Teachers' Algebraic Reasoning while Learning with Web Book

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Teachers' Algebraic Reasoning while Learning
with Web Book
Beba Shternberg The Center for Educational
Technology, Israel ATCM 2004, Singapore
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Based on

• "Visual Math Functions " The Function Web Book
  Yerushalmy, M., Katriel, H., and Shternberg, B.

Published in Hebrew and English by CET, Ramat
Aviv, Israel. www.cet.ac.il/math/function/english
(2002)
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About Web Books

• The availability of interactive web texts and the
  search for interactions that would deepen the
  involvement of students and teachers in
  investigative activities, have created a lot of
  expectations for a new type of books web books,
  sometimes called interactive books.
• Within the context of academic publishing, there
  is not an adequate definition for the term
  web-book, and this is a source of confusion and
  therefore a barrier to uptake.
• Not many electronic publications actually make
  full use of the possibilities of the media to
  support interactivity and dynamic options.

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Our Goals in Developing The Function Web Book

• To develop a prototype of a new kind of
  mathematics textbook.
• To develop a media that allows educators to study
  various facets of interactive electronic writing.
  
• To develop an environment that improves habits of
  mind of the students.
• To develop an environment that improves
  professional growth of teachers.

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Design Considerations
Our design considerations were based on two
assumptions

• Teachers need a collection of Web activities that
  represent a coherent approach to algebra.

(thought a huge collection of activities is
available on various Web sites, teachers trying
to draw from the various sources face a real
problem.)

• Teachers and students need software that would
  allow wider use, easy access from home, and would
  support learning in and out of the classroom.

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Major Features of Our Web Book

• The Web book environment consists of a large
  database of algebra activities based on the
  function as the central concept.
• The collection of activities is organized in two
  main units linear and quadratic functions.
• Each unit is divided into several topics.

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The Structure of Each Unit

• Each unit includes activities of two types
  Analytical activities Modeling based
  activities
• Each activity consists of software tools (Java
  applets) and of tasks of various scopes, ranging
  from libraries of exercises through focused
  explorations to writing an essay.
• When a unit is chosen, the learner is given with
  an overarching exploration and a corresponding
  writing task.
• This provides a framework for what the reader
  should try to achieve. However, actual
  achievement is constructed by the learners
  themselves through the more detailed choices that
  they make.

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Teachers Algebraic Reasoning while Learning with
the Web Book
An example
During the 2002-2003 school year, our web book
has been the core of a distance learning
professional development program involving about
30 algebra teachers working with grades 7 to 9.
In this presentation I will analyze briefly those
teachers performance of one specific task -
Vertex paths.
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The Task

• Equivalent quadratic expressions Vertex paths

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Analysis of Teachers Performances
We have found it effective to adopt The Van Hiele
Model of Geometric Thinking for analyzing
teachers answers by focusing on their algebraic
reasoning which was based on graphical
representations.

• The Van Hiele Model of Geometric Thinking
• In the 1950s two Dutch educators, Dina and Pierre
  van Hiele, suggested that children learned
  geometry along lines of a structure of reasoning
  they developed. They defined five levels of
  development.

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The Van Hiele Model of Geometric Thinking.
Level 0 (Basic Level) Visualization Students recognize figures as total entities (e.g. triangles, squares), but do not recognize properties of these figures (e.g. right angles in a square).
Level 1 Analysis Students analyze components of figures (e.g. opposite angles of parallelograms are congruent), but interrelationships between figures and properties cannot be explained.
Level 2 Informal Deduction Students can establish interrelationships of properties within figures and among figures. Informal proofs can be followed, but students do not see how the logical order could be altered, nor do they see how to construct a proof.
Level 3 Deduction Students understand the significance of deduction as a way of establishing geometric theory within an axiom system.
Level 4 Rigor Students can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen as an abstract system with a high degree of rigor, even without concrete examples.
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Our (Van Hiele?) Model of Teachers Algebraic
Thinking
Level 0 (Basic Level) Visualization Teachers recognize graphs as total entities (e.g. a parabola, a line with a peak), but do not recognize properties of these figures (e.g. intersection points, vertices, symmetry).
Level 1 Analysis Teachers analyze component parts of figures (symmetry, parallelism), but interrelationships between figures and properties cannot be explained.
Level 2 Informal Deduction Teachers can establish interrelationships of properties within figures and among figures .Informal proofs can be followed, but teachers do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.
Level 3 Deduction Teachers understand the significance of deduction as a way of establishing a theory within an algebraic system. Teachers see the interrelationships and the role of undefined terms, definitions, theorems and formal proof. They see the possibility of developing a proof for a new phenomenon..
Level 4 Rigor Not defined yet.
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Level 0 (Basic Level) Visualization
Analysis of Teachers Answers

• Teachers recognize graphs as total entities (e.g.
  a parabola, a line with a peak), but do not
  recognize properties of these figures (e.g.
  intersection points, vertices, symmetry).

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Level 0 VisualizationExample
Orit changed the parameter m in the product
representation f(x)m(x-r)(x-s) of a parabola.
Orit When m is positive, the parabola smiles,
and when m is negative, the parabola is sad.
When we increase m, we get a parabola with a
wider wings spread.
Using the tool Orit gets various graphs and just
describes verbally what she sees. But she does
not recognize any properties of the graphs.
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Level 1 Analysis

• Teachers analyze component parts of figures (e.g.
  symmetry, parallelism), but interrelationships
  between figures and properties cannot be
  explained.

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Level 1 Analysis. Example
Tali changed the parameter b or c in a systematic
way in the polynomial representation
f(x)ax2bxc of a parabola.

• Tali Changing the value of the parameters b or
  c we get the fallowing graphs.

