|
The Power of a Single Problem Working on one problem for an entire class is uncomfortable for many teachers. It feels like they aren’t working hard enough. But that is the problem with the traditional American lesson design; in that design the kids often aren’t working hard enough and the teacher is doing all the thinking. The traditional lesson design is the teacher teaches an idea or strategy. They often will model it in some way and then the students practice this in the same way. For a while, I thought if I followed this lesson design and used manipulatives it would be constructivist. It seemed that if the kids could physically manipulate something that it would force them to see the underlying math concepts. This wasn’t true for many kids. For instance, if we were using Base Ten Blocks, I could see that the flat was representing 100 and the long 10 and so on. But if the student hadn’t previously learned that concept, this manipulative could just be seen as a bunch of blocks. One definition of a successful activity in the classroom is one that teaches what we hope the kids will learn. A big idea we try to teach is number combinations. One activity we use for this is the two colored beans. Students are given a set number of beans and they toss them on their desk or on the floor. They then record how many are red and how many are white. The idea is that by doing this they will see that red + white = total beans. It is hoped that by playing games like this kids will become fluent with their facts. But could a student do this activity successfully by simply counting. The answer is yes. They count the white beans and then they count the red beans and if necessary they then count a third time and count all the beans. While counting is an important skill all children need to learn how to do, it is not the intent of this activity. This activity is trying to foster number fluency and using the above definition this well-meaning activity could be a failure for many students. If we change the activity a little bit we can eliminate this opportunity for counting. Instead of giving the student the beans, put a set number in a brown paper bag and label it with the number of beans. This forces the child to create a mental image of how many based on the number label. The number of beans you give a child could be different based on their level of development, so this one activity can easily be differentiated to meet the various levels within your classroom. A bag with 14 beans is very different than a bag with 7 counters. A bag with a mixture of tens and ones from the Base Ten Blocks would be even more challenging. This third bag is working towards the idea of multi-digit addition and subtraction. The child would reach into the bag and take out a handful of beans. They could be asked to estimate about how many they think they have, and then check or they could simply count how many they have. They would record what they have and then using this information they need to figure out what is still in the bag. This is a version of the game called “What’s Hidden?” Most adults think of it as a subtraction activity. Most kids think of it as a missing addend activity. For example, if we had 7 beans in the bag and drew out 4. Most adults would solve what’s missing by doing the equation 7 – 4 = 3, but most kids think 3 + o = 7 and count up. It is only over time that they come to see the connection between addition and subtraction strongly enough that they naturally use the inverse operation to solve for the unknown. But this connection can be taught and too often we don’t teach it. We just assume students will get it if we give them enough repetitions. This isn’t true for many students. They never see this connection and we, as teachers must make a point of making sure that they do because it is an underlying skill for Algebra. So even a simple activity like the red and white bean activity has the potential to teach important mathematical concepts. A single problem also has potential but in this case it isn’t so much to teach an important mathematical concept, as it is to show the connections between many concepts. A recent problem I worked on with students in Grades 3 – 5 is a good example of this. The problem asked the students to calculate how many laps they would need to run around a soccer field with the dimensions 75 yards by 115 yards to run at least a mile. It was amazing how many math concepts evolved from our discussions around this one problem. The first idea that emerged was that we needed to know how many yards were in a mile. This led to a mini-lesson in many classes on how do we get necessary information. We found this information in many places: dictionary, student planner, Ask Jeeves, useful information charts. Once we knew 1760 yards = 1 mile most students were off and running. I should mention that the lesson design I used was a Problem Solving Workshop. This is different than the traditional lesson plan in that I pose the problem without any front-end instruction. The students work on the problem in any way that makes sense to them using whatever tools they think will be helpful. It was interesting that most students drew a diagram of the field, but some just had a mental image of it.
One of the big ideas that naturally came up was perimeter since one lap was the perimeter of the field. What was interesting though was the variety of ways that students calculated the perimeter and this enabled me the opportunity to model some math notation.
