Beyond the Machine: Students’ metaphors for function and linear transformation

Beyond the Machine: Students’ metaphors for function and linear transformation A continuation of the RUME group: Michelle Zandieh, Jess Ellis, and Frances Henderson

Overview • Reminder of data • How we parsed the data • Overview of metaphors • Nonmetaphors • Metaphors • Description of each metaphor with pure example • Actual Data • How the metaphors appeared layered in the data

What is an utterance? Following Bakhtin (1986), Rasmussen, Marrongelle, and Kwon (2009) explain that “an utterance is not a conventional unit, like the sentence, but a unit nonetheless in the sense that it is marked out in the boundaries of speech”. *We will mark a change in utterance by “||” (0:00:45.8)Brent: In the context of high school algebra, explain in your own words what a function is? A process that changes an input to an output. || Number 2, in the context of linear algebra, explain in your own words what a transformation is? A process that changes an input vector or matrix into an output.

Nonmetaphor: Properties (P) How to Label:We use P to indicate that a student has reasoned about one of our interview questions by referring by name to a property of a function or linear transformation. Their complete response may or may not include analyzing the nature of the relationship between elements of the function / linear transf using one of the metaphors. We will note what property is being use in parenthesis next to the code. For example, P (ld) for when a student refers to linear dependence. Indicative references: the vertical line test, the horizontal line test, making a statement about the shape of the graph, referring to the column vectors of the matrix spanning or being linearly in/dependent.

PURE Examples: Properties (0:06:04.9)Greg: …And the way to make a matrix not able to be invertible would be to have it be dependent. P(ld) (0:04:02.1)Diane: [reads] “A parabola is not 1-1. Does not pass the horizontal line test” P(HLT) (0:06:48.7)Diane: I said that was one-to-one because it's linear independent, and so. Because if it's linear dependent, that means all the points lie on the same line, so there's infinitely many solutions to that problem.  P(li), P(ld), P(same line), P(infinite sol) 

Nonmetaphor: Computational (C1,C2) How to Label:We use C to indicate that the utterance refers to a computation. Computations may include but are not limited to addition, subtraction, multiplication, division, row reduction, computing an inverse or any other symbolic manipulation. There are two main types of computations for the purpose of this research: (C1) Computations that are done to carry out the function/transf, i.e., to get from the input to the output or to get from the starting value to the ending value. (C2) Other computations are “side calculations” in the sense that they are outside the function relationship itself, but are relevant to that relationship, perhaps done to note a property.

PURE Examples: Computation (0:13:47.9)Landon: From matrix multiplication, it's true…But basically just multiply this x by the 1st column, and the y by the 2nd column and split it up. C1 (0:16:00.3)Greg: [writes 3(x-3)] Do that first. [writes 1/(3x-9)] 3 times x, you just do 1/3 of the input, then plus 9 [writes 1/3x+9]. Because you're multiplying by 3, and you're subtracting 9. So if you divided it by 3 and then added 9, it should be good. C2

Metaphors • Input/ Output (I/O) • Sending/ Traveling (S/Tr)

Input/Output (IO) Metaphor:The complete metaphor includes an input, an output and therefore at least an implied space/entity that things are going into and out of. How to Label:Not all of these need be described in an utterance for that utterance to be labeled as I/O. Indicative Phrases:input, output, put in, get out, take out, plug in, accept, receive, returns **(accept, returns, receive) come from the perspective of the function or machine

PURE Examples: Input/ Output (0:01:36.3)Greg: …function is like the black box, you put something in, it comes out the other end. (0:00:29.4)Landon: Number 1, In the context of high school algebra, explain in your own words what a function is? And I wrote, A method that takes an input and spits out an output. (0:00:44.9)Greg: The question is, in the context of high school algebra, explain in your own words what a function is? And I said, a function is an equation that acceptsan input and returnsan output based on that input.

