The Integration of HandHeld Technology in the Classroom ITC:

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Greater Cleveland Council of Teachers of
Mathematics Nov. 29, 2006Cleveland, OH

• The Integration of Hand-Held Technology in the
  Classroom (ITC)
• Lessons learned,
• New students results what to learn from this
  trend
• Research findings implications for classroom
  practice,
• Antonio R. Quesada
• The University of Akron
• Department of Theoretical Applied Mathematics
• Akron, Ohio 44325-4002 aquesada_at_uakron.edu

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Scholastics use to define the terms before a
dialogue, so as Alice sayslets start at the
beginning

• What is Mathematics?
• The science of structure, order, and relation
  that has evolved from elemental practices of
  counting, measuring, and describing the shapes of
  objects.
• Encyclopedia Britannica
• A large majority of our population have serious
  misconceptions about what is mathematics and what
  is it that mathematicians do.
• Mathematical notation (symbols) the syntax
  (rules) needed to manipulate this notation have
  traditionally been the big obstacles for people
  to get to know mathematics.

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Public vision Math as a Foreign Language
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Mathematics is the language of the sciences!

• True, but lets not forget that notation is to
  mathematics like grammar is to writing.
• Grammar and notation are the necessary evils!
• You cannot be a good writer without grammar, but
  knowledge of grammar does not make you a novelist
  or a poet.
• Similarly, to do mathematics you need to be able
  to do calculations and to express your analysis
  algebraically/analytically, but mathematics is
  much more than that.
• Regrettably, many good students are lost in the
  notation and never get to know mathematics!

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Who can do Mathematics?

• Another generalized misconception is that either
  you can do math or you cannot! You are born with
  the gift!
• Thinking in this way is like saying that nobody
  can play basketball unless you play like an NBA
  player.
• The reality is that many students can do a lot
  better in mathematics, but like in any area of
  human knowledge you have to believe you can do it
  being willing to work hard consistently.
• Genius is 1 inspiration 99 perspiration (A.
  Einstein)
• I often remind my students of Polyas words,
  problem solving involves frustration, otherwise
  you are doing an exercise not a problem...

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What is mathematics about?

• Mathematics is about problem solving (modeling
  real-life and abstract situations to answer
  questions that we or others raise, finding and
  explaining patterns, exploring)
• Hence, it permeates not only all sciences but
  every area of human knowledge!
• Often our solutions are templates or models that
  apply to completely different examples.

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Prior to the Integration of Technology in the
Classroom (ITC)

• The accessibility of various topics in secondary
  level was fairly linear. In the US, problems were
  categorized as belonging to algebra I, algebra
  II, geometry, precalculus or calculus. Thus,
  optimization problems for example were first
  studied in calculus, since they typically
  required trigonometry and differential calculus.
• The introduction to functions and their
  properties was mainly analytical, graphs were
  better addressed in calculus, and very little was
  done numerically.
• Topics studied were in part limited by the
  constraints imposed by the extent of the
  algorithms/calculations needed and time they
  required. Therefore topics such as
  approximations, non-linear regression, recursion,
  and matrix applications were not (and could not
  be) included in secondary level.
• Mastering algorithmic calculations was essential
  to finish problems correctly. Hence, mechanics
  often took precedence over conceptual
  understanding, applications, and exploration and
  discovery.
• Teaching tended to be more teacher centered.

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Goals of this presentation

• To share some lessons learned from our research
  practice over the last 15 years on ITC,
  particularly on the use of handheld technology
  (HT).
• To review key research results and
  recommendations from these results about the use
  of handheld graphing technology (HGT).
• If time allows we will answer this question
• Can secondary students do new mathematics?

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In the beginning there were paper and pencil,
then
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Technologys Impact on Mathematics Instruction

• The integration of technology in the teaching
  learning of mathematics impacts every aspect of
  instruction course content, teaching methods,
  classroom activities, and assessment.
• Assumptions about mathematics curricula made in a
  time prior to the integration of technology in
  the classroom (ITC) are, in some cases, no longer
  valid.
• Thus, topics such as optimization, matrix
  applications, linear and nonlinear regression,
  recursion etc. are now accessible to students in
  earlier grades (prior to calculus).

