Saturday, July 24, 2010
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.
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.
Friday, July 23, 2010
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
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:
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:
- Solve by Graphing
- Solve by Substitution
- 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
The primary website is www.pictorialmath.com.
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:
- Written symbols
- Descriptive written words
- Pictures or diagrams
- Concrete models / manipulatives
- Concrete / Realia
- Spoken language / Oral representation
- School word problems
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.
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
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.