Annin, S. A., & Lai, K. S. (February 2010). Common errors in counting problems. Mathematics Teacher, 103(6), 402-409.

This article was about the easily confusing differences within algebra combinatorics especially those in combinations and permutations. Giving 3 examples of problems where kids that have been taught the material and understand what is being asked of them but misread the problem and have faulty logic when creating solutions. It concludes with the idea that a variety of problems can help students gain a better conceptual understanding of the problems and become familiar with potential errors that they are then mindful of creating consistency and success in solving these problems.

This was a great article in the field of combinatorics and the presentation of the problems given. The main point of the paper was clear and interesting with a great answer to the problem posed; it was a great insight into student understanding and posited a simple solution to the main point. It stated that combinations and permutations are taught to be categorized into 4 separate groups: Permutations where repetition is allowed, permutations where repetition is not allowed, combinations where repetition is allowed, and combinations where repetition is not allowed. But in 3 given scenarios these 4 clear-cut categories are breached where the problem given could fall into several different groups, thus the solution method is not obvious. However, by knowing what is required from the problems, students could solve them based on understanding rather than even needing to categorize the problem. So I thought this was a great article on it's topic and would recommend it to anyone.

A Proof Vignette

Otten, S., Herbel-Eisenmann, B. A., & Males, L. M. (March 2010). Proof in algebra
reasoning beyond examples. Mathematics Teacher, 103(7), 514-518.

This paper is written for the purpose of convincing its readers that the abilities of proof technique can be made at an earlier age than they are now. Taking a classroom of students from an early algebra class, the teacher attempts to help students learn about proof technique through fraction equivalence. By taking two seemingly different fractions and proving that they are in fact equal; this is accomplished by the cross-product rule and given a few examples that this works, the teacher asks the students how one might prove a more general fraction equivalence. The class then creates the fraction problem a/b = 2a/2b and then extends this to a/b = na/nb. Then by using the cross product rule that they know they show that anb = bna thus completing their proof. Having given enough evidence that they students created this solution on their own with little guidance the paper does indeed show that kids are able to develop an understanding of proofs earlier than thought.

While students are required to give proofs in Geometry they do not understand what they are doing when they do so and they do not develop the required abilities of analysis from their proofs, so this paper attempts to convince the reader that students are able to create proofs earlier than they do now. However, in this paper there is given only one example of kids doing such and in the example the teacher helped guide the kids in their thinking. Yes, the students came up with the solutions to the steps on their own, but the teacher led their thinking in the direction it needed to go; without her, this would not have worked.
From my own experience of proofs in Geometry I found that it was very difficult to do so because the teacher gave little more than one example, without several instances of what needs to be done, kids cannot do the work well on their own.
I also found this paper to be more or less useless; while it does give an example of early proof, it is simply "a vignette" and does not cover more than one classroom. Therefore the paper can lead to nothing but a hypothesis and does not show that it is indeed true in any concrete way. So I found this paper to be somewhat lacking in its presentation despite the interesting idea that is brought forth.