It’s not unusual to see social media posts like this one:

“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?”

Conceptual understanding is sometimes misunderstood. Let us help you understand why the manipulatives ARE important, and at what point in your studies their importance will diminish.

The operative word is “conceptual.” Students who truly master math are able to understand the concepts behind what they are doing. There are two basic ideas students must grasp 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 8-block, purple is associated with the 6-block, and so on.

To adults, these concepts seem so basic that it becomes tempting to skip them or cover them too quickly. Please recognize that the manipulatives are a powerful tool. It is essential that these understandings become automatic if the child is to experience success using the manipulatives to learn math. You can find suggestions for memorizing the block colors in your student’s Instruction Manual.

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 is now to understand the optimum time to stop using them.

Fading: Ending Manipulative Dependency

Now that we’ve discussed the importance of the student developing an intuitive, automatic understanding of the numbers that the blocks represent, let’s discuss ways to capitalize on that understanding and effectively move them from block use to straight computation.

It is 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, they should be using them only as models. 

Consider adding 2 + 3: with Math-U-See, the student reads the written numeral 2 (associated with the orange 2-block), the numeral 3 (associated with the pink 3-block), and the addition symbol, indicating that the two blocks are “smooshed” together. They will use this process for several problems in the workbook. When they appear to understand, it’s time to teach it back to you.

Here are two examples of what that teach-back might look like:

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 5-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 they may understand what addition means, there is nothing in their answer to show that they aren’t just parroting an action that they have seen demonstrated. The second student, on the other hand, used words like “means” and “stands for”, which indicates a grasp of the underlying concepts. 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 understanding. If they 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. One additional step may be necessary to give them the confidence to move on to independent work.

When a student uses manipulatives repeatedly, they begin to form a mental picture of the process that is taking place. In the previous example, as the student continually models 2 + 3, they will eventually “see” the 2-block and the 3-block “smooshing” together in their mind. This is what enables them 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. 

The next time you pull out the manipulatives for a math lesson, pull out a set of crayons or colored pencils also. After your child builds the problem, have them draw a picture of the blocks they used. Continue doing this until they are 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 they can look at the problem, close their eyes, and picture the blocks in their mind. (If necessary, they can even tell you out loud what they see in their head.) Eventually, your child will not need to close their eyes to picture the problem; they will automatically “see” the blocks in their mind and be able to solve problems without any additional support.

Manipulatives are a powerful tool for visualizing and modeling mathematical concepts. As you guide your student with the Math-U-See manipulatives, they will develop a strong understanding of the fundamentals of math and be able to perform computations confidently and accurately, moving from manipulatives to mastery.

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