We see that when we change the values of the
parameters, we either get reflection of the
vertex on both sides of the X axis, or vertices
lying on a line parallel to the Y axis.
It seems that we can predict forward the vertexs
path from the representation of the parabola.
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We can see that Tali analyzes component parts of
the figures (symmetrical with respect to the axis
or parallel to it), but she does not explain any
interrelationships between the figures she got.
She seems to be glad with her findings and
convinced they are correct, and does not try to
explain them, despite the fact that she was sure
that the phenomenon she found in one specific
function was common to all quadratic functions.
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Level 2 Informal Deduction

• Teachers can establish interrelationships of
  properties within figures (e.g. because of the
  symmetry of a parabola, the x coordinate of the
  vertex is an average of the x coordinates of the
  intersection points of the parabola with the X
  axis) and among figures (e.g. the width of a
  parabola depends on the coefficient of x2).
• Informal proofs can be followed,

but teachers do not see how the logical order
could be altered nor do they see how to construct
a proof starting from different or unfamiliar
premises.
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Level 2 Informal Deduction Example
Ricky's solution refers to the "product" form of
quadratic functions (where r and s are the
roots of the function).
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Level 2 Informal Deduction

• Ricky establishes interrelationships of
  properties she tries to convince us that the x
  coordinate will not change, only the y
  coordinate, and suggests that we look at the
  shape of the graphs while changing the
  parameter m in the drawing.

• However, she finishes with this shape. Ricky
  does not see the vertex path as an entity - an
  object, and thus she does not try to find out
  what kind of object it will become. And she does
  not see any logical connection between the
  regular change of the parameter and the vertex
  path.
• Finally she does not see how to construct a proof
  that all the vertices lie on a straight line!

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Level 3 Deduction

• At this level teachers understand the
  significance of deduction as a way of
  establishing theory within an algebraic system.
  Teachers see interrelationships and the role of
  undefined terms, definitions, theorems and formal
  proof. They also see the possibility of
  developing a proof of a new phenomenon.

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Level 3 Deduction Example
Michal inquires the vertex path dealing with the
"product" representation of a quadratic function
f(x)m(x-r)(x-s).

• Michal Relying on the symmetry of the parabola
  we can suppose that the extreme point of the
  parabola is the arithmetical average of its
  intersection points with the X axis (sr)/2. In
  our trials we discovered that if we change only
  one parameter (lets say r), then the vertex path
  is also a parabola whose vertex is determined by
  the second parameter (that we keep constant). In
  addition it is interesting to note that if the
  given parabola has a minimum, the "path parabola"
  has a maximum.

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• Michal We will try to explain these findings. As
  we noticed, the x coordinate of the parabolas
  vertex is (rs)/2, and therefore it is clear that
  if we change one parameter (for example r) the
  vertex's location will change.We will explain
  the change of the parameters

Substituting the expression for s in the
expression for the y of the vertex we can
actually find the vertex path
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• And then Michal summarizes
• Michal We found that the vertex path is given
  by .
• It explains the other findings. If the given
  parabola has a maximum, meaning agt0, the vertex
  path parabola has a minimum becausealt0, and vice
  versa. The vertex of the path parabola is at r,
  and it touches the X axis at the point r.

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Level 3 Deduction

• Obviously Michal understands the significance of
  deduction as a way of establishing her theory
  about the vertex path in this specific case. She
  makes the connection between the known properties
  of parabolas and the properties of the parabola
  that she got by changing the given parabola.
  Although the tool performs the changes
  accurately, and Michal even finds successfully
  the expression for the path parabola in the
  specific case that she inquired, she does not
  stop there. The drawing serves only as a sketch,
  and she is determined to develop a formal proof
  for her conclusion. As a result, no conjecture is
  left without a formal proof.

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Summary
In this presentation I have demonstrated one
activity in which secondary school algebra
teachers experimented with a new kind of
mathematics textbook The Function Web Book,
developing which we tried to make powerful use of
technology to support visualization and
qualitative thinking.
We saw that the vertex activity encouraged the
teachers to visualize new interesting
mathematical aspects of the parabola - a
mathematical object well known to them. We have
discovered that not all teachers feel a need to
explain their finding formally.
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Our Benefit

• For us the developers and researchers the
  examination of the web activities reveals new
  aspects of mathematical thinking and promotes
  innovating of well known tools for their
  analysis.

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The end. Thank you.
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Level 0 VisualizationAdditional Example
Changing the parameter b in the polynomial
representation f(x)ax2bxc of a parabola..

• Alon While changing b alone, the parabola moves
  on a right diagonal when b is negative, and on a
  left diagonal when b is positive.A straight line
  connects all the vertices when b is positive, and
  another line connects them when b is negative.

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Alons conjecture is wrong.
One could refute it even by activating the
vertexs steps and decreasing the difference in
the change of the parameter. Both options are
available in the applet as we can see in the
following figure
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Yael speaks about the polynomial representation
of the quadratic function ax2bxc
Level 2 Informal Deduction Additional Example
Yael according to the formula
and since the vertex lies on the symmetry line),
the y coordinate of the vertex is
Thus every change in the parameter a changes the
location of the vertex and thus changes the shape
of the function graph.
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Although Yael feels a need to explain the first
statement about the change in the vertex
location, she does not give any clue as to why
the shape changes as well, and she is satisfied
with the drawing as evidence. Connecting changes
in the parameter a with changes in the vertex
coordinates, Yael establishes interrelationships
between properties of the quadratic function, and
almost proves the connections formally. However,
despite the fact that the task is named the
vertex paths and the learners are asked to find
different paths of the vertices, Yael does not
mention the path at all, but finishes with the
argument that the vertices will move and the
shape of the graph will change, and it can be
seen in the drawing.