Diana Davis

Winter Study 2005

Seventh Grade Math

When I thought about teaching at Mount Greylock, I anticipated that there would be a lot of problems with discipline, because it is a public school with young children who are not always excited about doing math. However, it turns out that the true problem is with learning.

I believe that it is best to teach math in a way that makes it relevant to the students, and that builds upon their preexisting knowledge. Relevance is key to holding students’ interest, and new math must build upon previous math simply because students must have the skills to do the new math that they are learning. I was disappointed to find that most of the math that students do is rarely made relevant to their lives, and the concepts that they learn are seldom made clear.

Consider the way that students in the regular math class[1] are taught how to find the greatest common factor (GCF) of two numbers. Immediately before they learn this, they learn to find all of the factor pairs for a number. For instance, 24 can be expressed as:

1 x 24
2 x 12
3 x 8
4 x 6

Figure 1. The ways to factor 24

Then they learn to do prime factorization. For instance, 24 = 23 x 3. To find this, they use a “factor ladder.” This can be done in two ways, depending on which factor a student chooses first:

Figure 2. Two possible factor ladders for 24

I had never heard of a factor ladder before coming to Mount Greylock. Everyone I have talked to about prime factorization learned prime factorization using “factor trees.” (see Figure 3).

Figure 3: Three possible factor trees for 24

The problem with a tree, I learned, is that students might think of different factors to start with. One student might start by breaking down 24 into 3 and 8, and then break the 8 into 4 and 2, and the 4 into 2 and 2. Another student might choose to break 24 into 4 and 6 first, and then break down the 4 into 2 and 2 and the 6 into 2 and 3. This would create too much confusion, and it is much better to make sure that everyone has the same thing written down on their paper so that they can easily tell if they are correct; thus, we use a ladder so that everyone has exactly the same answer as everyone else.

After students were relatively facile at breaking down a compound number into prime factors, we introduced the GCF. The procedure for finding the GCF is as follows (see Figure 4):

  1. The problem says, “find the GCF for 24 and 16.”
  2. Draw a Venn diagram with two circles, and label one of the circles 24 and the other 16.
  3. Draw factor ladders for 24 and 16, and write out the prime factorization of each.
  4. Under the ladders, write 2x2x2x3 and 2x2x2x2, respectively.
  5. Figure out what factors they have in common. When you find one, cross it out and write it in the center part of the Venn diagram. Here you would cross out three 2s on each side and put those three 2s in the middle of the diagram.
  6. Whatever is left, put it in the part of each circle that is not intersecting with the other circle. So in the outside of the “16” circle you put 2, and in the outside of the “24” circle you put 3.
  7. The GCF is in the center region. Multiply the 2s together, and you get 8. That is the GCF.

Figure 4. The procedure for finding the GCF of 24 and 16

This is a fine method for finding the GCF, with many merits. For one, it is visual, with circles to draw, numbers to cross out and move, and clear “boxes” (i.e., the parts of the circle) to put things in. However, this introduction lacks an important element: an explanation of what the GCF is. The students know (i.e. have preexisting knowledge of) what a “factor” is: a number that goes into another number evenly. The teacher could explain that a “common factor” is a factor that both of the numbers have, because “common” means that they share it. Then the greatest common factor is the biggest one of these factors that the two numbers share. One could write some numbers, such as 8 and 12, on the board, and ask what factors they have in common. Someone would surely say 2, and someone else would certainly say 4. You might have to point out the 1, but hopefully someone would think of it without a hint. We would write these numbers on the board, and explain that these are the common factors. Then we would ask, “what is the greatest of all the common factors?” and 4 is the biggest, so that would be the GCF.

Such a discussion would provide some guidance and intuition as to what the students are actually looking for. Without this concept – “I am looking for the biggest number that goes into both numbers evenly” – students are merely practicing a procedure, not figuring out something that they understand. Then what happens if students are asked to find the GCF of 16 and 35? There is no number to put in the circles’ shared region, because the numbers share no factors. Logically, and this is indeed what happened, the students assume that the GCF must be 0 in this case, because the region is empty. But if the students have discussed, or have watched and listened to their classmates discuss, that 1 is always a common factor of two numbers, they will be better equipped to realize that the “missing value” in the center region is 1.

