the order dual of every Boolean algebra, or, equivalently, the algebra obtained by exchanging land and lor, is also a Boolean algebra. In general, any law valid for Boolean algebras can be transformed into another valid, dual law by exchanging 0 with 1, land with lor, and ≤ with ≥.

The operators of Boolean algebra may be represented in various ways. Often they are simply written as AND, OR and NOT. In describing circuits, NAND (Not AND), NOR (Not OR) and XOR may also be used. Mathematicians, engineers, and programmers often use + for OR and · for AND (since in some ways those operations are analogous to addition and multiplication in other algebraic structures and this notation makes it very easy to get sum of products form for people who are familiar with normal algebra) and represent NOT by a line drawn above the expression being negated. Sometimes, the symbol ~ or ! is used for NOT.

Every Boolean algebra (A, land, lor) gives rise to a ring (A, +, *) by defining a + b = (a land b) lor (b land a) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a * b = a land b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a * a = a, for all a in A; rings with this property are called Boolean rings.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x lor y = x + y + xy and x land y = xy. Since these two operations are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f: A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent.

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x lor y in I and for all a in A we have a land x in I. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. An ideal I of A is called prime if I ≠ A and if a land b in I always implies a in I or b in I. An ideal I of A is called maximal if I ≠ A and if the only ideal properly containing I is A itself. These notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A.

The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x land y in p and for all a in A if a lor x = a then a in p.

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