Insights Into Algebra 1: Teaching for Learning
Variables and Patterns of Change Teaching Strategies: 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:
I think the purpose of using the manipulatives is to get the big picture. What is happening in the problem? What do we have to do? I try to transition them through questioning and through our class discussion – “Okay, now that you’ve had examples, and now that you’ve had practice and you understand the big picture, how can we do this on a regular basis? How can we do this and apply this to other problems, without having to go back to the manipulatives?” We use this as our concept attainment … Then, as a class, we come up with the general rules and strategies that we would apply to a problem that will help them without the manipulatives.
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:
Some of my students have told me that they are tactile learners; they like to be able to touch things, see things. They are visual learners. They like to be able to manipulate and move things around so that they can really see what’s going on. If I’m just up in front of them instructing them, half of them are probably not paying attention. Watching them do it and circulating around the room, I can see if they are getting it or not … So it’s helpful to them, and it’s also helpful to me.
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 x2geometrically 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:
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. And I think that helps them to visualize the different arrangements of the tiles and the different sizes of the pool based on the dimensions. By using the tiles, they’re able to use their counting or any other arithmetic operations that they feel comfortable using. Other students who might not need access to a manipulative, they’re still seeing what’s being done, but they’re able perhaps to visualize mentally the different arrangements of the dimensions of the pool.
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:
The manipulatives are going to help primarily those that may still be struggling [to figure out] how this works, or why this works … It’s providing for the learner [who is] not quite at the symbolic algebraic level. It’s providing them another step, another strategy in their back pocket, so that the student can do it algebraically. If it feels really comfortable, they can do it with manipulatives and make that a stepping-stone, make that the bridge to going symbolically.
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:
I think that so often students do not realize that in talking about an equation, we have a mathematical sentence of equality. If you want to keep the equality, whatever happens on one side better happen on the other. If you are going to add five chips on this side, you have to add five chips on the other side. If you are going to distribute chips into two cups, you are going to have to take the chips from one side and put them into two cups, but you still have to distribute them … you will have two cups on one side and, on the other side, chips partitioned into two sets. You still have that [equality] that has remained intact. The manipulatives show that beautifully.
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:
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. Janel [the teacher] was able to infer that they understood what they were talking about. Developing proper language in mathematics is a critical job of the teacher – to model it, and then to help students develop it.
(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
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. It was only when they used the tiles and actually played with them that they realized that they were wrong … One of the students thought that because we had a fixed area, the perimeter would also be fixed. He thought that the first perimeter of 30 would go for every pool with an area of 36 square units. But because we had the manipulatives for him, he saw that he was actually incorrect.
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:
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 … they can see the concrete demonstration, they connect that to the symbolic, and they begin to learn the terminology mathematically as they listen. So the concrete provides a tangible way of making the transition from their understanding the concept of the equation pictorially to symbolically, but at the same time adding onto their knowledge of the English language.
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.
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. The lesson is a continuum where you have the students working with the manipulatives in order to lead them to understanding, building their understanding of the symbolic algebraic sentence. The students are modeling what the equation says mathematically with the manipulatives. Making those connections, making that bridge from the concrete, going all the way to the pictorial and then to the abstract symbolic, it’s key to understanding the mathematics. You cannot do manipulatives for their own sake. They really should not be done, I believe, one day … and then three days later, tie it to the symbolic. You need to do it as the students are working on this. The day may culminate on doing the symbolic, but they need to be able to go to and from it very freely.
Authors Rita Ross and Ray Kurtz give four guidelines for using manipulatives effectively:
- Manipulatives must support objectives.
- Clear expectations and procedures for using manipulatives must be presented to the class.
- Every student must be involved.
- 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:
The purpose of using the manipulatives was to help the students better understand the problem. In the beginning, they were taking guesses that were just wrong. They were only wrong because they could not see what was truly going on until they used the manipulatives. It was very helpful.
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 factors into fractional parts, , 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:
My rule of thumb is: Use the model with the manipulatives or a pictorial model whenever I have a simpler problem, and as I extend their facility with the algebraic solution, then they can let go of the physical model and go on to other number sets.
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?
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.
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 filefor students and have them cut out the pieces.
Read what Jenny Novak says about access to manipulatives:
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. You can use algebra tiles, you can use just simple Dixie cups and coins, poker chips, whatever you want as a manipulative. It’s very easy to get, it’s very easy to use, it’s very easy to put together in packets for the groups. And I think it’s pretty successful when we do the lesson.
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)
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 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.
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?
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.