1. No category

12th International Congress on Mathematical Education
Program Name XX-YY-zz (pp. abcde-fghij)
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do
not change this part.)
PROBLEM MODIFICATION AS AN INDICATOR OF
DEEP UNDERSTANDING
Cristian Voica
University of Bucharest, Romania
[email protected]
Florence Mihaela Singer
University of Ploiesti, and Institute for Educational Sciences, Romania
[email protected]
We proposed to a group of 4th to 6th graders to modify a problem. We found that the students who
remained close to the given problem proved deep understanding of the mathematical context
compared to others, who have had a more creative approach. The students of the first category
understood the patterns of different size configurations that satisfied the constraints of the initial
problem and they were able to make generalizations, while the others resorted on concrete
representations showing merely superficial perception of the mathematical content.
Key words: problem modification, creativity, deep understanding, cognitive novelty, gifted students
INTRODUCTION
In this paper we investigate to what extent a problem-modification (PM) context can
stimulate in-depth learning. The focus of this study is on how gifted students from grades 4 to
6 approach problem modification. Specifically, we explore if there is a relationship between
the nature of the modifications made by a student and the mathematical apparatus he/ she has
involved in posing a new problem by modifying a given one.
Problem modification is considered to be a component of problem posing (Silver, 1994).
Some researchers have reported a positive relation between mathematics achievement and
problem posing abilities (English, 1998; Leung & Silver, 1997). Other studies (e.g. Cai &
Cifarelli, 2005; Singer, Pelczer, & Voica, 2011; Singer, Ellerton, Cai, & Leung, 2011) argued
that instruction that includes problem-posing tasks (including problem modification tasks),
can assist students to develop more creative approaches to mathematics. A component of
creativity might be cognitive novelty (Furr, 2009; Spiro, Feltovich, Jacobson, & Coulson,
1992), which refers to developing new proposals/ideas that are far from the starting element.
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Consequently, we analyzed the relationship between the amplitude of the changes made by
students when developing a new problem starting from a given one, and the mathematical
relevance and coherence of the newly obtained problems.
METHODOLOGY
This paper reports a qualitative research based on case studies developed with some of the
winners of the final round of a contest with 51,979 participants. During the summer camp
organized for these winners, the students received the task to develop a new problem starting
from a given one. 54 students from the 270 participants of the summer camp voluntarily
responded to this call.
Task
For this study, we proposed students to modify the following multiple choice problem:
A squared kitchen floor is to be covered with black and white tiles.
Tiles should be placed on the floor so that in each corner is a black
tile, there are only white tiles around each black tile, and the
number of black tiles has to be the biggest possible (in the picture
there are two examples of such coverage). How many white tiles
will be needed when 25 black tiles will be used?
A) 25 B) 16 C) 81 D) 56 E) 72
The problem above was given to a total of 32,003 students from the 5th and 6th grades (11-12
years) in the first round of the contest.
Method
To better understand students’ individual approaches, we selected from students’ posed
problems: correct problems, ill defined problems, problems with interesting/ correct/ wrong /
incomplete solutions, easy and difficult problems, etc. Each time we looked for more hints
into students’ insights. We took away the clones (problems in which only the background
theme was changed) and proceeded to analyze all the other proposals.
Further, we selected 17 students from 4th to 6th grades for interviews. During the interview
sessions, the students could re-read their proposals, and then they responded to various
particular questions within the protocol interview. Each time, we focused students’
explanations concerning the content and the solutions of their posed problems. The questions
asked with the purpose to see how students understand the consequences of modifying some
parameters of their posed problems highlighted students’ distinct ways of thinking.
Task discussion
To better understand the nature of the given task, we further detail the categories that structure
the text of a problem, using the definitions of Singer and Voica (2012). In general, an explicit
formulated problem contains: a background-theme, (numerical) data, operators (or, in
complex problems, operating schemes), constraints over the data and the operating schemes,
and constraints that involve at least one unknown value of a parameter.
