Narrative: Setting and scenario
 
The scenario on the following page describes a typical tutoring session 
at the math roundtable in the Teaching-Learning-Computing center of Polk 
Community College, located in Winter Haven, Florida. Most PCC students 
are adult learners with jobs and family responsibilities. Many are taking 
math classes because they need them for their jobs. A few are taking them 
as lifelong learning, because they are interested in what their kids are 
learning in school, and don't want their kids to know that they never 
mastered math themselves. 
 
The main problem these students face is that they haven't learned the 
"translation" and "schema development" parts of Mayer's problem-solving 
process. In Winter Haven, teachers push the students through school 
without making sure they have the basic skills. The approach is 
traditional: teachers emphasize drill-and-kill. Also, geometry is not 
taught in high school.
 
These students are highly motivated, because learning math has real-life 
applications for them. They come to the tutoring center either because 
they are referred by their teachers, or they come voluntarily. Tutoring 
is free. Much tutoring is done by second-year students for a small 
stipend. There are two part-time math tutors and one part-time English 
tutor. I was one of the math tutors for two years.
 
Students bring their workbooks to tutoring sessions. On p. 54 of the 
algebra workbook is a sort of glossary. It is intended as a cognitive aid 
to help students with the translation process, by providing a summary of 
the required linguistic knowledge. It has a two column table: the first 
contains an English phrase, the second contains a mathematical operation. 
For example, if the first column contains "more than", the second column 
contains "subtract". Students haven't the foggiest notion why the 
correspondence works the way it does; they simply memorize the table and 
hope to solve algebraic problems by plugging in words just as they would 
plug numbers into a formula. It does not work! Students are not being 
taught conceptual competence, so they cannot translate word problems into 
mathematical formulas by using their own understanding of the principles 
underlying the problem sets.
 
I'll relate a typical story on the next page; then I'll critique it.
 
---------------------------------------------------------

A typical problem-solving scenario at the math roundtable
 
	Joe has 3 more marbles than Tom.
	There are 7 marbles on the table.
	How many marbles does each boy have?
 
What are we given?
Tom has some marbles, Joe has 3 more than what Tom has, and there are 7 
marbles all together.
What are we to find?
How many each boy has.  
What kind of problem is it?
It's a parts-and-whole problem.
OK, let's draw a picture of the parts.

 
Joe	-----------------		Tom		---------
	|		|				|	|
	|		|				|	|
	|		|				---------
	|		|
	-----------------
	lots of marbles					less marbles
 

Let's use a symbol to represent this relationship
(write down)
Joe > Tom
Understand what this symbol means?
Yes. The symbol is "more than".  Joe has more than Tom.
Ok, if Joe has more, then how many more?
The word problem says 3 more.
How many does Tom have?
I don't know.
OK, so let's call that unknown number "Tom" - we just give it a name, 
that's all.
OK
Then if Joe has 3 more than Tom, how many does Joe have?
Tom + 3
So let's fill in the boxes now.
(first box - write in Tom + 3; second box, write Tom)
OK, how many marbles are there all together?
7
Now what exactly are we given?  Can you tell me more now that you have 
been through this process?
We can see that Tom+3 is greater than Tom, and that has to account for 
all the marbles.
OK, how do we write that?
Tom+3 + Tom = 7.
You got it.  Now solve it?
That's easy - I can do arithmetic, once I figure out what they're looking 
for.
(student plugs in values and solves equation step by step; then checks 
answer.)

-----------------------------

Annotation and interpretation
 
Students have a good grasp of basic skills, and can easily solve 
algebraic problems when they presented as formulas or systems of 
equations to solve. But they have a terrible time with the translation 
process. That is where the tutoring session begins. But first, some 
theoretical background. Intuitively, I've usually followed the process 
used by Mayer (1987), i.e., to break the problem-solving process into 
four steps:
 
(1)	Translation or comprehension (do you understand what the problem 
is asking?)
(2)	Schema training (what general type of problem is this?)
(3)	Strategy training (can you set subgoals, solve the problem a step 
at a time, show your intermediate steps, and then check the answer?)
(4)	Execution (can you use your prior math skills--hopefully 
automated--to solve the problem?)
 