Sending/Traveling (S/Tr) Metaphor: The complete metaphor includes a starting location, an indication of movement, and an ending location. How to Label: Not all of these elements need be present to label as S/Tr, but there needs to be a clear indication of movement. Indicative Phrases:gets sent, goes to,change in the location, reach, go back, get to

PURE Examples: Sending/ Trav (0:06:04.9)Greg: When you're in transformations, you'll always be able to get back. If a matrix is invertible, you should be able to go both ways. (0:00:44.9)Greg: And question two is, in the context of linear algebra, explain in your own words what a transformation is? I said, a transformation is an equation describing a change in the location of a set of coordinates relative to a scale. (0:02:07.8)Roger: …Let's say you have the line or the vector 1,1. You put it through a transformation of 1,3,2,1. Then when you multiply it through, you get 3,4,3,4. Woops, 3. It's going to be the same thing. Here you have. 3 and 4. It's 1 matrix. This is wrong. So you have your 1,1 here and then it goes to 3,4 like that [graphs the vectors (1,1) and (3,4)].

Morphing (Mor) Metaphor: The complete metaphor includes a beginning entity (or beginning state of an entity), an ending entity (or ending state of entity) and the implication that some transformation occurred to turn the beginning entity into the ending entity. How to Label:Not all parts of the metaphor must be specified by the utterance, but there must be a clear sense that the beginning entity did not simply move to the new location (ending entity), nor was it replaced by the new output (ending entity), but that there was actually a metamorphosis of the beginning entity into the ending entity. This may be a description of one initial entity morphing in one ending entity, but it might also refer to more than one initial entity transforming into one or more than one ending entities. Indicative Phrases: transform, change, become

PURE Examples: Morphing (0:01:17.7)Landon: …And that's essentially what a transformation I would say is, because it transforms something into something else. (0:01:21.4)Diane: Linear transformations to me are more or less something that changes from one thing to another. (0:11:02.7) Greg: So if you have a dependent matrix, it could map multiple points, to maybe the same point, collapse dimension or do weird stuff. (0:00:07.3)Roger: …a transformation is a multiplication of matrices that leads to a new image produced from the original matrix or vectors in the matrix. 

Mapping (Map) Metaphor: The complete metaphor includes two entities that are connected to each other by a mapping, or two sets of entities that are connected by a mapping (i.e., a series of individual maps between pairs of elements). How to label:The statement refers to a description of how entities are matched up. Indicative Phrases:assigns, per, for

PURE Example: Mapping (0:07:12.9)Landon: …Because it's dependent, I'm going to end up with infinite solutions, which essentially says that there are an infinite amount of x's for each b. So there are, for every b, or every solution, there's an infinite amount of x's possible, since it's dependent. Which would contradict this, because there's only supposed to be 1 x for every b. NONexample: (0:02:51.4)Greg: A function is one to one if it maps each input given to no more than one output P(map)

Machine (Mach) Metaphor: The complete metaphor includes a beginning entity of state, an ending entity or state, and a reference to a tool, machine or device or some kind that causes the entity to change from the beginning entity/state into the ending entity/state. How to Label:Reference to a machine or something that produces, or creates something from the original entity. Indicative Phrases:acts on, produces

PURE Examples: Machine (0:09:46.5)Greg: Basically, you could do [writes a,b;c,d times 1,2;2,4]. That's why it's going to be valid. Pretty much anything you toss in here, this is still that transformation should be able to act on it, as long as they're both square matrices. (0:02:05.2)Diane: Because I think a function is what you have at the beginning, and then transformation is something you use to act upon the function to get something, an end result. (0:01:17.7)Landon: Because it essentially does the same thing. So it's like, how I have here a rule that assigns, essentially a function is the same thing, you put in an input, and it manipulates that input and turns it into an output

Metaphors

Mixed Example#1 (0:01:37.3)Brent: I just remember when I was in middle school or elementary school or whatever, learning about functions, and learning about them as a machine, you put something in, and ittransforms it to something else. Machine: as a machine, it transforms Input/ Output: put something in Morphing: transforms it to something else

Mixed Example #2 (0:01:17.7)Landon: Because it essentially does the same thing. So it's like, how I have here a rule that assigns, essentially a function is the same thing, you put in an input, and itmanipulatesthat input and turns it into an output. Mapping: assigns Input/ Output: put in Morphing, Machine: it manipulates, turns it into

In Conclusion… • What we have done: • Refined and expanded coding • Identified clean examples of pure and layered metaphors • What we would like to do: • RUME: Theoretical paper discussing these different ways of thinking • Journal paper? • Continue to look into the correlation between these metaphors and student success (using rubric) • See how these metaphors played out in students’ discussions of inverses

Thank You • Questions? • Input?

Beyond the Machine: Students’ metaphors for function and linear transformation