Teacher resources and professional development across the curriculum

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Insights Into Algebra 1 - Teaching For Learning
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Topic Overview Lesson Plans Student Work Teaching Strategies Resources
Workshop 1 Variables and Patterns of Change Teaching Strategies
Teaching Strategies:

Cooperative Learning

Manipulatives
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NCTM Standards


Manipulatives

The Benefits of Manipulatives
Using Manipulatives Effectively
Various Forms of Manipulatives

An old Chinese proverb states:
I hear, and I forget;
I see, and I remember;
I do, and I understand.
Manipulatives are hands-on tools that allow students to do mathematics. They provide physical representations of abstract concepts. As students often explain, "They help you see and touch the numbers" and "They let me see what's in my head."

In addition, manipulatives allow students to see what's not yet in their heads - the tools for solving equations, visual representations of algebraic factoring, and a multitude of other skills and concepts. Manipulatives can introduce mathematical topics and reinforce conceptual understanding in powerful ways.

Read what Jenny Novak has to say about connecting concrete to abstract:

Read transcript from teacher Jenny Novak
I think the purpose of using the manipulatives is to get the big picture... Read More

The Benefits of Manipulatives

Manipulatives help to lay the foundation for developing abstract thinking in mathematics. Algebra is often described as "the generalization of arithmetic," so manipulatives and algebra are a natural fit.

Jenny Novak, the teacher in Workshop 1, Part II, likes to use manipulatives because they allow students to interact with the mathematics and see what is happening. In addition, they provide her with instant feedback; if the students understand, they use the manipulatives correctly to represent expressions and solve equations.

Read some of Jenny's reasons for using manipulatives:

Read transcript from teacher Jenny Novak
Some of my students have told me that they are tactile learners; they like to be able to touch things, see things... Read More

A classroom research project compared the algebraic abilities of students who learned using manipulatives with students who learned without manipulatives. Veteran pre-algebra and Algebra 1 teacher Michaele F. Chappell, who conducted the research, identified several differences in understanding between the groups of students:

1. Ability to represent algebraic expressions
Students who learned in a traditional manner had difficulty expressing 2x and x2. For instance, they often represented the expression 2x as x + 2 or x2. On the other hand, students who learned using manipulatives expressed 2x as two equal quantities, x + x. Students who used manipulatives also expressed x2 geometrically as a square with dimensions x by x.

Read what Fran Curcio says about using manipulatives to represent situations in the video for Workshop 1, Part I:

Read transcript from teacher educator Fran Curcio
The use of the manipulative allows students who are at a concrete level of understanding to demonstrate, to represent the problem in a concrete way... Read More

2. Ability to evaluate and interpret expressions
Traditional students, when asked to evaluate 2x for x = 3, often found the value of 2 + 3 instead of 2 3. Students with manipulatives were more likely to create and evaluate expressions.

Read Miriam Leiva's thoughts about using manipulatives to help struggling students:

Read transcript from teacher educator Miriam Leiva
The manipulatives are going to help primarily those that may still be struggling [to figure out] how this works, or why this works ... Read More

3. Ability to make connections between concepts when solving linear equations
Students in a traditional classroom often viewed concepts in isolation, unconnected to other topics. They viewed the process of solving linear equations as several unrelated steps. On the other hand, students who used manipulatives viewed the process of solving linear equations as one concept with many interconnected steps.

Read what Miriam Leiva has to say about using manipulatives in teaching students to solve equations:

Read transcript from teacher educator Miriam Leiva
I think that so often students do not realize that in talking about an equation, we have a mathematical sentence of equality... Read More

4. Ability to communicate mathematically
The students who used manipulatives in the classroom were able to express their mathematical thinking, whereas students in the traditional classroom typically discussed only the algorithms and did not reveal deep conceptual thought.