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Some lessons learned from our research for the
last 15 years on the use of HT

• Calculators facilitate the use of the scaffolding
  method at every level

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Some lessons learned The scaffolding method
used at every level via technology allows to

• Bridge over cumbersome calculations when that is
  not our focus, facilitating
• Access to relevant algorithms traditionally
  excluded from our curriculum,
• More problem solving, more applications, more
  exploration
• A reduction on the time spent and numerical
  mistakes made in the implementation of
  algorithms.

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Some lessons learned Examples of the
scaffolding method

• In elementary school
• 1st level
• Learn to multiply (using manipulatives),
• memorize rules (using calculators via guess
  check),
• practice by paper pencil
• then 2nd level
• Explore patterns Solve many applied problems
  (using calculators and estimation)

• In secondary school
• 1st level
• Learn algorithms to solve equations or systems of
  equations, practice by paper pencil
• then 2nd level
• Explore Solve many applied problems (using
  calculators)
• Or, learn to solve problems bypassing the
  calculations input data, observe tendency, test
  different models until finding the best.

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Some lessons learned

• The rule of four
• Use algebraic, graphical, numerical, and verbal
  methods
• Pay attention to how students learn (Hands-on,
  exploring, conjecturing, communicating)
• Use multiple approaches to a concept

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Some lessons learned

• The teacher can always find the way of asking
  questions such that the students need mainly
  conceptual understanding to answer, no HT. Thus,
  one may ask
• In how many different ways can Joe, Mary, Paul
  seat in a row?
• Find the zeros of p(x)(x-a)(xb)2
• Find a polynomial of degree 3 whose zeros are 1,
  2, and -5
• Of course, there is nothing wrong by saying
• Memorize the trigonometric ratios for 300,450,600
• (No calculators allowed in tomorrows quiz!)

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Some lessons learned
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Some lessons learned If a calculator is
allowed, then we should require at least 2-digit
precision!

• But boss, l Just left out a decimal point.
  Don't I get at least partial credit? THOMA
  S

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Some lessons learned Visualization complements
the traditional algebraic explanation for many
problemsimproving students understanding

• Traditional textbook solution of

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Graphical explanation(A good picture is worth a
thousand words!)
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Some lessons learnedSolving inequalities force
us to think graphically!

• In 1992 less than 10 of Precalculus textbooks
  included some inequalities involving functions
  other than quadratic and rational functions
  (unpublished). The situation has not improved
  much (Quesada Smith, 2006)
• However, we know that thinking graphically
  algebraically increases conceptual understanding
  while reducing calculations and errors.
• Thus after teaching the algebraic approach to
  solve a new kind of equation, ask the students to
  solve inequalities. Ask them to solve

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Some lessons learned

• HT facilitates studying families of functions via
  transformations. Thus the graph of
  is obtained from the graph of the parent
  function by performing

a horizontal shift to the right followed by a
vertical shift down
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Some lessons learned

• Multiple representations provide a global
  approach for solving equations. (Solving an
  equation involving a continuous function is
  finding the zeros of the function.)
• Can we estimate from the table or the graph the
  solution of this anonymous equation of a
  continuous function?

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Some lessons learned HT creates the need for
awareness about the limitations of technology.
Consider a students question The two graphs
represent the given function, how is this
possible?
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Some lessons learned
The misleading 2nd graph is obtained by using a
window that exceeds the precision of the
calculator, producing truncation
error! Exposing students to these
limitations of technology, helps to demystify it!
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Does this create more work initially?In the
mid-nineties, when I commented about the amount
of additional work required from in-service
teachers to update in content, methods,
assessment, and integration of technology a
friend sent me this
Somewhere, someone, works more than teachers do
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Some lessons learnedRecommendations for
teachers interested in integrating HT into their
courses

• Start little by little! One or several chapters
  at a time. Learn, correct, and increase the scope
  of your change. Dont try to change your course
  in one semester.
• If possible work as part of a team. Share ideas
  and tasks. Network with colleagues.
• Learn from those preceding you (avoid
  rediscovering the wheel).
• Remember There is a wealth of knowledge
  available on the net!

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Some lessons learned On the use of inquiry in
the mathematics classroom

• We strongly recommend using inquiry-based
  activities at every level. However, our research
  shows that understanding and/or being familiar
  with a topic does not automatically guarantee
  successful application nor retention of key
  ideas. For that to happen, students also need to
• Verbalize what they have learned (asking them to
  write in a journal the key concepts properties
  learned after each inquiry-based activity helps)
• Memorize key definitions, properties, and
  algorithms
• Practice (it is hard to memorize algorithms
  without practice! HOMEWORK is essential!)
• Example We are all familiar with coins, but do
  we know the answer to these questions?
• Is the head on a penny looking to its left or to
  its right? Where is the date imprinted?
• Can you generalize your answers to a nickel, to a
  dime, or to a quarter?

