Manipulatives provide a hands-on approach to making sense of mathematics. Students who are not naturally abstract thinkers are more successful in math when they have had an opportunity to formulate concepts and rules through discovery and exploration. 

For example, when students are first learning about multiplication, it is better to begin with 3 groups of 5 instead of the algorithm 3 x 5. Provide a manipulative (i.e., beans, cubes, skittles) and ask the child to place the items in ‘3 groups of 5’ and count the total. Ask the child to rearrange the items in ‘5 groups of 3’ and find the total. After doing this a few times with different factors, the child should “discover” that the product remains the same regardless of the order of the factors which is the associative property of multiplication. 

Quick vocabulary lesson: In mathematics, the answer to a multiplication problem is the product and the numbers being multiplied together are called factors. 

When a child is beginning to learn about multiplication, I don’t recommend using these terms right away in the exploration phase. Manipulatives are super important because they allow children to start discovering some of these concepts in a concrete way before moving on to represent their thinking with pictures before moving on to more abstract thinking involving algorithms.  

This process is known as the CRA or Concrete Representation Abstract model. Here, I break down the multiplication example from above into these three steps:

Concrete:

Representation: 

Abstract: 

3 x 5 = 15

Quick note: Multiplication is repeated addition. It provides an efficient way to add the same number over and over again. With that said, automaticity is super important because it allows students to be more efficient, especially when solving problems requiring more than one mathematical operation. It is critical to build foundational understandings first through the use of manipulatives with the goal of moving toward more abstract learning. 

Let’s further explore this idea with a similar concept like division. Division is the inverse or opposite of multiplication. Division is all about taking a bigger number and separating it into as many equal groups as possible. 

For example, you have 30 objects that you want to divide or separate into equal groups of 3. Like the multiplication example above, get 30 objects such as beans, beads, counters, etc. and separate those objects evenly. 

In this example, you are dividing 30 into equal groups of 3. You could subtract 3 from 30 over and over again (repeated subtraction) until you can no longer subtract another group of 3. This strategy works and provides for great concrete understandings. 

Quick vocabulary lesson: In mathematics, the answer to a division problem is the quotient and the larger number being divided is the dividend. The number that you’re dividing by is called the divisor.  

Let’s go back to the repeated subtraction strategy for division. While this strategy works, it is not very efficient. For example, say that you want to divide 1540 into equal groups of 5. You could subtract 5 from 1540 over and over again but that would take a really long time. A more efficient strategy would be subtracting ‘100 groups of 5’ at once. While a student may decide on a number other than 100, I chose 100 here because most students can multiply by 100 with no problem.  

When the number of groups are added up 100 + 100 + 100 + 8 the sum is 308, which is the quotient (answer). This strategy is known as partial quotients because you are finding “part” of the quotient at a time, which is why you have to put those parts together at the end to get the whole quotient. A more abstract thinker will be able to do this using the long division strategy. 

In conclusion, math is flexible parents must be flexible in their thinking when providing extra support at home to supplement instruction. Consider strategies like these that allow children to see the “why” behind 30 ÷3 or 1540 ÷ 5 before jumping right into long division. For any child struggling to understand mathematical concepts and ideas – try remediation that involves concrete instructional practices. 

If you need providing your child with educational support beyond the school walls, I would love to help. You can schedule a free 15-minute consultation here. Math can be hard, but it does not have to be. Let’s work together. 

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