Welcome to Part 4 of MathBait™ Multiplication. In a change of pace, this article does not contain lessons, activities, and games. In Parts 1-3, we focused on helping students develop a strong conceptual understanding of multiplication and provided over a dozen digital games for students to amp up their fact fluency through play. Parts 5-7 will continue to build single-digit fluency while also helping students to tackle multi-digit multiplication.
But how do you teach multi-digit multiplication? There are an astonishing number of approaches and methods. Which methods are grounded in solid pedagogy? Which are not worth our time? Should students learn more than one method? Before we dive head first into more fun and exciting resources, we thought it was important to take a moment to address the many methods of computing multi-digit multiplication and provide our best advice, techniques, and hacks to support your students on this journey. This article is for the parents, the teachers, the aids, and the tutors to help you sift through the noise and find a solid way to help your students attack multiplying larger values.
Understanding New Math
Before we discuss the many methods for multiplication, let's take a look at why so many methods have been popping up in the classroom.
In the last 50 years, our collective view of mathematics education has shifted. This is a good thing. With so many changes in our way of life, advances in technology, and better understanding of the science of learning, we want our educational system to adjust with the times and provide students the knowledge they need for tomorrow's landscape.
This has been dubbed "new math" and it drives everyone crazy. First and foremost, there is no new math, math is basic truth and beauty, and while we have made advances in our understanding, the math we learned 50 years ago isn't different than the math students are learning today. However, how students are learning has changed.
Decades ago calculators in your pockets or access to Google, Bing, and AI from anywhere in the world didn't exist. Thus, teaching how to compute has become less important. In order to prepare our students to be the problem solvers of tomorrow, we don't need to train human-computers, we need to help students build the skills they need to program the computers and this has been the catalyst to update our curricula.
We won't dive deep into this topic. MathBait™ founder Shayla Heavner published an article on the topic in the Journal of Computers in Mathematics and Science Teaching titled "Building Conceptual Understanding: Improving Mathematics Education in Online Courses". Her research was one of the big drivers for the creation of MathBait™ as she found evidence to support a better way of learning. But we do want to name the struggle – understanding and dealing with "new math" is a thing – and help parents and educators navigate this updated way of approaching mathematics. It isn't about the answer, but about the process. In understanding how things work and why they work, students are building the logical skills they will need to be successful.
Problem solving is key for today's students. However, that doesn't mean students need to master every method. By providing multiple perspectives, students gain a better understanding of why and how something works and then pick the method they like best or a method that makes most sense in the problem they are solving. This means many parents find their child's homework nonsensical, confusing, and riddled with steps and procedures they never learned. If this sounds familiar, this article is for you. We are lifting the veil and explaining the most common methods step-by-step while also providing our best insights on if the method should be taught, if so when, and how best to introduce each method to help your students soar.
What is Multiplication (and Multi-Digit Multiplication)?
We certainly can't talk about how to multiply if we don't have any understanding of what multiplication really is. Math is funny that way. Many of us have been multiplying our entire life yet still have a very limited view of what we are actually doing.
To make matters worse, there isn't just one thing multiplication is. This makes the operation both powerful and confusing. The most common views on multiplication are:
Repeated Addition
Area
Scaling
Each viewpoint can provide us with valuable insight on how to multiply.
Repeated Addition
This is the best way to first introduce multiplication for students. As they already know how to add, it builds on previous knowledge and it is easy for students to understand the resourcefulness of this tool.
If I need to add many of the same thing, it's easy to see why multiplication is helpful. For instance, a box of taco shells contains 9 items. How many shells will I have if I purchase ten boxes? I could write out,
9+9+9+9+9+9+9+9+9+9
but that is time consuming and tedious. It is much easier to consider (and write) 9×10. With an understanding of commutativity, students now have a strategy allowing them to pick the easier and quicker approach. Counting by 9's is tedious,
9 18 27 35 45 54 63 72 81 90
but counting by 10's is simple,
10 20 30 40 50 60 70 80 90.
Further, students with an understanding of place value can ascertain that 9 tens is nothing more than a 9 in the ten's place and zero ones, which quickly becomes 90.
