Problem posing

“Without posed
problems, there are
no problems to solve.”

FREQUENTLY ASKED QUESTIONS

Who
For

poses mathematics problems?

whom are mathematics
questions posed?

Dunker

described problem
posing in mathematics as the
generation of a new problem or
the formulation of a given
problem. (Dunker, 1945).

Silver

described problem
posing as it is refers to both the
generation of new problems
and the re-formulation of given
problems, posing can occur
before, during or after the
solution of a problem (Silver,
1993).

Stoyanova

has defined
mathematical problem posing
as the process by which, on the
basis of concrete situations,
meaningful mathematical
problems are formulated
(Stoyanova, 1996).

SKILLS IN PROBLEM POSING
1) Use problem-solving strategies to investigate and solve the
posed problems.
2) Formulate problems from every day and mathematical
situations.
3) Use a proper approach for posing problems up to the
mathematical situations.
4) Recognize relationships among different topics in mathematics.
5) Generalize solutions and strategies to new problem situations.
6) Pose complex problems as well as simple problems.
7) Use different subjects' applications in posing mathematical
problems.
8) The ability of generating questions to improve problem posing
strategies like:
- How can I finish the problem?
- Can I pose another questions?
- How many solutions can I find?

HOW ARE PROBLEM-POSING SKILLS
RELATED TO PROBLEM- SOLVING SKILLS?
 Silver

and Cai found that students’ problemsolving performance was highly correlated
with their problem- posing performance.
Compared to less successful problem solvers,
good problem solvers generated more, and
more complex, mathematical problems.

BENEFITS OF PROBLEM POSING
 It

frees learners from the one-answer
syndrome.
 It enables learners to view common things in
uncommon ways.
 It legitimizes asking questions.
 It fosters the predicting, conjecturing, and
testing of hypotheses.
 It builds a spirit of adventure, intellectual
excitement, and class unity.
 It demonstrates the spiraling nature of inquiry
learning. No problem is really solved.
 It develops a sense of personal ownership and
responsibility for mathematical investigations.

WAYS TO CHANGE A PROBLEM
Some ways to change a problem to create new
problems
 Change the numbers.
 Change the geometry.
 Change the operation.
 Change the objects under study.
 Remove a condition, or add new conditions.
 Remove or add context.
 Repeat a process.

CHANGE THE NUMBERS


This is the most obvious way to change a problem.
Give your students one or more problems and ask
them to identify any stated or implied numbers.



When considering numerical changes to a
problem, many different domains and
representations can prove interesting.

CHANGE THE GEOMETRY


Any problem with a geometric setting is ripe for
new variants. The simplest problem-posing
maneuver is to change the shapes involved.



Different categories of shapes that suggest
possible substitutions include polygons and their
number of sides, regular versus non-regular
polygons.



Changes of dimension can yield exciting
challenges and patterns.



Continuous and discrete spaces (e.g., the lattice of
points with integer coordinates) usually require
distinct methods of solution and offer contrasting
conclusions

CHANGE THE OPERATION


Algebraic: We can switch between addition,
subtraction, multiplication, division,
exponentiation, and roots. We can also change the
order of operations.



Geometric: We can change between scaling,
translating, rotating, and other transformations.
We can construct medians rather than
perpendiculars. We can trisect or n-sect rather
than bisect an angle, segment, or area.



Analytic: We can change the function involved
(example make it exponential rather than linear)



Probabilistic: We can substitute a predictable
behavior for a random one). For example,
Juancho, a fifth grader, altered the Connect the
Dots problem so that the jump size was chosen
randomly for each step;

Start with a circle with 16 points, equally spaced:
You are going to make a shape by jumping around
this circle. Pick a number, j, that determines how
far you will move around the circle with each
jump.
Start at the top point (labeled 0), and draw a
segment to the point j steps away.
Begin each new jump where the previous one ends
and continue this process until one of your jumps
returns to the starting point, 0.

CHANGES THE OBJECTS UNDER STUDY


Rather than just look at real numbers, we can
consider vectors, matrices, or functions (e.g.,
polynomials) as the operands.



For example, elementary school students often
discover that 2 + 2 = 2 * 2, but find no further
examples (except perhaps 0 and 0).

REMOVE OR ADD CONTEXT


If a problem comes with a particular setting, we
can make it abstract by removing any nonmathematical details.



Alternatively, we can add a story to an otherwise
abstract problem. For example, it is easy to dissect
a rectangle into four equal pieces, but when that
rectangle becomes a cake and there are four kids
who each want their fair share, a whole realm of
new and difficult mathematics problems emerge



You can turn each problem-posing method into a
practice activity. Ask students to superimpose a
story or context on an abstract problem of their
choosing (e.g., a geometric construction or a
system of equations they find by scanning through
a textbook).

REPEAT A PROCESS


iteration can lead to surprising and beautiful
mathematical questions and results.



We can repeat any operation, such as squaring a
number, bisecting a side, or rotating a figure, to
yield ever more complicated objects or sequences
for study.

