Posted at Nov 02/2018 03:21PM by Stu 4:
I was having a conversation with Julia earlier regarding the difference between a concept image and a schema, but I still don't have a solid image in my head.

Colloquially speaking, I would call the concept image a student's individual "knowledge base," referring to the sum total of their self-accepted ideas about mathematical concepts. These ideas all make sense to the student, but may not make sense to anyone else or be in line with formally accepted definitions. But each of these ideas exists independently of all others (if we were in analysis I would consider an ambient space where every subset within the space is isolated) and are not connected (meaning that recalling one idea in the concept image does not necessarily evoke the recollection of another).

In fact, it would be impossible to even describe all the ideas within a student's concept image. We could only speculate on what exists in a student's concept image after observing an artifact created by a student. I'm sure that there is a way to describe my ideas with the term "meaning," but I am not sure that I will use it correctly so I have not attempted.

A schema seems to be (once again, colloquially speaking) how a student "perceives" an "object" (whether tangible or intangible). These schema are not isolated or disjoint and collectively describe how the student views reality. Our collective schema are constantly being agitated as we continue to experience the world, and we must reconcile these agitations. Thus, I would classify the process of analyzing this reconciliation (as it occurs in real time) as describing the student's "thinking," which has been the focus for most of Pat's papers that we have read so far.

I'll admit, when I first starting reading the Tall paper, I thought "Oh! This is easy... concept image is just a synonym of schema." But now I'm almost positive they are very different.

Posted at Nov 02/2018 05:25PM:
Pat: If you are interested, see Schemes for Thinking with Magnitudes. In this article you'll get a short but intensive course on schemes, assimilation, and accommodation.

Posted at Nov 03/2018 02:02PM by Stu 8:
Adding on to what you were saying about concept image, @Stu 4, I get confused when Tall and Vinner talk about a student's concept definition being personal to the student (not necessarily a formal definition accepted by mathematicians in general) but different from the student's concept image.

So, when talking about how students think about concepts in mathematics, Tall and Vinner suggest the constructs concept definition and concept image, which from what I've deduced seem to be (resp.) how a student articulates in words his understanding of a concept and a collection of his ideas about different features of that concept in different contexts.

Maybe the definition is a part of the student's concept image; it is one aspect of explaining how he thinks about it. I can see what you mean in thinking the image is similar to a schema.

Posted at Nov 03/2018 03:12PM by Stu 9:
@Stu 8, yes the concept definition is how the student defines a concept in their own words. As I understand it, there is a generally accepted "correct" concept definition for limits, continuity, etc. by the mathematical community at large, then there is the specific concept definitions for each student, as you and Stu 4 have both described. It is only when a student experiences a cognitive dissonance between their definition and the widely accepted definition, by "experts" in the field, that they see something incorrect in their concept image. As Stu 4 has drawn attention to, there is a similarity to what Tall and Vinner are discussing and the perturbations that students experience which was discussed by Hackenburg in her paper we read two weeks ago. I agree with Stu 4, there seems to be some similarities to constructs and ways of thinking that we have discussed thus far. I am curious as to why they are in fact different.

Posted at Nov 03/2018 03:49PM by Stu 3:
I think it is important to note that there is the potential for dissonance between two aspects of the concept image (and not just with the concept definition accepted by the mathematics community). Recall that different parts of the students concept image may be in conflict with each other and the student may use whichever one is evoked by a particular situation without seeing the conflict unless both evoked simultaneously.

A concept image includes everything the student associates with a particular concept. It is specific to a particular concept. For example: Any image, personal definition, examples, ... a student may invoke when considering functions. Depending on what they are working on they may invoke different images, or examples (which may be in conflict with each other as we saw in the articles). I think schema is the structure (cognitive framework)by which students organize and interpret information. I think it is not specific to a particular concept (like functions). I do believe there are similarities in that both concept images and schema are personally conceived.