Read Fran Curcio's comments about manipulatives and mathematical communication:

Read transcript from teacher educator Fran Curcio
In some cases, students might have been referring to dimensions using times - "9 times 4" as opposed to "9 by 4" - but what they meant was clearly communicated in the way they were using the manipulatives... Read More

(Source: Chappell, Michaele F. and Marilyn E. Strutchens. "Creating Connections: Promoting Algebraic Thinking With Concrete Models." From Mathematics Teaching in the Middle School. Reston, VA: National Council of Teachers of Mathematics, September 2001.)

Manipulatives also help students to recognize their mistakes. For instance, a student who incorrectly believes that area and perimeter change proportionally would benefit from exploring area and perimeter with a set of unit squares.

Listen to Janel Green's description of what happened in her lesson:

Listen to audio clip of teacher
Janel Green
The manipulatives were very helpful in the beginning of the lesson, because when I asked the students to take a guess, many of them guessed incorrectly... Read More

Manipulatives are often very helpful for struggling students, and especially for those students who have difficulty connecting the concrete ideas of arithmetic to the abstract ideas of algebra. An equation and an algorithm provide a symbolic representation, which may be difficult for students to conceptualize, but manipulatives provide a tangible representation.

Manipulatives also help students who are English language learners. In addition to providing a concrete representation, they allow students to increase their vocabulary. Demonstrations using chips and cups or other hands-on tools enable students to visualize the concept; with a teacher's help, they may come to understand important words by pairing terminology with the concept.

Read what Miriam Leiva has to say about using manipulatives with English language learners:

Read transcript from teacher educator Miriam Leiva
Here is an opportunity with the pictures and the models - in particular, the concrete, with the cups and the chips - to reach the second language learner... Read More

Reflection:
Consider manipulatives that you have used, or would like to use, in your classroom. List three ways that manipulatives can help deepen students' mathematical understanding.

record your thoughts in your journal


Using Manipulatives Effectively

While manipulatives can be incredibly helpful in promoting understanding, misuse and improper implementation can be detrimental. It is important to understand the potential pitfalls in order to avoid them.

Many opponents claim that manipulatives often serve as nothing more than toys for students, and they provide a distraction from, rather than access to, learning. However, the proper use of manipulatives does enhance learning.

Listen to Miriam Leiva's comments about connecting manipulatives to symbolic representations:

Listen to audio clip of teacher
educator Miriam Leiva
The use of the manipulative is not for manipulatives to be used for their own sake... Read More

Authors Rita Ross and Ray Kurtz give four guidelines for using manipulatives effectively:
  1. Manipulatives must support objectives.
  2. Clear expectations and procedures for using manipulatives must be presented to the class.
  3. Every student must be involved.
  4. There must be a procedure for evaluating students.
(Source: Ross, Rita and Ray Kurtz. "Making Manipulatives Work: A Strategy for Success." Arithmetic Teacher [National Council of Teachers of Mathematics], January 1993; issue 40: pp. 254-258.)

Read about Janel Green's purpose in using manipulatives in her lesson:

Read transcript from teacher Janel Green
The purpose of using the manipulatives was to help the students better understand the problem... Read More

Manipulatives, though often wonderful, have their limitations. They cannot represent every situation. For instance, it is often difficult to use manipulatives to represent an equation that involves fractional or decimal coefficients. Instead, try using manipulatives to represent situations in which the coefficients are integers. Such an introduction should improve students' success when they move on to more complicated equations.

This example serves as a reminder that manipulatives will not, by themselves, improve mathematical understanding. Teachers need to use them judiciously to help students transition from concrete representations to symbolic proficiency. Because manipulatives often do not allow for representations of complex situations, students have to master symbolic procedures before the teacher removes the manipulatives.

As an example, consider the trinomial x2 + 3x + 2. Students can use algebra tiles to represent the factored form of this trinomial, (x + 2)(x + 1), as shown below. On the other hand, the trinomial equation factors into fractional parts, equation, which students cannot easily represent with algebra tiles. They could perhaps conceive a representation that involves half sticks and half units, but only after they have attained a conceptual understanding of trinomial factoring.


