Saturday, July 24, 2010

Bloom's Taxonomy

I am relatively new to the teaching profession, so the first time I heard about Bloom's Taxonomy, it made sense to me. But I knew little about its origins. Like many things in education, there are strong feelings both for and against many educational concepts. There are those educators who are true believers and avid users of Bloom's Taxonomy and those who abandoned its use years ago.

One proponent of Bloom's Taxonomy states that it is the first idea taught in educational classes and the root of most ideas in education. One commenter to this blog notes a "love-hate" relationship with Bloom, as every lesson plan had to be written with elements of Bloom's.

One opposing view of Bloom's Taxonomy by Dr. Brenda Sugrue argues that Bloom's taxonomy is not supported by any research on learning. Sugrue also argues that the six levels in Bloom's taxonomy cannot be consistently applied to learning objectives. As an alternative to Bloom's, Sugrue presents a Content-Performance Matrix.

I believe that utilizing Bloom's taxonomy in algebra and calculus tasks can be very beneficial. When students come out of these classes, the ultimate goal is that the mathematics that they learn will be applicable to their lives in the "real" world. So, students should not only be able to recall and repeat algebra and calculus formulas, but they should be able to apply them. So once students have basic definitions and comprehension (Bloom's levels 1 and 2), they should be able to move to the higher levels of thinking. For example, in Algebra, once we finished the three different methods of solving linear systems, I engaged my classes in a discussion as to which of the three methods they liked best. For most students, it amounted to "I like method X because it's the fastest / easiest". Even with that type of statement, these students have moved to the Analysis level of Bloom's Taxonomy. As one moves to more complex mathematics such as Calculus, the thought processes can become more abstract and much of the work will be in the upper levels of Bloom's taxonomy.

So, overall, I believe that Bloom's taxonomy has as much usefulness in any level of mathematics as it does in other subject areas.

Wiki World!

Today was my first venture into editing a page within Wikipedia and I found it to be a relatively easy learning curve. Wikipedia is amazing in that anyone in the world can edit a Wiki page with seemingly nobody to review / approve edits before they are posted for the world to see. It is a self-policing community.

I updated the page created by Maria Drujkova on Multiple Representations. In particular, I added links to half a dozen other Wikipedia pages to make Maria's page a bit richer.

Great experience!

Friday, July 23, 2010

Problem Solving Article Review

I read the article “Teaching Students Math Problem-Solving Through Graphic Representations at the following website: Math Problem Solving. The premise of the article is that graphic representations can be used to teach students with learning disabilities to be effective problem-solvers. While the article focuses on students with learning disabilities, these techniques can be used for a more general student population.

The first phase of problem solving is for students to identify the problem type and then to organize and represent the key information in a graphical way using schematic diagrams. The second phase is to solve the problem by selecting and applying the appropriate mathematical operations based on the problem type. The article shows that using graphic representations to emphasize conceptual understanding can help children to solve problems. After the students have created their graphic representations, they put together a plan to solve the problem.based on the type of word problem being solved.

The article then goes on to advise teachers on how to evaluate students’ problem solving performance. Teachers are encouraged to examine students’ completed tests not only for correct solutions but also for strategy use. If students are consistently getting stuck at one point, more instruction should be provided.

The reported results of this problem solving method for students with disabilities in elementary and middle schools have been good. Students have shown dramatic improvements in problem-solving scores, have maintained their new skill set for up to four weeks, and have also had more positive attitudes toward strategy instruction. Teachers have also reported more effective problem-solving in their classes.

Again, though this article targets students with disabilities, I believe that these skills are more far reaching than just for students with disabilities in elementary or middle schools.

Thursday, July 22, 2010

Linear Systems - Exercises vs. Problems

Linear Systems

Definition: A System of Two Linear Equations in two variables x and y, also called a linear system, consists of two equations that can be written in the following form:

Ax + By = C Equation 1
Dx + Ey = F Equation 2

A solution of a system of linear equations in two variables is an ordered pair (x,y) that satisfies each equation.

In Algebra 2, there are three different ways to solve linear systems:

  1. Solve by Graphing
  2. Solve by Substitution
  3. Solve by Elimination

Here is a link to an applet used to graph and solve linear systems: Linear Systems Applet. The screen shot is here:

Here are some textbook “recipes” for solving linear systems by two of the three methods:

The Substitution Method

The Elimination Method

Here is a problem which requires applying the knowledge learned by solving linear systems. Note that this also addresses the area of multiple representations, discussed in Week 1.

Thursday, July 15, 2010

Multiple Representations

The following PowerPoint presentation explains the significance of multiple representations in the teaching of mathematics:

The primary website is

This PowerPoint presentation gives an overview of multiple representations. For example, it discusses eight widely used representational systems in the teaching and learning of mathematics:

  1. Written symbols
  2. Descriptive written words
  3. Pictures or diagrams
  4. Concrete models / manipulatives
  5. Concrete / Realia
  6. Spoken language / Oral representation
  7. Experience-based
  8. School word problems
The paper then goes on to give a specific example of a math multiplication problem in three different representations. Finally, the author provides a "Top Ten" List of reasons to use multiple representations in the teaching of mathematics.

This is an excellent paper. What I find interesting is that, with the exception of the three examples presented, most of this PowerPoint presentation is targeted towards oral learners, as there are LOTS of words and very little else used in this presentation. As an oral learner myself, I find this appealing, but I have found that even adding a few pictures to slides makes them more attractive and helps many students to learn better.

Blockfest Webinar

I attended the Blockfest webinar on Wednesday July 14th. The primary presenter discussed an organization which hosts "Blockfests" in various places around the country. A Blockfest is an event targeted towards children ages 2-8 and their parents. The children are broken down into smaller play groups, each of which has a trained moderator to work with the parents and help to relate the "block play" to mathematical concepts. The idea is to begin planting the seeds of mathematics in students at early ages. The events have been successful. "Success" for the parents means that they are learning how to interact with their children and begin to plant the seeds of mathematics concepts at an early age. "Success" for the children means they have had fun playing with blocks and their new friends for the duration of the event.

Though I teach high school mathematics, I believe that making math relevant and fun at early ages is a wonderful way to plant seeds of learning that make math fun. The more opportunities to make math fun at early ages, the more likely students are to engage in mathematics in elementary school and high school ages. I see many students who come into class with the attitude "I can't do math" from Day 1 of the year. I believe an event like Blockfest is one good way to counteract this kind of mindset.

Wednesday, July 14, 2010

Math Education Goals

Five Important Goals of Secondary Math Education

1. Teach students basic concepts and skills of the course which they are studying.

2. Teach students in engaging ways that they will remember in the future.

3. Teach students to apply math skills to real world problems.

4. Teach students problem solving methods which go beyond the mathematics that they are learning.

5. Teach students to work in collaborative work to simulate real world work situations.

Monday, July 12, 2010


Hi Everyone!

No musical interlude, but welcome to my blog site. I am still working on a theme song, so stay tuned ...