Powerpoint of Valeria Aguirre Holguin Un

University students’ behavior working with newly introduced mathematical definitions

  Valeria Aguirre Holguín Presenter: Annie Selden

  ICME-13

  July 24-31, 2016 Hello! Regards from Mexico.

  What behaviors undergraduate students present when working with definitions new to them to

  Initial research question:

• – evaluate and justify examples and non- examples,
• – prove statements, and
• – evaluate and justify true/false statements?

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).
• The purpose of the course is to teach proof construction.

Background (continued)

• 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

  

Methods of data collection

• Five definitions (not yet covered in class) were selected from the course notes:

  function; continuity; ideal; isomorphism; group.

• Individual semi-structured task-based interviews were conducted (5 handouts

  

Data collection (continued)

• On each definition 4-5 students were interviewed individually.
• Each student thought aloud and used a LiveScribe pen. I took field notes.

  Data collection (continued)

• 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)).

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.

Big picture Function Continuity Ideal Isomorphism Students Handout 1 Handout 2 Handout 3 Handout 4 Handout 5

  

First pass

Second pass

  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,

  the less was the interference of inappropriate previous knowledge.

  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 .

  

Everything that I know, I guess, relates to a graph... y= sin, y= cosine,

y=tan... [goes one drawing graphs and bringing previous knowledge].

  Results (continued)

• I observed a tendency to ignore the given definition and use only their concept images and prior knowledge (not necessarily mathematical).

  “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] one input has three outputs, it fails the vertical line

  ɋ

  

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 words in between the symbols or what they're

  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

  examples were inappropriate.

  

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 inappropriate previous knowledge.

  Results (continued)

• 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.

  I was able to identify five stages:

Stage 1: Awareness

  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

  

Stage 2: Contextualization

  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 into graphing here.

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...

  

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 homomorphism, one-to-one, and onto.]

  

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.

  

International Journal of Mathematical Education in

Science and Technology, 41(2), 199-215.

  Thank you!

  Handout 1 Handout 2

Handout 3

  Handout 4 Handout 5