Contextual Situations vs. Concrete Examples

[Ed Laughbaum]

Hello Listers,
 
The New York Times article “Study Suggests Math Teachers Scrap Balls and Slices” by KENNETH CHANG, posted at http://www.nytimes.com/2008/04/25/science/25math.html?_r=1&oref=slogin is based on research at The Ohio State University. The researcher indicated that “The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning, …”  Also see a news release at http://www.eurekalert.org/pub_releases/2008-04/osu-ced042108.php
 
“The problem with the real-world examples, Dr. Kaminski [OSU] said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems."
 
This suggests to me that teaching one type of word problem does not translate to understanding another type of word problem where the mathematics is the same. But this is not related to teaching the mathematics needed to work the problems.
 
I would like to respond to the research, and to clarify teaching using contextual situations vs. the “multiple concrete examples” concept. An obvious problem is the meaning of “teaching students multiple concrete examples.” I would guess this has nothing to do with teaching mathematics, but rather, using known mathematics in applications – where all needed mathematics has previously been taught.
 
Teaching in context is different than teaching “with” concrete examples. Certainly the train problem is not intended to teach any new mathematics, but instead, it is used to APPLY the mathematics already taught.
 
In teaching mathematics using contextual situations (that make sense to your students), you must immediately and directly get to the abstract mathematics you are intending to teach. Because:
 
“… when neurons fire simultaneously, their synaptic connections become stronger, raising the chance that the firing of one will trigger the firing of the other.” “The two neurons that meet at the synapses become locked in a sort of physiological embrace. This is the physical basis for the formation of functional circuits during brain development, for learning and memory, …”
Bagley, S. (2008). Train your mind: Change your brain, 30-31. Ballantine Books. NY, NY.
 
Contextual situations can be used to TEACH mathematics, but I also have the sense that many among us think of “contextual” problems as simply being motivational tools for the mathematics they intend to teach. So we present a problem to motivate, and then teach the mathematics needed to solve the problem – never using the problem to teach the mathematics. So we teach mathematics abstractly and then apply it to contextual problems.
 
As a simple example of teaching in context, suppose I want to teach the concept of the zero of a function. In the example below, students have previously learned/used (through contextual situations) the representations of functions, and the connections among them. So, students understand what A = 1000 – 2.5t is and how we generated it. But they have never found zeros before, nor do they know their meaning.

A 1000-ml I.V. bag is being used to administer medication to a hospital patient at a drip rate of 2.5 ml per minute. The function that models the amount of I.V. fluid left is A = 1000 – 2.5t, where t is time in minutes. When will the I.V. bag have no fluid left?
 
We immediately go to the graphing calculator and graph the function on an appropriate window and start tracing, stopping at numerous points to see if any fluid remains. We discover at 400 minutes, there is nothing left. We discover that this number is the x-intercept. We then go to TABLE and check for the amount left every 10 minutes or so. We discover the same thing that at 400 minutes, there is nothing left. In both cases the function is zero at 400 minutes. So we call this number the zero of the function.
 
The contextual situation has given meaning to the mathematics, and we have now taught the intended mathematics. The concept of a zero now has meaning. BTW, we use more than one contextual situation and discover the same characteristics each time – so we generalize.
 
Why is meaning important? 
“When a child has a personal stake in the task, he can reason about that issue at a higher level than other issues where there isn't the personal stake. … These emotional stakes enable us all to understand certain concepts more quickly.”
Greenspan, S. I. & Shanker, S. G. (2004). The first idea: How symbols, language, and intelligence evolved from our primate ancestors to modern humans, 241-2. Da Capo Press. Cambridge, MA.
 
“In general, how well new information is stored in long-term memory depends very much on depth of processing, … A semantic level of processing, which is directed at the meaning aspects of events, produces substantially better memory for events than a structural or surface level of processing.” “… all that is required to produce a durable long-term memory is perception of a meaningful stimulus event.”
Thompson, R. F. & Madigan, S. A. (2005). Memory, 33-34. Joseph Henry Press. Washington, D.C.

“New Information becomes more memorable if we “tag” it (that is, associate it) with an emotion [like a familiar real-world context].” “We understand something new by relating it to something we've known or experienced in the past.”
Restak, R. (2006). The naked brain. 164. Three Rivers Press. New York.
 
