Macduff Dot Math Paper

7/26/2019 Macduff Dot Math Paper

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Mind, Mathematics & Science

Modeling an Integrated Approach

Rob MacDuff, Ph.D.

Director of Research

ASU Modeling Program

[email protected]


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ASU Modeling Program

modeling.asu.edu

David Hestenes - Director

Jane Jackson - Co-director


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Mathephobia

! Definition

A degree of mathematical maladjustment typified by

excessive anxiety or indecision.

! Symptoms

Me:Hi, I see you are taking courses here at ASU, whatare you enrolled in?

Student:I am taking religious studies.

Me:What does religious studies prepare you for?

Student:I wanted to take engineering but I am not very

good at math. I had a terrible math teacher in high

school.


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Mathephobia - Implications

! Percentage of students are afflicted by Mathephobia?

Quick estimate:

! 36% of grade 3 students do notmeet or exceed theArizona State AIMS test.




! 75% of 60 grade 12 of a AP physics students couldnotdifferentiate between a fraction and a ratio!


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Student performance:

Only 1/4 of 15-year-old American students are at

proficiency level 4 or higher. (PISA Executive Summary).

At Level 4 students can work effectively with explicit modelsfor complex concrete situations that may involve

constraints or call for making assumptions.

They can select and integrate different representations,

including symbolic ones, linking them directly to aspects

of real-world situations. Students at this level can utilize

well-developed skills and reason flexibly, with someinsight, in these contexts.

They can construct and communicate explanations and

arguments based on their interpretations, arguments and

actions.


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Research indicates 60 - 70% of adults suffer from math

anxiety and much of it is rooted in childhood experience.Our quick estimate suggests 75 - 85% is a betterestimate.

! Suppose we wanted a program to create Mathephobes!

! Would you find fault with choosing the current program?


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Mathephobia - Solution

! At this time are you afraid of math. Why or why not?Has this changed from your first day here?

! Oh my god LOL no not at all. But I was all 4 yearsand I never got math, and using dots us soooo great, it

really dose help for me. I am glade you guys invited thisprogram. I love it.

! I am not afraid of math anymore because when I used todo math the only thing we used was numbers and inthis math we use dots and it is much easier with dotsthen with numbers.

! Originally at the beginning of this year I was very afraidof math. Now I feel that it is easy and I do not have aproblem anymore.


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Mathephobia - Solution

! Your daughter did a very good job on last week'smidterm--82%

! You know she actually enjoys math right now. For thelast 4 or so years, we have had full out battles athome with math books flying, calculators beingthrown etc. during homework. She has HATED mathbecause she has never ever understood it. She is avisual learner and no matter what I/we did with hershe could not do it. She has has 3 different tutorsover the summers as well. She would never order fastfood (by herself) because she could not add things up

in her head and know if she had enough money withher. I have been so so proud of her new attitudechange in math. She seems to get whatever it is youare doing there. She even added something up in thecar the other day and blew me away (I almost had topull over).


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Blooms Taxonomy - Goals

! Knowledge: observation and recall of information

! Comprehension:the ability to understand information

! Application: use oflaws, theories, rules, principles, etc.

! Analysis: break down material into its component parts

! Synthesis: put parts together to form a new whole

! Evaluation:judge the value given a purpose


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A Few Main Ideas

! Knowledge: But though all our knowledge begins

with experience, it does not follow that it all arises

out of experience.Kant

! Understanding: To know an object is to act on it. To know

is to modify, to transform the object, and to understand the

process of the transformation. Piaget

! Comprehension: The ability to construct a mental model

of some particular situation. MacDuff


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Although the names may vary from author to author, the

essence of the process being described remains the same:

there is some iterative or recursive means by which the

concepts are interpreted, modified, transformed or refined.

Process of Model ConstructionSpecial Theory of Cognition

MacDuff

Lakoff, G. -- Metaphorical thinking where conceptual

systems reside in the unconscious.

Craik, K. -- Mental models mirror real world situations.Johnson-Laird, P. N. -- Mental model based deductive reasoning.Schank, K. -- Activation of mental scripts.Kant, I. -- conceptualization as being done in

accordance with the laws of the understanding


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A Few More Terms

! Model- a representation of structure

! Structures- relationships between the elements in a

a)

system (static)Or the change in the elements of the

b) system (dynamic)

! Structuresare perceived using conceptual systems

and expressed using representational systems in the

form of a model encoding inscription (MEI)

! Contextprovides constraints on types of models that

can be constructed


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Making Mental Models

! What are theelements?

! What are theoperations?

! What are theresults?

! What are therelationships?

What is the Model Encoding Inscription ? What is the System?


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I didn't buy any bread when I went to the store.IC DC.

When I went to the store, I didn't buy any bread.DC, IC.

I went to the store; however, I didn't buy any bread.IC; IM, IC.

I went to the store, but I didn't buy any bread.IC, CC IC.

I went to the store; I didn't buy any bread.IC; IC.

I went to the store. I didn't buy any bread.IC. IC.

What are the elements?

What are the operations?

What are the relationships?

What are the results?

What is the MEI?

What is the System?

When it comes to writing about not getting any bread at the store

MEI MEI


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Is it a system?

Elements?

Operations?

Relationships?

Results?

V I R

/ x

V

II Rx

V

R

R

I

VIR

Current(I)

Resistance

Voltage

?

Symbolic MEI?Graphic MEI?


