Chapter 5
Insight in the Pyramid, Level 3
Some key words and statements for problems labeled Level 3 are:
 Mathematical thinking and reasoning
 Communication
 Generalization
 Develop new strategies and own models
 Distinguish relevant information from redundant information
 Give mathematical arguments or proof
 Incorporate skills and competencies of other levels
 Make assumptions for information that is lacking
 Pose your own questions, not only solve those of others.
The problems on level 3 are hard to solve but it is also hard to design problems that fit
this level. Level 3 problems are often found in so called multiple question items or super
items. Examples of these multiple question items can be found in chapter 8.
In general these are unfamiliar problems for students but of course they should be able to
handle the mtahematics involved.
1.
A storm is expected. Is it possible that one of the trees behind the house will cause
damage to the house if it is knocked over by the wind? Explain how this question can be
answered if you are in this kind of situation and give mathematical reasons why your
strategy could work.
Classroom Assessment, Truus Dekker, Nanda Querelle, Corine van den Boer
2000
1
Possible student answers:
Note: Students may make drawings to clarify their reasoning. No examples of these
student pictures are shown
 Measure the distance from a tree to the house. Make a photograph of a friend
standing next to the tree. Estimate the height of the tree using the known length of
your friend. If the length of the tree is less than the distance to the house there should
be no problem.
 Measure the distance from a tree to the house. While the sun is shining, measure the
length of the shadow of the tree and also your own shadow. The ratio of your own
length and your shadow is the same as the ratio of the tree’s length and its shadow. So
now you can find the length of the tree. Compare this length with the distance to the
house.
 Take a known distance from the tree, for instance 50 meters. Measure the angle under
which you see the top of the tree. By using the tangent, estimate the length of the tree.
Compare this length with the distance to the house.
 ……………
Age: 12, 13 , 14
Level: 3
Content: space and shape
Context is relevant, situation daily life
Typical for many Level 3 problems is that it is not always easy to identify the content as
algebra, geometry, number etcetra. The next problem gives an example. It can be solved
in a number of ways; either geometrically, by drawing bars or pies and dividing those, or
by using number sense. Students are free to choose their own mathematical tools but the
problems also should encourage the possibility of choosing a more algebraic or more
geometrical way to solve a problem, according to capabilities or preferences of the
student. Moreover, if students solve the problem in different ways it gives you an
opportunity to see how a particular student is thinking. It also enables the teacher to share
different results with the whole group of students and discuss why one way could be
more appropriate in this case than the other.
2.
Your younger brother wants you to explain why 4/5 is bigger than 4/7
How would you do that?
Possible student answers:
 I made a drawing of two custard pies of the same size, one divided in 5 pieces, the
other one divided in 7 pieces. If you may take 4 servings of each, from the first pie
you get a bigger share.
 4/5 = 28/35 and 4/7 = 20/35; 28 is more than 20
 I will tell him that 1/5 is more than 1/7 If you share a heap of marbles with 5 children
you get more marbles as when you share the same number with 7 children.
Next, 4/5 = 4 x 1/5 and 4/7 = 4 x 1/7 so that does not change the idea.
Classroom Assessment, Truus Dekker, Nanda Querelle, Corine van den Boer
2000
2
Age: 11, 12
Level: 3
Content: Number, Geometry
No context
The expression “mathematization”, used for Level 3 questions is used in two ways.
1. Horizontal mathematization
Students need to find the specific mathematics in a general (realistic) context. Then
the problem can be solved using mathematical tools. They discover regularities and
relationships and recognize similarities between different problems. In chapter 2
some examples about linear relationships were shown on this last topic.
2. Vertical mathematization
Once a problem, posed within a context is solved, a translation is needed from the
mathematical back to the realistic situation. This may mean the mathematical model
has to be refined or adjusted or combined with other models. Students are expected to
prove regularities and give mathematical arguments for their views.
One could argue that mathematization occurs in all contextualized problems, since
students have to find the relevant mathematics at all levels. On Level 3 however this goes
beyond simple recognition of well known problems and requires integration of many of
the skills needed for the lower levels.
