Our overarching question:
What do we mean by mathematical understanding?
Some subquestions and thoughts:
How does mathematical understanding relate to being able to follow and execute an algorithm?
How does mathematical understanding relate to our bodies -- gestures, kinesthetic experiences, senses, physical models (large and small), our memories and imagination of physical experiences,...?
How does mathematical understanding relate to language and to symbolism? If a concept cannot (yet) be stated in formal mathematical symbolism, does that mean that it is not fully understood? (See this Dance Your Dissertation video in mathematics...) Are formalisms an artefact of technologies like the printing press? Are they still relevant in contemporary times with photos, animations, videos and the internet?
Is there such at thing as 'grokking' a mathematical concept -- understanding it completely and intuitively, beyond words?
Sophie writes: "I constantly find myself gaining more understanding of topics I supposedly mastered years ago". Thurston offers at least 37 definitions for 'derivative' and says that the list continues, and there's no reason for it ever to stop. Is it ever possible to get to the bottom of mathematical understanding of a concept -- to reach the ultimate understanding of it (even a very elementary concept like addition or multiplication)?
How can we know if a learner is developing a deeper understanding of a mathematical concept? What do we actually mean by a deeper understanding?
David H. writes: "I cannot think of a single math problem that can be identified as 'exclusive' under Romero and Mari's definition. Even the simplest of problems can be solved numerous ways. One plus one may be solved with use of a number line, pictures, manipulatives, etc. " Is there any mathematical concept that can be understood in only one way?
What do we mean by mathematical understanding?
Some subquestions and thoughts:
How does mathematical understanding relate to being able to follow and execute an algorithm?
How does mathematical understanding relate to our bodies -- gestures, kinesthetic experiences, senses, physical models (large and small), our memories and imagination of physical experiences,...?
How does mathematical understanding relate to language and to symbolism? If a concept cannot (yet) be stated in formal mathematical symbolism, does that mean that it is not fully understood? (See this Dance Your Dissertation video in mathematics...) Are formalisms an artefact of technologies like the printing press? Are they still relevant in contemporary times with photos, animations, videos and the internet?
Is there such at thing as 'grokking' a mathematical concept -- understanding it completely and intuitively, beyond words?
Sophie writes: "I constantly find myself gaining more understanding of topics I supposedly mastered years ago". Thurston offers at least 37 definitions for 'derivative' and says that the list continues, and there's no reason for it ever to stop. Is it ever possible to get to the bottom of mathematical understanding of a concept -- to reach the ultimate understanding of it (even a very elementary concept like addition or multiplication)?
How can we know if a learner is developing a deeper understanding of a mathematical concept? What do we actually mean by a deeper understanding?
David H. writes: "I cannot think of a single math problem that can be identified as 'exclusive' under Romero and Mari's definition. Even the simplest of problems can be solved numerous ways. One plus one may be solved with use of a number line, pictures, manipulatives, etc. " Is there any mathematical concept that can be understood in only one way?
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