Their Work vs. My Work

“Always start by celebrating their work!” I love that bit of advice that my math teacher gave us and hope to remember to focus on their work when I am a full-time teacher. I am learning how important it is to let it be their work instead of mine. I need to let my students give me their strategies to do a problem, rather than only offering them an “official” solution. We learn by doing, and that applies to thinking as well. If I do not give them a chance to figure things out for themselves, they won’t learn how to think critically. My job is to teach children to think for themselves, not mimic me.

One of the fun things we have been doing in my math class is to see how many different ways a math problem can be done. For instance, we were given the problem 12 x 15. Because we had been practicing mental math, simplifying calculations so we could do the problems in our heads, I approached this problem using this same concept. I remembered that when you multiply a number by 10, you just need to add a 0 to that number to come up with the answer. Thus, 12 x 10 = 120. That left 12 x 5 to be solved. I knew that 5 is half of 10, so I just divided the 120 in half and added the 60 to the 120 and got 180. It was that simple!

Others in my class did it differently but used similar mental math methods. Some multiplied 15 x 10 to get 150. Then they multiplied 15 x 2 to get 30 and added it to the 150 to get 180. Another started out the same way with 15 x 10, but then added 2 x 10 and 2 x 5 to get 30. Again, it ended up the same, but the process was longer. Even longer was the FOIL method, but it worked as well: (10 x 10 + 10 x 5 = 150) plus (2 x 10 + 2 x 5 = 30). One in our class even did the traditional place value multiplication method and multiplied 12 x 15 in her head. She began multiplying 2 x 5 to get 10 and then carried the ten and did mentally what most write on paper.

We even made arrays in a lattice model, also called a four-square model. We multiplied the numbers similar to a FOIL method, but in a four square place value format.

To make a long story short, I know this sounds complex, but it was actually quite enlightening to see how many different ways a small class can solve the same problem. The key is to let the students think about it first, and then have them share their different solutions. Students learn from students, and again we celebrate their work!

Launching a Problem

Word problems strike fear in many a child, not to mention some adults – like me, for instance. Whenever I hear a problem about one train leaving a train station at say - 4 pm, and another train leaving a different station at say - 5 pm, I cringe because I know I’m going to have to find out what time they will meet, and I can never remember how to figure that out. What would happen if I saw these kinds of problems differently? That’s the rationale behind launching a problem.

Interestingly enough, the goal of launching a problem is to inspire those creative juices – rather than finding a solution. Fortunately, I’m not teaching those train problems, but I did observe a group teaching third graders how to divide a “bunch” of lollipops. First lesson I learned is that third graders need to learn what a “bunch” means. If they don’t speak the language, it won’t work. After learning, or perhaps seeing what a bunch is, “a group,” we may begin. Note - when launching a problem, make sure they understand the terms being used.

Next, how do you keep the students attention? I think showing them bunches of lollipops would be a great way to begin personally. Then ask them how many bunches of five, in this problem anyway, they have to purchase in order for 27 students to each get one. It was amazing to see how complicated this was for the third graders. Knowing that five bunches would only feed 25, they understood. But they had 27! They knew six bunches would be too many, so they were confused. Can you blame them? What about those other three lollipops? They knew you don’t throw away lollipops, and it would be unfair for one to keep the extras. They have learned their lessons on ethics well.

I wonder, though, if the class had pretended to be at a store and given play money and told to actually buy enough real lollipops for 27 people if this would have been easier. I personally think it might have been, but it wasn’t the way we tried it that day to launch this word problem. If that doesn’t work, my other suggestion is to give them an example of a similar problem, perhaps an easier one like how many buses does it take to transport 40 students when one bus can only hold 30? I think modeling how one would do a similar problem is always a great beginning. I know I’m going to need another model before I get another train problem.

Choral counting

In music, a group singing in unison is called a chorus. In mathematics, one can be part of a similar group but it is choral counting, and instead of singing it is counting together. I had the opportunity to participate in choral counting recently in my mathematics class. My professor began the class by explaining that choral counting is like “skip counting” or saying a multiplication table. The difference, however, is that we were to say them together as an entire group - and were to do the 19s. I was not the only one who gasped audibly at first, never having practiced the 19s. I was pleasantly surprised, however, how it went.

She gave us a few minutes to prepare silently on how we would say our 19s, and we collectively began to search our minds for patterns, having been preconditioned to the existence of patterns by our previous lessons. Fortunately, it was not that difficult to find one after due consideration. When we began the count, our class was able to make a decent showing, having discovered that 19 is very close to 20. One could mentally add 20 and quickly subtract one to equal 19.

As we orally counted, my professor wrote the numbers down on her flip chart for us to see. The surprise came when she stopped us after we had only counted five 19s and asked us to look for patterns. We quickly stared at two obvious patterns. She had written 19, 38, 57, 76, and 95. The ones read: 9, 8, 7, 6, 5, and the tens read 1, 3, 5, 7, and 9. We were anxious to see how this pattern would continue. She did ask us to continue, and each time we went a little further in our count, though we always started at our beginning number 19 in order to keep the rhythm.

