Blog Entry #7

Teuscher, D., & Reys, R. E. (2010). Slope, Rate of Change, and Steepness: Do Students Understand These Concepts? Mathematics Teacher, vol. 103 (No. 7), 519-524.

Dawn Teuscher and Robert E. Reys' article, "Slope, Rate of Change, and Steepness: Do Students Understand These Concepts?" Teuscher and Reys bring up an issue that occurs with many students, in the fact that they don't understand the similarities and differences between slope, rate of change, and steepness when it comes to graphs of functions. They performed studies on tests that reflected questions that would appear on the A.P. Calculus exam, to see how students answered questions about slope, rate of change, and steepness. Unexpectedly, there were many students who believed that rate of change was the absolute value of the slope, or how "steep" a graph was, and gave incorrect intervals for answering questions about graphs and their rates of change. Teusher and Reys summed up the article by stating that teachers needed to do a better job of using the terms, and clarifying their meanings. This, in turn, will give students a better base for higher level mathematics, and prepare them more for future studies.

Teuscher and Reys bring up an excellent point that teachers need to better convey the concepts of slope, rate of change, and steepness, and explain their meanings, and how they are related. Doing so really will help students become better prepared for exams, like the A.P. exam, and for future, more advanced, mathematics. If students are allowed to continue with the incorrect ideas of slope and rate of change, when they take higher level classes, and subjects like physics, they will have an extremely difficult time, because they will have to change their views, and will not be able to connect their learning to the "knowledge" that they have about slope and rate of change. If teachers, however, explain these concepts thoroughly, it will make Calculus and Calculus based subjects much easier for students to grasp.

Bridging the Math Gap (Blog Entry #6)

Switzer, J. M., (2010). Bridging the Math Gap. Mathematics Teaching in the Middle School, 15(7), 400-405.

J. Matt Switzer's article "Bridging the Math Gap" addresses the fact that there is a discontinuity between the math that is taught in elementary school and the math taught in middle school. Switzer states that there are many times when teachers within middle schools don't understand specific algorithms and mathematical methods that students had been taught in elementary school. The teacher then cannot understand exactly what students are thinking, and how they are solving problems. This creates a large gap in the students transition between the two schools, because the teacher cannot make connections to what students have already learned. Switzer then goes on to give a proposed solution to this problem, stating that teachers should strive to learn and understand what is being taught in the elementary schools, and then tie those ideas into the newer concepts that they are teaching. In this way, students are taught for understanding, and are better able to see the connections in mathematics between what they already know, and the new things that they are being taught. To demonstrate, Switzer then uses the concept of partial products to illustrate his point.

Switzer's observations on the transition between elementary school and middle school are very well stated. Middle school math teachers have a huge responsibility of connecting what students already know to the things that they need to teach them. This would help to eliminate the thoughts among Middle School students that they are doing an "entirely different kind of math." To accomplish this, three main things would need to happen. First, teachers from elementary schools and teachers in middle schools would need to get together and work with each other in order to facilitate a smooth learning experience for the students, as Switzer states in his article. Second, teachers would also need to be flexible and able to adapt quickly. When teachers come across something that they have never seen before, they must study it and understand it so that they can make better connections with students who already have that knowledge. Third, teachers would need to consistently seek out new algorithms and methods of teaching the concepts that they need to teach in order to find the best way to connect with the minds of the students. Switzer's article does an excellent job of demonstrating these facts, and his points are valid for anyone teaching transition math classes.

MthEd 117 Blog #5

In Mary Ann Warrington's paper, "How Children Think about Division with Fractions", she gives examples from her class of students at the Atrium School in Watertown, MA, to illustrate the fact that children can develop their own mathematical formulas and ideas when you give them the opportunity to discuss them within the classroom. Warrington believes strongly that "children construct knowledge on the basis of what they already know," and that the best way to teach them is to let them teach themselves through discussion in the classroom. Warrington does not even tell her class the correct answers, but lets them guide themselves to discovering the concepts themselves. The greatest example she gives is when she asked the class what 4 and 2/5 divided by 1/3 was. Many believed that it was 13 and 1/15, but one student disagreed and brought up the idea of 13 and 1/5. As Warrington states: "This response was not only logical and mathematically correct but also a shining example of the autonomy that children develop when encouraged to think for themselves." Giving children the opportunity to think this much and develop their own mathematical ideas and theories can be quite a wonderful way to allow children to explore the world of mathematics.

However, there are flaws with teaching this way. The first is the time constraint. In public schools, this method of teaching would not work in any way. Warrington had actually been teaching fractions to these students for FIVE months, which, in a public school, is the time given to teach MANY different topics. Second, the issue arises, what if that one student hadn't had those thoughts, and nobody had thought that 4 and 2/5 divided by 1/3 is 13 and 1/15 was wrong? Would Warrington give them the correct answer, or would she let them go on thinking the way they were? Third, what about those who just don't have the ability to think about mathematics in this way. For some, math just does not ever click. And last, Warrington said that "children can and do invent ways to do sophisticated mathematics," but sometimes those ways are just plain wrong. Take Benny for an example. He had many ways of doing math, but only some of them were actually logical and correct. Warrington's method of teaching is a wonderful way of letting children discover the world of mathematics, but it would really only work in a private school institution, and never in a public school with the curriculum they have to cover.