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The background-theme represents the general context in which the problem happens or is
described. In general, the background-theme expresses ”what is about in the problem”. The
background-theme of a problem is characterized by one or more parameters. In our case, the
background-theme refers to paving the floor of a kitchen and the parameter is the kitchen size.
The data are (numerical or literal) values associated to the parameters. In the proposed
problem, the (numerical) data is 25.
The operating schemes are actions suggested by the text of the problem (mathematical
operations, but they can also refer to activities such as ”plot”, “draw”, “trace”, “intersect”,
“cut”, etc). In our problem, the operating scheme refers to placing tiles based on a set of rules.
The constraints imposed to the data and the operating schemes are restrictions that state the
relations of the background-theme with the data and the operators. For the discussed problem,
these constraints over the data and the operators are: In every corner there is a black tile;
Around each black tile there are only white tiles; and The number of black tiles must be the
biggest possible.
The constraints that imply at least an unknown value of a parameter are those restrictions that
state the relations among the data, operating schemes and the problem question. In our case,
this is the relationship inferred for the white tiles by the existence of the 25 black tiles.
When changing the text of a problem, each of the above components can be modified. We
found all these types of modifications among the students’ new posed problems.
Preliminary data
A statistical analysis of the answers given to the starting problem by 32,003 students from
grades 5 and 6 is shown in Figure 1.
Figure 1. Statistical results on choosing the distracters of the starting problem (NR means
non-answer, and X means canceled answers – because of ticking more than one distracter.)
Before analyzing the results, it is important to mention that the authors highlighted possible
errors of reasoning in setting the distracters. Thus, for example, the choice of distracter A (25)
can show that the student has not decoded the text of the problem because he or she just
repeated the number given in the text, while the choice of the answer B (16) may mean that
the student has focused on the visual information of the last drawing, without understanding
the need for developing a pattern.
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Almost 16% of the respondents chose the distracter A and more than one fifth (21%) made the
option for the distracter B. The correct answer D was chosen by almost 31% of the students.
In addition, we found that most students who chose the answer C have actually developed a
fair argument, but did not sufficiently focus on the problem (they calculated the total number
of tiles instead of the number of white tiles). Based on these data, we considered that the given
problem was of medium difficulty, because about 40% of the participants made (total or
partial) correct reasoning.
RESULTS
The problem given to students as a starting point has a double code: verbal (the text) and
iconic (two geometrical drawings); the iconic element could be missing, but all together were
meant to facilitate better (and easier) decoding of the problem. Many of the students of the
experimental sample combined the linguistic and iconic codes in explaining their posed
problem and its solving. The focus on the iconic component is not surprising, taking into
account the students’ age.
In the next, we analyze some of the problems posed within our sample. (The students’ texts
can be found in Appendix). As we will see, in general, the students that relied on the iconic
code have formulated incomplete problems or “open” problems, but without noticing that
those problems have more than one solution, while the others who focused on the text posed
more mathematically coherent problems.
Victor’s posed problem
The problem posed by Victor (grade 4) is a graphical transposition of a pavement problem
(see Appendix). Victor kept only the context of the problem (squared network) and focused
on the use of graphic marks on a drawing (Figure 2).
Figure 2. Graphical explanations accompanying Victor’s posed problem and his drawn
solution
We can observe that Victor is far from the initial problem. During his interview, Victor said
that his problem is more interesting than the original because it is more complicated: “it
requires points and lines, and has more colors”.
However, although Victor is very proud of his problem, his proposal is actually no more than
a list of instructions for drawing and coloring, based on some verbally expressed rules and an
iconic example. To better understand how Victor came to this proposal, we investigated how
he solved the initial problem. We found that, although he obtained the correct result, he
solved that problem also through an iconic analysis (he drew all the configurations described
in the text and counted the white squares on the last drawing). We concluded that Victor did
not actually understand the initial problem in depth.
