A7: Bloom’s Taxomony of Educational Objectives

ED 253D introduced me to a theory that strongly resonates with my view of learning: Bloom’s Taxonomy of Educational Objectives. Depicted here, it is a model for how a student’s mastery of a subject improves with practice. Bloom proposed that a student’s proficiency develops in a fixed sequence of cognitive stages — from “Lower-Order Thinking Skills” to “Higher-Order Thinking Skills” — as follows:

  1. Remembering. First, students must be introduced to the new knowledge, and they must remember it.

    • Here is the mathematical recipe for how matrix multiplication works.

  2. Understanding. Next, students must get a sense of the new material: Explain it plainly and relate it to other familiar concepts.

    • The formula for matrix multiplication amounts to taking rows of this matrix and dotting them with columns of that matrix. It is just a way of doing arithmetic in batches. Here are some examples of matrix multiplication…

  3. Applying. Students practice the knowledge and get a feel for using it.

    • Now you try. Multiply these two numeric matrices. Next, try multiplying these symbolic matrices.

  4. Analyzing. Now that students can “blindly” apply this new knowledge, they need to integrate it with their existing knowledge.

    • Matrix multiplication is a generalization of normal multiplication of real numbers. Many of the rules for algebra on the real numbers work for matrices as well. For example, multiplication distributes over addition.
    • You can use matrix multiplication to do certain types of transformations on vectors — rotate them and flip them.

  5. Evaluating. Students explore the material more deeply, finding extensions and shortcomings of it.

    • Unlike a real number, a matrix is not always have a multiplicative inverse. What properties does a matrix need to have in order to have an inverse?
    • A matrix can rotate a vector, but not translate it. How would I have to augment a matrix if I wanted to rotate and translate it?

  6. Creating. Students invent their own way to apply this knowledge.

    • Here are some example areas where matrices arise: … Now, invent of a real-world math problem that can be solved using matrices and matrix inversion.