Many people wonder how they should use the Math-U-See manipulatives to teach math. If you don’t use them properly, they are not going to be very helpful. They are such an integral part of the Math-U-See program that it might be useful to understand Steve Demme’s intention in creating them in the first place.
Steve believes that math is a topic that is learned to be applied in real life. This real-life application is usually demonstrated in curriculum in the form of word problems. In order to use math in real life, it’s important that students not only know formulas and procedures, but that they also know the concepts behind the procedures, and the formulas. In other words, to understand “how” to solve a problem, but also “why” we solve it that way, and even “when” the solution requires multiplication, division, even algebra.
Steve and Math-U-See use a number of little “tricks” that help students memorize math facts, such as acronyms, mnemonics, and visualizations. But if the student doesn’t have a healthy grasp of the whole concept of math, they are unable to be good problem solvers when faced with real life scenarios.
Related blog post: Make Memorizing Math Facts Fun With These 10 Activities
So WHY Use Manipulatives For Teaching Math?
First, in the decimal system, place value is an essential concept to master. No matter how small or large the number, we only have the digits 0-9 to represent it. We can use those digits in conjunction with place value to demonstrate every number that we can show, regardless of size. The physical characteristics defined for the Integer Blocks and associated with digits carry through the Algebra/Decimal Inserts and the Fraction Overlays.
For instance, consider the number 134 (see the image above). That would be one hundred, three tens, and four units. We write the number 1 and, because it is in the hundreds place, it is telling us how many hundreds we have – one hundred, or 100. The same is true for the 3 in the tens column – three tens, or 30. And in the units column, we have four units, or 4. (Notice we don’t call it the “ones” column – one, or 1, is a number. We use the term “units” to clearly differentiate between the place and the number.)
Studies have shown that American students often don’t truly understand place value. An example was done comparing Asian students with American students. When presented with a set of manipulatives, the students were asked to demonstrated the number 42. Asian students selected four 10s (40) and two units (2). American students selected 42 units placed in a long line, demonstrating a lack of understanding of the power of place value.
Place value, using our base-10 system, is essential for truly understanding math concepts. So Steve created the Math-U-See Integer Blocks. Not only can they demonstrate place value of integers, but also place value of the unknown, or x. By adding the Algebra/Decimal Inserts, the concept of x is illustrated and polynomials can be represented in a concrete way. We use the same colors as the Integer Blocks, but add inserts that present a smooth surface. The colors remind students of the proper placement with regards to place value: units are still the smallest, then x, then x2 and so on. If a student has been taught base 10 with our manipulatives, students can build on what they’ve already mastered and algebra becomes much more understandable. The relationship between arithmetic and algebra becomes clear. While we might think they are two separate disciplines, they’re definitely not. Using the manipulatives demonstrates the connection.
The Algebra/Decmial Inserts are effective for teaching algebra for several reasons. They’re synchronized with the base 10 Integer Blocks: same colors, same shapes. Our Fraction Overlays start with the green unit block and block it up big, to five inches by five inches, so that when it’s cut into smaller sections, they’re still visible and understandable. The colors of the Fraction Overlays mirror the colors of the Integer Blocks: halves are orange; thirds, pink; fourths, yellow; and so on.
So, there a reason behind the sizes, colors, and shapes of all of the manipulatives that carries through each level.
An example of a technique learned early on that can be applied in later learning is using the blocks to create a rectangle when demonstrating multiplication. This concept can then be extended to geometry concepts such as base times height for the area of a rectangle, or 1/2 base times height for a triangle, or determining the perimeter of a shape. Visuals such as the multiplication rectangle make it easier to see where the formulas come from. Many students in other programs memorize the formulas with no understanding of what is actually happening in the problem. We use blocks to demonstrate the Pythagorean theorem or to factor trinomials or polynomials. By illustrating the concept, students understand the bigger picture of where the formula comes from and can retain and apply the information when appropriate, rather than just memorizing with no conceptual understanding.
But HOW Do You Use Manipulatives to Teach?
The definitive answer is: it depends. The manipulatives are an illustration of a concept. The goal is understanding. Steve says, “Math-U-See really means Math-U-Understand.” To understand a concept, many kids have to see it. Some students can learn from talking abstractly about things, but many more benefit from actually seeing an example and having the opportunity to model concepts themselves. Research shows that the more different ways one is exposed to a concept, the more likely they are to understand and retain it. Math-U-See has always looked for concrete, real-life examples and illustrations of math concepts, from using a belt to explain circumferences to using ceiling tiles to demonstrate area. Using only our three different manipulative kits, math concepts can be demonstrated from Primer through Algebra 1.
To use the manipulatives to demonstrate a concept to a student, it’s important that you, the teacher, understands the concept and how the manipulatives illustrate it. Watch the lesson video first; they’re designed for the instructor and model how to use the manipulatives. Read the lesson in the Instruction Manual. Take the examples and, using the appropriate manipulatives, demonstrate the concept over and over until your student says, “Oh, I see!” or the equivalent. This will look very different with each student, or even with the same student learning different concepts. Some will quickly achieve that “aha!” moment, while other concepts may take more time.
The Build, Write, Say, Teach It Back approach is very effective with Math-U-See. Using the manipulatives to build a problem, students then write the math problem and say, or explain, what is happening in the problem. When students say, “Oh, now I get it,” it’s tempting to just take their word for it. That’s why we ask them to teach the concept back to us and demonstrate their mastery. That way, we can be sure that they’ve grasped the concept. Math is sequential and builds from concept to concept, so it’s critical that mastery and understanding is achieved before moving on.
As Steve Demme says:
Most people will solve a math problem by writing it down. Most people will see a math problem by having to read it. Prices are on the menus at Wendy’s. You have to read numbers. So writing is the predominant way that math is expressed and solved.
Math-U-See adds two additional components to writing the problem: Build It and Say It. The more different ways you interact with a concept, the more likely it is that you will not only understand, but also learn and retain the knowledge. We emphasize building with the manipulatives and explaining verbally what we’ve built, so that we’re using our eyes, ears, mouths, touch, at the same time that we write. Later, when we write a problem, we are visualizing the blocks and hearing in our heads how to do the problem. Finally, we add one more component: Teach It Back, to demonstrate mastery of the concept. Once mastery is demonstrated, your student is ready to move to the next concept.
The manipulatives are not a crutch. They are simply a concrete way to illustrate a concept. In Math-U-See, we illustrate abstract concepts in a concrete way that is new to most people, whether students or instructors. This leads to many “aha!” moments for parents, not only for students. As students master the material and internalize it, they will begin to wean themselves away from the blocks or other manipulatives. Because all children are unique, there is no single foolproof way to use the manipulatives. We recommend beginning with the plan we provide, but experiment until you find the best approach for each child. Be open to adapting to their learning preferences. Learn what their “aha!” moments look like. If you use the manipulatives how they’re intended to be used, you may be pleasantly surprised at how easy teaching and learning math can be.