Chapter 6
Context
How to choose appropriate contexts for problems to be used in a test.
Why do we want to use contextualized problems in a test and why don’t we use just
questions on the mathematical content that was taught? In chapter 4 it states that by
posing a question in context, it makes the problem a little harder to solve. So why bother
and make things more complicated for students? It is hard enough to build on basic skills,
is it not? The reader may ask the question: “Is there a difference between being able to
subtract two numbers and finding the difference in price between two candy bars?” At
least for students there is and from our experience we can tell that solving realistic
problems can drive skill building and actually does so very effectively. We want our
students to be mathematically “literate”, which means that we are not just concerned with
mathematics at some level of understanding but also with using mathematics in a whole
range of situations. These situations are called the context of the problem. Note that this
context could also be drawn from mathematics itself.
Students hardly ever meet bare problems in everyday practice. Moreover it is sometimes
hard for our students to recognize the mathematics in real world problems. But there are
also students who may be weak in mathematics but are nonetheless capable of solving
diffidcult problems when shopping or while doing practical work as a carpenter or a
bricklayer. As soon as there is a real need to solve a practical problem, students show
more skills then teachers expected them to master. This means that we not only need bare
problems that make it possible to show the mastering of basic skills but problems that
have a link with to the real world as well. It does not mean we could do without bare
problems at all, students need to practice and get accustomed to routine procedures.
“Modeling” is the key word in this concept. There is a real problem in the real world that
needs solving. The first step taken will often be making a mathematical model of the
situation. Than the problem is solved within the model with the aid of mathematical tools.
Afterwards the results are ‘translated’ back to the real world and the results are adjusted
accordingly. For instance, the perimeter of a circle can be rounded to three decimals. But
when it is an answer to the length of an ornamental string to be sewn at the hem of a
round tablecloth, it must be rounded upwards to the nearest ten (centimeters) if this is the
way the string is sold. In this chapter we will discuss the choice and the different roles
and functions of the contexts used in tasks for assessment.
A summary of the reasons to use context problems:
 To introduce a new subject or a new concept in mathematics. By using examples
within a context, the mathematical content involved becomes clear.
 To practice a new concept or procedure. By doing many different context problems
with the same mathematical content the students learns how to use and apply this
content.
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To show the power of mathematics. By understanding that different context problems
are based on the same mathematical content.
To show a student masters the mathematical content. By using an unfamiliar context
in a test that is based on the same mathematical content that was used during previous
lessons.
To get students involved in the problem. By using real life problems students can
show they are mathematical literate and know how mathematics is used to solve
practical problems showing up in real life situations.
The role of the context can also be different in different classroom situations or when
used in a test. An overview of these different situations is given below.
Relevance and Role of Context
Relevance, zero-order context
The context used should be relevant for solving the problem. Always ask yourself: “Why
would anyone want to know the answer to this problem”.
The first problem provides an example of an irrelevant context, referred to as ‘zero-order’
context.
1.
The cross section of a wine glass has got the shape
of (part of) the following sinusoïd:
y = 3sin(x + ½) + 3
Determine the volume of this glass.
Note: Since this problem is considered a bad example, no other data are given about it.
The context is clearly used to “dress up” the mathematical problem. You could not really
use a wine glass with this shape. So why bother to calculate the volume?
Another example:
2.
Rain is falling in an angle of 48 degrees because of the wind. Next to the building, a dry
patch is found, width two meters. Using this information, calculate the height of the
building.
Note: Since this problem is considered a bad example, no other data are given about it.
There must be an easier way to find the height of the building than first finding (how??)
the angle in which the rain is falling and next using the tangent in order to find the height.
We call this type of problems “nontext”-problems!
First order context
The real first order use of the context is found when the context is both needed and
relevant to solve the problem and make judgments about the answer.
3.
For a parent’s evening at school 150 parents are expected. At each table four chairs can
be placed.
How many tables are needed?
Show how you found your answer.
Answer: 38 tables
Possible explanations:
 student made drawing(s) of chairs and tables
 student tallied groups of four and counted
 4 – 20 – 40 – 80 – 120 – 140 parents
1 – 5 – 10 – 20 – 30 – 35 tables
for 10 extra parents you need 3 extra tables
 150 : 4 = 37, rest 2 So you need one extra table
Age: 7
Content: quantitative reasoning, number sense
Level: 2 or 3
Context is relevant, situation: daily (school) life
The problem is really different from: Compute 150 ÷ 4 =
Second order context
The context is needed and relevant to solve the problem and make judgments about the
answer. Moreover you have to make a model, in other words ‘mathematize’ the problem.
This may sound poshy, but it is good to realise that for students it is not easy, nor obvious
that you have to find the necessary mathematics needed to solve the problem.
4.
A ladder, length three meters, is placed against the wall, one meter from the bottom of
the wall. Up to which height on the wall does the ladder reach?
Answer:
The ladder reaches up to about 2.8 meters.
Possible explanations:
 h2 + 12 = 32;
h = 8  2.8
or
 student made a correct drawing to scale
Age: 14, 15
Level: 2
Content: Space and shapes, calculations in geometry
Context is relevant, situation daily life
The student has to choose the appropriate mathematical tools to solve the problem. It is
possible to make use of a right triangle and the Pythagorian Theorem, but making a
correct drawing to scale will also provide an answer that is accurate enough in this case.
The next problem can be used in different ways. The way shown in the example starts
with a level 3 question because of the complexity of the information. The student must
translate the model into mathematics and realize that first you need to find the diameter
of the tablecloth. Afterwards the answer found within the model must be translated
backwards into the real life situation and the student needs to reflect on the context in
order to find the width of the available cloth that meets the conditions. Than (s)he has to
mathematize again to calculate the costs.