75 + 115 + 75 + 115 = 380 (75 + 75) + (115 + 115) = 380 (75 + 115) * 2 = 380
The kids were able to understand why I used the parentheses because they seemed logical in this context. It made sense to use them because that was how they solved the problem.
Another idea that came up was how students did this computation. Interestingly enough many did it as mental math using a left to right method. {100 and 100 is 200 and 15 and 15 is 30 so 115 and 115 is 230.} One student did it this way and then wrote it on his paper like this:
1 115 + 115 230
When I asked him why he did that when he already knew the answer he told me because this was the right way to do it. When I pointed out the ideas was to accurately get the answer and the mental math procedure did that perfectly well, he got quite animated in explaining to me that he needed to show his work and this was the right way to show it.
Once students had calculated the perimeter, most solved it using an adding up strategy. Some did this as a series of equations and others made a chart.
380 + 380 = 760; 760 + 380 = 1140 and so on.
Others used a guess and check strategy. I think it is 4 laps. 4 * 380 yards is 1520 yards. Too small I need to adjust up.
Since the mile falls somewhere in between lap 4 and lap 5 students needed to decide if they should round up or down. This opportunity to make choices isn’t often presented in traditional math word problems where the numbers usually seem to work perfectly. In fact this potential messy situation of rounding up or down led to some of the best discussion of all. I was able to share more math notation by recording:
4 laps < 1 mile < 5 laps.
Again, the students had no problem with these symbols because they made sense in this context.
Some students wanted to know exactly how much more than 4 laps was a mile. This was solved in two ways. Some students started at lap 4 and figured out how much to add to get to a mile: 1520 + o = 1760, while others started at lap 5 and figured out how much to take away to get to a mile: 1900 - o = 1760. This second method led to some confusion, because they thought the answer was how many yards more they would run than 4 laps and didn’t realize right away that it meant the yards less than 5 laps. This was good though because it led to a discussion of what was actually going on and we referred to the diagram quite a bit. This was interesting because even though these students did the computation correctly they had the wrong answer. Since they were going beyond the problem posed, I assumed these were strong students and I got the sense they were not used to getting things wrong. It was good for them to have to struggle with the idea that the numbers worked but the conclusion was wrong.
Another really good discussion occurred in one Fifth Grade classroom. One student calculated the yards beyond lap 4 by using the integer divide key: 1900 ÷ 380 = 4 r 240 Another student did the same calculation using the regular divide key: 1900 ÷ 380 = 4.65
This led to a discussion of fractions and decimals and why we even need fractions and decimals.
One idea that didn’t come up that I wish had is the role estimation could have played in this problem. One teacher shared with me that his method was to round 380 to 400 and 4 laps was then about 1600 yards, so he concluded it must be 5 laps. This is really a nice way to solve this problem because it is so simple. I wish I had thought of it (or even better one of the students had thought of it) earlier in the week because we don’t often provide real opportunities to showcase when estimation is a useful strategy. The Problem Solving Workshop allows for this much more than the traditional lesson plan, which usually teaches estimation with naked numbers as a way to see if the computation is reasonable. Students don’t see much value in that, it is just more work. Because problems naturally provide a context, they are open to the possibility of using estimation to solve them. I wonder if no one thought of using estimation simply because we are not used to using estimation in Math class. Perhaps over time, that will become a natural strategy for students and teachers.
So as you can see, this one problem led to a lot of math. Yes we only did one problem, but what a lot of work those kids did. And the best part is they thought it was fun. They asked me when I was coming back with another problem. Go figure! Kids asking for problems.
References:
Burns, Marilyn. About Teaching Mathematics: A K–8 Resource, Second Edition
Fosnot, Catherine Twomey and Dolk, Maarten. Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction
Hiebert, James and Stigler, James. The Teaching Gap.
Van de Walle, John. Designing and Selecting Problem-Based Tasks |
