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.

With or Without Algorithms

Warrington presents an interesting paper wherein middle-school aged students from all backgrounds find solutions to given fractions without the aid of any 'tricks' or techniques. And they do so with accuracy and without a lack of speed. Indeed, Warrington's class demonstrates the benefits of non-algorithmic learning in that "responses were forthcoming", and "[children could take] a straight computation problem and assign meaning to it by creating a word problem." Fractions seem to make more sense this way, and they have meaning. I also think that by invoking continual classroom discussion like Warrington has done in her classroom that the children will not be so timid to ask questions in the future and are more willing to other people ideas, other points of view for the same problem thus expounding upon their own knowledge and making it stronger.

In contrast however, students without any set of rules they can refer to result in (more often than not) nonefficient work and prone to occasional failure. Warrington admits that it's a lengthy process when she explains that for the understanding of a single problem (4 2/5 divided by 1/3) "went on for some time, and for many it carried over into recess." Also, without a way to check one's answer like you'd be able to with an algorithm or two, answers are less consistent. The obvious example is that given the same problem stated above, the children in Warrington's classroom all got the answer wrong except for one. So although the children were undoubtedly smart, they didn't have a solid way of doing fractions, only ways they came up with themselves.

Constructing Constructivism

Constructivism is a theory of learning by von Glaserfeld where knowledge is not simply received by the senses but is built up on an individual basis by the learning person. It is also the idea that the world we live in may not be reality, but is merely the creation of the subject's experience and observation; when one finds something to be 'true', it enforces the concept that it is real, however there is no way to prove that it is indeed reality. This is why it is theory rather than fact, indeed Glaserfeld points out his own paradox by saying, "to asses the truth of your knowledge you would have to know what you come to know before you come to know it." So you build knowledge upon experience and accept it as truth, whether it is fact or not.

von Glaserfeld then turns to the teacher and states that, "two things are required for the teacher [to build conceptual knowledge in students]: on the one hand, an adequate idea of where the student is and, on the other, an adequate idea of the destination." What Glaserfeld means by this is that teachers must be cognizant of each student and how their progress in understanding the mathematics is going for it is the student who has to do the conceptualizing and operating not the teacher. So the teacher's job is to generate understanding, rather than train specific performance. Because of this if I were to teach using a constructivist method I would encourage and accept student input and participation. I would also encourage students to inquire about what they are learning by asking thoughtful, open-ended questions and encouraging students to ask questions to each other. By promoting individual involvement and interaction each student will learn their own way and will see the perspectives of others further solidifying the understanding of the concepts.

Benny Mix-Up

S. H. Erlwanger's essay Benny's Conception of Rules and Answers in IPI Mathematics gives a striking example of the necessity for Mathematical understanding through the example of one clever -- but ill informed -- boy named Benny. By looking at Benny's progress in the IPI program, he is successful and quick, though when Erlwanger interviews with him he finds astonishing minunderstandings saying things such as 2/1 + 1/2 = 1. However, despite Benny's confusion, Erlwanger attempts to help the boy grasp mathematics for several weeks and Benny does indeed realize that something like 29/10 = 2.9; but Benny is still under the impression that 2/1 + 1/2 equals a whole. This then exemplifies that an understanding in mathematics must be built early on, or it is very hard to teach later.

Erlwanger further argues that teacher participation is required. That the teacher must be aware of the understanding that each student has. In IPI, "[Benny's] teacher can only become aware of his problems if he chooses to discuss them with her." And in this method of learning the teacher is more of a robot simply correcting answers by the key regardless of anything else. Benny realizes this himself and complains that "they have to go by the key... what the key says." To be sure, if the teacher isn't actively involved in teaching the curriculum then their is no understanding with their 'learning', because when one doesn't know something, the person with experience helps make it clear and the same is true for math.

Skemp Analysis [Blog 2]

Understanding can be broken up into 2 different types according to Richard Skemp:

• Relational Understanding, and
• Instrumental Understanding.

Relational understanding denotes a grasp of why something works the way it does and Instrumental understanding denotes a grasp of the simple 'how to' do it.

They are both types of understanding, and both aim for the learner to answer a set of problems. Furthermore, Relational understanding seems to encompass Instrumental understanding in that when one understands the 'why' of something, then the ability to do it and therefore 'how to' do it will also be present.
However, they are opposed in their methods. Relational understanding is both easier to remember and more adaptable to new tasks but it is more difficult and takes more time to achieve. While Instrumental understanding has more immediate rewards and gives answers more quickly than a relational mode would; however, it is also more superficial and more rules must be added for more scenarios.

Therefore, as an advocate of Relational understanding I believe that through the old adage, "practice makes perfect" explaining why something is the way it is to a student several times is preferred to a simple, single explanation or rule for how to do the same thing.

Blog Entry Numero Uno

1. Math to me is a puzzle, a challenge, a game. It's a way to manipulate and understand abstract structures ranging from numbers to graphs, to sets, to so much more. It's a neat way to understand the world.
2. Simply, the way I learn math best is doing it. Sitting in the classroom is great, and so is studying. But the way I learn best is sitting down and actually putting the pieces together.
3. I believe my students will learn best by seeing the math done clearly and concisely. After helping others through math and tutoring, I've found that shortcuts and tricks and cool facts are all just clutter to most. And a consistent, memorable way of doing any set of problems is the best way to do it.
4. Note taking! Foremost I believe this achieves so much because everyone learns differently, and on top of that, in class examples are given and explanations are given that students can copy and refer to later.
5. Clutter! Sooo many math teachers clutter up the material with confusing side-notes or underdeveloped ideas. Before moving a step further, the subject matter should be clear. That's what I think.