**“Compared to What?”**

Several studies show that comparison is beneficial to students. However, the writers of the article saw a gap in literature involving school-aged children. As the title suggests, their main goal was to see what comparisons provided maximum benefit for learning in 7^{th} and 8^{th} grade students. The issue being brought to light is the fact that, although we know comparison is beneficial and implement comparisons in a teaching environment, we may not be effectively utilizing the strategy. In many cases, what is being compared misses the mark of what *should be* compared.

What exactly are we comparing? What should be compared? Which types of comparisons are most effective? The authors look into three types of comparison for their study which varied in solution methods and problem features. One type compared similar problems with the same solution. Another type compared moderately similar surface features with the same solution method. The last type compared different solution methods to the same problem. They hoped to find out what type, or types, would be most effective in order for teachers to use to improve their use of comparison. They wanted to hit on the “three critical components of mathematical competence” which involved “procedural knowledge, procedural flexibility, and conceptual knowledge”.

After completing the study, the researchers found that, “comparing solution methods generally led to greater conceptual knowledge and procedural flexibility than comparing equivalent or different problem types”, but it “did not lead to greater procedural knowledge”. Being able to solve a problem using different solution methods indicates a certain level of understanding beyond purely getting an answer. It creates connections to simpler levels of calculating expressions with variables. They also found that in order to compare, it might be better to have something known to compare to. Going into the study, the students had some experience with equations and the steps to solving them. They did not go into the lessons completely blind. When comparing, it seems that you need some prior knowledge, or at least something worth comparing to. For example, it might be beneficial to be somewhat comfortable with a certain solution method before being exposed to another one rather than being thrown two comparable methods without introduction.

They concluded that comparing two different solution methods seemed to be most beneficial to learning if carefully implemented by teachers. Problems and solution methods must be chosen carefully. Also, comparing problem types might be a good form of comparison if the goal is flexibility of solution methods. It seems that the two types combined could potentially expand benefits if properly executed. Another conclusion was that comparison is only effective with careful support. Questions must be designed to support comparisons; they must encourage certain responses. They show that comparison is beneficial, but teachers must be careful as to what exactly is being compared. You can’t just compare anything and hope it works. In all, future research should be geared toward what is being compared in order to maximize benefits to students.

As I read through the study, I kept thinking, “this is a good idea, but what about other groups?”. The limitations of this study were addressed, but I think it runs deeper than classroom conditions and mathematical ability. In either case, comparison wouldn’t harm learning, but the issue lies in the fact that this study cannot completely be generalized to many other students. The only thing this study says is that a particular group of 7^{th} and 8^{th} graders, the majority of which are white, benefited more from the methods approach of comparison. Other parts of the study suggest that the authors pushed the results in that direction in regard to how they formulated the packets. Other factors need to be taken into consideration. There needs to be more research in order for this to have a bigger impact, at least, in my opinion. The study is far too limiting, which they made mention of. Overall, there needs to be more research done. There are so many other questions to be answered in regard to benefits: who will benefit? Does the research vary cross culturally? Does it vary depending on age? Does it vary based on environment? And the list of questions goes on.

While I think this was a great study to conduct and that they made great points, it further shows that more information needs to be obtained. There are many different possible combinations of studies that can be run, and that may be where the problems arise in retrieving data.

**Common Core: High School Algebra**

Personally, after reading this, I couldn’t help but think, “this was how I learned algebra”. Overall, we hit upon everything on the list. However, there were some parts about theorems that I am certain were not covered. After having gone over them throughout many college courses, I think that those theorems would have been great additions to the curriculum, at least in more depth. (Such as the Binomial Theorem and methods of proof.) Also, since technology keeps improving and is a constant in mathematics, it might be beneficial to have a technological component to solving algebraic problems. (At least being able to graph more complex functions using certain programs. However, only after graphic representation is understood. Otherwise students will come to rely on such methods without understanding what it means.)

I like the fact that it seems like the curriculum is not just geared toward pure computation. It seems like it is geared toward understanding particular concepts as well by building on certain basics, such as with equations and inequalities. The focus seems to be on getting a firm grasp on what it being taught. However, it does not always work that way given other outside factors that have become the norm among students, such as studying for the test rather than for purposes of learning.

Another problem is that many kids are thrown into algebra without prologue. Many classrooms are not uniform in the material being taught or the sequence in which the material is presented. All of the material presented seems fair to follow… on paper. However, prior knowledge, or lack of knowledge, may get in the way of understanding. Many of these topics should be taught sooner as a way of easing some students and encouraging others. Also, students might benefit from more proof within algebra. There seems to be an introduction into it, but not much of a focus. I’m not saying to have a whole month dedicated to proof, but it should be interwoven and given a bit more attention as an introduction to what may be learned in geometry. I believe that conceptual focus might be key to rooting certain mathematical topics: more answers to why questions instead of how.