“The human mind, we see, is not equipped with an evolutionarily frivolous faculty for doing Western science, mathematics, chess, or other diversions. … The mind couches abstract concepts in concrete terms.”
Pinker, S. (1997). How the mind works, 352-353. W. W. Norton & Company. New York.
 
Once students understand the concept of a zero, we now immediately use abstractions such as.
Find the zeros of:
y = x + 3
y = x – 3
y = x + 4
y = (x – 3)(x + 4)
What do students do with these? Experience shows that they look for the x-intercept AND learn that they don't need the graphing calculator since the zero is the opposite of the constant. The process now continues with functions like y = 2x + 5, etc. At a later time, when appropriate, we will find zeros using pencil-and-paper techniques.
 
My point is that “teaching students multiple concrete examples” has various meanings, and I suspect that the research did not use the process as described in this posting.

Best Regards,

Ed

[Alain Schremmer]

On Apr 29, 2008, at 9:23 AM, Ed Laughbaum wrote:

> Hello Listers,
>
> The New York Times article “Study Suggests Math Teachers Scrap  
> Balls and Slices” by KENNETH CHANG, posted at http://
> www.nytimes.com/2008/04/25/science/25math.html?_r=1&oref=slogin is  
> based on research at The Ohio State University. The researcher  
> indicated that “The motivation behind this research was to examine  
> a very widespread belief about the teaching of mathematics, namely  
> that teaching students multiple concrete examples will benefit  
> learning, …”  Also see a news release at  http://www.eurekalert.org/ 
> pub_releases/2008-04/osu-ced042108.php

Well, if nothing else, this shows that we do agree on occasion: I  
posted this very link on April 28, 2008 9:16:24 AM EDT!!!

> […]


> My point is that “teaching students multiple concrete examples” has  
> various meanings, and I suspect that the research did not use the  
> process as described in this posting.

So do I and, bluntly, as I wrote when I posted the link, I suspect  
that "(the above research is full of holes)".

The particular hole I was interested in is that they seem to take for  
granted that there is only one way that "examples from the real world  
[can be used] to teach abstract concepts". They do not seem to  
realize that examples might in fact be necessary in a completely  
different context, namely the model theoretic context—no connection  
with modeling.

To give a rough idea of what a model-theoretic approach is, let  me  
quote from the very first paragraph of the text I recently uploaded  
on http://www.freemathtexts.org/

To put it as briefly as possible, Arithmetic and Algebra are both about
developing procedures to figure out on paper the result of real-world  
pro-
cesses without having to go through the real-world processes themselves.
To make this a bit clearer, here are two examples from Arithmetic the
Algebra counterpart of which we will deal with in Part III of this book.

and, a few pages later

As a matter of fact, this is most likely how, several thousands of years
ago, Arithmetic, got started when, one may imagine, Sumerian merchants,
faced with the problem of accounting for more goods in the warehouse
and/or money in the safe than they could handle directly, decided to  
have
both the goods and the money represented by various scratches on clay
tablets so that they could see from these scratches the situation  
their busi-
ness was in without the inconvenience of having to go to the warehouse
and/or to open the safe.

Thus, in this approach, it is hard to see how Kaminski's "“The  
problem with the real-world examples was that they obscured the  
underlying math, and students were not able to transfer their  
knowledge to new problems." could possibly apply. Moreover, it does  
not seem to have occurred to Kaminsky that the culprit here might be  
the "problems approach", a particularly noxious variant of the  
"topics approach".

While I have no idea how the model theoretic approach relates to  
Laughbaum's viewpoint, one difference is that only one "situation" is  
needed to establish a one one-to-one correspondence between "real-
world items" and "paper names", "real-world relationships" and "paper  
verbs", etc

As I think I already mentioned, this seems to intrigue some people in  
the English department and, this coming Fall, there might be a "link"  
in which the reading material in the English developmental course  
will be the text used in the Math developmental course from which the  
above is excerpted.

As for whether the model theoretic approach "works", I think that it  
does, whatever that may mean, but I have only circumstantial evidence  
and no hard systematic data.

Regards
--Schremmer
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