Read what Miriam Leiva has to say about the thoughtful use of manipulatives:

Read transcript from teacher educator Miriam Leiva
My rule of thumb is: Use the model with the manipulatives or a pictorial model whenever I have a simpler problem... Read More

Reflection:
What rules and procedures can you put in place in your classroom to ensure that manipulatives are tools for learning and not a distraction from learning?

record your thoughts in your journal


Various Forms of Manipulatives

Manipulatives come in many shapes and sizes. Manipulatives can be made from inexpensive materials bought at a local supermarket or hobby shop. In many cases, online applets and computer applications provide virtual tools that students can use to explore mathematics. Alternatively, myriad resources are available commercially from retailers.

Teacher-made Manipulatives

Often the most effective manipulatives are those teachers create themselves. Indeed, classroom teachers developed many of the manipulatives that are now available commercially, and companies began mass-producing and marketing them because they were so effective.

The cups and chips manipulative is an example of an effective, yet inexpensive, manipulative that can be used in an algebra classroom. For linear equations, the number of cups represents the coefficient of the variable, and the number of chips represents the constant terms. For instance, -2x + 3 = 7 can be represented by cups and chips as follows:


Algebra tiles are another example of inexpensive manipulatives for the algebra classroom. You can purchase algebra tiles or simply print out copies of this file for students and have them cut out the pieces.

Read what Jenny Novak says about access to manipulatives:

Read transcript from teacher Jenny Novak
When I first heard about [the cups and chips] activity, I was excited because I didn't have to go out and buy all new materials... Read More

Virtual Manipulatives (Applets and Computer Programs)

The National Council of Teachers of Mathematics (NCTM) strongly promotes the use of technology for student investigations into mathematical topics. For algebra, the NCTM states:

Students' examination of graphs of change ... can be facilitated with specially designed computer software. Such software allows students to change either the number of minutes used [on a cell phone] in one month by dragging a horizontal "slider" (see fig. 6.13) or the cost per minute by dragging a vertical slider. They can then observe the corresponding changes in the graphs and in the symbolic expression for the relationship. Technological tools can also help students examine the nature of change in many other settings. For example, students could examine distance-time relationships using computer-based laboratories ... Such experiences with appropriate technology, supported by careful planning by teachers and interactions with classmates, can help students develop a solid understanding of some fundamental notions of change.
(Principles and Standards for School Mathematics, NCTM, 2000, p. 229)

Fig. 6.13. Computer software can help students understand some fundamental notions of change.

For an online version of the software referenced above, see the NCTM Illuminations Web site at http://standards.nctm.org/document/eexamples/chap6/6.2/index.htm.

Online computer simulations, or applets, serve a variety of purposes. In many cases, they replicate hands-on manipulatives, such as cups and chips, geoboards, and balances. However, because they appear online, they are available to all students at any time. While it would be impossible to list all sites with math applets, a quick search on the Internet will reveal many of them; see the Resources section for a short list of well known applet sites.

Commercial Manipulatives

Commercial manipulatives are available in abundant varieties - numerals, dice, money, pattern blocks (shapes that show both geometric and fractional relationships), thermometers, dominoes, dry-erase clocks with moveable hands, base ten blocks, algebra tiles, hundreds boards, and many, many, others. A quick search on the Internet will identify many retailers. Manipulatives come in many shapes and sizes, for a variety of purposes, and for teaching a multitude of mathematical concepts. You might want to peruse the manipulatives that are available commercially before attempting to reinvent the wheel, or you may want to look at what others have made to get ideas for manipulatives that you can create yourself.

Reflection:
List at least one manipulative that you've used or seen used from each of the three categories described above: teacher-made, virtual, and commercial. Which manipulative on your list was most effective in promoting student understanding? Why do you think that particular manipulative was effective?

record your thoughts in your journal

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