Manipulatives, thought by some to be useful only for young children, are appropriate for students of all ages and should be used whenever new concepts are introduced.

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The use of concrete materials, such as manipulatives, helps students visualize abstract concepts and builds understanding. Consider, for example, using base-ten blocks to build an understanding of multiplication.

The standard multiplication algorithm is probably the most difficult of the four algoritms when students have not had numerous opportunities to explore their own strategies first. As with other algorithms, as much time as necessary should be devoted to the conceptual development of the multiplication algorithm using concrete- or semiconcrete models, with the recording or written part coming later. (Teaching Student-Centered Mathematics, 2018. Van de Walle, Karp, Lovin, Bay-Williams)

A Progression Designed for Understanding

While the standard algorithm is still the end goal for multi-digit multiplication, using concrete and visual models emphasizes the importance of place value when multiplying. This leads to better understanding, which leads to greater student success.

Direct modeling involves using manipulatives to represent a problem. Using direct modeling, the problem can actually be solved without any written record. Consider the early stages of teaching multi-digit multiplication. Base-ten blocks are used to model 4 x 23 as shown below–four groups of 23.

Even without being told to do so, students will likely group the tens together and the ones together to determine the value. They can skip-count by tens to find the value of the tens and then count on by ones to find the product, or they can count the tens, count the ones, and combine the tens and ones. Either will arrive at a product of 92. A written record could be added showing 80 + 12 = 92. Notice that no regrouping of the ones is needed to solve the problem in this way, although the understanding that 92 can be composed of 9 tens and 2 ones or 8 tens and 12 ones is now a part of most place value standards for good reason.

So what’s next then? Moving from equal groups to the area model creates a stronger connection to the standard algorithm and also emphasizes place value. An area model for 4 x 23 modeled with base-ten blocks is shown in the picture below. You can see that using the area model involves decomposing the 2-digit factor into place value parts. You still see 4 groups of 23, but now they are rows. As with the last model, students can determine the product by simply combining the value of the tens and ones.

A written record can be combined with the area model to introduce the partial products method.

After ample experience using the base-ten blocks to build the area model, students can transition to a more symbolic representation, as shown below.

I want to talk for just a minute about distance learning. It’s important that we don’t bypass the concrete stage due to remote learning. It’s important to use resources that incorporate movable pieces to provide that visual experience. The websites listed below also feature free virtual base-ten blocks. Each of these websites allows you to turn the pieces horizontally to more easily create an area model (some others don’t).

• The Math Learning Center (has the option to join and break apart the pieces) 
• Coolmath4Kids
• Didax

I hope you find these resources useful! If you have any others to share, please drop them in the comments.