However, I've capitalized on my knowledge of visualization in programming 
to flesh out steps 1 and 2, by having students draw a graphical 
representation of the problem before attempting to solve it.
 


Joe has 3 more marbles than Tom.
There are 7 marbles on the table.
How many marbles does each boy have?

 

The marble problem I have presented here is typical of the problem class 
described by Gelman and Greeno (1989) in their paper on the nature of 
competence. Mathematical competence refers to the ability to understand 
the propositions presented in the problem statement, as well as the 
question posed, to make decisions about what inferences to make or 
subgoals to pursue, and to determine a set of goals for planning actions 
that can provide an answer to the question (p. 150). Foremost among the 
required skills is conceptual competence, conceptual understanding of the 
domain of mathematics. However, as Gelman and Greeno note, 
 
Conceptual competence does not contain recipes for successful 
behaviors...what conceptual competence does is provide those constraints 
the planner must honor if it is to generate a successful plan of action 
(p. 168).
 
In their analysis of the problem "How many more birds than worms are 
there?", which is very confusing to students, the authors replace the 
problem statement with a new one, "There are how many more birds than 
worms?" (p. 162). Their reasoning is that, by restating the problem, the 
students don't have to analyze the syntax of the problem; students use 
the "how many" as a numerical determiner, just like numerals, except that 
the number is translated as "?". I do not use this precise technique, but 
I do use the concept of a numerical determiner, such as "<" and ">". 
 


What are we given?
Tom has some marbles, Joe has 3 more than what Tom has, and there are 7 
marbles all together.
What are we to find?
How many each boy has.
What kind of problem is it?
It's a parts-and-whole problem.
 


I start by having the student restate the problem in his/her own words, 
until they can specify all the "givens". Next, I ask them what they are 
"to find". For beginning students, I have them write these "givens" down. 
If necessary, as the discussion progresses, I have them cross out any 
information that is irrelevant to the problem solution. If I were to 
follow the system I was taught in sixth grade, the next step would be to 
write down "rule". However, this takes a phenomenal jump of the 
intellect--a jump these students are simply incapable of performing. 
Instead of "rule", I ask them what type of problem this is. Mayer refers 
to this as schema training. Students know how to solve river problems, 
parts-and-whole problems, financial problems, mixture problems, and the 
like. Once they can characterize the problem, we can then move on to the 
visualization process.
 
Chi and Bassok (1989) caution us about students who characterize problems 
based on surface features. Bruer (1993) does likewise, in the example of 
the naive students' rules for solving balance beam problems. This becomes 
important for my more advanced students, once they have to deal with 
dynamics problems, but I have found that it's not a major problem with 
the algebra students.
 


OK, let's draw a picture of the parts.
(student draws a pictorial representation of the problem: large box for 
Joe, small box for Tom.)

 

I learned about visualization from Professor Johnson when I took Data 
Structures (Advanced Pascal) at Mass Bay Community College. Data 
structures are complicated objects of computer code, but they can usually 
be depicted by a simple picture. I also refer to a paper by Rohr (1986), 
in which she analyzes the use of icons vs. words for students who are 
given the task of writing simple computer programs. Needless to say, the 
students who used the iconic representations, such as flow charting and 
moving icons about the screen, did much better than those who used the 
verbal representations. 
 
We assumed that the retention of objects, operators, and their structural 
overall interrelations and consequently their understanding, is easier 
for subjects under the icon condition because they can build up a visual 
spatial overall picture of the structure of the system components. 
Subjects under the command word condition, in contrast, have to memorize 
possible sequences of operations. This will stress memory mechanisms (p. 91).
 
