Transcript Mathematical Approaches that Support K-12

Mathematical Approaches
to Support K-12 Student Learning in
Mathematics
Rosalie A. Dance
University of the Virgin Islands
[email protected]
• Mathematical models of real world phenomena:
investigations that
o lead to mathematical concept learning and
o reap understanding of phenomena of importance and
interest to the student.
• Geometric illustrations of number, relations,
numerical operations, algebraic operations,
algebraic concepts,
– including tangents, limits, area under curves,
equilibrium values, rate of change, ….
– Rolle’s theorem for 10th graders.
• Play, experience, touch, see
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Support for development of intuitive understanding
of advanced concepts from earliest learning.
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2nd grade: What if I DID subtract a bigger number from a small one? What would
happen if I tried to take 3 away from 2? 2 – 4? 2 – 5?
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5th grade?: What number could go in the box?
3 + █ = 12.
3 + █ + █= 12.
3 + 2█ = 12 (given 2█ means two boxes of the same thing)
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When students have studied graphical representations of linear functions and
learned how to use them to solve linear equation: use a graph to solve non linear
equation.
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Look for patterns. E.g., the differences between consecutive terms in the
sequence of perfect squares is the sequence of odd numbers. why IS that?
1,4,9,16,25,36,…
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Notice limits. E.g., 1 + ½ + ¼ + 1/8 + …
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Communication, spoken and written.
• Students talk about Mathematics
to share their understanding of it,
to explain real world phenomena using it.
Students enjoy discourse, and they profit from it.
It helps them develop their thinking, and it helps them
recognize that mathematics is a human endeavor.
• Students should write mathematics, simply,
clearly.
• Students should also write about mathematics
they have done: letters, stories, essays.
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Problem solving. Do it!
Use Polya’s approach.
• Give instruction in how to
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read problems,
take note of what information is given in the problem,
take note of what is asked,
connect what is given and what is asked,
work forward; and work backward,
think and work geometrically where possible,
express everything clearly, simply, in algebraic terms,
check reasonableness of a tentative result,
verify results.
• Give positive feedback generously.
-Encourage pride in small steps and large ones.
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Integrate history of mathematics and
related history
• Especially include history from non-western
cultures and from under-represented groups
• Include women
• Include men
• Include Africa, Asia, Europe, the Americas; the
North, the South, East and West
• Connect to times and places they study in
history and literature
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Connections
Connect mathematics to the students’ world
AND to other mathematics
• CONTEXT.
Sickle cell & malaria investigation. Alcohol,
caffeine, heavy metals elimination from the bloodstream.
Population issues. Life expectancy variations in different
populations. Epidemics (the spread of communicable
diseases.
Students should uncover significant information
about topics that interest them by doing
mathematics.
Investigations associated with the mathematics
provide a hook for the math concepts in the
learner’s memory.
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Connections-2
• CONNECTIONS TO OTHER MATHEMATICS:
Connect geometric thinking and algebraic language
Connect new concepts to old ones whenever
possible.
Give hints about mathematics that lies ahead as it
relates to mathematics students are doing now
o Slopes in elementary algebra→what calculus is about.
o Areas →what else calculus is about
o Volumes in geometry class → volume problems in
calculus course
o Number patterns → function concept
o Number patterns and intuitive notion of convergence
of sequences
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Technology
• Use technology to support and enhance
conceptual understanding, thinking and
problem solving.
– Numerical methods
– Graphical methods
• Use methods of operation that help students
– to enjoy and appreciate their own skills and
– to eschew any use of calculating devices that
actually inhibits efficiency.
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Overcome early obstacles
• Teach number sense
– Fractions, decimals. (Eat 1/3 of your pie and give 2/3 to your mom to
share with your dad.)
– Signed numbers (Above and below sea level; gin rummy; in and out of
debt.)
• Play
– Sing the multiplication tables.
– Use the zero concept to understand addition of signed numbers.
– Act out un-doing addition and multiplication to develop inverse
operation concept
– Act out graphs; dance trig functions
– Act out recursive functions to discover their rule (e.g., Tower of Hanoi)
– Roll dice; what do you expect?
Etc.!
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