5.
Marian has got a small round coffee table, height 60 centimeters, diameter tabletop 50
centimeters.
She wants to sew a round tablecloth for this table, which just touches the floor. It should
be made from one piece of cloth and as economical as possible.
The hem will be 1 centimeter.
In the shop Marian found cloth in the right color, available in two different widths:
Width 90 centimeters at $9.50 per meter, measured per 10 centimeters.
Width 180 centimeters at $17.50 per meter, measured per 10 centimeters.
1. Find out which width of cloth Marian will choose and how many centimeters in
length she will buy.
2. How much does Marian have to pay?
Answer: 1. diameter of the circle that represents the table cloth: 60 + 50 + 60 = 170 cm
Marian will choose a piece of cloth width 180 cm, length 170 + 1 + 1 = 172
cm
Marian needs 180 centimeters, you need to round upwards
2. 180 cm = 1.80 m
1.80 x 17.50 = 31.50, Marian pays $31.50
Age: 14, 15
Content: Space and shapes, 2D, measurements, number sense
Level: question 1, level 3
question 2, level 1
Context is relevant, situation: daily (professional) life
If students are not used to solving problems in contexts and a teacher wants to teach them
to do so, it is helpful to cut the questions into a number of smaller ones to make them
realize what is done here:
1. Make a sketch of a cross section of the table with the cloth.
2. What is the width of the cloth needed?
3. Taking into account that a 1 centimeter hem is needed, what is the length of the piece
of cloth?
4. Which size is appropriate, the one with width 90 centimeters, or 180 centimeters if we
want to be economical?
5. How many centimeters of cloth does Marian buy, rounded off properly?
6. How many meters are equal to the length in centimeters?
7. What does Marian have to pay?
One has to realize that by cutting the problem into this number of questions, the level of
each separate question changes to Level 1!
Third order context
This type of context serves as an example for the construction or re-invention of new
mathematical concepts. The bus model in the next problem is used as an informal way to
introduce addition and subtraction. The teacher tells a story about a bus driver who keeps
track of the number of people in his bus. People are getting on and off the bus at
differents stops. Students play the story walking around the classroom. One students gets
a drivers cap and other students hop in and out of the “bus” while calling aloud the
number of people in the bus. Afterwards the students do problems on paper. When the
students master the bus model, a notation with arrows is used. This arrow notation refers
to the bus situation but it can also be used for events in other contexts than that of the bus,
for instance about a game of marbles. Students can make strings of arrows and are
encouraged to tell their own story.
6.
Here you see a bus problem.
1. Use arrows for this problem.
Answer:
+4
3 7
2. Now make your own problem.
An example of student work:
+2
-3
+1
(1  3  0  ………)
Age: 6, 7
Level: from I to III
Note: This problem is labeled I to III since this depends on whether they are familiar with
the bus model or not.
Content: Number, basic operations
Context is relevant, situation daily life.
In the previous part, contexts were classified according to the different roles they can
play in a problem. Another distinction is also possible.
Real versus artificial versus virtual context
Virtual context
A virtual context may contain elements that are not real itself but are drawn from reality.
Real problems often are too complicated to use in a mathematics context so we simplify,
idealize or generalize.
On a real map you will hardly expect to find this kind of contour map:
7.
At which height is the climbers’ hut situated?
Answer:
The hut is situated somewhere between 350 and 400 meters. (You cannot give a more
exact answer, since the contour lines go up by 50 meters).
Age: 15,16
Level: 1 or 2
Content: geometry
Context is relevant, situation daily life.
The situation is simplified and idealized. Though in general students will probably not
have difficulties in answering this particular question, one should always be aware of the
influence this ‘simplifying the situation’ might have on students.
Artificial context
Sometimes a context can be stylized or generalized but that is not what is meant here.
When an artificial context is used, the situation for the context problem is a fantasy
world. Students are not always willing or able to co-fantasy but sometimes this type of
situation can be justified, especially for younger students.
8.
Seven gnomes work in the woods. They stop for lunch. Is there a toadstool for every
gnome?
Answer: No, there are seven gnomes and only six toadstools.
Age: 5, 6
Level: 2
Context is relevant, situation fairy tale.
Mathematical context
Though the content of the problem is taken from mathematics itself, it is not a bare
question. Moreover, the question is an unfamiliar one where subject matter that was
taught during the previous lessons is now applied to a new situation.
9.
Below you see a graph of y = 2x. The straight line goes through both (0,0) and another
coordinate pair consisting of whole numbers.
a. Write down one coordinate pair of whole numbers on the line y = 2x
There are many straight lines through (0,0) and another coordinate pair of whole
numbers, for example y = 125x
b. Write down one coordinate pair of whole numbers on the line y = 125x
c. Is it possible to make a graph of a straight line through (0,0) which nowhere has a
coordinate pair of only whole numbers?If your answer is “yes”, provide an example, if
your answer is “no”, explain why not.
Answers:
a. Many answers are possible, examples are (1,2); (-2,-4); (10,20)
b. Many answers are possible. Formulas should have the form y = mx, where m is a
whole number. Examples are y = -5x; y = 8x; y = 237x
c. Yes, it is possible. Examples are y = 3x; y = -17x; y = x
Note: Only full credit is given when a correct example is provided.
Age: 13, 14
Level: a. 1; b. 2; c. 3
Content: algebra, linear functions
Context: mathematics
This problem is meant for a test. When the answers are discussed with the students
afterwards, the following question could be added:
d. Suppose I could draw all the straight lines through (0,0). Would my paper become
totally black?
Answer: Yes,because each pair of coordinates, whole numbers and other ones, will be
reached by some line of this whole range.