Why Use Manipulatives?

As I was talking to instructors in preparation for writing this manual, they told me that they found many barriers to using manipulatives. Some said their programs didn’t have any, and there were no funds to buy any; some instructors themselves weren’t comfortable using them, so they weren’t comfortable using them with students; some were afraid that the students using manipulatives might get into places that the instructors had no explanation for; many said that their students resisted using them. These are major barriers, and the results of using manipulatives had better be worth the time and mental energy it takes to overcome them.

I think most people would agree that manipulatives such as base ten blocks or fraction pieces provide a model of mathematical operations to supplement verbal explanations (themselves abstract) of abstract processes. For example, the two cubes that show 1 and 1 000 are clearly not the same size, and illustrates physically the difference in the value of the digit 1 in the numbers 1 and 1 000.

Manipulatives slow down the action

Going further, however, I find that using manipulatives slows down the process of explanation so students have more time for understanding. For example, in showing 24 – 19 using blocks, the time required to exchange 1 of the tens of the 24 for 10 ones is much longer than the time required to say “Regroup the 24 into 1 ten and 14 ones” or “Borrow 10 from the 20 and write 14 in the ones column.” This extra time gives the student a moment to absorb what is happening before you go on to the next step.

The student controls the pace of the work

Furthermore, if the student is using the manipulatives himself, and not just watching, the student controls the pace of the work. Students who are not sure of themselves move more slowly. If you watch students working with manipulatives, you get a sense of their understanding. You can’t see what’s in their heads, and often they cannot articulate their thoughts, but you can tell by their hands where their understanding is faulty.

Students get just as frustrated as anyone does when the manipulatives take more time than is necessary, and that spurs them on to making up shortcuts. To continue with the example above, when doing subtracting problems with the blocks, eventually students get tired of counting out 10 ones in order to exchange them for a ten-rod. They may set aside a little group of 10 ones instead of putting them back in the general pile, so they don’t have to keep counting them out again and again; they may start to count off the pockets on the back of the ten-rod instead of dealing with the loose pieces, or they may invent some other short cut. All of these are indications of a growing understanding of the operation of subtraction, and of a movement towards abstraction, which will allow students to do the operations using only pencil and paper, or in their heads.

Yet the word “necessary” in the first sentence of the previous paragraph is key. When the student is controlling the pace, she will continue to use the full process as long as it is necessary for her understanding. When she has not fully grasped the fact that subtracting 19 from 24 requires 1 of the tens in the 24 to be changed to 10 ones, counting out the 10 ones, exchanging them for 1 ten from the big stash, checking that she now has enough ones to take 9 ones away, then setting them aside and counting what is left—all these are absorbing and necessary tasks, and do not seem to take up “unnecessary” time. As she begins, through repetition, to become familiar with the process, as her understanding increases, her impatience with the physical process increases, because it is no longer a necessary piece of the action of subtracting. Her impatience will lead her to a shortcut, and eventually to doing the question without using the manipulatives.

Manipulatives help students remember

The use of manipulatives provides memory cues for different kinds of learning. The movement of the arm in operations of addition or subtraction, multiplying or dividing, reinforces the meaning of the operations for kinesthetic learners. In the operation of addition, the arm sweeps two or more groups of blocks together. In subtraction, the arm moves to separate a part from the whole. In multiplication, the arm moves repeatedly to add a group a particular number of times. In division, the arm makes the sharing-out movement that card players or parents of small children recognize immediately. The shapes, colours and sizes of the manipulatives provide cues for visual learners, and they carry those images with them when they move to working mentally or on paper. Finally, since talking seems to go with using manipulatives, auditory learners get to tell themselves stories about what they are doing, and hear others talk about the processes they are demonstrating, and this verbal rehearsal of the process is committed to memory.

Students get the right answer

Most important, the students nearly always get the right answer when they use manipulatives. If you ask a student to add 1/3 + 1/6 using manipulatives, the answer will never come out to 2/9, which is a common error students make when adding fractions on paper. This means that the instructor deals with a student who has the correct answer. Rather than having to deal with an error, you can work on extending understanding, or helping the student articulate the concepts. The benefits in terms of student self-confidence are evident.