WITH STARTING POINT IN SYMBOLIC SCHEMES
The teacher gives the formula a+b=x and requires as many
examples of exercises as possible. Then the teacher asks for the
formulation of varied problems. The position of the unknown is
changed ( a+x=c; x=a+b ; x+b=c ; etc.) with the same requirements
(proposing various exercises and problems). The same procedure
is carried out, starting from one of the models a-b=x, a+b+c=x, a-bc=x, etc., or from graphical models, diagrams, tables.
Children were asked to do the tasks described above in a gradual
progression of internalizing, which emphasizes recurrent cycles of
understanding: orally, mentally, in writing (without or with
minimum verbalization, and the result is required for checking).
Letters are to be used just accidentally, or gradually, depending on
the students’ level and teacher’s knowledge about their
appropriate use. Usually, instead of a,b,c,x, other symbols – more
familiar to children – were used as “boxes” or “shells” for the
substitutions.

EXAMPLE 1


1+1=?



Assume the student has already figured out that
the result is 2.



By increasing a term on the left by 1 the total on
the left is increased.



In order to get the equality back, one should
increase by 1 the right side as well.

Applying the enlightening idea repeatedly, the
student may even conceive of the notion that
adding any number on the left can be balanced by
adding the same number on the right. The left
hand side is a sum of two terms.
 A second observation can be made to the effect
that it does not matter to which of the terms the
number has been added. And then a third one that
the number does not have to be wholly added to
one of the terms - it can be split in any way
imaginable.


EXAMPLE 2


One of two brothers is 7 years old while the other
is only 5.



As you can see, the sum of their ages is 12.



Please invent another problem and solve it.



Even if the following was obtained as (7 + 1) and
(5 - 1)



One of two brothers is 8 years old while the other
is only 4. As you can see, the sum of their ages is
12.



One of two sisters is 7 years old while the other is
only 5. As you can see, the sum of their ages is 12.

DIFFERENT KINDS OF PROBLEM POSING

PROBLEM

The mathematics teacher assigns an assignment
to her students. The boy asked the Monster to
answer his assignments in his behalf.

PROBLEM POSING WITH PLOT
Examples:
List actions in a logical order
Clues to identify a part of the story
Compare actions
Make predictions based on evidence
Connect the plot to a math concept
Redesign the plot to include mathematics
A “math walk” through the story
Explore conjectures, estimations, generalizations

MATHEMATICAL TASK WITH PLOT

Let’s pretend that
the Monster in our
story charges the
boy 35¢ for each
multiplication
problem and 60¢ for
each division
problem he solves.

How many types of each
problem did the Monster
solve if the boy’s bill is
$15?

How many different
solutions can you find?

PROBLEM POSING WITH CHARACTER


Make the Character(s) come to life



Put the Character(s) in the students’ classroom,
bedroom, kitchen, or community park



Make the story real to children: what open-ended
tasks can characters face?

MATHEMATICAL TASK WITH CHARACTER
Let’s pretend that the
Monster’s Magic Calculator
has broken.
Only the 5, the 2, the ×, the
−, and the = buttons work.

How did the Monster
use the Magic
Calculator to get the
homework answers of:
-10
1
3
10
24
100

PROBLEM POSING WITH ILLUSTATIONS
Illustrations can:
Capture students’ interest and imagination
Visualize the mathematics
Enhance the details of the narrative
Place mathematics in a meaningful context

MATHEMATICAL TASK WITH ILLUSTRATIONS
How many different
monster faces can you
make from 4 different sets
of scary eyes, 3 different
eerie noses, and 3 different
sets of pointy teeth?

PROBLEM POSING WITH SETTING
Consider how the students can relate to the setting.
Settings help to create integrated teaching units.
Students can begin to see the mathematics in the
everyday.

MATHEMATICAL TASK WITH SETTING
Respond to our class survey
about the places where we do
our math homework.
1)At the kitchen table
2)In my room at home
3)At the library
4)At a friend’s house
Draw a graph to display
the results.

PROBLEM POSING WITH OBJECTS


Familiarity (and unfamiliarity) with objects in the
story provide learning opportunities.



Objects in stories can be compared, contrasted,
sorted, and classified. (Got Van Hiele?)



Objects in stories can be described or constructed
with 2D and 3D shapes

MATHEMATICAL TASK WITH OBJECTS
To do the boy’s math
homework, the Monster
requires a very special type
of pen.

• Which store has the
better buy, Pens-R-Us or
Pen-Mart? How do you
know?

Pens-R-Us sells 2 pens for
$1.40

• Which store would you
go to to buy 24 pens?
Show your thinking in
words, numbers and
pictures.

Pen-Mart sells 3 pens for
$1.99

PROBLEM POSING WITH TIME FRAME
The TIME FRAME of a story:


Builds understanding of time measurements



Creates the necessity for standard units



Can bridge a students imagination and reality



Allows for comparisons and contrasts

MATHEMATICAL TASK WITH TIME FRAME

Before the boy’s teacher
realizes that the Monster is
doing the math homework,
the Monster has spent
1,725 minutes doing
homework.

• How long is this in
hours?
days?
weeks?

