Teachers' Algebraic Reasoning

while Learning with Web Book

Beba Shternberg The Center for Educational Technology, Israel

ATCM 2004, Singapore

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)

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.

About Web Books

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.

Our Goals in Developing The Function Web Book

Design Considerations:

Teachers need a collection of Web activities that

represent a coherent approach to algebra.

Our design considerations were based on two assumptions:

(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.

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.

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.

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.

Teachers’ Algebraic Reasoning while Learning with the Web Book

An example

Equivalent quadratic

expressions

Vertex paths

The Task

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.

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.

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.

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).

Analysis of Teachers’ Answers

Level 0 : VisualizationExample

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.

Orit changed the parameter m in the “product” representation f(x)=m(x-r)(x-s) of a parabola.

Using the tool Orit gets various graphs and just describes verbally what she sees. But

she does not recognize any properties of the graphs.

Level 1: Analysis

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

Level 1: Analysis. Example

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

It seems that we can predict forward the vertex’s path from the representation of the parabola.

Tali changed the parameter b or c in a systematic way in the “polynomial” representation f(x)=ax2+bx+c of a parabola.

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.

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.

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.

Ricky's solution refers to the "product" form of quadratic functions : (where r and s are the roots of the function).

))(( sxrxm

Level 2: Informal Deduction Example

Ricky: When we change m, the function becomes narrower or broader. The x coordinate of the vertex does not change and it stays the same during all the

changes:

since the function's zeros are r and s, and since the x coordinate of the vertex is their average). Yet when we change the parameter m, the y coordinate of the vertex changes. An increase in the absolute value of m causes an increase of the absolute value of y.

2

sr

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!

Level 2: Informal Deduction

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.

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: (s+r)/2. In our trials we discovered that if we change only one parameter (let’s 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.

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: We will try to explain these findings. As we noticed, the x coordinate of the parabola’s vertex is (r+s)/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:

)2(4

)(4

)2

)(2

()2

)(2

(

22

222 rrssa

rsasrrs

assr

rsr

ay

rxs

srx

k

k

Substituting the expression for s in the expression for the y of the vertex we

can actually find the vertex path:

22222 )()2()2(2)2(4

rxarrxxarrxrrxa

yk

22222 )()2()2(2)2(4

rxarrxxarrxrrxa

yk

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 a>0, the vertex path parabola has a minimum because–a<0, and vice versa. The vertex of the path parabola is at r, and it touches the X axis at the point r.

2k )rx(ay

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.

Level 3: Deduction

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.

Summary

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.

Level 0 : VisualizationAdditional Example

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.

Changing the parameter b in the “polynomial” representation f(x)=ax2+bx+c of a parabola..

One could refute it even by activating

the vertex’s 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:

Alon’s conjecture is wrong.

Yael speaks about the “polynomial” representation of the

quadratic function : ax2+bx+c:

a2

ac4b

a2

b 2

a4

ac4b2

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.

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.