We tried to convey some of this insight. We pointed out that if you multiply together all of the factors in the circle labeled 24, you get 24, and the same for the circle labeled 16 (which is the idea behind the circles). If students understood this, then it would be clear that if you had a 0 in the middle, then the product of everything in both circles would be 0. However, this takes more of a mental stretch because a student must remember what you get when you multiply together everything in the circles, and a student that could do that could certainly think of the idea that the GCF of two numbers that are relatively prime (i.e., have no common factors) is 1.

Clearly, no matter how much a teacher explains, some students are not going to understand. Although we explained that when you multiply together everything in the “24” circle you get 24, some students still crossed out a factor on each side that the numbers had in common and then put two of them in the central region, rather than putting one there (see Figure 5).

Figure 5. A common error in the GCF procedure

The discussion about multiplying everything in the 24 circle together should have prevented such an error, because clearly 3x2x2x2x2x2x2 is not equal to 24, but some students, whether it was because they were still busily copying down the ladders and the circles, or because they were talking to a friend or looking out the window, or simply because they did not follow the thread of the discussion, did not understand this explanation, and thus did not benefit from the discussion. However, if all students were well-versed in what the GCF actually is – “the biggest one of all of the factors that 24 and 16 have in common” – then it would be clear that 64 cannot possibly be the GCF of 24 and 16, because it is not a factor of either one. This should be obvious because all factors of 24 must be less than or equal to 24 – which they know from practicing the factoring skill shown in Figure 1 – but there was a lack of understanding of the concept of the GCF.

This, I think, is a major problem. Granted, math does not come easily to many students, so they need class time to practice the math they are learning. It might seem like a waste of time to try to teach students to understand the concepts behind the math they are using, because they will only need to compute, rather than explain, to pass the MCAS. In fact, if they learn how to compute well, they will get all of the problems right, so it doesn’t matter if they understand what they are doing. Taking this attitude conveys a lack of trust in the student, and a lack of desire to push a student to understand everything that he or she is capable of understanding, and it deprives students of a tremendous opportunity to learn.

After all, despite practicing the computational procedures day after day in class, on homework, then in class again, then on quizzes,[2] half of the students still have major difficulties with the math. Often, this is because they have made a mistake similar to the one in Figure 5: a merely procedural error. This demonstrates that they have only memorized the procedure, and forgot a small part thereof; furthermore, it shows that they never understood what this procedure actually accomplishes, or what they are doing. It is understandable that they make mistakes, because the procedure has no meaning for them. This is analogous to when a student told me that the rule for a number being divisible by 10 is that it is divisible by 2 and 3, and the rule for being divisible by 6 is that it ends in 0: She knew the rules, but she didn’t know why they worked, so it was just an exercise in memorization (and in this case matching). In other words, students are practicing these skills – “drill and kill” – day after day, and they are still not learning them well.

This is a sign that something must be done. I sympathize with these students: how should we expect them to pay attention in class and stay interested when the skills they are practicing day after day are not relevant to their lives and seem meaningless or useless? Luckily, there is some intrinsic pleasure in being able to break down a big number like 528 or 1000 into its prime factors, and it is somewhat engaging to draw circles and put numbers in them, and every so often there is a worksheet where solving the problems gives you the answer to a clever riddle, all of which help to keep students engaged. But if they could be interested in the math itself, that would be a significant achievement.

The difference between doing real math and merely learning a procedure manifests itself in the terminology used to describe what the students are doing. In class, the methods they are using, such as the Venn diagram to find the GCF, are actually called the “procedure,” as in: “Your procedure for finding the GCF is to do the factor ladder, write out the prime factorization, and then cross out the prime factors they have in common and write them in the middle.” I have always heard the word “procedure” in science when you are doing experiments and everything needs to be done in an exact and particular way, but I have never, ever heard it in math. I have heard “method” and “strategy,” which carry connotations of thought and creativity, used to describe the mathematical process, but never the dry and impersonal term “procedure.” If all they are supposed to be learning is a procedure, then there is no point. A computer might as well do it for them. Math class is not engaging unless it requires true thought, or is relevant to life.

Relevance to life, as I mentioned at the outset, is crucial. It is admittedly hard to make the GCF relevant to life. The best “practical” examples involve things like putting equal ratios of girls and boys on teams, or figuring out how many cookies to put on a plate.[3] These examples are not exactly something about which students will say, “Oh! I was just wondering how to figure that out, and now I know!” Thus, when I taught the class, I asked students to provide the numbers whose GCF we would find. I would ask a student for his or her favorite number, and then I would figure out a number to pair with it that would provide for an interesting GCF. This way, there was a connection, albeit somewhat trivial, between the student and the factoring.

