“the definition of insanity is doing the same thing over and over again and expecting different...
TRANSCRIPT
“The definition of insanity is doing the
same thing over and over again and
expecting different results.” - Albert Einstein
Presenter: Tony Hollowell
The 80/20 Principle
20% of the inputs cause 80% of the outputs
“The trivial many vs. the critical few”20% of motorists cause 80% of accidents20% of your carpet gets 80% of the wear20% of the population has 80% of the wealth20% of a businesses’ customers determine
80% of the total profits
What are the “critical few” for the Core 40 Exam? Write down all the things your students do
in a given year in the classroomTasks, projects, activities, etc.
The “Critical Few” for the Core 40 Exam
1. Example Problems• Students do not need to do more problems. They
need to do more specific types of problems. • Quality and consistency are important, not quantity
2. Graphing Calculator Activities/Enrichment
3. Assessments (summative)
Graphing Calculators in the State Standards
What I’m Proposing (and what I did in my own classroom) Students should have daily/weekly access
to graphing calculators in an Algebra I classroom
Why introduce graphing calculators in Algebra I? Preparation for future math courses and
assessments (AP Calc, SAT, ACT) Benefits
Reduction of computational errorsReal data analysisHands-on interaction with the contentMultiple representations of functionThey are allowed on the exam!!
The Fundamental Law of Graphing Calculators… Graphing Calculators cannot think for the
student. They can only supplement what they already know.
Corollary to the Fundamental Law of Graphing Calculators: Your students need to know Algebra for a graphing calculator to be useful
Introduction to Graphing Calculators Graphing Calculator Scavenger Hunt
Fundamental Law of Technology
My students, as a whole, will always know more about technology than I do individually
Graphing Calculator Activities
Three major content strands todayLinear EquationsSystems of Linear EquationsQuadratics Work on them individually or with a partner
Break!!
Questionnaire
Discussion of the Activities
Build critical thinking Involve real data Can be used to cover ALL major content strands
in Algebra I Easy to make!
NCTM Illuminations: TI Activities Exchange:
Core 40 Questions with a Graphing Calculator What you may do to solve these problems
may not be the same as someone else The calculator is only a tool (usually just
helps check an answer)
Graphing a Line
Using equations that are in slope-intercept formEx: Graph the function y = -2x – 3
Corollary: learn to get equations in slope-intercept form!
Finding Intercepts
Find the x and y intercepts of the equation 2y + x = 4
Slope through two points
Write the complete equation of a line that goes through the two points (-3,4) and
(-5,10)
Systems of Equations
Which value of b will make the graphsy = -2x +1 and y = x + b intersect at the point (-2, 5)?
A. -2B. 7C. 5D. 2
Exponents
Simplify the expression 2
3
5
10
Quadratics
Which of the following are solutions of the equation (t + 6)(2t – 9)= 0 ?
I. t = 5 II. t = -6 III. t = 4½A. IB. IIC. IIID. II and III
Quadratics
How many solutions are there to the equation x2 + 8 = -3x?
Authentic Testing Environment
Creating tests that mimic the expectations on the Core 40 examAllow formula sheetAllow graphing calculatorsTimed (50 minutes)Use of computers depends on what your
students will do when they take the ECA How often is this feasible at your school?
Building Assessments that remove Graphing Calculator dependency Students should learn how to think without
their calculator SAT and ACT allow for a graphing calculator
throughout the entire exam the AP calc exam does not.Enforces the idea that technology is only a
tool
Building Assessments that remove Graphing Calculator dependency Graphing Calculators are allowed on
almost every assessment (mine and standardized tests)
Test items are designed KNOWING that graphing calculators are available
Requires some creativity (usually the inclusion of variables, but not always)
Building Assessments that remove Graphing Calculator dependency Standard A1.4.2
Find the slope, x-intercept, and y-intercept of a line given its graph, its equation, or two points on the line.
Construct a question for this standard that will assess student comprehension irregardless of the use of a graphing calculator.
A note on critical thinking
If we do not teach critical thinking, our country is in trouble.
Any skill that is downloadable, wikipediable or googleable is not a skill that will provide economic viability
Teaching memorization is easy, teaching critical thinking is hard.
A note on critical thinking
Content should ALWAYS be subservient to critical thinking/analysis
This is nothing “new” “The ‘missing link’ in all systems of education
today may be found in the failure of educational institutions to teach their students how organize and use knowledge after they acquire it.”-Napoleon Hill, Think and Grow Rich (published in
1920)
Assessment Construction
Review the assessments in the handouts Do you think allowing a graphing
calculator for these problems will cause calculator dependency?
Implementation
A year-long process Which type of graphing calculators to use? How to get funding?
First, you need to be convinced of their use and effectiveness
80/20 analysis revisited
If the “critical few” is only 20% of your time, what do you do with the rest of your time?
Get Creative! Play Games! Analyze! Do Projects!
Side Note: If what you are doing is working, you don’t need to change much
Stop the insanity. Do something different. “The definition of insanity is doing the
same thing over and over again and expecting different results.” - Albert Einstein
Transforming your classroom
Algebra vs. the Cockroaches Finger painting and solving equations Colored pencil math 36 cards and puzzles If you think of something but you are not sure if it
will work, just imagine handing them a worksheet.
REFUSE to do things that are ineffective
My “Critical Few”
1. Example Problems• Students do not need to do more problems. They
need to do more specific types of problems. • Quality and consistency are important, not quantity
2. Graphing Calculator Activities/Enrichment
3. Assessments (summative)
"It is not the critic who counts: not the man who points out how the strong man stumbles or where the doer of deeds could have done better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood, who strives valiantly, who errs and comes up short again and again, because there is no effort without error or shortcoming, but who knows the great enthusiasms, the great devotions, who spends himself for a worthy cause; who, at the best, knows, in the end, the triumph of high achievement, and who, at the worst, if he fails, at least he fails while daring greatly, so that his place shall never be with those cold and timid souls who knew neither victory nor defeat."
–Teddy Roosevelt
A note on leadership
Leadership is not a position There is a difference between “being in
charge” and “leading”
Remember…
Your students are not a number The Core 40 Exam cannot measure every
important skill our students need to be successful in the world