### Have we got outlines

I thought that presenting three outlines might give you an idea of where I've been going over the past couple of years. I could make myself more crazy than I am trying to get these outlines to format properly, but I'm not going to.

The first is the outline for my original presentation. The presentation very closely followed this outline.

Development of Mathematical Thinking and Its Effect on Teaching Math to Adolescents

April 20, 2004

I. Introduction

II. Definitions

III. Evolution & Biology

IV. Biological Hardwiring

V. Is there a Math Gene?

VI. From Arithmetic to Mathematics

VII. How we create and do mathematics

VIII. How the embodied mind brings mathematics into being

IX. Mathematics and the embodied mind

X. Conclusions

XI. Teaching Mathematics

XII. If Devlin and Lakoff/Nunez are right . . .

XIII. The Mathematician’s Brain (this was a joke)

The second is the original outline for a paper I wrote during the spring 2005 semester (completed just a week less than a year after the first presentation). This was a starting point for the paper, but the end result was somewhat different - a lot more stuff from anthropologists and archaeologists as well as a brief analysis of mathematics as language. However, you can still see where I was going a year later.

I. Introduction

a. Appearance of Abstract Symbols

i. Where

ii. How common are they?

b. Do they indicate cognitive processes?

c. Do evolutionary psychologists address this question?

II. Presentation of research on Paleolithic cave art

a. What appears in the art

i. Figural/animal representations

ii. Types of abstract symbols

b. Frequency of representations

III. Presentation of research on Paleolithic artifacts

a. Counting bones

b. Sculpture and marked stones

IV. Evolutionary Psychology

a. Presentation of research

V. Conclusion

a. Do Paleolithic symbols indicate the start of mathematical cognition?

b. Does the literature support this conclusion?

And finally the original outline for my project proposal written in December 2005. As I started writing the paper, this outline began to resemble more of a sketch than an outline. As the paper has gone through various revisions and additions, the whole process has become more detailed and included more information from people who actually know what they're talking about (unlike me).

I. Introduction

a. Mathematical cognition – theories

i. Majority look at number sense and numerosity

ii. Lakoff and Nunez (2000) – theory of embodied cognition applied to mathematics

1. metaphor

2. schema

3. blending of constructs

b. specific construct

i. containment

ii. containment as class

c. underlying cognitive construct for set theory

d. previous studies focus

i. most are linquistic

1. studies show existence

2. studies are based on interpretation of English language texts

ii. study of containment concept in young children (Casasola & Cohen, 2002)

1. evidence for in children at ages 10 months and 18 months

2. study shows development from ages 9 – 11 months to 17 – 19 months

iii. study by Fischbein & Beltsan (1998)

1. flaws

a. using collection instead of containment – different metaphor

b. testing set knowledge instead of cognitive process

2. study results

a. in students

b. in teachers

c. shows correlation between student knowledge and teacher knowledge

e. present study

i. investigate presence of containment and containment as class metaphors

1. general student population

2. math majors before and after mathematics class that includes set theory

II. Methods

a. Participants

i. Approximately 40 students

1. total determined by number of students enrolled in math-225 (includes introduction to set theory)

2. have equal number of students not in math program

ii. half of subjects are math majors enrolled in math-225

iii. half of subjects non-math majors – control group

1. must not have taken math-225 or equivalent

b. Materials

i. Test designed to show evidence of the containment and containment as class metaphor

1. test to be created

2. reliability study – test-retest on general student population

3. validity study – method to be determined

ii. watch or stopwatch

c. procedure

i. two subject groups tested independently at start of semester and again at end of semester with same procedure

ii. provide informed consent

1. if professors have agreed, inform subjects of extra credit for participating

2. explain that test will be given a second time, may opt out at any time

iii. describe test without indicating what looking for – do not want to prime subjects

iv. give test to each group

v. note time to complete (start to completion by last subject) for each group

vi. collect tests

vii. thank subjects

viii. repeat steps ii through vi for retest at end of semester

ix. analyze data and write-up findings for presentation

1. apply findings to math education

x. meet with each group

1. explain cognitive process looking for

2. present brief description of results

3. thank subjects for their participation

III. Expected findings

a. Analysis

i. Results of test and retest to be analyzed between groups and within groups for evidence of containment and containment as class

ii. Use a factorial design (group x test) – look at results within and between students and control group

b. Hypotheses

i. Test(math-225) < Retest(math-225)

ii. Test(math-225) = Test(control)

iii. Retest(math-225) > Retest(control)

IV. References

So there you have it, the growth of an obsession. *grin*

The first is the outline for my original presentation. The presentation very closely followed this outline.