The other value of the repeated addition viewpoint is that it can be extended beyond multiplication and into exponents. Exponents can be viewed as repeated multiplication and as multiplication is repeated addition, we have summed up PEMDAS as nothing more than adding. How cool is that? The main operations we use are all just adding. This perspective of math, as nothing more than numbers and adding, can be powerful and take mathematics from what seems like an overwhelming subject back down to child's play. But we can't stop there. Only viewing multiplication as repeated addition is limiting.
The Area Model
An understanding of repeated addition leads perfectly into the Area Model. If I have three squares, and I continue to stack identical 3-square blocks, my total number of blocks can be found by repeatedly adding 3.
This view not only helps students transition into two-dimensional thinking, it will support larger products, partitioning, distribution, and binomial multiplication. It's a must for students to learn how area relates to multiplication.
The Area Model not only supports a better understanding of multiplication and products, but it is also the best way to help students understand commutativity. From the image above, they can quickly see if we count the stacks rather than the rows, we have 3 stacks of 4 blocks each. The way we count doesn't change the number of items. Thus, we can conclude 3×4=4×3.
Area is all around us. Planting a garden, building a house, painting a room. The Area interpretation gives students an understanding grounded in the world around them rather than an abstract idea.
Scaling
Allowing students to play with scaling is not only fun, but supports their foundation for working with ratios and fractions. Further, scaling is the most powerful view of multiplication as it will allow students to later see that with only multiplication can we create any value.
At MathBait™ we present scaling as a shrink ray or growth laser to better understand algebraic manipulations. For instance, if I need to make Dino 6 times his size, I must multiply him by 6. But just like in Honey, I Shrunk the Kids we cannot pick and choose what scales. If we set off our laser, everything in the path is enlarged or shrunk.
As students progress, this view of multiplication will greatly benefit their algebraic endeavors. For instance, in solving the equation 2x+4=6, students can multiply by 1/2, casting their shrink ray on everything: (1/2)(2x+4=6). Making everything half its size, the 2x becomes x, the 4 becomes 2, and the 6 becomes 3 to find x+2=3. This multiplicative view can greatly enhance a student's ability in algebra. They don't need to guess between operations, "Multiply? Divide? Add? Subtract?" as we have thrown out the need for division. With scaling, we can reach any value and make any number larger or smaller as we please.
Each of these views of multiplication is powerful. And while students should eventually have an understanding of each one, it is ideal to spread them out over time. Skip counting is a great introduction to multiplication as repeated addition. Once students are confident in their skip counting, they can begin to explore multiplication through shapes and blocks leading to the area model. Finally, as students build fluency, introduce scaling as growth. This will provide a solid foundation to re-examine scaling in upper-elementary and middle school.
Method 1: The Area Model
Method 2: The Lattice Method
Method 3: Partial Products and the Standard Algorithm
Method 4: Chinese Sticks
Method 5: Vedic Multiplication
Method 6: The MathBait™ Way
Conclusion
It is vital that our curricular material moves away from calculation. Students will need to develop problem-solving skills to be competitive in college and the workplace. This is the "new math" that many fear and misunderstand. The many many methods of multiplying larger values can be mind boggling. In this article we broke down each method to help parents and teachers better understand the origins and mathematics behind the steps and procedures. So many of us were taught "how to" and never saw behind the curtain to understand why.
We find the Area Model to be a strong method to introduce students to multi-digit multiplication. While the Standard Algorithm is useful, the MathBait™ Method leans on some of the benefits of the Standard Algorithm while also helping students to develop a stronger understanding of place value, as well as an understanding of structure that will greatly benefit them in later years and as they begin to study algebra.
Next week, we are providing an amazing tool that will continue to help students develop fluency while also allowing them to play with place value and multiply large numbers with ease (when a 6-year old can find 146789×567 in only a few minutes you know it's good!).
Take your students on an adventure through math with Marco the Great! Our unique storytelling approach is sure to engage your students and provide a new perspective of mathematics which is both enjoyable and easier to understand.
Try it today!
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