Whit Ford left this comment on my post Curriculum Change and Understanding by Design: What are they solving? He makes a lot of sense. I just have to share it.
I believe the method of planning lessons is less important than WHAT you are asking the students to think about. Most Algebra I and II texts I have come across suffer severely from “elementitis” (see “Making Learning Whole” by David Perkins), which makes it very challenging for teacher to convey “the whole game” to students while still following the text. For example…
A teacher who is talking about how to “collect like terms” is not going to motivate the students as much as one who succeeds in relating this to a more interesting and complex problem which is related to student’s daily lives in some way. This is a HUGE challenge when teaching mathematical abstractions, one I am struggling with as I prepare to teach the first semester of Algebra I using a traditional text. However, it does lead to some interesting potential exercises:
– Ask students to give you examples of two objects in their lives (or in the room). Chances are you will get answers like two apples, two desks, two eyes, etc. Note these on the board as students mention them, then ask… so do you ever come across “two” all by itself? The answer is NO – “two” is an abstract concept, one which we apply constantly in our daily lives, but an abstraction nevertheless.
– So how do we come up with “two” of something? By finding them, collecting them, putting them together, etc. The abstraction of this process is what we have called “addition”. But what kinds of things can you add together and have it make sense? A foot and another foot – certainly. An apple and a pear – only if you recast each as “a fruit” – then you have “two pieces of fruit”. A meter and three centimeters – only if you recast each in the same units – then you have “103 centimeters”. So what is to be learned from this process? We can only add “like” things together, or quantities that are measured in the same units, if the answer is to make any sense. Addition certainly lets us add the quantities of one apple and one pear… 1+1=2, but 2 of what? The answer must make sense in the real world, and the abstract process of adding abstract quantities does not always result in a useful answer.
– So what about 2x+3y? We have two of “x” and 3 of “y”. Can we simplify this abstract expression? Until we know what “x” and “y” represent, until we have been given values for each of them (with units), we don’t even know if adding them together will produce an answer that makes any sense (see apples and pears above). Furthermore, since we have differing quantities of each, we will have to postpone combining them until we know values for each variable (since one value must be doubled, while the other must be tripled). On the other hand, if the problem were 2x+3x, we are being asked to assemble two and three of the same quantity “x”… intuitively, this MUST be 5 of the same quantity “x” – no matter what quantity and units “x” represents, since the units of both terms will always be the same.
I am hoping that such approach (extended considerably with more examples and practice) will begin to build both a conceptual and a practical understanding of the mathematical abstraction “like terms”, along with how to combine them when they occur… yet, this is just ONE of the many topics covered at a very procedural level by most Algebra I texts. Our challenge is to get students to understand the forest, when the textbook spends most of its time talking about trees.
His post Learning the Game of Learning is a good read, too.
You may also want to check-out my post on combining algebraic expressions . It links conceptual and procedural understanding and engages students in problem posing and problem solving tasks.
