If someone were to force me to identify the one mathematical gift I would choose to bestow upon students, I would whip out my magic wand and shout, "NUMBER SENSE! NUMBER SENSE! NUMBER SENSE!" (Come to think of it, that might just work! I'll have to try it later).
Why number sense?
Number sense is actually quite intuitive. So why teach it? Why reinforce something that comes naturally to students? That's a good question. And I have a good answer. The traditional teaching methods employed in mathematics have actually turned students AWAY from their inherent sense of number. Memorizing "steps" and focusing on traditional algorithms and/or formulas is about as far away from the development of number sense as are the temperatures in Death Valley and Antarctica.
In order to develop a sound sense of number, students must focus on place value, composing and decomposing numbers, understanding the relationships between and among operations, acquiring automaticity and fluency with facts and operations, observing the magic of mathematical properties, comparing and contrasting whole and rational numbers, and more. Students with a strong sense of number are able to estimate well and determine the reasonability of any given answer with ease.They can communicate clearly and effectively about their thinking and can use words, pictures, and numbers to defend their answers. They are extremely FLEXIBLE thinkers and learners and are equally at ease using mental math as recording their math in written form.
The research and data is definitive that number sense with regards to rational numbers (fractions and decimals) is where our students are falling the farthest behind. This is not just a Brassfield problem, nor is it even a problem exclusive to Wake County or the state of North Carolina. Students across the country struggle more with their conceptual understanding of rational numbers than of any other single concept in math.
Please consider taking a gander at this fantastic 2010 article by the National Counsel for Teachers of Mathematics entitled, "Using Number Sense to Compare Fractions."
http://www.ileohio.org/materials/Documents/Using%20Number%20Sense%20to%20Compare%20Fractions.pdf
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