Development of Mathematical Thinking and Its Effect on Teaching Math to Adolescents

April 20, 2004

I. Introduction

II. Definitions

III. Evolution & Biology

IV. Biological Hardwiring

V. Is there a Math Gene?

VI. From Arithmetic to Mathematics

VII. How we create and do mathematics

VIII. How the embodied mind brings mathematics into being

IX. Mathematics and the embodied mind

X. Conclusions

XI. Teaching Mathematics

XII. If Devlin and Lakoff/Nunez are right . . .

XIII. The Mathematician’s Brain (this was a joke)

The second is the original outline for a paper I wrote during the spring 2005 semester (completed just a week less than a year after the first presentation). This was a starting point for the paper, but the end result was somewhat different - a lot more stuff from anthropologists and archaeologists as well as a brief analysis of mathematics as language. However, you can still see where I was going a year later.

I. Introduction

a. Appearance of Abstract Symbols

i. Where

ii. How common are they?

b. Do they indicate cognitive processes?

c. Do evolutionary psychologists address this question?

II. Presentation of research on Paleolithic cave art

a. What appears in the art

i. Figural/animal representations

ii. Types of abstract symbols

b. Frequency of representations

III. Presentation of research on Paleolithic artifacts

a. Counting bones

b. Sculpture and marked stones

IV. Evolutionary Psychology

a. Presentation of research

V. Conclusion

a. Do Paleolithic symbols indicate the start of mathematical cognition?

b. Does the literature support this conclusion?

And finally the original outline for my project proposal written in December 2005. As I started writing the paper, this outline began to resemble more of a sketch than an outline. As the paper has gone through various revisions and additions, the whole process has become more detailed and included more information from people who actually know what they're talking about (unlike me).

I. Introduction

a. Mathematical cognition – theories

i. Majority look at number sense and numerosity

ii. Lakoff and Nunez (2000) – theory of embodied cognition applied to mathematics

1. metaphor

2. schema

3. blending of constructs

b. specific construct

i. containment

ii. containment as class

c. underlying cognitive construct for set theory

d. previous studies focus

i. most are linquistic

1. studies show existence

2. studies are based on interpretation of English language texts

ii. study of containment concept in young children (Casasola & Cohen, 2002)

1. evidence for in children at ages 10 months and 18 months

2. study shows development from ages 9 – 11 months to 17 – 19 months

iii. study by Fischbein & Beltsan (1998)

1. flaws

a. using collection instead of containment – different metaphor

b. testing set knowledge instead of cognitive process

2. study results

a. in students

b. in teachers

c. shows correlation between student knowledge and teacher knowledge

e. present study

i. investigate presence of containment and containment as class metaphors

1. general student population

2. math majors before and after mathematics class that includes set theory

II. Methods

a. Participants

i. Approximately 40 students

1. total determined by number of students enrolled in math-225 (includes introduction to set theory)

2. have equal number of students not in math program

ii. half of subjects are math majors enrolled in math-225

iii. half of subjects non-math majors – control group

1. must not have taken math-225 or equivalent

b. Materials

i. Test designed to show evidence of the containment and containment as class metaphor

1. test to be created

2. reliability study – test-retest on general student population

3. validity study – method to be determined

ii. watch or stopwatch

c. procedure

i. two subject groups tested independently at start of semester and again at end of semester with same procedure

ii. provide informed consent

1. if professors have agreed, inform subjects of extra credit for participating

2. explain that test will be given a second time, may opt out at any time

iii. describe test without indicating what looking for – do not want to prime subjects

iv. give test to each group

v. note time to complete (start to completion by last subject) for each group

vi. collect tests

vii. thank subjects

viii. repeat steps ii through vi for retest at end of semester

ix. analyze data and write-up findings for presentation

1. apply findings to math education

x. meet with each group

1. explain cognitive process looking for

2. present brief description of results

3. thank subjects for their participation

III. Expected findings

a. Analysis

i. Results of test and retest to be analyzed between groups and within groups for evidence of containment and containment as class

ii. Use a factorial design (group x test) – look at results within and between students and control group

b. Hypotheses

i. Test(math-225) < Retest(math-225)

ii. Test(math-225) = Test(control)

iii. Retest(math-225) > Retest(control)

IV. References

So there you have it, the growth of an obsession. *grin*

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