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.