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Check your answers!
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Lesson learned The impact of HT is enhanced by
using teamwork!Prospective employers, i.e.,
business industry have asked for it, research
favors it!
Students learn from each other and learn to work
together!
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Nowadays even pirates have retirement problems!

• A pirate hid a treasure in a tiny island
  when he was young. The island had a palm tree and
  two big rocks one was falcon shaped while the
  other looked like an owl.
• To hide the treasure, the pirate counted his
  paces as he was walking straight from the palm
  tree to the falcon-shaped rock. He then turned a
  quarter circle to the right and walked the same
  number of paces placing a stick in the ground. He
  returned to the palm tree and repeated the
  process, counting his paces while walking
  straight to the owl-shaped rock, turning a
  quarter circle to the left, and walking the same
  distance before placing a second stick in the
  ground. Finally he connected the sticks with a
  rope and buried his treasure beneath the
  midpoint.

Years later, concerned about the SS reform, the
pirate returned to the island looking for his
treasure and found that the two rocks remained
but the palm tree has long since died. Can the
reaches still be unearthed?
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After the ITC

• The distinction between activities appropriate to
  students at various courses and ability levels
  becomes less clear.
• The ability to bridge over cumbersome
  calculations via technology allows students at
  various levels to
• use technology to meaningfully explore concepts
  and problems previously proposed only to the most
  advanced mathematics students,
• and to extend the breadth and depth treatment of
  these concepts.
• It is feasible to change the focus to a more
  conceptual one, with relevant applications, and
  increased exploration.
• Teaching is becoming more student centered, with
  inquiry playing an increasingly bigger role.

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The mere formulation of a problem is often far
more essential than its solution, which may be a
matter of mathematical or experimental skill. To
raise new questions, new possibilities, to regard
old problems from a new angle requires creative
imagination and marks real advances in science.
Albert Einstein
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Are ordinary secondary students, i.e., other
than geniuses such as Gauss or Pascal,capable
of finding new results in mathematics?
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• 2. Some new results produced by students during
  the last 10 years

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There are numerous examples of mathematical
discoveries by secondary students during the last
ten years!

• We may wonder about not having these kind of
  students
• Or we may ask
• Are there common underlying factors on these
  students discoveries?
• Am I creating/promoting these factors in my
  classes?

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After looking at these findings and interviewing
some of the people involved I found the following
threads

• Common thread 1 Use of HT in particular of
  Dynamic Geometry Software (DGS). But, why?
• When properly used HT DGS facilitates the
  inquiry-based approach promoting this model

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Common thread 2 Students who find new
results invariablyhave been challenged by their
teachers!

• There are many ways of consistently challenge
  our students
• Do we ask our students to try to generalize their
  solutions to the problems we give them?
• Do we encourage our students to ask new questions
  and try to solve them, rewarding them for these
  efforts?
• Do we dare to ask to our students true
  challenging questions, questions for which we may
  not have an answer?
• If we dont challenge them, risking not knowing
  the answer to some of their own questions, we'll
  be perpetuating the myth of the teacher knows
  everything we will hardly be rewarded with
  their discoveries!
• We need to learn to say I dont know, let me
  think about it!
• (I can attest to the fact that the world does
  not end when you say this)

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RememberTeach your scholar to observeyou will
soon raise his curiosity. Put the problems before
him and let him solve them himself. Let him know
nothing because you have told him, but because he
has learned by himself. Undoubtedly the notions
of things acquired by oneself are clearer and
much more convincing than those acquired from the
teaching of others

