Algebra is one of the five content strands of Principles and Standards and is increasingly appearing as a strand on state standards lists. Exactly what this means from a state viewpoint is not at all uniform so you may be influenced in how you approach this topic by what your state requires. One goal of this chapter is to help develop a useful concept of what algebra or algebraic thinking is about at the K to 8 level.
In my search for an appropriate perspective on K-8 algebra the writings of the late James Kaput consistently offered what I believe is the most useful organization of algebraic thinking is found in the five forms of algebraic reasoning that he described (T-79). Kaput emphasized that these are not separate components of the curriculum. He saw the making of generalizations and a meaningful use of symbolism as supporting all of algebraic reasoning and mathematics in general. These ideas are addressed in the first section of the chapter with specific attention to the equal sign, variable, and equation solving. The next two components - structure, and patterns and functions - Kaput saw as separate strands to be addressed specifically. In this chapter you will find a full section on helping students make structure explicit and beginning to prove generalizations based on structure. The remainder and bulk of the chapter develops ideas of patterns and functions. Kaput's fifth form of algebraic reasoning, mathematical modeling, is "a web of languages and permeates all the others" (Kaput, 1999, p. 135). While modeling is explicitly mentioned at the end of the chapter, much of this aspect of algebraic reasoning is found in the sections on functions.
And so, although Kaput's description of algebraic thinking guided the organization of this chapter, it is not explicitly found in the headings used throughout. You may have to help your students see how these components build together as you work through the chapter.
Generalizations and Symbolism
A correct understanding of the equal sign is the first form of symbolism that is addressed. The development follows many of the ideas found in the work of Tom Carpenter and his colleagues in the book Thinking Mathematically (Heinemann, 2003). The failure of the curriculum to construct an accurate understanding of the equal sign has been well documented in the research dating back to the 1970s and yet little has been done to correct this failing. Students from grade 1 through middle school continue to think of = as a symbol that separates problem from answer. The ideas in this first section of the chapter should not be viewed as belonging only in the early childhood curriculum. The methods and ideas are appropriate for and adaptable to any level.
Out of an exploration of open sentences an understanding of variable can and should develop. In this chapter I have restricted the uses of variable to two: as an unknown value and as a quantity that varies. The correct use of the equal sign with variables that vary are essential symbolisms used throughout the remaining components of algebraic reasoning - especially in the areas of functions and mathematical modeling. The use of variables as representing specific unknown values leads to a need to solve equations and techniques there of.
Making Structure Explicit
A recent focus of research in algebraic reasoning is on structure or generalizations based on properties of our number system. Generalization is again the main interest. For example, when students note that 4 x 7 = 7 x 4, what helps them to understand that this is true for all numbers? Further, how do students go about "proving" that properties such as the commutative property or relationships on odd and even numbers are always true? By helping students make conjectures about the truth of equations and open sentences, they can then be challenged to make decisions about the truth of these conjectures for all students.
Extending, inventing and observing patterns, and being able to match patterns formed of different physical materials yet logically alike (isomorphic), is certainly doing mathematics. To the extent that these patterns can be generalized and described symbolically (A-B-B-A-B-B-…) place them squarely in the realm of algebraic reasoning. Examining pattern in numeric situations such as number sequences or the hundreds chart are another form of early pattern exploration.
Growing patterns now command a larger presence in the intermediate and middle-grades curriculum. Although mathematically interesting themselves, I see growing patterns as a vehicle for developing early concepts of function. The discussion distinguishes between recursive relationships and functional relationships. It also introduces the notion of different representations for the functional relationships found. These relationships are explored further as the concept of function is expanded in the rest of the chapter.
Functions and Representations of Functions
This section expands to a more general concept of function than the restrictive discrete functions of growing patterns. The thrust of this discussion is that there are five different representations that can be used to help make the function concept meaningful to students (T-86). It is important that students see functions in all of these representations and are able to see how each is a different way of seeing the same functional relationship.
In your exploration, technology should be used whenever possible. The graphing calculator is more than adequate and I strongly suggest that every student have access to one if possible even if just for this chapter.
Sources of Functional Relationships for the Classroom
I have made an effort to show teachers of the middle grades a wide variety of ways to bring function into the classroom. Not only do different contextual situations provide interest, each different context is another example of mathematizing our world - mathematical modeling - the fifth type of algebraic thinking described by Kaput. Growing patterns provide a first example of the functional relationships. Other functional contexts include the following:
Real-World Contexts. Several other categories have "real" contexts. What sets these apart is that all the
examples result in clearly defined formulas. This is in contrast to scatter plots of real data.
Proportional Situations. Of course these are also real world but it is useful to separate these out because
they make an important connection to the key idea of proportional reasoning. Note again that all
proportional relationships have linear graphs that pass through the origin. This was also noted in Chapter
18. Formulas and Max/Min Problems. Here the connection is to measurement and provides some interesting examples of functions that students can get their hands on. Scatter Plot Data. The idea of a best-fit line is developed completely in Chapter 22. It can just as well be introduced here. A best-fit line is a functional relationship that approximates real-world phenomena that do not strictly obey the relationship. If found with a graphing calculator, then the equation and chart are also immediately available, even for non-linear relationships. An important idea here is that the function found as the best-fit line is a mathematical model for the real-world situation. It is important for students to see that mathematical models often do not reflect the wide degrees of variation that occur in real relationships. Contrast these situations with those in the section on Real-World Contexts. Fun Experiments. There are no new ideas here, only a collection of fun things to do in the classroom. Depending on the situation, there may or may not be a nice theoretical model to match the experimental data.
Generalizations About Functions
Rate of change has been an underlying concept in all discussions from growing patterns forward. In this section, this idea is developed more explicitly as students begin to explore the rise and fall of graphs without equations. The slope concept is then used to create the general category of linear functions. Slope is defined in the usual manner found in early algebra courses. Here the intent is to keep the discussion informal but you can make it as technical as you wish.