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Teofil’s posed problem
Teofil (grade 6) proposed two new problems in which he kept the constraints between the
data and the operators of the initial problem. He changed all the other elements of the
wording: background-theme (one of his problems refers to a shape of a chess board, and the
other – to the location of the trees in his grandfather’s garden); numerical data (63 and
respectively 45); operators (paving a structure with stones, planting trees in a given
configuration, respectively); and the constraints imposed to a parameter (the sizes of the
rectangles). Teofil also used drawings to solve these problems (Figure 3).
Fig.3. The figures made by Teofil for solving his posed problems. In the left: the (wrong)
solving of his first problem. In the right: the figure for the second problem (Teofil uses “M”
for apple tree and “P” for pear tree).
Teofil’s solutions to his problems were merely partially correct. He could not argue why the
figures he made must exactly have the dimensions he used for the drawing and he was not
aware that one of the original problem constraints ("the number of the black tiles should be
the biggest possible") is essential for solving. So we see that Teofil is also influenced by the
iconic information and that he takes into account constraints that are not in the text; in fact, he
examined a particular case that does not answer the problem uniquely.
Cosmin’s posed problem
Cosmin (grade 5) proposed a problem with a pavement made of tiles of two different
dimensions (Figure 4). He changed the initial problem quite a lot: for example, he introduced
other constraints between the data, another configuration of the tiles and other numerical data.
During the interview, it again appeared that Cosmin was based on a concrete geometric
representation. He made a figure, set up a measure unit and matched the numerical data with
the figure.
Fig.4. The figure and the solving of Cosmin’s problem
Asked in what way he could change the numerical data of his problem (so that a new posed
problem have a solution), he highlighted only a weak condition (the white squares should
have the sides smaller than the black ones), without noticing the need of divisibility
conditions. Therefore, although Cosmin succeeded to make a problem quite different from
the initial model, his understanding remains at a superficial level.
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Radu’s posed problem
Radu (grade 6) posed a problem that mostly preserves the original text; he only changed the
relationship between the data and the unknown (instead of the number of black tiles he
indicated the total number of tiles), and he replaced 25 with a much bigger number – 16,129.
Apparently, this last change is just a detail. But it leads to change the problem-solving
strategy. In fact, the big number proposed by Radu implies a generalization of the relationship
between the elements involved in the problem (number of white and black tiles).
During the interview, we wanted to see if the choice of the number 16,129 is or not accidental:
Radu: I already had this in my head and I wanted ... since I've learned I already thought,
come on, the initial problem required black squares and I thought how can I
make it oddly enough? More ... to ask to find the number of white squares,
as I realized that the white are hard to know and have to use a concept that I
didn’t learn in school, but I didn’t know how to explain....
Interviewer: What concept?
Radu: Here I used the square root ...
Interviewer: And what’s the problem? What does square root mean? Haven’t you learn it at
school?
Radu: No, we haven’t ... the root of x2 is x ... and I don’t know how to explain ...
Interviewer: How did you get this number, 16,129?
Radu: Well, you’ll notice that this number, the square root of ... must be odd, you’ll see ...
Interviewer: Let's see!
Radu: If we have black corners, they will be more black tiles than white tiles ... I mean on a
column of the square ... because we have black tiles in the corners, too...
Therefore, we can see that, although Radu does not have the formal knowledge of the
involved concepts, he is aware of various restrictions deduced from the text - for example, the
total number of tiles must be an odd number, perfect square.
Mihai’s posed problem
Mihai (grade 6) kept the same requirements as the initial problem. He chose to formulate the
most general case of this problem, considering a square of sizes (2k – 1) × (2k – 1). The
drawing made by Mihai (Figure 5), in which the model continuation is suggested by a dotted
line, shows that he predicted the general case. However, Mihai’s solving is connected to the
particular case of the problem. Mihai calculated the number of white squares on each column,
then he made their sum and thus he answered the question, instead of calculating the
difference between (2k – 1)2 and k2. Thus, we can see that he generalized the pattern, but he
kept the same idea of solving – a "local" computing. This is understandable, because, being in
sixth grade, although he shows good mathematical training, he obviously does not have much
practice in algebraic computing.
