Conceptual Layering

The success of the Teacher to Teacher Press company has been largely built on the concept of Conceptual Layering. This is a teaching strategy that allows all students to be successful in their mathematics classes, and provides all teachers the tools to reach all their students.

Most Conceptual Layering lessons begin with concrete, manipulative, visual examples to provide students the maximum opportunity to be initially engaged. The numbers used are normally positive integers. A complete explanation of the lesson is given so students see the entire process being described. As students gain familiarity and skill, extensions are provided, including negative integers, rational numbers, then real numbers and complex numbers as appropriate.

One of the key aspects of a successful lesson provides students multiple ways to “see a concept”. An example of this can be illustrated with Patterns and Functions. A linear relationship between two variables, typically x and y, can be shown in one of five related ways:

• A visual or physical manipulative
• A description using language
• A t-table
• A graph
• A formula

These representations are not isolated, but interrelated, and deep understanding resides in the ability to see how one representation relates to the others.

Here’s a more complete treatment by my coauthor about this topic:

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Conceptual Layering

A Teaching Strategy for Inclusion

By Brad Fulton

Too often, students “fall off the algebra train”.  It is as if the train comes by at full speed and students are asked to hop aboard.  Those that can run fast may make the jump, but most fail to ever get on board.  Carrying the analogy further, it would seem reasonable that if the train began more slowly, more could hop aboard.  Then the train could come up to speed without losing passengers.

In fact, this can be done.  The underlying principles of algebra are not difficult to comprehend.  Students as young as 4^th grade have been able to work with variables, simplify expressions, solve linear equations, understand slope and intercepts, and solve systems of equations.  They not only have solved such problems, they have shown that they understand them.  Even primary students have been able to understand concepts such as combining like terms, solving equations, and using implied multiplication with coefficients and variables.  If students can grasp the concepts of algebra at these young ages, we must believe they can master Algebra 1 in the middle grades.

Conceptual Layering is a teaching process that allows students to first establish a conceptual foundation for their algebraic thinking.  Then upon this solid foundation, they build a solid structure of algebraic rigor.  Conceptual layering starts with the simplest form of the concept being taught.  Higher and higher levels of understanding are gradually presented until the student is performing at the desired level. For example, to teach the distributive property, we might begin thinking about food ordered at a restaurant.  If h represents the cost of a hamburger, and f represents the cost of an order of fries, then this expression represents three orders of a hamburger with fries:

3(h + f )

We might ask the students what food will be prepared.  Clearly the restaurant needs to give the customer three hamburgers and three orders of fries.  This leads us to a very basic application of the Distributive Property.

3(h + f ) = 3h + 3f

This approach also alleviates the confusion students often encounter of not distributing the coefficient across all the terms in the parentheses.  The following error is common when students think of distribution solely at symbolic levels.

3(h + f ) = 3h + f

Now the student is asked to practice this concept until they have a sufficient mastery.  This will likely take only two or three examples as opposed to the two or three dozen practice problems that might be assigned traditionally.  Then the teacher can increase the complexity (accelerate the speed of the train) by adding some cheeseburgers or sodas to the order.

4(2c + f + 3s)

Once this level is mastered, the teacher can introduce letters that are not typically associated with food, such as x and y as shown here.  This will be a gradual transition in the mind of the learner.

4(x + y)

Next the teacher might use a negative coefficient, or a decimal value.  If the students have already learned how to multiply variables, the distributive property can be expressed with variables alone.

-4(x + y)

.4(x + y)

x(x + y)

Notice that none of these stages requires extensive practice time.  A few examples are all that is needed.  When complete, the same total amount of work has been done, but the concepts were incrementally layered so that the increase in complexity was gradual.  In conceptual layering, the teacher decides how far to take the lesson, as some students do not need to get to the final level until they are in the Algebra 1 course.  While some students may get off the train sooner than the teacher would want, they make it a lot further toward the destination because they were allowed to get aboard in the first place.

Unlike scaffolding which is used to sequence concepts and units of study as they are taught, conceptual layering looks within those concepts to determine the most natural and efficient way to present them to the brain of the learner.  Conceptual layering reflects the brain’s natural learning style.  Our brains seek complexity and challenges, but they prefer this to be done in an incremental way.  We are intrigued by anomaly and incongruity.  We are attracted to the interestingly unusual.  If I told you I was thinking of the following pattern, {1, 2, 3, . . .} you would probably suspect that the next number is four and you’d probably be bored by its simplicity.  However, if I told you that it was five, you would be more likely to be intrigued.  You would want to know what I was thinking.  It might even bother you if I refused to explain it.  Once you see that adding two adjacent numbers yields the next (The Fibonacci sequence: 1+2=3, 2+3=5, etc.), then you understand and have a sense of relief.  Your brain would then go on to seek other stimulation.  Our brains are attracted by a mild degree of confusion.

Textbooks fail to take advantage of this natural learning mode of our brains.

In general, when a new concept is introduced, the initial examples and practice problems will entail decimals, fractions, and integers.  The learners focus so much mental effort on the arithmetic, they often fail to comprehend the overarching concept being taught.  Unfortunately the typical algebra curriculum offers too great a degree of confusion to satisfy and engage most learners.  The solution, to remediate the child with a rehashing of previous year’s mathematics fails to challenge their brains at all.  Conceptual layering is the solution to both of these errors and offers the surest way for students to achieve success in mathematics.