A similar technique has been used successfully by Nakaji for teaching 
students how to solve physics problems. Nakaji had students draw the 
"pictures or images they were seeing in their minds as they were choosing 
the most appropriate frame of reference from which to analyze the 
physical situation" (p. 80). Since my tutoring also involved the calculus 
and physics students, I used the same visualization techniques with those 
students, by expanding the static representations used with the algebra 
students to dynamic visualizations for the calculus students. 
 
For the algebra students, such as the one in the student-teacher 
interaction described here, I use simple pictures: vectors or directional 
arrows for speed of river current and speed of boat; beakers to contain 
amounts of milk and water; boxes to contain principal and interest; and 
boxes to contain counted items such as the marbles in this problem. The 
key to the river problems is to point the arrows in the correct direction 
(for addition or subtraction) and, for the counting problems, to make the 
boxes different sizes (to represent greater or smaller amounts of 
whatever is in them). Here, Joe's box is clearly larger than Tom's box. 
This pictorial representation clarifies the problem statement that 
originally confused the student.
 
Next, we move on to the symbolic processing, as Gelman and Greeno 
suggest. Here is where I begin to introduce symbolic operators other than 
the usual ones used for addition, subtraction, multiplication, and 
addition. 

 

Let's use a symbol to represent this relationship.
(write down)
Joe > Tom.
Understand what this symbol means?
Yes. The symbol is "more than". Joe has more than Tom.

 

Here, I am in firm agreement with Chi: it is very important to have 
students use self-explanations, even if I have to write the symbolic 
representation down the first few times. Visually, I have them relate the 
symbol to the diagram. That makes a connection between the incoming 
knowledge (the symbol ">") and the relative sizes of the boxes. I do the 
same for directional arrows, although I also have to add a sign ("+" or 
"-") in addition to the inequality symbol. I think that's why students 
have so much trouble with river problems; there's not only an inequality 
involved, but also a directionality. There are two attributes to the 
variable (size and direction), whereas the marble problem has only one 
(size).
 
Having dealt with the translation and schema training, we now move on to 
the strategy training and planning. It's important that the student 
understand how to relate the representation back to the givens, so that 
he/she will plug the right value into the equation later on. This part 
has to be done slowly and carefully. It involves laying out a matrix. The 
more complicated the problem, the more rows we need to add to the matrix. 
For example, a river problem or a financial problem would need the third 
row. The use of the matrix as a strategy is shown in Figure 1:

 
(student draws a matrix like a tic-tac-toe board, then fills in

Joe 	> 	Tom   	Total

in the first row of cells) 
 
Figure 1. Starting the matrix.
-----------------------------
 
Next, we begin the comparison process and the conversion of the words and 
pictures into mathematical language.
 


OK, if Joe has more, then how many more?
The word problem says 3 more.
How many does Tom have?
I don't know.
OK, so let's call that unknown number "Tom"--we just give it a name, 
that's all.
OK
 


An important part of strategic knowledge is determining the unknown. 
Metacognition, which involves careful selection of domain-appropriate 
strategies, comes into play here. Students must be able to devise and 
monitor solution plans. To begin to plan, they must be able to clearly 
identify what they know and what they don't know. Larkin (1989, p. 297) 
refers to this transition as moving from the basic representation 
(containing objects that are mentioned in the problem statement and that 
are often identifiable in the real world; hence, the picture) to the 
computational representation (consisting of equations or other formal 
mathematical statements). Here, the students must know how to choose the 
mathematical operators according to what difference between the goal and 
the current state they will reduce. (p. 287). They've already dealt with 
the ">" sign, and now must generate another symbol to represent the 
unknown quantity, namely, how many marbles Tom has. This is not 
intuitive, and must be specifically taught. 