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Figure 5. The drawing made by Mihai, attached to the posed problem and the solving he gave
to this problem. In the right side: “We have k2 black squares ⇒ there are k black squares on
c1 and l1 respectively (c1 means the first column, and l1 means the first line – a.n.) ⇒ there are
k black squares on c1, c3, c5, …, and on l1, l3, l5, … respectively, and no black squares on c2x
and on l2y ⇒ there are 2k – 1 white squares on c2, c4,…,c2k-2 and k–1 white squares on l1, l3,
…, l2k-1 (here is a mistake – a.n.)⇒ we have (2k–1)(k+1) + (k–1)k=…=3k2 – 4k+1 (white
squares – a.n.) for k2 black squares.
Emilia’s posed problem
Apparently, the problem proposed by Emilia (grade 4) is a clone. Emilia explicitly mentions
all the conditions of the initial problem (even the constraint "the number of tiles ... should be
the biggest possible" – a condition neglected by most of the respondents). However, her text
contains elements that show deep understanding of the original problem. Thus: her text is
made of short sentences with minimal language; the constraints are reformulated and
reordered to be more easily handled in solving. Emilia changes only minimally the
background-theme (the colors of the tiles) and the numerical data of the problem (36 instead
of 25). Within the interview we wanted to see to what extent Emilia controls that latter
(numerical) change.
Interviewer: What is different (in your problem – a.n.) compared to the model?
Emilia: Well, I said there are pieces of green and blue tiles ... I changed the numbers from
25 to 36 green tiles.
Interviewer: But why you didn’t change to 26?
Emilia: Because ... I wanted 36 ...
Interviewer: But if you were to do another problem, what could you put instead of 36?
Emilia: 81.
Interviewer: Why?
Emilia: In order to be divided evenly ... that 9×9 = 81, and the wall must be squared.
Interviewer: I see...
Emilia: ... or 4×4 ... 6×6 ...
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To our surprise, Emilia proved that she is aware that the number in the problem has to be a
perfect square. Obviously, in the fourth grade, the notion of perfect square is not used.
Although it is clear that Emilia did not know the concept, she had a firm intuition of it and of
its appropriateness for the problem (she said "I wanted 36 ..."). We see that Emilia has a
thorough understanding of the context generated by the initial problem. Although she does
not propose a generalization, she predicts the pattern that can generalize the problem. We
believe that, in her case, this is a manifestation of deep understanding.
CONCLUSION
We might think that a student who poses a problem almost identical with the starting model is
either uncreative or unsure of his/her ability to propose something new compared to the
starting problem. Such reasoning we did when we have excluded the "clones". However, at
this point we conclude that the analysis of the students’ posed problems should be much more
nuanced.
Surprisingly, we found that a few students who remained close to the problem text reveal
deep understanding of the mathematical context compared to others who have had a more
creative approach, posing a problem that was farther from the initial text. A disturbing
question emerges from here concerning the balance between creativity and problem posing.
The purpose of this study is not to answer this question, but the simple fact that this issue has
emerged from the data is important and deserves a focused analysis in other studies.
The overall data resulted from this small experiment show that, in modifying a given
problem, the 4th to 6th graders proceed, roughly speaking, in one of the two following ways:
1. The student imagines a certain configuration and, based on it he/she formulates the
problem, which is further checked by the chosen configuration. But in most cases, the
proposed text does not uniquely determine the configuration, a fact that the student does not
seem to be aware of. It seems that the anchoring in drawings obstructs the understanding of
the relationships among the elements of the problem. Throughout the cognitive route, the use
of drawings as metaphors to better convey the mathematical content (Gardner 1993) has been
precarious among the sample. The metaphor (the drawing) has not made a full loop to get
back to the abstract concept and adequate reasoning.
