University students’ behavior working
with newly introduced mathematical
definitions
Valeria Aguirre Holguín
Presenter: Annie Selden
ICME-13
Initial research question:
What behaviors undergraduate students
present when working with definitions
new to them to
–
evaluate and justify examples and
non-examples,
–
prove statements, and
4
A bit of background
•
This study takes place within the context of a
transition-to-proof course for undergraduate
students with a variety of backgrounds (math,
engineering, pre-service teachers).
•
This course contains topics from
different areas of mathematics such as
set theory, functions, real analysis, and
abstract algebra; so that students can
experience a variety of different kinds of
proofs (Selden, McKee, & Selden,
2010).
•
23 students volunteered to be
interviewed.
6
Methods of data collection
•
Five definitions (not yet covered in class)
were selected from the course notes:
function;
continuity;
ideal;
isomorphism;
group.
•
On each definition 4-5 students were
interviewed individually.
•
Each student thought aloud and used a
LiveScribe pen. I took field notes.
•
Order of the 5 handouts:
Handout 1: definition,
Handout 2: extended interpretation,
Handout 3: example and non-example,
Handout 4: proof,
Handout 5: true-false statements
(inspired somewhat by Dahlberg and Housman (1997)).
9
Analysis
•
Iterative analysis of the data.
•
Two passes: first
I analyzed each
definition across the 5 handouts of each
student individually; second, I analyzed
each handout across all definitions and
all students.
•
Looked for, and categorized,
commonalities of responses and
actions.
12
Results
•
Students
’
previous knowledge can strongly influence,
not necessarily in a beneficial way, the acquisition of
new concepts.
•
Evidence suggesting that
the newer the definition
to
the student, and the less related to everyday language,
Results (continued)
Going back to this definition... what do you understand from it? What do you think of when you see this?
Well... y= mx+b, y being the function, I know how to find the slope for that, find a point on a line, I know how to do that kind of stuff, but the stuff that is saying in here I have no clue about. As far as functions go, I know this is one of the functions for a straight line; you could have like parabolas with the x square, whatever, you know. I know that there is
different types of, you know, sine, cosine and stuff like that but, again, I'm not understanding exactly what this definition is trying to tell me.
14
•
I observed a tendency to ignore the given
definition and use only their concept images
and prior knowledge (not necessarily
mathematical).
Results (continued)
“
a function is like a machine
, you
put something
in to get something out
. Like a machine that
makes copies of a newspaper in English and
Spanish
…”
[commenting on extended
interpretation]
ɋ
one input has three outputs, it fails the
vertical line
test
, it is not a function
Ɍ
[reasoning about a
Results (continued)
•
Students tend to neglect the details of a definition
in constructing a proof, they are not fully aware of
why they are provided. Very few of them seemed
to try hard to follow what the definition states.
There is an
if
and a
then
part, you see?
OK…yes, and they mentioned that in the classroom,
those are keywords
, this and this [remarking the "if"
and the "then"] ...
I guess I didn't notice that
,
I need
to pay attention to that, the words, the simple
Results (continued)
•
Students were initially reluctant to provide examples, but if I
probed a little further and provided time, students were often
able to provide an example, or at least to realize that their
Results (continued)
•
Observations suggest that
the newer the
concept (to them), the harder to provide
examples
or non-examples
but also the newer
the concept the less the interference
of
•
Many students in this study tended to
consider the details of mathematical
definitions
(only)
when they tried to construct
a mathematical
proof
,
but seldom when
they evaluated examples
, non-examples,
and true/false statements.
•
When their attempts to use the definition
were unsuccessful, they went back
to their
previous (mathematical or everyday)
knowledge.
Stages in learning to use mathematical definitions
As a product of the iterative analysis of data,
students were classified in different stages in using
mathematical definitions properly.
These
stages are not intended to be a definitive
set of steps through which a student must pass
in order to use definitions appropriately, rather they
describe and categorize the different behaviors
observed amongst the 23 participants of this study.
20 20
1. Understand there is a difference between
dictionary/everyday definitions and mathematical
definitions (as Edwards and Ward (2004)
suggested).
Can you think of a particular example? Can you tell me the properties that object would need to have in order to be a
group
?OK, so you have to have an element, or actually each… you have to
have a semigroup, each element has to have a subset... for example, I don't know, this is what I can picture, like the school. The school is a
[with emphasis] school, as a whole, but it has different colleges, the university has colleges, there is colleges that belong to the university and each college has departments, like for example like a g', so departments, colleges, you know, are composed for and there... you know, the university, which equal university. That's the way I see it.
Stage 1: Awareness
Stage 2: Contextualization
2. Understand when, and where, to use
mathematical definitions.
I have never seen this definition before [...].
I
have heard the word
function
from previous
math classes, calc I, II, III, trig and precalc […]
It helps with graphing,
but we are not going
Stage 3: Implementation attempts
3. Recall, look for, and attempt to use/follow
definitions, not necessarily with success.
So what is your approach? What are to trying to prove?
OK, so I'm trying to prove this function is continuous. So you just let out your specifications, let R be the real numbers, let it be a function defined by the 2x+3...and so... there is going to be... I want to use this definition... the definition for continuous and just sort of work that, so I didn't know if I should... suppose... that there is... but...[whispers reading the definition] I don't know if I need to prove that there is an 'a' also in the reals or... or not, or should that... yeah, I guess I'm not really quite sure... we have the reals over here and they are following this function 2x+3, and there is an element in here that is mapping to this element in here and...so that... I want to see...oh! I guess that would be, I guess I should use this... OK... so to me, maybe I wanna
assume that there is and "a" in R such that this and that is true but... yeah... I... I'm not quite sure Valeria [whispering]... I don't know I'm
Stage 4: Accomplishment
4. Use the definition successfully (within a
given context).
I have to prove that it
[
n --> 2n
]
is an
isomorphism
.
So
to be an isomorphism it has to be all of the above
. So
I want to prove that going from S equals integers and T
equals even integers that... both under addition... that...
that... the function theta of n equals 2n is an isomorphism,
meaning that...
[
He writes down: theta is
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References
Dahlberg, R. P., & Housman, D. L. (1997). Facilitating
learning events through example generation.
Educational Studies in Mathematics
,
33
(3), 283-299.
Edwards, B. S., & Ward, M. B. (2004). Surprises from
mathematics education research: Student (mis)use of
mathematical definitions.
The American Mathematical
Monthly
,
111
(5), 411-424.
Selden, A., McKee, K., & Selden, J. (2010). Affect,
behavioral schemas, and the proving process.
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