We recently started in on sequences and the book opens the chapter with problems that require students to create a table and come up with a pattern. The first one was about multiplying bunnies. The problem stated that we start off with two bunnies and, each month, each pair of bunnies has two babies. The problem wanted students to figure out how many bunnies there would be at the end of 12 months, as well as to determine the pattern and what family of equations described the growth.
I had one student who fairly quickly completed the problem. One of the extension questions that the textbook suggested was for students to attempt to write an equation that represented the situation. This student, after he graphed the growth, recognized that it wasn’t a linear growth pattern. When asked if the situation could be represented by a straight line, he said no, it was something else. However, when he was set to the task of determining an equation for the situation, he always came up with a linear equation. When I asked him what the graph of his equation would look like, he knew it would be a line. When I showed him the basic exponential equation (y=b^x), he recognized that was the form he needed to use, but even by the end of class all of the equations he was creating were linear equations.
I didn’t expect him to come up with the correct equation, but it was interesting to me that he was constantly writing linear equations. If I had been able to sit with him, he probably could have been guided to finding the correct equation. It just wasn’t possible for me to do that. I know that it is a topic covered later on in the chapter, and in another chapter in the textbook, so I know that he’ll eventually have the opportunity to work with the concept more.
Something I’ve noticed for a while in both my math class and my chemistry classes is this aversion to graphing. In both cases, the graphs were simple linear relationships, but students will go out of their way to not graph.
I noticed it with my math students in a station activity that I had them work on. Of the nine stations, three of the stations were ones that required them to graph. When I reviewed the work papers later, almost all of the students had ignored the directions and simply solved algebraically. When we began sequences and were working to build rules for either arithmetic or geometric sequences, students still had no desire to graph. They would happily create a table, but were missing on what kind of relationship there was until they saw it graphically.
In my chemistry class, it became noticeable when there were two consecutive labs that had a graphing component. One student flat out asked me why we had to graph (though I think the underlying comment was “why do we have to do all this work?”). One of the questions on the second lab asked them to determine which value of the x-intercept was more accurate: the one from the calculations or the one from the graph. At least 3/4 of the students said that the graph provided the more accurate answer, and many of those responses came from the students who were vehemently against generating the graph in the first place.
I think what I find the most puzzling is why students have such an aversion to graphing. In math, of the many methods we discuss regarding solving a system of equations, graphing is certainly the easiest. In chemistry, graphing your data is often much easier than any of the required calculations, even when you have to create a best-fit line to go with the data. Perhaps students missed out on the discussion of why we graph data or given equations. I think it is important that students see the benefit of graphing and that it does tell you something, either about an equation or a given set of data.
I’ve found it interesting how much difficulty some of my students have with multiplying and dividing by 1. I think they do know, at some level, that one times any number is just that number. But it seems like when it is applied in someway, they have a hard time seeing it.
For example, there was one problem that we did on the daily warm up. We were creating an estimate and had rounded our divisor to 1, and the dividend to 0.25. When I asked what my quotient would be I got a lot of confused looks and shrugs. With enough prompting, they could see that the quotient was 0.25, but I’m not 100% certain that all of them really saw it.
I’m not sure if this is because they feel overwhelmed by the fact there was a decimal in the answer, or if they really don’t have a solid grasp of that particular math fact.
One thing I’ve found interesting in student thinking is how students decide what operation a story problem calls for. When I was asking one student why he had decided one problem was a multiplication problem, it was because one of the items in the problem had units of something per something. Despite having worked to not connect specific words to specific operations, and stressing that you need to think about what the problem is asking, he still saw the word “per” and automatically assumed the problem was asking for multiplication (it was really a division problem). This was even with a specific set of notes, with a very similar problem, where we went step by step and solved it as a division problem.
It makes me wonder if there is a way to teach students how to look at story problems that allows them to move away from latching onto specific words and helps them to be more successful solving the problems. We’ve attempted having them write their own stories, in the hope that if they are writing a story using that operation, it would make it easier to identify that operation in a problem they are given. It seems to help some students, but other students still have difficulty in identifying the correct operation. I know it was something that even I had problems with up until college, when engineering courses present you with only story problems. I feel like there has to be a more efficient way, rather than hoping that constant exposure will make all the pieces click into place.
We’ve spent this last week covering decimal multiplication. The biggest issue my students have had is trying to line up the decimal points to do the multiplication. Students (and even I) find it confusing that, in addition and subtraction, you line up the decimal points to perform the operation, but multiplication and division are more based in whole number operations.
After we had covered solving decimal multiplication with both the standard algorithm and by using fractions, I provided students with 6 practice problems and allowed them to choose the method that worked best for them. The majority chose the standard algorithm, but many were still having problems with the multiplication because of the decimal point. One student called me over for help and said “I just don’t know what to do, I don’t know how to multiply these.” I rewrote the problem without the decimal points, he looked at me and told me “oh I can do this!”
I’m hoping that giving students a skills sheet that has multiple problems to practice with will help them. It’ll be interesting to see how many will still try and line up the decimal points to solve the problem.
At the beginning of the decimal unit, we included a variety of addition and subtraction problems on the warm-up for practice.
One day we had an addition problem, one that we were using to ensure that students knew how to “carry” when the sum in one column was a double digit number. The standard way to solve decimal addition is to line up the decimals, then to add from right to left. “Riley” chose to solve the addition problem a different way. He split each number into it’s whole number and fraction parts. He added up the fraction parts first, then added up the whole parts. Then he added the two results together to get his final answer.
If the problem was 12.37+4.86, he first added together 0.37 and 0.86 to get 1.23. Then he added together 12+4 to get 16. Finally he added 16+1.23 to get his final answer of 17.23.
It was a unique way to solve the problem. It isn’t the most efficient way to solve the problem and it opens up more opportunities for error, but it was fascinating to see how he view the problem and how to solve it.
During the ratio and rates unit, we provided a variety of warm-up problems that allowed students to practice the math. One of the unit rate problems was to find the price for one apple if 5 apples cost $8.35.
At the beginning of class during the warm-up, we paused the time and had students line up in the hallway. As we were walking out, “Michael” asked me what the problem was. I told him the problem, he thanked me, and we went out in the hall. After a few minutes, we returned to the classroom and everyone resumed working.
As I went around to check on students’ progress, “Michael” stopped me. He asked me to check his answer to the apple unit rate problem. As I looked over it, he told me that he did all the math in his head as we were standing out in the hallway.
It was impressive to me that he could do that kind of mental math. Most of my 6th grade students need to write out all of their division problems and that he could do the division without needing to see it in front of him was really interesting to me.