*Pure mathematics is, in its way, the poetry of logical ideas.* ~Albert Einstein

**I never did very well in math – I could never seem to persuade the teacher that I hadn’t meant my answers literally. ****~Calvin Trillin**

**Introduction**

Manipulatives are proving to be effective tools for teaching certain math concepts to all students (Berkas, 2007). Students learn best through active experiences rather than lectures or passive listening. Visual images stick with people longer and better than abstract or language based concepts. (Stover, ret. 2011) Manipulatives are objects that can be touched and moved by students to introduce or reinforce a math concept. Since 1940 to the present day, the National Council of Teachers of Mathematics (NCTM) has encouraged all grade levels to use manipulatives in daily math instruction. Many concepts in math are abstract. Manipulatives are useful in helping students move from the concrete to abstract level. (Hartshorn, 1990) Students are being encouraged to use manipulatives to demonstrate understanding in representational and abstract stages (Berkas, 2007). They are beneficial for students who have been diagnosed with a learning disability. Manipulatives provide additional reinforcement of basic math skills and enables them to complete math problems independently. (Schreiner, 2010)

**Manipulative Toolbox**

Once a child is familiar with a manipulative it should be made available to use at any time to help them think, reason and solve problems. After a while, they begin to develop a great toolbox of ideas for solving problems. (Rudnicki, 2010)

**Positive Impacts**

Manipulatives show positive impacts when combined with:

- virtual manipulative software
- reflective practices
- cooperative learning
- activities that are exploratory and deductive in their approach

(Berkas, 2007)

**Types and Uses**

Unifix Cubes (colored linking cubes)

- learn measurement using nonstandard units
- visualize concept of area and geometric shapes
- percent, probability and fractions

Two-Color counters

- teach equivalence
- number concepts/ strategies for addition and subtraction
- create visuals showing ratio, proportion, fractions
- understand odd/even numbers
- create patterns

Playing cards, dominoes and number cubes

- illustrate decimals or whole numbers
- mental math games (addition, subtraction and/or multiplication war)
- understand probability and logical reasoning

Pattern blocks and tangrams

- understanding of spatial sense and geometry
- create shapes with certain number of sides, angles, complex patterns
- solving word problems

Measuring cups, volume containers, rulers, scales, and clocks

- concept of measurement

(Stover, ret. 2011)

**Critique**

Research has shown that long-term use of manipulatives is more effective than short-term use (Hartshorn, 1990).

**Interesting Facts**

“Some state and local systems have mandated the implementation of manipulatives through policy, law, or curriculum documents. Some have also provided funding.” (Hartshorn, 1990)

Manipulatives help hold the attention of the inattentive learner (Stover, ret. 2011).

**Multiple Intelligences**

Logical Mathematical

Using manipulatives in math help students to understand and analyze concepts with a concrete purpose.

Bodily Kinesthetic

In terms of mental and physical activities, manipulatives allow students to use their mind and parts of their body to solve problems. To show students how much a gallon is, start with cups of water, then pour those cups to make pints, then quarts to the eventual gallon. This is both a physical and mental process that show students the meaning of a gallon.

Interpersonal

Often times in the fields of math and science we work with a group of people to solve a problem. Using manipulatives in groups allow students to work together and to listen to the ideas of their peers to come to a conclusion.

**Universal Design for Learning**

I. Provide Multiple Means of Representation

1. Provide options for perception

- Options that customize the display of information
- Options that provide alternatives for visual information

2. Provide options for language and symbols

- Options that illustrate key concepts non-linguistically

II. Provide Multiple Means of Action and Expression

5. Provide options for expressive skills and fluency

- Options in the tools for composition and problem solving

III. Provide Multiple Means of Engagement

7. Provide options for recruiting interest

- Options that enhance relevance, value and authenticity

8. Provide options for sustaining effort and persistence

- Options that foster collaboration and communication

**Great Links**

**“7 Musts for Using Manipulatives”**

**National Library of Virtual Manipulatives**** **

**Resources**

Berkas, N. & Pattison, C. (2007). “Manipulatives: more than a special education intervention.” National Council of Teacher of Mathematics.

Hartshorn, R. & Boren, S. (1990). “Experiential learning of mathematics: using manipulatives. ERIC Digest

Rudnicki, A. (2010). “Why use manipulatives to teach math.” http://www.ehow.com/print/about_664188_use-manipulatives-teach-math_.html.

Schreiner, E. (2010). “How to use manipulatives to teach math to ld students.” http://www.ehow.com/print/how_5991197_use-teach-math-ld-student.html.

Stover, E. (ret. 2011). “About the use of manipulatives in math.” http://www.ehow.com/print/about_5048097_use-manipulatives-math.html.

Stover, E. (ret. 2011). “How to use math manipulatives in the classroom.” http://www.ehow.com/print/how_5031920_use-math-manipulatives-classroom.html.

Renae Deschenes-Desmond

March 20, 2011 at 11:06 pm

Hello Michael!

Reading your post brought back memorable memories (both good and bad) from when I was in elementary school! If it weren’t for math manipulatives, I guarantee that I would still have a strong dislike for math, as this was always a challenging subject for me. Something I found interesting in your post was the statistic from the National Council of Teachers of Mathematics. You said how the NCTM has been supporting the use of manipulatives in all grade levels for seventy plus years, yet some teachers with whom I have worked with personally in grades K-6, are still not exposing children to them. How do you think we can send home the message that these manipulatives are an effective way to be teaching math concepts? One of the articles you provided was wonderful. Marilyn Burns offered some great suggestions in using math manipulatives in the appropriate way. I especially liked her idea of allowing children free exploration time when a new manipulative was introduced. It just makes sense! I have one of her books About Teaching Mathematics, which is a K-8 resource, and I highly recommend it. It’s not only about how to use manipulatives but informative strategies to teach these difficult math concepts. The other link you provided is excellent! Thank you for sharing that, as I have another trick to add to my bag! I never knew that there was a free website dedicated to virtual manipulatives. Since I have only had experience with hands-on manipulatives, do you have a preference of using the virtual manipulatives versus the hands-on materials? A final thought I have after reading your post is the importance for all teachers, especially if they have students who have learning disabilities in their classroom, to make sure that they have access and sufficient time to work with these manipulatives.

Amy B.

April 5, 2011 at 7:34 pm

Hi Michael!

I very much appreciated your blog and the suggestions for use of the many different kinds of math manipulatives. I currently teach a small math class of students that are 2-3 years below grade level. As I discover more about the learning styles of the students that I work with, I can appreciate that lessons where I use math manipulatives are much more effective than those with paper and pencil tasks. My district recently employed a math expert, Mahesh Sharma, whose philosophy focuses on the use of math manipulatives to increase students’ number sense. I have become increasingly interested with making math “real” to students and steering clear from worksheets whenever possible. As we know, fitting in curriculum standards and leading students to achieving high test scores is a high priority in most schools. I am wondering what your thoughts are on taking more time for students to explore math concepts with experimentation with manipulatives, even if this valuable time (proven by research to be effective!) compromises some of the standards that we are required to teach?

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