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Manipulating Math: Concrete vs Abstract Thinking

Outside the Box Teaching Ideas


Math ManipulativesWhen I was a kid, math instruction consisted learning all the important steps and rules. I was not handed manipulatives, given an explanation for the rules, or the meanings behind the math. You did fractions “this way” because that was the way to do it, multiplication tables were memorized because the teacher said to memorize them. It was just that way. The“old school” way.


Today this is not the case. With new calculators, assistive technology and innovative approaches to math instruction, there is a new math in town.


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Now, math instruction focuses on understanding math concepts through the use of hands-on activities, manipulatives, real life applications, and “how to” essays. A far cry from the days of “copy the board and do page 65!” Students are more actively engaged in learning math and group problem solving while in previous years we were given the task and expected to complete independent of others.


In trying to keep up with new teaching methods, I read many articles about the integration of manipulatives in teaching math to elementary grades. I have also learned there are new methods of using manipulatives to teach various math concepts.Ways I didn't even know existed! 


This got me thinking...

  • Would using manipulatives more often in the classroom make a difference in how my students learn math?
  • Are manipulatives really so great and wonderful, or are they being overused? 
  • Do they prevent students from developing abstract thinking skills, memorizing and learning multiplication facts, recalling facts quickly, restrict their imagination, decrease their ability to solve everyday problems  and critical thinking?
  • Have you ever been to Dunkin Donuts and the young cashier was unable to count out how much change to give you back?

What Are Manipulatives?

Manipulatives are defined as concrete objects (things you can touch and move around) that aid in classification, patterning, counting, equations, fractions, multiplication, and other math tasks. They include  attribute blocks, geometric shapes, base ten blocks, unifix cubes, fraction bars, and plastic counting cubes. They may also include everyday items such as buttons, soda caps, bingo chips, beans, dice, straws, craft sticks, food items such as jelly beans, and marbles.


Virtual manipulatives can come in endless varieties such as pictures, puzzles, and games. Virtual manipulatives may include engaging colors and sounds and  most virtual manipulatives create a game out of the math, which tends to keep the student's interest alive while teaching concepts that may seem too challenging.

Benefits of Manipulatives in the Classroom

The National Council of  Teachers of Mathematics (NCTM) strongly advocates the use of manipulatives in the classroom. The NCTM believes manipulatives allow students to actively construct their own understanding of math.

  • Through manipulatives, students are able to explore, develop, test, discuss, and apply ideas of math concepts through the use of manipulatives.
  • Manipulatives make math concepts interesting and engaging for students (especially the younger grades).
  • Computers can also be used as a manipulative - contributing to your technology standards.
  • Many resources can be found online and incorporated into lesson plans easily.
  • Manipulatives encourage active learning and class or small group participation.
  • For early elementary students, manipulatives are great for patterning, counting, sorting, and organizing.
  • In middle and upper elementary grades the use integer bars, grid paper, geometric shapes, compass/protractor, solids, 3D objects are all great manipulatives that engage students and aid in instruction.

Concerns About Manipulatives in the Classroom

While I absolutely agree with the value manipulatives play in the classroom and have used them, I am concerned that manipulatives may prevent students from learning abstract thinking skills.


Struggling students are found to benefit from instruction that is explicit and systematic. Not all children learn in the same way or react to physical materials in the same way. What works for one may not work for all.


What if more time is spent on the manipulative while less time is spent on the abstract thinking part of math?

Would students still be able to both understand  math concepts, and complete pencil to paper computations? I am all for math being “fun” but overuse of the manipulative could cause students to become too accustomed to the activities of the lesson and forget the actual lesson.


What about virtual manipulatives that can be accessed almost anywhere?  Is it possible that students might learn to  become too reliable on using objects to solve a problem? What happens when they are in a situation where they need to solve an problem quickly, and they do not have the tools with them to do so?


Also, not all concepts can be related in concrete form.  Students should still learn how to problem solve, calculate, and think abstractly. How do we transition from manipulatives to abstract thinking without causing more confusion?

Tips to Best Integrate Manipulatives into Math Instruction

I suppose the first step to answering this question would be to ensure we are using the manipulatives properly. Many teachers (including myself) are not properly trained to use manipulatives. At no fault of their own, it just was not a main focus in college math classes in earlier years.


If manipulatives are used improperly, it could confuse the students. There are hundreds if not thousands of resources online as well teacher guides and other resource books that will list methods and examples of using manipulatives.


Here are some things to consider as you're planning manipulatives for your math class:

  • Be careful not to overuse manipulatives or try to utilize them for all concepts. Not all manipulatives can be used to meet curriculum expectations. Teachers may need to develop new ideas or teaching methods for their students that may or may not include manipulatives.
  • Determine which manipulative to use in the classroom that will contribute to the lesson. Some manipulatives are designed for specific math topics and can not be used across the math board. Having the learning outcome in mind and considering the child(ren) when planning the lesson will help in selecting the best manipulative.Manipulative materials must be selected that are appropriate for the developmental level of the students.
  • Encourage students to share what they have learned orally or in written form. Most students benefit from presenting this explanation as it helps organize their thoughts. You can also assess the use of the  manipulative based on the student's explanation. If the manipulative aids in understanding, check for understanding when the manipulative is not being used.
  • Behavioral challenges: For me, this is a big deterent from using any manipulative for a lengthy period of time. Many students will begin to become distracted or will use the pattern block as a flying missile (believe me, it has been done). Quickly outlining rules at beginning of a lesson as well as allowing for some “free play” when first introducing the manipulative (primarily with younger students) will help deter them from wanting to play when they need to be learning.
  • Cost and time considerations: I have many manipulatives that I have purchased from ebay, yard sales, or other teachers who were retiring. Some manipulatives can be made but that will be time consuming especially if you have a large class. With pattern blocks, for example, you are going to need several blocks for each student. That is a lot of cutting- although you can print a pattern and have students do the cutting. Sometimes making the manipulative is a math lesson in itself. This is also where virtual manipulatives are a plus- if you have a smart board or an in class computer.

Balancing Concrete (Manipulatives) & Abstract (Pencil & Paper) Thinking Skills

After reading the various pros and cons and studies completed on teaching math using manipulatives, I am still not completely sold on manipulatives as being the BEST method to teaching and learning math skills, but it is a growing trend in the education world and certainly has its benefits.


I admit, I am still a little “old school” in my methods of teaching math, but it is only because I want to ensure my students can think both abstractly and concretely. However, if this can be accomplished through pencil and paper as well as the use of manipulatives, then I am definitely open to learning how to properly use ALL the tools in my math toolbox. 


What are your thoughts to using manipulatives in the classroom? How do you transition from concrete teaching to abstract teaching? What steps do you use? Share in the comments section!