Jean-Jacques Rousseau
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Problem A developer is building a new mall P
close to Akron, Barberton, and Cuyahoga Falls.
Find the location of the mall, such that the sum
of distances to the three cities is
minimal.(You may assume that the appropriate
place to build the warehouse is vacant!)
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Traditional Solution
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Solution by Bridget Connie (Arnie Egerbrensten
students)Reflect one vertex B upon the segment
connecting the centers of the equilateral
triangles constructed on sides
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The GlaD Construction Charles H. Dietrich
teacher at Greens Farms Academy, pose to his 8th
grade honor students Dave Goldenheim Dan
Litchfield the well known problem of how to
subdivide a given segment in n equal segments.
However, he added the condition of doing it
without using a compass. (June, 1995)
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Dave Goldenheim Dan Litchfield solution
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Can we find points on the real plane whose
coordinates are the real and imaginary parts of
the complex solutions of a quadratic equation?
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Solution by Shaun Piper, 12th grade, St.
Pauls School, Concord, NH

• Reflect the parabola upon y5 (line of symmetry)
  
• 2. Find the zeros of the new parabola
• 3. Rotate 90 the segment determined by these zeros

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Frank D. Nowosielski (Patapsco H. S., Baltimore
Cty, Mariland) asked his 9th grade students to
confirm MARION WALTERS THEOREM. If the
trisection points of the sides of any triangle
are connected to the opposite vertices, the
resulting hexagon has area one-tenth the area of
the original triangle.
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Ryan Morgan, after verifying the theorem, became
interested in finding out what would it happen if
the sides of the triangle were n-sected
(partitioned into n equal parts)
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• Ryan Morgans conjecture
• For n odd, if the central n-section points
  of the sides of any triangle are connected to the
  opposite vertices, the ratio of the area of the
  original triangle to the area of the resulting
  hexagon is
• (9n2 - 1)/8 to 1

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• 3. Research findings implications for classroom
  practice.

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Interesting Facts on the use of HT in the
classroom

• More than 25 of what was taught in mathematics
  changed after the introduction of the scientific
  calculator.
• As of the year 2000, over 80 of high school
  teachers in the US used HT in their classrooms
• Very controversial topic in education, however
  research on HT is still sparse
• Uncertainty and many unanswered questions still
  exist

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Differences in how HGT is used in classrooms and
in how its impact is measured contribute to
serious disagreements about the role of HGT in
mathematics education, its effect on students

• understanding,
• ability to perform routine procedures,
• facility with algebraic skills.
• attitudes toward mathematics
• as well as its pedagogical implications.

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The Three Meta-Analyses used as references for
this presentation

• Handheld graphing technology in secondary
  mathematics Research findings implications for
  classroom practice.
• Gail Burrill et all, 2002
• Sources chosen from 180 reports and 43 were
  used.
• A Meta-Analysis of the Effects of Calculators on
  Students Achievement and Attitude Levels in
  Precollege Mathematics Classes.
• Aimee J. Ellington, 2003
• Sources chosen from 86 studies 54 were used.
• The Graphics Calculator in Mathematics
  Education
• A Critical Review of Recent Research.
• Marina Penglase Stephen Arnold, 1996
• 103 studies used.

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Framework for synthesizing the research included

• How teachers students use HGT?
• What beliefs, knowledge, skills are learned
  applied?
• What is gained by HGT use?
• The existence of a treatment and control group
• What impact does HGT have on the performance of
  students from different gender, racial,
  socio-economic status, and achievement groups?

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Synopsis of the resultsI. How do teachers use
HGT and how is this use related to their
knowledge and beliefs about technology,
mathematics, and teaching mathematics?

• Teachers Philosophy guides calculator use
• Rule-based teachers are likely to perceive HGT by
  the affective reaction of students not as an
  enhancement to instruction
• Non-rule based teachers perceive HGT as an
  integral part of instruction and focus on the
  cognitive student reactions
• Teachers' beliefs methods highly influence how
  students use technology
• There is a shared belief that there are
  limitations to HGT and the importance of
  understanding the meaning of the numbers in an
  equation

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Synopsis of the resultsII. With what kinds of
mathematical tasks do students choose to use HGT?

• Students used HGT as a tool for
• computations
• transformations
• data collection and its analysis
• moving among different representations
• checking their answers!
• Students primary use of HGT was to graph,
  minimal use on tasks that did not require
  graphing.

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Synopsis of the results III. What
mathematical knowledge and skills are learned by
students who use HGT?

• Students who used HGT with curriculum materials
  supporting its use had a better understanding
  (than those who did not use the technology) of
• functions,
• variables,
• solving algebra problems in applied contexts,
• creating and interpreting graphs
• Students gains are directly proportional to the
  time they spend using HGT
• Students tend to use HGT with the methods taught
  and preferred by their teachers