 

Then if Joe has 3 more than Tom, how many does Joe have?
Tom + 3
So let's fill in the boxes now.
(first box - write in Tom + 3; second box, write Tom)

 

At this point, we start adding information to the matrix, as shown in 
Figure 2:


(student starts to fill in the second row of cells; all 
operators go on the lines delimiting the cells in each row)

Tom + 3		Tom	(blank cell)

Figure 2. Starting the solution process.
---------------------------------------
 
The next step is to attempt to generate a solution or formula from the 
information in the matrix--the execution process. Here, it is critical 
that the student engage in verbal self-explanations. That will uncover 
any misconceptions before he/she starts to write down any further 
information.
 


OK, how many marbles are there all together?
7
Now what exactly are we given? Can you tell me more now that you have 
been through this process?
We can see that Tom + 3 is greater than Tom, and that has to account for 
all the marbles.
OK, how do we write that?
Tom + 3 + Tom = 7
(student fills in remaining boxes)


 
Figure 3 shows the completed matrix:
 

(Student completes second row of cells:)

Tom + 3		+	Tom	=	7 
 
Figure 3. The completed matrix.
------------------------------ 

You got it. Now solve it.
That's easy. I can do arithmetic, once I figure out what they're looking for.
(student plugs in values and solves equation step by step; then checks 
answer.)


 
From this point, the strategy is correct, and the student finally moves 
on to the execution process. The students at my math roundtable have 
good, automated arithmetic skills, and can solve simple equations of this 
sort. However, to continue with the self-monitoring process, I do require 
them to write down each and every intermediate step. This is critical 
when they start moving quantities to the other side of the "=" sign, 
because the values change sign at that point. Here is where most of the 
subgoal errors are made.
 
Finally, when they arrive at the solution (which, for this problem, is 
that Joe has 5 marbles and Tom has 2), they must plug in the solved-for 
value of "Tom", namely 2, and verify that the equality holds. This is the 
assessment part of metacognition, and it is very important. Often, when a 
student arrives at an answer, he/she is so happy to have solved the 
problem, that he/she neglects to check the answer and see that it is, 
indeed, correct!
 
In conclusion, I have found that, although I send my algebra students to 
the kids' section of the bookstore to buy and study 6th grade math 
workbooks, they don't seem to mind doing the extra work.  And, when they 
come to me with good marks on their tests and thank me for teaching them 
"a system", I am happy to know that this type of instruction succeeds.


References
 
Bruer, J.T. (1993). The mind's journey from novice to expert. American 
Educator, 6-15, 38-46.
 
Chi, M.T.H., & Bassok, M. (1989). Learning from examples via 
self-explanations. In L.B. Resnick (Ed.), Knowing, learning, and 
instruction: Essays in honor of Robert Glaser (pp. 251-282). Hillsdale 
NJ: Erlbaum.
 
Gelman, R., & Greeno, J.G. (1989). On the nature of competence: 
Principles for understanding in a domain. In L.B. Resnick (Ed.), Knowing, 
learning, and instruction: Essays in honor of Robert Glaser (pp. 
125-186). Hillsdale NJ: Erlbaum.
 
Larkin, J.H. (1989). What kind of knowledge transfers? In L.B. Resnick 
(Ed.), Knowing, learning, and instruction: Essays in honor of Robert 
Glaser (pp. 283-305). Hillsdale NJ: Erlbaum.
 
Mayer, R.E. (1987). Learnable aspects of problem solving: Some examples. 
In D.E. Berger, K. Pezdek, & W.P. Banks (Eds.), Applications of cognitive 
psychology: Problem solving, education, and computing (pp. 109-122). 
Hillsdale NJ: Erlbaum.
 
Nakaji, D.M. (1991, Summer). Classroom research in physics: Gaining 
insights into visualization and problem solving. New directions for 
teaching and learning, 46, 79-87.
 
Rohr, G. (1986). How people comprehend unknown system structures: 
Conceptual primitives in systems' surface representations. Visualization 
in Programming, 5th Interdisciplinary Workshop in Informatics and 
Psychology (pp. 89-105). Scharding, Austria: Springer-Verlag.
 

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