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How is the mathematics that we teach changing?
Johnny W. Lott
Past President
National Council of Teachers of Mathematics
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Is mathematics curriculum changing?
What was your first grade class like?What math did you learn?Has that changed?Should we be changing?Why?
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Is the mathematics curriculum changing?
What was your middle school like?What math did you learn?Has that changed?Should we be changing?Why?
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Is the mathematics curriculum changing?
What was your high school like?What math did you learn?Has that changed?Should we be changing?Why?
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How did you answer?If your school is very much like today’s
school, then…
It was very progressive when you were there, or…
Today’s school is very backward, or…You’re very young!
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From grade school
• How many different ways did you learn to add?
• Did you learn geometry?
• Did you learn about statistics?
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From middle school
• Did you learn how to add, subtract, multiply and divide whole numbers every year?
• Did you learn how to compute with fractions, percents and decimals?
• Did you learn geometry?
• Did you learn about statistics?
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From high school
• Did you learn to multiply binomials?• Did you learn to factor?• Did you learn to do two-column proofs?• Did you learn to do flow proofs?• Did you ever see a proof?• Was geometry anything but proofs?
• Did you learn about statistics?
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Your age can be judged by your answers, or…
• You went to a progressive school, or…
• Your school was backward, or…
• You lie, or…
• You truly have Alzheimer's.
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How should we be changing today?
• Students still have to learn the basics!– Do we want them to be able to add,
subtract, multiply, and divide?
– Silly question--of course!
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Do we want them to learn by rote?
• That is a possibility!
• So is learning different methods of computing.
• So is using technology to learn about numbers and computing.
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Fact:
• Research tells us that students learn when four-function calculators are used.
• They still learn to compute as well as they ever did.
• They are better problem solvers.
• Hembree and Dessart, 1986 and 1992
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What else can they be learning with numbers that wasn’t done in the past?
• They can be learning why they are doing what they are doing at the same time!
• They can be learning algebraic thinking at the same time!
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Computation problem
• What happens when you multiply any two-digit number by 101?
• This one is for the purists!
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Computation problem
• No-Cal candy bars cost $2.09. If the original price each was $0.12 but the price has been inflated, how much does each cost now?
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Message
• Problems have changed and teaching strategies have changed!
• Basic basic skills did not!
• But basic skills have changed!!!!
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New/old Problem
• What is the pattern below if it continues?
7, 13, 19, 25, 31, …
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What is the pattern?
• Add 6.
• Skip count by 6.
• Arithmetic sequence with a difference of 6.
• Others?
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What is a formula for the pattern?
• First term is 7
• Next is current + 6
• Recursive formulas were not heard of when I was in school!
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Questions
• Would you accept this as a formula for the pattern?
• Up to grade 3?
• Up to grade 6?
• Up to grade 9?
• Up to college seniors?
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What did many of you expect?
• Remember: 7, 13, 19, 25, 31, …
• What is the common “add 6” pattern?
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6, 12, 18, 24, 30, What is the pattern here that 3rd graders know?
Number of term with term below1 2 3 4 5 6… n6 12 18 24 30 36…7 13 19 25 31 37…
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Does this process work for any arithmetic sequence?
• This is algebraic thinking!
• Does this type of process have a use in today’s world that did not exist when I was in school?
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Think
SPREADSHEETS
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What else might be changing?
• Transformational geometry--flips, slides, turns, and glide reflections
• Matrices
• Data Analysis– Did you know that only bar graphs and
circle graphs existed when I was in school?
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Some graphical representations are true infants!• Stem and Leaf Plots
• Box plots
• Dot plots
• Every grade school child should know how to use these by the end of middle school.
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Where do we go from here?
• I don’t know;
• NCTM doesn’t know;
• Mathematicians don’t know;
• No one does, but…
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All know that we can’t go backward.
• Well, nearly all!
• Can we sit still?
• Unfortunately, no!
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What can we do?
• Realize that WE must continue learning.
• Realize that FAMILIES must continue learning.
• Realize that Principles and Standards is the most visionary document on the math scene today.
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Realize that neither NCTM nor you can sit still.• Go to conferences.• Read the literature.• Read newspapers.• Read literature of today.
• LEARN HOW THE WORLD AROUND YOU IS CHANGING.
• YOUR CLASSROOM WILL BE AFFECTED.
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Johnny W. LottDepartment of Mathematical
SciencesThe University of Montana
Missoula, MT [email protected]