Two Algebra Texts on WA math standard A1.4.A “Write and solve linear equations

1. Two Algebra Texts on WA math standard A1.4.A “Write and solve linear equations and inequalities in one variable.” Prentice Hall Algebra I Pearson Prentice Hall isbn-13: 9780133659467 (2005 ed)‏ -vs- Discovering Algebra Key Curriculum press isbn-13: 9781559537841 (2007 ed)‏

2. Of all the skills a student must learn in a first Algebra course, simple algebraic manipulation is paramount. Without this skill, further advancement in mathematics is not possible. Standard A1.4.A addresses the state's requirement for this skill. This presentation illustrates the difference in how this skill is taught through in two texts written from two entire different pedagogical perspectives. Many in the educational community prefer Discovering Algebra, and it is on the cusp of being approved for use in the Seattle Public School system as I write this. Algebraic manipulation skills come as a result of training and practice in the recognition and use of the fundamental properties of real numbers. When a student learns these properties and gains skill in the power over all things numerical that they make possible, they fade into the background, just as rules of grammar, syntax, and punctuation fade from view for one who has learned to write in their native tongue Even though we forget their names they remain the foundation of what we build on top of them. Those that don't learn them build on a foundation of dubious integrity.

3. Prentice Hall gets off to a good start with the fundamental properties of real numbers with the Identity and Inverse Properties of Addition on page 56. All highlighting is in the book. I have not added anything to these shots. The idea that the sum of a number and its additive inverse equals zero is clearly shown on a number line. I suppose that this is what Drs King and Bright derisively call “naked mathematics.”

4. Page 70 continues with the Identity property of Multiplication, etc.

5. Page 72 presents the Inverse property of Multiplication, and the introduction of the concept that every non-zero real number has a multiplicative inverse.

6. Page 73 completes this development by defining the reciprocal, and introducing the important idea that division by a number is equivalent to multiplication by its reciprocal. This explains the rationale behind the old “invert and multiply” procedure. It's not rote if one can understand and name the properties that lead to it.

7. On page 86 important properties are summarized.

8. Practice identifying the properties in use solidifies understanding of how they are used.

9. On the next page an example shows use of the properties in the simplification of an expression. The presentation of algebraic manipulation as a step-by-step procedure - in which no step is a guess - is extremely important for the development of students' habits if they are to become facile with this skill.

10. By page 118, students have had some practice simplifying expressions, and are ready to solve equations. The term “solution” is defined, and the general goal of isolating the variable on one side of the equation is described. The word “undo” occurs here in the context of the use of inverse operations to get the variable “alone on one side of the equal sign”, but never again. The use of the fundamental Inverse Operations, defined earlier, and the Properties of Equality are all that are needed to solve linear equations. No games, no tricks, no made-up terms.

11. Page 144 covers proportions and the important technique of cross-multiplication. This is not a trick, nor is it presented as such. It is developed with the Multiplication Property of Equality, introduced much earlier and familiar by now.

12. Interestingly, Discovering Algebra starts out with an introduction to proportions before even beginning to solve equations, on the same page where the idea of a variable is introduced. This is on page 97. In step one of this investigation, the author commiserates with the student that it is “hard to guess” what the value of the variable might be. It is suggested that one might multiply by 19? Why? Why indeed! The text doesn't not explain how or why this is a legitimate thing to do. Below, a curious argument is presented that similar smoke and mirror techniques might yield values for variables in the denominators. The student is once again asked why this works, but how could they know? The fundamental properties have not been presented yet. This presentation of mathematics as a series of “tricks” is pervasive throughout the text.

13. Here's the next page, showing only samples of “three student papers”. The student is invited to make up other methods of solving this problem, but hasn't been given a coherent rationale for any of them. The process of learning math through discovery this way sets students up to guess at why a particular “trick” works in one situation but not another. Guessing and then finding out they guessed wrong fosters doubt in what many kids believe is an ability that they either have or do not. Repeated disappointment and frustration leads these kids to conclude “I'm just not good at math.” This kind of foolishness fosters this, and therefore does more damage than good.

14. Almost fifty pages later, after a lot of unrelated fluff, we're back to working on ways to solve simple equations. In the story that introduces this “investigation” math is presented as a “trick”. It's clear enough how this method works, and not inappropriate as a way to introduce the standard, efficient, and general method, but as with the lattice method for multiplication, it takes up a lot of space and time. For this introduction, the student is asked to make up a sequence of operations with randomly chosen integers. Then they build the table as described

15. The result of the made-up expression is placed in the lower right, and the “secret number” is revealed by following the arrows. The student is told that this will work with any equation.

16. The terms “equation” and “solution” are correctly defined here, but we're still on the whole “undoing” table bit! Where are the Properties of Equality? Have they been repealed for 21st century math? At the bottom of this page, the author suggests that this technique might serve a student perfectly well out in the “real world”.

17. Sure enough, here's a real-world problem, solved with a table. Ok, this book has examples, but how about some examples of something worthwhile?

18. Roll forward another fifty pages. Would you expect some real equation solving by now? I would, but no. Instead, we are presented with another alternate presentation of algebra, using the old “balance beam” principle. This might have been an appropriate way to introduce the Properties of Equality, but the best place for that was a hundred pages back.

19. More of same here, but look at the text at the bottom of this page. Aha! A definition! In passing, where most students would miss it, the concept of “like terms” is defined. Now we're getting somewhere, right? Well, no. The example used here is with like terms that are simple integers. I combed through the entire book and could not find a single example or exercise where a student would need to combine like terms that contained variables as factors. Now that's rigor. Discovery style.

20. The top paragraph of this page is cute: '...you won't need the pictures once you get the idea of doing the same thing to both sides of an equation. And pictures are less useful if the numbers in the equation aren't ”nice” '. “Nice” numbers? Where was that defined? And “doing the same thing to both sides of an equation” is as close to a statement of the additive and multiplicative properties of equality as the book gets in this chapter on Linear Equations.

21. Not until page 243, in chapter 4, are the fundamental properties of real numbers presented in standard form. Up to this point, students have been expected to discover these on their own. How many did so and did so correctly? How many, by this time, have developed , practised and ingrained fallacious “tricks” in their place? It is too late for the authors to present these to do any good. Why did they even bother?

22. This page of review problems cracks me up. Here we have some very simple one-step equations to solve, using the method that “you like best”. Then a balance problem. For review. And this is a high school text? And then a matrix arithmetic problem? Why in the world are our children being taught matrices instead of algebra? This is madness!

23. The authors of Discovering Algebra are so proud of their novel new “undoing” method that they put it in the glossary.