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Challenging our Understanding

! The KEY to mental development is the ability to

ask questions that challenge your own

understanding.

" Let abe the number of questions that we know

the answers to.

" Let bbe the number of questions that we do not

know the answers to.

" For most of us, b>>> aa

b

Level of Knowing00 1

a

b


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+

+

=

= ?

If fractions are rational numbers,

then why cant we add them?

1

4

a

b d

c

1

4

1

4

b d (b d)+


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2 !3

6 !in

2in !3in

= 6

= 6in

= 6in2

How many different types of multiplication are there?

In what sense are they multiplication?

SystemModel Encoding

Inscription

wgf

w w

g

g

f f=x

x


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Model Construction - Cognitive

! A mental modelis a representationofthestructureineither a static or dynamic system.

! A systemis constructed from selected components of aconceptual system:

a) Elementsb) Relationships

1) Simple/Comparative (=, >,


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! A mental model is a representationofthestructure ineither a static or dynamic system.

! A mental representationis constructed in a conceptspacedefined by:

a) Background

1)Concept space defined by cognitive metrics

b) Foreground

1)A particular location in the space

c) Meaning

1)Relationship to other concepts2)Constraints imposed by external referents

d) Representation

1)Unknown


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A model encoding inscriptionis a symbolicrepresentationofstructure in a mental model.

A symbolicrepresentationis a model encoded inscriptionconstructed from selected components of a

representationalsystem.a) Elements (tokens)

b) Operations

c) Relationships

d) Results

1) Truth value (well-formed)


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Model Construction - Semiotics

! Syntactics

! Symbols without syntax

Torid fliss nipilate swittle noyl toof

! Symbols withsyntax

The torid fliss nipilated swittles with a noyl toof.

! Semantics

The tall man painted sceneries with a bristle brush.

Six divided by two plus two equals five

How much comprehension if all ones visualizes is 6/2 + 2 = 5

Thetorid fliss nipilatedswittleswith anoyl toof.


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Comprehension, Reasoning &

Understanding - redefined

! Comprehension: construction of multiple mental

model

! Reasoning: determining the relationships acrossmodels

! Understanding: being able to connect the symbols

(quantitative, relational) to their mental model and

the external referent.

! Meaning:is associated with the ability to constructmodels and express those models in multiple model

encoding inscriptions.


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Making Models (Comprehension)

1

4

d

b

b

d

a

c

"

r

Type Model

Count

Area

Line

Angle

4r(evolution)

1

"

c(ircumference)a(rc)


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Relating Models (Reasoning)

1

4

b

d

a

c

"

r

Models

= =

1

4d

4 bb

b =

Model of

Geometric

Structure

Model of

Arithmetic

Structure

Coordination of

Different

Structures

Relational

Proportional Reasoning

Quantitative

Algebraic Reasoning

b

d

a

c

ac

d

ad

c

1

4 d

=1

4

=bd

ac

"

r14

= =


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Proportional Reasoning

b

d

=

Model of

Geometric

Structure

Model of

Geometric

Structure

a

c

Algebraic Reasoning

b =a

cd

b = ad

c

Relating Models (Understanding)

Coordination of

Different

Structures

1

4b

d

a

c

"

r

Models

b ar

2=

a "rrad

=


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Something to Think About!

! My daughter Einat (7.5 years old, secondgrade) asked me to train her in arithmetic.

! G: "How much is 14 divided by 7?

! E: (quickly) "2." (She knows that division is

the inverse of multiplication, and she knows7 times 2 is 14).

! G: "How much is 12 divided by 4?

! E: (quickly) "3." (Same logic, again).

! G: "How much is 13 divided by 4?" (It justisn't fun to drill without throwing in somethought provoking questions, isn't it?)

! E: (after some thought) "I don't know."(there is no number she can multiply by 4to get 13).

! G: "What does it mean to divide 12 by 4?

! E: "Create groups of 4." (I was surprised - Iwas expecting "divide into four groups". Ithink I told her that once, and she picked itup magically...)

! G: "Very good. And if we create groups of 4from 13, what happens?" (Mimicking arepeating taking away gesture with my

hand, as if we're taking 4 objects at a timefrom a group of 13).

! E: "There is one left that we cannot group.

! G: "Exactly! We call this 'a remainder', and

we say 13 divided by 4 is 3 with a

remainder of 1. Now how much is 14

divided by 4?

! E: "3 with a remainder of 2.

! G: "15 divided by 4?

! E: "3 with a remainder of 3.

! G: "16 divided by 4?

! E: "4.

! G: "Very good! You didn't fall in my trapand didn't say '3 with a remainder of 4'!

! E: "Please give me some chain exercises.

! G: "How much is 8 plus 4 plus 2 ?! E: "14

! G: "How much is 8 divided by 4 plus 2?

! E: "I know! It's 2 with a remainder of 2!

! G: "What?!

! E: "We have 2 groups of 4, and then weadd 2, which do not form a group...

! What would YOU say at this point? Guy.


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Model ConstructionSpecial Theory of Cognition

MacDuff

Knowledge: But though all our knowledge begins with

experience, it does not follow that it all arises out of

experience. Kant

Understanding: To know an object is to act on it. To know is to

modify, to transform the object, and to understand the process of

the transformation. Piaget

Comprehension: The ability to construct multiple mental models

of some particular situation. MacDuff

Originally at the beginning of this year I was very afraid of

math. Now I feel that it is easy and I do not have a problem

anymore.

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