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© 2014 Pacific Crest
17
The Challenge
Trouble reading mathematically-based literature
The Solution
Step 7
of the LPM (information) is an important part of preparing for a meaningful mathematical learning
experience, as should become obvious later in this chapter. Helpful tips include making a list of key
vocabulary, focusing on what is boxed, highlighted, or derived (examples and tools), trying to explain
how and why you can perform each step of an example, reviewing the summary and discussion points
to make sure that you agree to what you have learned and returning to recheck content if it seems that
something is missing.
The Challenge
Lack of or ineffective critical thinking skills
The Solution
TheMethodology for Elevating Knowledge (
LPMsteps 9,10, and 11
) requires that we explicitly sequence
our thinking, moving from directed to convergent and finally to divergent thinking and questioning.
Additionally, the processing of mathematical learning requires that we use cognitive domain skills that
match the level of learning we are working to construct: Level 1: Information Processing Skills; Level 2:
Constructing Understanding Skills; Level 3: Applying Knowledge Skills; Level 4: Problem Solving Skills.
Many learners fail to appreciate that, if they want to build the ability to apply knowledge (which is a
Level 3 skill), they must use thinking skills at that level. The more basic Level 1 Information Processing
Skills (such as recognizing differences or following directions) aren’t enough.
The Challenge
Not understanding methodologies or not being able to
generalize
The Solution
Identify and analyze generalized models (abstractions) of process knowledge (when they are available).
You can create your own by comparing and contrasting the examples from the book and laying out
the steps of the process for each of the examples.
Step 9
of the LPM (models) is critical for learning
mathematics because it is through the analysis of the methodologies and/or examples that you work
towards clarity and generalization of process knowledge.
The Challenge
Lack of understanding and clarity
The Solution
Step 10
of the LPM (thinking critically) is focused on the development of understanding through
analyzing methodologies and models using critical thinking (in the form of critical thinking questions or
CTQs). The goal of a critical thinking question is to produce a solid foundation for the knowledge by
clearly activating each cornerstone of meaning, making strong connections, addressing specific issues
in the use of steps, and clarifying special conditions or limitations with the use of the knowledge.
1.2 Applying the LPM: Math and Metacognition