Log 8

Teaching Secondary and Middle School Mathematics Chapter 7

General notes on chapter 7 - Teaching Specific Mathematics Content:

The bulk of this chapter was a sort of hodge-podge of various lesson ideas for teaching Algebra, Geometry, Data Analysis, and Discrete Mathematics.  The chapter also started out again discussing the value of an integrated curriculum (as if the author hasn't already driven that topic into the ground).

For Algebra it gives a nice example of creating a function to describe burning a candle, and discusses how a function can be represented by a graph, a table, an equation, or a verbal description, and how combining all of them gives a richer experience for the student.  This leads to a discussion of how the NCTM Principle and Standards states what all students should be able to do in Algebra:

• Understand patterns, relations, and functions
• Represent and analyze mathematical situations and structures using algebraic symbols
• use math models to represent and understand quantitative relationships
• analyze change in various contexts.

The Geometry section starts with a discussion of fractals and then talks about the NCTM standards for Geometry:

• Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
• Specify locations and describe spatial relationships using coordinate geometry and other representational systems
• Apply transformations and use symmetry to analyze mathematical situations
• Use visualization, spatial reasoning, and geometric modeling to solve problems

The Data Analysis section discusses data in the media.  The NTCM standards are:

• Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
• Select and use appropriate statistical methods to analyze data
• Develop and evaluate inferences and predictions that are based on data
• Understand and apply basic concepts of probability

The chapter ends with a discussion of what constitutes discrete mathematics, and how many students find this topic to be especially motivating because of its real world applications.

Question 3 - Use the Internet to find how students learn probability, examples of common misperceptions, and classroom teaching strategies.

I found a white paper at a New Zealand website http://www.stat.auckland.ac.nz/~iase/publications/3/2989.pdf where they discuss some of the common misperceptions as being:

• Representativeness - Not understanding that the size of a sample effects its quality for representing the population at large.
• Equiprobability - Not understanding that past events don't influence future random events (i.e., rolling a 5 does not decrease the probability of getting a 5 on the next roll).
• Outcome Approach - Expecting a probability of above 50% as a "yes vote" and not understanding that there is still a chance of the event not occurring.

I found another pretty good white paper at http://www.amstat.org/publications/jse/v3n1/konold.html which was a little more informative, but it's still a pretty dense topic to summarize.

Anyway, it seems like there are two strategies for overcoming probability misconceptions:

1. Run experiments - This gives the students a little hands on time and may make more of an impression than more formal methods.  The problem is that it takes a lot of data to come up with reliable results.  This can be done via computer programs, but this again tends to remove the students from the "hands on" experience.
2. Formal probability calculus - This gives students a formal explanation for why experimental results end up as they do, but they may be a bit difficult for students to understand.

Obviously, the best approach is to combine all of the above.

Question 8 - Advantages and disadvantages of integrated curriculum versus traditional

The biggest disadvantage of an integrated curriculum is that there is no agreed upon California standard of what is going to be taught at each class level.  Therefore, any student who has the misfortune to travel from one school to another (even within the same state) is not going to have the same math background as their fellow students at the same level.  Also, there is no way to measure how well one school is doing as compared to other schools (i.e., standards testing).

The disadvantage of an integrated curriculum to students who move has been kind of downplayed in the textbook, but I consider it to be a very, very serious issue.  Many Americans these days have to move around to make a good living, improve their living conditions, or care for a sick parent, and we shouldn't penalize their children for this.  If the government could decide at a national level how to standardize an integrated curriculum, this would not be a problem, but that is certainly not the case right now.

On the other side, it seems like an integrated curriculum would do a better job of teaching students in a way that the information would stick for the long term, rather than learning Algebra to pass the final at the end of the year, then forgetting it the next year.  Also, students would be able to relate better the different branches of mathematics.