Posted at Nov 09/2018 05:00PM by Stu 4:
I wanted to put out a few thoughts as a sanity check for myself regarding mathematical objects.
In the first paper by Sfard, I'm not sure she explicitly defines "mathematical object." The closest I can get to a definition from the paper is that an object is "a metaphor which makes a mathematical construct in the image of a material thing" (pg. 60 at the bottom).
Thus, to me it seems like things like geometric objects, sets, functions, spaces, vectors, graphs, etc. are mathematical objects. On the contrary, things like lemmas, theorems, and individual properties would not be considered mathematical objects.
Also, I wanted to try and give an example of the transition from operational to structural conceptions. Given a triangle ABC (a mathematical object), I can say that any triangle similar to ABC would have two congruent corresponding angles to triangle ABC. The property of similarity is not a mathematical object, but after determining several triangles similar to triangle ABC (interiorization), a student might begin to realize that there are an infinite number of such triangles because there are an infinite number of constants of proportionality relating the sides of potential similar triangles (condensation), and in a moment of "inspiration" might conceive of the set of all triangles similar to triangle ABC as an infinite set that possesses many identical properties as the original triangle ABC (reification), and that could be acted upon with many similar operations (translations, rotations, etc). Most high school students would never need to consider the set of all triangles similar to triangle ABC, so there may not be a lot of relevance. But I wanted to throw an example out there to see if people agree with it.
Posted at Nov 10/2018 03:08PM by Stu 8:
From what you and others have said about mathematical objects, it makes sense to me to think of the set of similar triangles as a mathematical object. This initially made me think that mathematical objects were anything for which one could formulate a definition, since theorems and lemmas are statements mathematicians construct and judge to be true from their definitions, but they are not definitions themselves. Then it occurred to me that similarity has a definition (but I don't think we can regard similarity as an object), so I had to refine my concept image for mathematical objects due to the brief cognitive conflict this realization evoked. Then I thought of an analogy to language. In English, we have the different parts of speech to consider the different roles words can have in the structure of a sentence: nouns, pronouns, adjectives, verbs, ect. It seems that objects could be considered a certain class of mathematical nouns. Quantities seem to be nouns formed by adjectives; these adjectives are formed through verbs (mental operations). For example, two triangles are similar (an adjective) when the ratios between their side-lengths (ratios and side-lengths being quantities) are equal (another adjective; used to explain that the quantitative difference between these ratios is zero). To say one is additively comparing these ratios is to use a verb; additive comparison of ratios is also a mental operation.
This makes me think one can form a theory of mathematics knowledge based on nested parts of speech. When one undergoes an action, they use a verb; the result of an action is described with an adjective to state the property of a noun on which the action was performed. Describing the results of lots of these actions can allow one to form a class of adjectives which one can use to compare the results of these actions. Maybe using previously considered adjectives to assign adjectives to predict results of similar actions indicates formation of a process conception of mathematical reasoning. When one can formally classify these adjectives by some means of categorization, the class becomes an object.
I am not sure whether this has much coherence, but it's the inkling of an idea. Thanks for the food for thought, @Stu 4.
Posted at Nov 10/2018 08:12PM:
Pat: Keep in mind that the objects we care about are objects students form. You shouldn't speak of objects as if they exist apart from the persons conceiving them. Is a black hole an object? Well, yes, to people who have a concept of a black hole. But that conception, for different people, could range from an image of a dark splotch on white paper to an understanding of the system of equations that define a black hole by its relativistic properties.
Posted at Nov 12/2018 10:00AM by Stu 4:
I agree with Pat. After reading the second paper, it has become clear to me that "mathematical objects" are not things that exist independently of the human mind. I feel that an object is a mathematical idea that an individual feels is significant enough to have created a system of operations or structures related to that idea. Sure, there is an entire class of objects that mathematicians have agreed are useful and necessary. But I think Pat brings up a good point in that each individual has their own systems of meanings for what we deem to be "relevant objects."
This begs the question, "If mathematical objects are individual conceptions grounded in an individual's unique experience of 'reality,' can a researcher authentically describe what objects that individual is actually envisioning?"
This may be a little philosophical, but I'm curious as to how researchers determine that their theories and constructs regarding student thinking are a good enough estimate or approximation for the thinking that actually occurred in the mind of the student.
Posted at Nov 12/2018 10:23AM:
Pat: @Stu 4 said, "If mathematical objects are individual conceptions grounded in an individual's unique experience of 'reality,' can a researcher authentically describe what objects that individual is actually envisioning?" I'll be interested to see what the rest of y'all say about this!
Posted at Nov 12/2018 10:41AM by Stu 9:
@Stu 4, you have summarized a lot of my thoughts in a very concise question.
My question to your question (to avoid attempting to answer it) is what do we mean by object? Which definition of mathematical object is more conducive to imputing knowledge and ways of knowing to students?
One of the issues in my mind is also a question of which theoretical construct of "object" is being used by the researcher? Currently you are assuming the theoretical opinion "mathematical objects are individual conceptions grounded in an individuals experience of reality". However, I would present another theoretical definition of mathematical object.
Since the second article is a discussion about mathematical objects between Dr. Sfard and Pat, below is an excerpt from Pat's paper we read a while ago, Development of Speed as a Rate:
"The “things” reasoned about are not objects of direct experience and they are not abstract mathematical entities. They are objects derived from experience—objects that have been constituted conceptually to have qualities that we call mathematical...If the person doing the reasoning about a situation is also a person who has reflectively abstracted systems of relationships and operations having to do with conceiving it, reconstituting them as mathematical objects (Harel & Kaput, in press), then, from that person’s point of view, the situation is composed of mathematical objects," (Thompson 1994).
Posted at Nov 12/2018 03:34PM by Stu 6:
Using Sfard's article my understanding of her usage of "mathematical object" is a mathematical concept that goes through a computational process by mathematical thinkers and doers to become a structural abstract object that exists independently and something that can be manipulated as a real thing. Thus, while in the computational process the concept depends on a person's conception until it is reified into a mathematic object.
Posted at Nov 12/2018 03:43PM:
Pat: @Stu 6 -- do you see the paradox in your description of Sfard's meaning of mathematical object? A mathematical object is in someone's head until it is outside anyone's head.
Posted at Nov 12/2018 04:19PM by Stu 6:
Yes! That's what's plaguing me at the moment. I'm stuck on her quote "their existence does not depend on human judgement or will". This would indicate that every human reifies an object the same. Thus, every person has the same understanding/image/representation of the object. I ask for leeway in use of those words as a means to grapple with the ideas under discussion.
Posted at Nov 12/2018 04:43PM:
Pat: @Stu 6 -- The quotation you cite is consistent with what is called a realist epistemology. You'll see in my and Anna's dialog a clash of epistemologies.
Posted at Nov 12/2018 06:06PM by Stu 2:
@Stu 9, you brought up a good point on which theoretical construct of object is being used. Even though, Sfard states that we need to be sensitive to the differences in authors contents and intentions, I have ground much of my knowledge about what "an object" or "a thing" is from Pat's perspective of a thing ,in which is reasoned, is not an object of direct experience or an abstract mathematical entity, but Im trying to wrap my head around Sfard notion of a function as a mathematical object. hahahaha.
P.S. @Pat, thank you for stating what kind of objects we should be focusing on "Students conception"