Formulas, Functions, and Solving Equations – Pedagogical Content Knowledge? David Pollack Youngstown State University Spring 2006 PMET Conference Kent State University Kent, Ohio David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 1 of 22
What is Pedagogical Content Knowledge? What I know for sure: How can mathematicians help? • See more than meets the eye • See where it’s going • See underlying principle • See analogies and natural groupings • See the conceptual underpinning David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 2 of 22
What’s in a formula? More than a way to order a sequence in a single processor. Claim: a careful look at expressions and order of operations can clarify concepts about equation solving and functions later. I introduced what I call “Tournament Diagrams” in M3767 at Youngstown State University – roughly “Algebra for future Middle Childhood Teachers.” Tournament Diagrams Example: − × 3 5 4 ( ) + 5 6 4 Most of us probably don’t see first a well- ordered, well-determined sequence in which the operations must be performed, we see a structure in which some things can be performed in parallel and others have a precedence structure. Like a tournament. David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 3 of 22
The “Tournament Diagram” for the expression − × 3 5 4 ( ) + 5 6 4 is: 3 5 4 5 6 4 + * − * / It makes the “parallel and precedence” structure explicit, as well as showing one how to evaluate. Experience shows me that my students find it revealing. With more vigor I have been told many times that the middle grades students find it helpful. Why? I don’t really know! David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 4 of 22
Example 1 continued – show how to evaluate the expression. (Typical comment to students: “By the way, you should be able to do all these calculations in your head, including the last one. Do you know a good way to do it?”) 3 5 4 5 6 4 + * 10 20 − * -17 50 / -0.34 Possible value: In order to do something well, we know to always learn more than just that immediate goal. What should the “more” be? Something that sheds light on and reinforces the immediate goal and is interesting mathematically itself. Perhaps this is an example of a suitable David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 5 of 22
“more” when the “immediate goal” is to evaluate expressions. Note: we discuss directly the fact that some symbols, such as fraction bars and square root signs, serve as both an indication of an operation and as a grouping symbol. Example 2. Don’t overly standardize representations. This time we’ll work horizontally. 3 3 * 2 + 20 ( ) − − 1 1 5 David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 6 of 22
20 + 3 * 2 b ^ e / 3 5 b ^ 1 e − -1 What do the “b” and “e” stand for? David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 7 of 22
Now evaluate the expression: 20 + 20.96 3 24 * 2 b ^ 8 e / 3 0.96 5 b ^ 25 1 e + 2 -1 David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 8 of 22
Another representation meant to show what mathematically literate people “see” is what I call Loop Diagrams. We draw loops around pieces of the expressions, starting with those parts that have to be evaluated first and then successively drawing larger and larger loops, each time including the parts that have to be evaluated next. With this representation we write a number on the edge of each loop showing the value of the quantity inside. − × 3 5 4 ( ) + 5 6 4 Now loop the parts in the numerator and denominator that have to be evaluated first and evaluate them: 20 − × 3 5 4 ( ) + 10 5 6 4 David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 9 of 22
Now make larger loops to get the next parts that have to be included: -17 − × 20 3 5 4 ( ) + 50 5 6 4 10 Finally we divide -17 by 50 so we can loop the whole expression and know its value: -17 -0.34 − × 20 3 5 4 ( ) 50 + 10 5 6 4 David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 10 of 22
I have used the looping technique with young children motivated by a very quick introduction to a few key features of a graphing calculator. All they need to know is • When you type an expression it appears on the left of the screen; after you hit “enter” the answer appears on the right. • When you press “3 sto B” the calculator stores the number 3 in the letter B. It shows the command with an arrow on the screen. From now on, until you change it, the calculator thinks that B is 3. I demonstrate with something like “B + 4”. You might want to do this (the store operation) if you needed to remember that each baseball costs 3 dollars. • 4B means 4 times B. For example, how much will it cost to buy 4 baseballs. This is about all you need to motivate some basic algebraic thinking. David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 11 of 22
Solve these problems by “looping”. David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 12 of 22
Operation numbering: Another way that we will show order of operations is to place a number by each operation in an expression. The number should show what has to be done first, what next, etc. If two operations are “in parallel,” i.e. they could be performed at the same time, use the same number. Note that some of the operations are indicated with familiar arithmetic symbols, some by fraction bars, and others by the space between two numbers. In order to avoid confusing these ordering numbers with the numbers in the original expression, write them in a different color. − × 3 5 4 ( ) + 5 6 4 David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 13 of 22
Now try the same thing with example 2. 3 3 * 2 + 20 ( ) − − 1 1 5 These new representations reinforce traditional symbolic skill. Write a sequence of lines beneath the original expression, each time replacing a piece of the expression with the answer to a calculation. The pieces that you work with are the same ones that we just circled. You should have the same number of new lines as the highest number that you got when you numbered the operations. Each line of the list corresponds to performing operations that were on the same “level” (literally the same level in a tournament diagram). David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 14 of 22
Example 2 again: 3 3 * 2 + 20 ( ) − − 1 1 5 3 * 8 = 20 + 2 5 24 = 20 + 25 = 20 . 96 Each line corresponds to a layer in a tournament diagram or to performing all of the operations with the same “level number” from the “operation numbering” task. David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 15 of 22
All of these representations bring the process of working backwards into sharper focus. This concept should be well developed (I think) before a student makes use of the idea of equivalent equations in a formal sense. Exercise in “working backwards”. Use a tournament diagram to solve for x in the equation ( ) − × x 2 3 8 × + = 6 5 35 + 1 4 Can you find x with a loop diagram? David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 16 of 22
Differentiated instruction: Yet another way of representing the order of operations in an expression used to be important in computer science (I don’t know if it still is). It is called post-fix notation, or reverse Polish notation. To do this, you make a list where the two inputs to a binary operation precede the operation sign. So 5 + 6 is written 5, 6, + and (5+6) – (3*5) would be written: 5, 6, +, 3, 5, *, –. Example 2 would be written like this: 20, 3, 2, 3, ^, *, 5, 1, -1, –, ^, / +. A related method, also useful to a computer, is called pre-fix notation. Again you get a list of numbers and operation symbols, only this time the operation symbol precedes its two inputs. With pre- fix notation example 2 would be written: +, 20, /, *, 3, ^, 2, 3, ^, 5, –, 1, -1. See if you can start with the post-fix notation or the pre-fix notation and come up with the original expression. Try some others. David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 17 of 22
Input vs. Parameter: − 20 5 Consider the expression , which equals - − × 4 3 2 7.5. We have seen many ways to illustrate this expression; one of them is the following tournament diagram. 20 5 4 3 2 − × − ÷ With the example above, think of the number 20 as an input that will change from problem to problem and think of the other numbers as parameters that will remain the same. With that point of view, rewrite the tournament diagram as an inline “flow diagram”. David Pollack, Youngstown State University, Kent PMET Conference, Spring 2006, page 18 of 22
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