2. The student largely keeps the same context and makes minimal changes such as a
numerical data or some constraints of the original problem. In these cases, the students
understand the patterns of different size configurations that meet the constraints of the initial
problem and they are able to generalize. The problems issued in this way are more coherent
than the resulted proposals of the first case.
Subsequently, the first category of students shows a superficial understanding of the
relationships and procedures involved in the problem, while the second shows a deeper
understanding that allows them to transfer/ extend/ develop analogous problems on other
configurations (iconic or numerical). Two issues arise from here, referring to the limits of PM
for learning, and, respectively to the efficient use of PM in learning.
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The lack of consistency of some of the problems posed by the students from our sample
shows that there are limits concerning the use of PM for learning. Solving a problem requires
first to decode the connections among the elements of the text. If this phase is not enough
clarified, PM can amplify confusions and misunderstandings. Therefore, if there is an
excessive focus on PM in the class, without a meta-analysis of the posed problems, the
conceptual and procedural ensemble needed for mathematical thinking can be altered.
However, it is important to resort to PM in learning for at least two reasons, which result
directly from this study. One of them refers to the development of student’s motivation: many
of the children who participated in this study have returned after the interviews with new
versions of their problems, and showed awareness and interest for going deeper into the
mathematical concepts involved in the problem.
On the other hand, the children who were able to develop generalizations showed that they
well understood the problem(s) and that their thinking has worked beyond the factual
available information. Therefore, PM more than PS can help teachers to detect which children
have a deep understanding of certain mathematical content area. In fact, many students from
our sample have achieved the correct result for the original problem, but did not pose a
coherent problem as a result of modification. Even more, their interviews revealed a
superficial understanding of that initial problem.
How can PM be organized, so to use the advantages and to overcome the risks? From the
above results, a PM training in which variations in small steps of each components of a
starting problem are used, coupled with the analysis of the consequences of each change on
the mathematical context of the problem seems to be an effective way to deepen
understanding.
APPENDIX: THE STUDENTS’ POSED PROBLEMS ANALYZED IN THIS PAPER
1. (Victor) Ana has drawn a 4x4 square. Andrei told her to color it as follows: "The squares in the
corners are dark gray, those on the diagonal are light gray, and the rest are white. If two squares of
the same color are next to the other, either horizontally or vertically, the line between them should
be thicker. If two same color squares are next to the other on the diagonal, place a dot and then a
line to show which the neighboring squares are.” Draw and color the square.
2. (Teofil) a. For a pavement one uses white and red stones. A red stone is surrounded by four
white and that there were used 63 red stones. Find the number of white stones that's been used.
b. Grandfather has apple trees and pear trees in his orchard. He planted the trees so that a pear tree
is surrounded by apple trees on all sides. Knowing that my grandfather planted 45 apple trees,
find the number of the planted pear trees.
3. (Cosmin) A square is made up of smaller squares: black and white. Each white square has the
sides of 5 cm, and the black has the sides of 10 cm. In the corners there are just black squares and
the white ones are surrounding them. If there are four black small squares, what is the surface of
the big square?
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4. (Radu) I need to place black and white tiles on the square kitchen floor. In each corner there
should be a black tile. Around each black tile there are only white tiles. The number of black tiles
should be the highest possible. The floor consists of 16,129 tiles. Find out the number of black
and the number of white tiles.
5. (Mihai) We have a square (2k –1) × (2k –1) divided into (2k –1)2 squares with the sides of 1.
We color black the maximum number of squares so that there are no two black squares with
common top or side. The drawing is in the figure, and there are k2 black squares. Find the number
of white squares as function of k.
6. (Emilia) On the square wall of the kitchen there are green and blue tiles. They are arranged so
that each green piece is surrounded by blue pieces, and the number of green tiles is the biggest
possible. The green pieces are in the corners. How many pieces are blue, if 36 are green?
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