Another topic we covered in the regular math class was “expanded notation”:

Standard form: 643.89

Expanded form: (6x102) + (4x101) + (3x100) + (8x10-1) + (9x10-2)

Standard form: 4002.07

Expanded form: (4x103) + (2x100) + (7x10-2)

Figure 6: Two examples of expanded form

If this seems obvious, perhaps it is (especially if students learned the place values, such as the “hundreds place” and the “tenths place” well), and in general students did not have much problem changing numbers to expanded form. I grant that there is some value in learning this, which is that hopefully it makes synapse connections about the place values and powers of 10, but it’s almost entirely useless. The main reason for students to learn it is that it is on the MCAS. Thus, when I reviewed expanded form with the class, again I asked for students to suggest numbers – if one student said 5 and another said 20 and another said 13, that would become 520.13 – and we translated that number into expanded form. Again, this is a rather superficial way to involve students in a potentially boring topic, but it is a small improvement.

The advanced class provided a better opportunity for relevant examples. The first class I taught was on ratios, rates, unit rates, and dimensional analysis. I incorporated relevance thus:

  • Ratios: There are 40 advanced math students out of 100 seventh-grade math students. This is a rate of 40/100, or 2/5. I also asked for other examples of rates, and one student suggested 4 paints (which I learned is a kind of horse) to 10 horses, or 4 paints to 6 appaloosas, or 6 appaloosas to 10 horses. This student was not quite clear on what could be a rate – whether it had to be the part to the whole, or whether it could be the part to the part or the “other” part to the whole, so we discussed all of these different ratios via her horse example.
  • Rates: I asked the students to raise their hands if they had a sled. Out of 20 students, there were 12 sleds, so we expressed this as 12 sleds / 20 students or 3 sleds / 5 students. One student didn’t understand the difference between a ratio and a rate,[4] so I asked the class for a volunteer to explain it to him, and then to make sure that he had understood what a unit rate was, he gave the example of 5 loaves of bread per 10 airplanes, which is indeed a rate.
  • Unit rates: We converted the rate examples to unit rates. 3 sleds / 5 students is the same as 3/5 sled per student, or 0.6 sleds per student. We discussed which would be a more useful way to express it – would you rather know that for every 5 students, there are 3 sleds, or would you rather know that there are 0.6 sleds per capita? The students had never heard the term “per capita,” which is admittedly an abstraction, so they preferred the former.

Dimensional analysis is the most unfamiliar topic of the four, and it is also the one that is the most interesting and useful. To introduce this topic, I created a (most wonderful, in my opinion) set of problems for homework the night before my class that would induce students to think in a “dimensional analysis way,” before they even knew what dimensional analysis was:

1

Six people are sharing nine pizzas.

(Don’t worry, they’re mini pizzas.)

(a)If each pizza is cut into four slices and everyone eats the same amount, how many slices does each person eat?

(b)If each pizza costs $5 and is cut into 4 slices, how much is it per slice?

(c)If each pizza costs $5 and everyone pays an equal amount, how much does each person pay?

(d)Multiply your answers to (a) and (b). Why is this the same as your answer for (c)?

(e)If you didn’t already, add units to your answers. For example, the units of (a) are slices/person.

(f)Redo (d), this time including the units when you multiply.

Answers

6(slices/person)

$1.25(dollars/slice)

$7.50(dollars/person)

6 x $1.25 = $7.50

Because the total cost paid by each person is the same as if they paid by the slice.

6 slices/person x 1.25 dollars/slice

= 7.5 dollars/person

(units on top and bottom cancel out)

1

Figure 7: An exercise to teach dimensional analysis

This worksheet has several important properties. First, it showcases a situation that might arise in everyday life, which students can thereby solve easily. (An example of this phenomenon is when a student is having trouble figuring out what 100 divided by 4 is, so you ask, “how many cents are in a quarter of a dollar?” and they can solve that just fine.) Second, because it might actually arise in a student’s life, it is relevant, and thus more interesting than a typical dimensional analysis problem that asks them to convert, for instance, five gallons into cups. Third, it presents the topic through familiar reasoning (they have certainly thought through a situation like those in (a), (b), and (c)), so that they see that the idea of dimensional analysis (parts (d) and (f)) is something that they have understood all along, and now they will just be learning a particular technique.