SILVER AND CAI (2005) IDENTIFIED THREE CRITERIA
THAT ARE COMMONLY APPLICABLE TO MOST PROBLEM
POSING TASKS:
1.Quantity - refers to the number of correct responses

generated from the problem posing task.
2.Originality - is also another feature of responses that

can possibly be used as a criterion to measure students’
creativity.
3.Complexity - refers to the cognitive demands of the

task. It can be categorized as low, moderate, or high.

Low complexity

• Recall or recognize a
fact, term, or property
• Compute a sum,
difference, product, or
quotient
• Perform a specified
procedure
• Solve a one–step word
problem
• Retrieve information
from a graph, table, or
figure

Moderate complexity

High complexity

• Represent a situation
mathematically in more
than one way
• Provide a justification
for steps in a solution
process
• Interpret a visual
representation
• Solve a multiple-step
problem
• Extend a pattern
• Retrieve information
from a graph, table, or
figure and use it to solve a
problem
• Interpret a simple
argument

• Describe how different
representations can be
used to solve the problem
• Perform a procedure
having multiple steps and
multiple decision
points
• Generalize a pattern
• Solve a problem in more
than one way
• Explain and justify a
solution to a problem
• Describe, compare, and
contrast solution methods
• Analyze the
assumptions made in
solution
• Provide a mathematical
justification

Task objective: From the information below, construct
mathematical problems, and solve them, to
demonstrate your competency in using the basic
rules for manipulating inequalities to simplify and
solve simultaneous inequalities or inequalities
involving linear, quadratic or modulus functions.

A gardener is planting a new orchard. The young trees
are arranged in the rectangular plot, which has its
longer side measuring 100m.

SAMPLE 1 OF PROBLEMS SHOWING LOW
MATHEMATICAL COMPLEXITY


If a fence around the orchard measures more than
330m, and the area of the orchard is not more than
7000m2, find the range of values of the shorter side.

SAMPLE 2 OF PROBLEMS SHOWING MODERATE


The gardener decides to divide the plot of land into
three sections for growing three different types of
plants. It is given that section C is representative of a
quadrant and the area of section C is bigger than that
of section A. The various sections require different
types of soils of different prices. If the gardener has a
budget of $400 for buying soil for the orchard, what is
the maximum value of x?

SAMPLE 3 OF PROBLEMS SHOWING HIGH


It is given that the width of the orchard is 50m.
Starting from point B, a worker P walked along the
edge in a clockwise direction and back to B at a speed
of 2m/s. Another worker, Q, started from point A and
walked along the edge in the clockwise direction and
back to point A at a speed of 1 m/s. What is largest
possible area of triangle BPQ?

EXAMPLE PROBLEM
Ann has 34 marbles, Billy has 27 marbles,
and Chris has 23 marbles. Write and
solve as many problems as you can that
uses this information

STUDENTS ARE ABLE TO POSE PROBLEMS SUCH AS
THE FOLLOWING:
How many marbles do they have altogether?
 How many more marbles does Billy have than Chris?
 How many more marbles would they need to have
together as many marbles as Sammy, who has 103?
 Can Ann give marbles to Billy and Chris so that they
all have the same number? If so, how can this be
done?
 Suppose Billy gives some marbles to Chris. How many
marbles should he give Chris in order for them to
have the same number of marbles?
 Suppose Ann gives some marbles to Chris. How many
marbles should she give Chris in order for them to
have the same number of marbles?


MATHEMATICS CONTENT: LINEAR EQUATION WITH
ONE UNKNOWN


Situation: A factory is planning to make a billboard. A
master worker and his apprentice are employed to do
the job. It will take 4 days by the master worker alone
to complete the job, but it takes 6 days for the
apprentice alone to complete the job.



Students’ Task: Please create problems based on the
situation. Students may add conditions for problems
they create.



Problem 1. How many days will it take the two
workers to complete the job together?



Problem 2. If the master joins the work after the
apprentice has worked for 1 day, how many
additional days will it take the master and the
apprentice to complete the job together?



Problem 3. After the master has worked for 2 days,
the apprentice joins the master to complete the job.
How many days in total will the master have to work
to complete the job?



Problem 4. If the master has to leave for other
business after the two workers have worked together
on the job for 1 day, how many additional days will it
take the apprentice to complete the remaining part of
the job?



Problem 5. If the apprentice has to leave for other
business after the two workers have worked together
for 1 day, how many additional days will it take the
master to complete the remaining part of the job?



Problems 6. The master and the apprentice are paid
4500Pesos after they completed the job. How much
should the master and the apprentice each receive if
each worker’s payment is determined by the
proportion of the job the worker completed?



Problem 7. The apprentice started the work by
himself for 1 day, and then the master joined the
effort, and they completed the remaining part of the
job together. Finally, they received 4900 Pesos in total
for completing the job. How much should the master
and the apprentice each receive if each worker’s
payment is determined by the proportion of the job
the worker completed?



Problem 8. The master started the work by himself
for 1 day, and then the apprentice joined the effort,
and they completed the remaining part of the job
together. Finally, they received 4500 Pesos in total for
completing the job. How much should the master and
the apprentice each receive if each worker’s payment
is determined by the proportion of the job the worker
completed?

THE END
(Merry Joy Ordinario BSED 4B)

Problem posing

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