Posted at Nov 03/2018 04:25PM by Stu 9:
@Stu 3, that is a really good point. Schema are the culmination of actions, processes, and objects, which are constructed in a non-linear manner according to APOS theory (Is this the same type of Schema?). I think dissonance occurs when someone attempts to interiorize an action or object into their scheme and the pieces don't quite fit. On the other hand, as you say, student's concept images already contain potential conflicting viewpoints, and only upon attempting to evoke the two potential conflict factors simultaneously do they become actual cognitive conflict factors. So, how does a concept image differ from an image in the Piagetian sense?

Posted at Nov 03/2018 05:35PM by Stu 4:

I have read part of the article "Schemes for Thinking with Magnitudes" and am pondering it. I have a few questions/thoughts:

1) Is the "image" in "concept image" referring in part to Piaget's description of image?

2) After reading some of the descriptions in the paper, I consider an image as a mental database derived from past experience of how objects behave and interact with one another. An individual also applies the image to predict the behavior of other objects. When it is applied to the individual it is strictly applied in the sense of hypothetical reasoning, but an image never as a whole initiates the real-time actions of the individual.

3) A scheme, on the other hand, is a collection of images, previous experience, etc. that the individual consciously or subconsciously draws upon when determining their course of action in real time. It is also anticipatory, meaning the individual believes that they know what the outcome of their action will be (whether or not the outcome actually happens or is even rational). Once the action has occurred and a consequence has been evoked, the individual reconciles the new information into a new or existing scheme.

Am I getting closer to a proper definition? I don't think I can define anything in a single sentence yet, but I'm trying to at least solidify my thoughts into a single entity....

Posted at Nov 04/2018 07:13AM:
Pat: @Stu 4—Don’t try reconciling APOS language with Piagetian language. APOS claims to be Piagetian, but IMHO it departs from Piaget at the word “object”. We can talk about this on Tuesday.

Guys, this is the world of math ed. Different theories using similar language with different meanings.

Posted at Nov 04/2018 12:24PM by Stu 1:
All, to be honest I am struggling to define constructs in Dr. Roh's paper. In the past papers that we have read, most authors have outright defined the constructs they use. In Dr. Roh's paper it seems as though she is applying constructs, but I don't feel she directly defines them. For instance, Dr. Roh discusses how students' image of limit are not compatible with the epsilon-N definition, but she never defines what she means by image. Are others experiencing similar thoughts, or am I completely missing the boat on this?

Posted at Nov 04/2018 12:47PM by Stu 1:
Please disregard my previous comment. I have since had an epiphany!

Posted at Nov 04/2018 02:20PM by Stu 6:
Stu 1, please share your epiphany. I was struggling with the same thing.

Posted at Nov 04/2018 02:22PM by Stu 6:
In Dr. Roh's paper is the "counting process" a didactic object and/or a construct?

Posted at Nov 04/2018 03:28PM by Stu 8:
I would think the epsilon strip is a didactic object and the counting process is a way to use the didactic object to help students apply the definitions she wants them to explore (def A, def B) and analyze.

I think the counting process could be used as a construct or an application of a construct if we consider what happens in a student's head during the process.

Posted at Nov 04/2018 03:41PM by Stu 8:
"She" refers to Dr. Roh. I had referenced her earlier before I revised that post.

Posted at Nov 04/2018 06:03PM by Stu 1:
I am struggling to understand what Dr. Roh means by the logical structure of the definition. Does any one have any insights? I am thinking this may have something to do with quantifiers and the order of the quantifiers within the inequality?

Posted at Nov 04/2018 06:27PM by Stu 3:
THe paper talks about the different way students interpret “for any epsilon-strip”. Some understood it as there had to be one, some understood it had to be for any selected one, ... Her categorization is (at least partially) based on how they understand that phrase. I think the logical structure of the definition has to do with how the student understood the meaning of “for any epsilon strip”.

Posted at Nov 04/2018 07:27PM by Stu 7:
I think it might also refers to the components such as the hypothesis and the predicate in an if/then statement

Posted at Nov 05/2018 03:37PM by Stu 1:
I would like everyone to know the epiphany I thought I had yesterday was no epiphany! :) Many of you have asked me about it and I too am still struggling with the concepts of Dr. Roh's paper. I will mention that reading the paper Pat sent this morning did help me understand the assigned paper on a deeper level. Thanks Pat!