Besides the patterns we first noticed, we saw others emerge to join them. The ones counted backwards from 9 to 0 and then started all over again with the same numbers. Likewise, in the tens the next five numbers repeated the numbers 1, 3, 5, 7, and 9, but then it changed. The next two sets of five were 0, 2, 4, 6, 8 and again repeated the same 0, 2, 4, 6, 8. Later, at home I continued with this pattern and noticed there was an extra transition number 9 and then it went back to 1, 3, 5, 7, and 9 for the next two sets of five numbers. I became so excited in my exploration; I looked further and found still another pattern.

Now I saw there were five different numbers in the tens, five different numbers in the one hundreds, five different numbers in the two hundreds, and then there were six different numbers in the three hundreds. Interested in how this would play out, I continued and found three more groups of five numbers and then another group of six numbers. I learned that different multiplication tables provide different patterns. I also had the opportunity to later facilitate a couple other choral counts for both second and third graders and saw that they too became very quick at noticing patterns, but that is another story…

Children’s Math Strategies

When I was in elementary school, I learned that there was only one way to do math problems, and that was the way the teacher taught us. In my current math class, I am learning the opposite. There are many ways to approach math problems, and often the way they’re done is more important than their answers. I would never have believed this a few years ago, but today I am learning to see math in a very different way.

I had the opportunity to do two different one-on-one assessments with two different third graders, in two different classes, and at two very different levels. I was told that I could not help the children and only evaluate their strategies for finding basic mathematical facts. We were given half an hour to do four questions, but they were all challenging ones.

My first child, Mr. J, was very friendly and worked very hard trying to please me, but he had a very difficult time understanding the basic directions and needed a lot of coaching to keep him on task. He exhibited a low attention span, but I loved his attitude. Though he was unsuccessful at solving any of the problems, he did his best when he was asked to divide 10 large cookies between 6 children. He would give each child one cookie and the other four to his “brother, sister, dad and mom.” It made a lot of sense to me, and I hated to tell him that he couldn’t give the other cookies to his family and could only divide the 10 cookies among the six children. From there, though, it went downhill. He then tried to divide the cookies all in half but couldn’t figure out what to do with them when he ended up with 20 half-cookies.

My recommendation for Mr. J was to work on early math facts using direct modeling. Because the story problems were too abstract for him, he wasn’t able to find a formula for them, and I recommended direct modeling using counters or manipulatives to help him to see the concepts.

My other child, Mr. C, was very different. His single-minded focus was to finish the work, and his focus was intense. On his first problem, he read it and immediately wrote a mathematical equation out and solved the problem. In fact when I asked him why he did the problem that way, his response was “Because that’s what you’re supposed to do.” That sounded like something I would have said and done when I was his age. Ironically, he was able to solve all the problems except the cookie problem. Like Mr. J, he too wanted to start by giving each child one cookie, but after that he differed. He worked very hard to divide the rest of the cookies between the children and finally resorted to drawing it out. Interestingly enough, he was able to draw the division accurately but did not know how to convert it into numerical measurement. I believe he is moving into the counting strategy. He further demonstrated this on another problem when he successfully divided 246 pieces of candy putting 10 in each box. He solved this by continuing to subtract 10 numerically from the total separately each time until he no longer had 10 to subtract.

Earlier Mr. C had told he did not like math. I could see why as I watched him struggle and work so hard to complete his work. I wondered how it could be easier for both Mr. J and Mr. C and think about my earlier blog on subitizing and how seeing patterns can make math easier. A different child in Mr. C’s class knew that there were 10 tens in 100. That meant there were 20 tens in 200 and 24 in 240. Knowing that would have saved Mr. C a lot of time and perhaps made math a whole lot more fun for him. I’m looking forward to teaching children how to see more of these patterns and helping them to see that math can be fun.

Subitizing

What are some of the different ways students can add up groups of dots? Finding that out is the intent of the exercise called Quick Images. Two of us worked with a small group of third graders assessing their abilities to subitize, find the total number without giving them enough time to count each one individually. These dots were on flash cards and each flash card showed a different grouping. There were groups of twos, threes, fours, and fives. We flashed them an image for only a few seconds and then asked them to visualize the dots in their heads. We then gave them a chance to check their answers by flashing the dots in front of them once more for another three seconds. After this we asked for their totals.

The students did have one advantage. This exercise had been demonstrated to them, and they were now alert for patterns, and they found them. I was amazed, however, at how many different patterns they could find. They circled the ways they had grouped them, so the rest of us could see how they were able to come to their answers. Because they were asked to explain their methods to the rest of us, such as dividing the dots into groups of threes or sixes, it became evident to the students while explaining themselves if they were correct or not, and we all learned from each other.

One interesting development I noticed was that the students who were struggling with other mathematical concepts were excelling in this exercise. These students saw this exercise as more of a game than an assessment, and they were having fun. This was evident in their joyful faces and the excited way that they participated. Gone were the anxious faces. They were in their element, and it was like the playing field had been made even for them at that moment.