My friend’s son, Kurt, was having trouble with his ninth grade algebra homework. He was given the equation,
P = -300 + 15N
and was asked to solve for N. The story in the textbook that went with this equation was,
“Jo has a lawn mowing business. This equation shows the relationship between the profit that Jo makes, P, and the number of lawns that Jo mows, N.”
I started by asking Kurt some questions to see if he could link the number sentence to the story about Jo. Kurt was able to tell me that Jo started off 300 dollars in debt, that N would be the number of lawns that Jo mows and that Jo charges 15 dollars for each lawn she mows.
We then moved on to solving for N. I asked Kurt if he had any ideas about how to start. Kurt said, “N is being multiplied by 15, so I could start by dividing by 15.” Kurt’s new equation was,
P÷15 = -300 + N
I responded as I imagine many algebra teachers would respond, “You divided the N by 15 and the P by 15, but what about the -300?” Kurt replied, as many algebra students will reply, “Oh, I have to divide -300 by 15 also.” His new equation was,
P÷15 = -20 + N
I said, “That was a great way to solve for N. With the numbers in this number sentence, starting by dividing by 15 worked well. Sometimes starting with division will be a little messy. Is there any other way you could have solved for N?” After some time, Kurt replied, “I could have added 300.” I asked him to do this. His new equation was,
P + 300 = 315N
The number sentence that Kurt wrote showed me that he did a good job generalizing the rule that I had just told him. I had told him that when he divided by 15, he needed to divide everything by 15. Now he that he is adding 300, and he thinks he needs to adds 300 to each term as well.
Why is it that
(-300 + 15N) ÷ 15 = -300÷15 + (15÷15)N
when you divide by 15 you need to divided each term by 15 where as
(-300 + 15N) + 300 ≠ -300 + 300 + (300 + 15)N
when you add 300 you don’t need to add 300 to each term?Mathematicians would explain the difference between dividing by 15 and adding 300 in terms of the distributive versus the associative property. You can think of dividing by 15 as multiplying by 1/15 and since multiplication distributes over addition both the -300 and the 15N need to be multiplied by 1/15 in order to preserve the equality relationship. When you add 300 to -300 + 15N, the distributive property no longer applies because addition does not distribute over addition. In this case we use the associate property,
(-300 + 15N) + 300 = (-300 + 300) + 15N = 15N
My response to Kurt when he forgot to divide the -300 by 15 was not at all helpful. It just another rule that Kurt needed to remember and he didn’t know when the rule applied and when it didn’t.
I knew Kurt as a third grader when he was in a CGI class. If I had asked him what 465 + 300 was, he would have said something like, “400 plus 300 is 700 plus 65 is 765.” He would have implicitly used the associative property of addition to solve this problem. Although he would not have written it like this, the idea that he used could be represented as,
465 + 300 = (400 + 65) + 300 = (400 + 300) + 65 = 700 + 65 = 765
He would not have added 300 to both the 400 and to the 65 as he did when he added 300 to -300 + 15N to get,
(-300 + 15N) + 300 = (-300 + 300) + (15+300)N = 315N
If I had asked third-grade Kurt to solve a problem like, “15 kids are sharing 180 jelly beans, how many jelly beans would each kid get?” He might have said something like, “150 divided by 15 is 10, there are 30 left, 30 divided by 15 is 2, so each kid gets 17 jelly beans.” This idea could be represented as,
180 ÷ 15 = (150 + 30) ÷ 15 = (150÷15) + (30÷15)
He would have known that both the 150 and the 30 would have to have been divided by 15. He would not have only divided one of the addends by 15 as he did when he first divided -300 + 15N by 15 and got,
(-300 + 15N)÷15 = -300 + N
Now it is 6 years later and what has happened? Why is it that the informal understandings that Kurt had as a third grader aren’t connected to the algebra he engages in as a 9th grader? According to Kurt and his mom, he has been in a lot of math classes where he has been expected to learn procedures that his teachers have demonstrated for him. Kurt often tells his mom that he doesn’t understand his math lessons. This is frustrating to Kurt because he works hard in school and he really wants to understand. Although Kurt’s mom has been concerned about Kurt’s lack of understanding, Kurt appears to be a good math student. He received mostly A’s in his middle school math classes and was recommended for algebra as an 8th grader (he took 8th grade math instead). To make matters even more complicated, Kurt’s middle school used a highly regarded reform math curriculum.
There are several things that elementary and middle school teachers can do to prepare students to learn algebra with understanding. Perhaps the most important is to
encourage students to use their own strategies to solve problems. This is true for complex problems as well as basic computation problems. We know that when students generate their own strategies to solve problems, they understand these strategies. Students may or may not understand the strategies we teach them. It might not matter whether we teach students standard algorithms or the more intuitive algorithms found in reform math curricula. Students can use the intuitive algorithms that they don’t understand just as they can use standard algorithms without understanding them. If a student learns to use a strategy that we have taught them, it is very difficult to assess if the student is using the strategy with understanding. When students use strategies that they have generated, we know they understand these strategies. The only strategies that can become a foundation for future learning are the strategies that students use with understanding.
A second way to provide students with a foundation for learning algebra with understanding is
to recognize when students intuitively use the fundamental principles of mathematics and highlight their use of these principles and draw attention to the use of these principles. The fundamental principles to watch for are: the associative, distributive, and commutative properties as well as the inverse relationships between operations. (See
Thinking Mathematically: Integrating Algebra and Arithmetic in the Elementary School, (2003) by Carpenter, Franke and Levi, published by Heinemann, Chapters 4, 8 and 9 for a discussion of these principles.) If you see a third grader using the associative property of addition, I don’t recommend saying, “You just used the associative property of addition, and let’s learn this property.” I do recommend that you draw attention to what the student has done to reinforce the idea for this student and others. If a child solves, 438 + 300 by saying,
“400 plus 300 is 700 plus 38 is 738”
we can draw attention to this intuitive use of the associative property in several ways. We can have the students share and make sure the rest of the class hears this strategy. We can ask questions such as, “why did you just add the 300 to the 400 and not also to the 38?” We can write the number sentence,
438 + 300 = 400 + 300 + 38,
on the board and ask other students if they think this number sentence is true.
It is possible for all students to learn high school algebra with understanding. Part of the responsibility for learning algebra with understanding clearly lies within the algebra class. Elementary and middle school teachers also play an important role in students’ learning algebra with understanding. How elementary and middle school teachers teach basic computation has a great impact, for better or worse, on students’ ability to learn algebra with understanding.
