From Math Manipulatives to Mastery
This blog post post will provide you with some guidance in determining how much and how often you should use math manipulatives and when it’s time to stop.
We here at Math-U-See are used to hearing it and seeing it on our social media posts. Generally the question goes something like this:
“My child can complete the worksheets when he’s using the blocks, but he can’t solve problems without them. Should I let him use the blocks on the tests? Can I move on to the next lesson? Has he mastered the material?”
Perhaps you’re in this situation yourself, or perhaps you’re concerned about your student becoming dependent on the blocks. Some parents approach the issue with the attitude of “when she’s ready, she’ll stop using the manipulatives.” There is an element of truth to this in that, once a student is fluent in solving problems, she often finds manipulatives to be tedious and time-consuming and “weans” herself on her own. The goal, then, is to help the student develop conceptual fluency so that she finds that manipulatives are no longer necessary.
Notice that the operative word is “conceptual.” Students who truly master math grasp the concepts, or the understanding, behind what they are doing. Using this operative definition of mastery, let’s look at the two basic ideas that a student must grasp in order to be able to use the Math-U-See integer blocks with understanding.
1. Each integer block represents a number of concrete objects (ex., the pink block represents three buttons).
2. Each integer block has its own identifying color. Brown is associated with the eight-block, purple is associated with the six-block, and so on.
To adults, these concepts seem so basic that it becomes tempting to skip them or cover them too quickly. However, it is essential that these understandings become automatic if the child is to experience success using the manipulatives to learn math. Here are some ways you can tell whether your student has mastered these fundamental concepts.
1. Place three small items in front of the student (pennies, buttons, paper clips, etc.). Ask him to find the block that represents this number of objects. While the student may need to stop and count the objects, he should immediately go to the pink 3-block. If there is any hesitation, or if he needs to count the “bumps” on the block, you need to spend more time with activities like this to help your child develop an intuitive understanding of the numbers the blocks represent.
2. Play any kind of game that associates the blocks with their colors. For example, you could place crayons of corresponding colors in front of the student and ask which block goes with each (ex., orange with 2, brown with 8). The goal is to develop an automatic association of the blocks with their colors, which will later help the child move away from the blocks altogether.
Again, the goal here, as with any concept in Math-U-See, is mastery. Don’t venture too much farther into the program until your child is able to identify the numerical value that each block represents and recall its color without any hesitation.
Suppose, however, that your student DOES understand what the blocks mean and is able to use them competently. The issue now is that she won’t STOP using them—in other words, she needs to use the blocks for every problem, even on tests.
Fading: Ending Manipulative Dependency
In the first part of this post, I discussed the importance of the student developing an intuitive, automatic understanding of the numbers that the blocks represent; in this part, I’ll discuss ways to help students build on that understanding so that they can move from using the blocks to straight computation.
First of all, it’s important to note that manipulatives are not meant to be a computational tool; their purpose is to serve as representations of a concept to help facilitate understanding of that concept. Therefore, once a student understands that the blocks represent numbers, she should be using them only as models. Consider the example of 2 + 3. In the Math-U-See program, the student learns to read the written numeral 2 (which she has associated with the orange two-bar), the written numeral 3 (which she has associated with the pink three-bar), and the addition sign, which indicates that the two bars are “smooshed” together. She uses this process to complete several problems in the workbook; then, once it appears that she understands the concept, it’s time for her to “teach it back” to you. What should the “teach back” process look like for this particular problem? In other words, how does the student’s response show mastery of the concept?
Consider these sample student responses:
STUDENT 1 – “First you take an orange block. Then you take a pink block. Then you put them together. Then you find the block that is just as long as the two blocks together. The answer is five.”
STUDENT 2 – “First I see a 2. That means I need an orange block because it stands for 2. Then I take a pink block for the 3. The plus sign means adding. Adding means you put things together, so I ‘smoosh’ the blocks together. I want to find out how many I have now, so I look for a block that’s the same. The five block is the same as the two and the three together, so 2 plus 3 equals 5.”
Do you see a difference in the two responses? The first student has simply described a process. While the student may understand what addition means, there is nothing in his answer to show that he isn’t just parroting an action that he has seen demonstrated. The second student, on the other hand, used words like “means” and “stands for”, which indicates a grasp of the concepts underlying the physical representations of number. If necessary, you may need to ask your student additional questions, such as “Why did you do that?” or “What does that mean?”, in order to probe the depth of his understanding. If the student cannot answer your questions, then more guided practice with the manipulatives will be necessary.
It may very well be, however, that your student shows complete understanding of the concept but has become dependent on the manipulatives to perform computations. There is one additional step that she may need to give her the confidence to move on to independent work.
When a student uses manipulatives repeatedly, whether she is aware of it or not, she begins to form a mental picture of the process that is taking place. In the previous example, as the student continually models 2 + 3, she will eventually “see” the two-block and the three-block “smooshing” together in her mind. This is what enables her to be able to perform the addition of 2 and 3 without the blocks. Over time, as the addition fact 2 + 3 = 5 moves into long-term memory and becomes automatic, the mental picture will fade. This process has given rise to the term “concreteness fading”, which simply refers to an educational technique that helps students move from physical manipulatives to pictures to abstract computations.
Here’s how the technique works. The next time you pull out the manipulatives for a math lesson, pull out a set of crayons or colored pencils, too. After your child builds the problem, have him draw a picture of the blocks he used. Continue doing this until your child is able to draw the picture INSTEAD of building with the blocks. Then, after your child has been drawing pictures for a while, put the crayons aside. See if your child can look at the problem, close his eyes, and picture the blocks in his mind. (If necessary, he can even tell you out loud what he sees in his head.) It may take a little while, but eventually your child will not need to close his eyes to picture the problem; he will automatically “see” the blocks in his mind and be able to solve problems without any additional support.
There is no question that manipulatives are a powerful tool for visualizing and modeling mathematical concepts. As you guide your student with the Math-U-See manipulatives, your child will develop a strong understanding of the fundamentals of math and be able to perform computations confidently and accurately. In other words, he will have moved from manipulatives to mastery.