This knowledge grid describes two affective (interest and confidence) aspects and six types of cognitive knowledge in mathematics. As one moves towards expertise level in the cognitive domains, the affective domains, interest and confidence, become more and more cognitive. Conversely, "more expert" interest and confidence play an important role in moving students toward expertise in the cognitive knowledge domains. Links lead either to fuller explanations with student data. The grid was adapted from the Science knowledge typology of R. Shavelson and the Model of Domain Learning (MDL) of P. Alexander.
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Affective
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Acclimation
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Competence
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Proficiency
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Students are motivated to learn by external (often grade-oriented)
reasons that lack any direct link to the field of study in general.
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Students are motivated by both internal (e.g., intrigued
by the problem) and external reasons. Students still prefer concrete
concepts to abstractions, even if the abstraction is more useful.
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Students have both internal and external motivation.
Internal motivation comes from an interest in the problems from the
field, not just applications. External interest may also come from problems
in the field.
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Confidence |
Students are unlikely to spend more than 5 minutes on
a problem if they cannot solve it. When given an explanation, they want
minor steps explained. They are unable to complete problems requiring
the combination of steps.
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Students spend more time on problems. They will often
spend 10 minutes on a problem before quitting and seeking external help.
They are more comfortable with sketches of arguments. They can start
multi-step problems, but may have trouble completing them.
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Students will spend a great deal of time on a problem
before going to text or instructor. Students will disbelieve answers
in the back of the book if the answer disagrees with something they
feel they have done correctly. Can solve multi-step problems.
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Cognitive
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Acclimation
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Competence
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Proficiency
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Factual
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Students start to become aware of basic facts of the
topic.
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Students have working knowledge of the facts of the
topic, but may struggle to access the knowledge.
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Students have quick access to and broad knowledge about
the topic.
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Procedural
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Students start to become aware of basic procedures.
Can begin to mimic procedures from the text.
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Students have working knowledge of the main procedures.
Can access them without referencing the text, but may make errors or
have difficulty with more complex procedures.
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Students can use procedures without reference to external
sources or struggle. Student is able to find missing steps in procedures.
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Schematic
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Students have begun to put knowledge and procedures
into packets. Uses surface level thinking to form schema.
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Students have working packets of knowledge that tie
together ideas with comon theme, method, and/or proof.
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Students have put knowledge together in packets that
correspond to common theme, method, or proof, together with an understanding
of the method.
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Strategic
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Students begin to apply schema based on some strategy.
Uses surface level features of problems to choose between schema.
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Students choose schema to apply based on some heuristic
strategies but not all,
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Students choose schema to apply based on many different
heuristic strategies.
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Epistemic
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Students begin to understand the common notions of the
field. They begin to recognize that a valid proof cannot have a counterexample,
they are likely to believe based on 5 examples, however, they may be
skeptical at times
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Students are more strongly aware that a valid proof
cannot have counterexamples. They use examples to decide on the truth
of a statement, but require a proof for certainty.
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Students recognize that proofs don't have counterexamples,
are distrustful of 5 examples, see that general proofs apply to special
cases, and are more likely to use "hedging" words to describe
statements they suspect to be true but have not yet verified.
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Social
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Students will struggle to write a proof, and few words
will be written, even if they say the words at the same time. Variables
will seldom be defined, and proofs lack logical connectors.
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Students at this stage are likely to use an informal
shorthand that can be read like sentences for writing a proof. They
may employ connectors, but writing lacks clarity often due to pronouns
or poor use of terminology.
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Students in this stage write proofs with complete sentences.
They use clear concise sentences and emply correct terminology. They
use variables correctly.
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For each of these squares, one can go in depth. For example, epistemic knowledge appears to break into two separate pieces, namely, creative epistemic knowledge, the epistemic knowledge of how the discipline discovers truth, and validating knowledge, the epistemic knowledge of how the discipline verifies what others have discovered.
A central idea in the model domain of learning is that moving toward expertise in any one of the lower six categories will almost certainly cause one to move back in other categories. For example, as students develop knowledge schema, they begin to pack factual and procedural knowledge in ways that decrease the amount of factual information working memory. On the other hand, as students learn more methods of solving problems (an increase in factual knowledge), they often become strategically weaker initially. These were borne out by student JaB in his think aloud, where he used an inefficient schema for trying to solve the problem (showing poor strategic knowledge). However, the schema chosen was more advanced than what students at an earlier stage could draw upon.