The next problems show that children at primary school can also be given interesting
Level 3 problems. For younger children teachers sometimes prefer to have arguments
given orally, because it is hard for them to write down their reasoning even if they are
perfectly able to tell you what they mean. By doing this type of problem in different
contexts, we hope students will learn to generalize and find out that it is basically the
same mathematical content, though the problems do look different. And that is an
important skill for students of all age groups.
The cow problem can also be used to discuss different solutions in small groups. This
helps students to express their opinion and share their views with others. Only one
solution is provided here. Some students used ratio tables where others, still working in a
more informal way, used a copy, cut the cows and pasted the same number of them in
part of the meadow.
Classroom Assessment, Truus Dekker, Nanda Querelle, Corine van den Boer
2000
3
3.
In which meadow does a cow have the largest grassy area?
meadow 1
meadow 2
Answer: In meadow 2
Possible student work:
I computed the area of both meadows and counted the cows in each meadow. Then I
divided the area of each meadow by the number of cows.
Age: 7, 8
Level: 2, 3
Content: Number sense, area, smart counting
Context is relevant but can easily be exchanged by a similar context, situation daily life
Classroom Assessment, Truus Dekker, Nanda Querelle, Corine van den Boer
2000
4
At Level 3 we expect student to pose their own questions provide answers for it and
argue mathematically about them, apart from the examples shown in their text books.
This gives the teacher an opportunity to find out whether misconceptions still exist, for
example when a student persistently uses “cube” instead of “square” or shows important
mistakes in the mathematical concept taught in this section of the book. By discussing
their problem with classmates, students are forced to compare their views with those of
others.
4.
a. A group of 90 children wants to go cycling on a pedal boat. Each pedal boat can hold
four children.
How many pedal boats are needed for the whole group? Show your work.
b. Now make a similar problem yourself, to be solved by the student sitting next to you.
Of course you must provide an answer yourself and compare this with your friend’s
solution.
Possible student answers to question a:
 A student asked his classmates to form groups of four, each for one boat. Then he
counted: for 20 students you need 5 boats, double and double again, 80 students for
20 boats. The other 10 students need 3 boats, 23 in total.
 Some students make drawings of pedal boats and passengers, (which however is very
time consuming!) You need 23 boats.
 Students make groups of four tallies, and 2 extra in the end
 Repeated counting of groups of 4 (4 + 4 + 4 + …., or 4, 8, 12, …) After 22 x 4 you
have an extra two students, so you need one extra boat, 23 in total
 Student used a ratio table:
students 4 8 80 88 For 88 students you need 22 boats, you need one
boats
1 2 20 22 extra boat for the two students, 23 in total.
Note: What can be expected from students answering question b, depends on their age
and their experience with similar questions. For a six year old it is already great when
(s)he is able to change the numbers appropriately. In their perception the situation really
changes when boats are used that can hold 2 children instead of 4.
Age: 7, 8
Level: 2, 3
Content: Number sense, smart counting
Context is relevant but can easily be exchanged by a similar context, situation daily
(school) life
By correctly making your own problem and discussing solutions with a friend, you can
show if you are understanding this type of problem and that you are able to generalize
and show insight in the number system. Some students find challenging problems that
show their teacher they are already capable of doing problems on a much higher level,
where others just copy the problem with other numbers, thus showing they need more
exercise in this type of problem.
Classroom Assessment, Truus Dekker, Nanda Querelle, Corine van den Boer
2000
5
The last problem in this chapter builds on the previous problems about the Pythagorian
Theorem on Levels 1 and 2 found in chapter 4, however it is now posed in its most open
way:
5.
Two neighbors share a brick wall that divides their front yards in two equal parts. (See
photo).
“This is not fair!”, neighbor A states “The wall is built oblique.” Neighbor B does not
agree.
Find a way to decide for these neighbors whether the wall is connected with the front of
the house in a way that both front yards are equal in area. You need to give mathematical
proof to settle the argument.
Possible student answers:
 Suppose (you cannot see that on the photograph) that the front yards are built as a
rectangle. Measure the two diagonals of each rectangle and if they are of the same
length, you can decide the wall was not built oblique.
 Take some measurements and use the Pythagoras Theorem to decide whether the
angle between the wall and the front part of the house is 90 degrees.
 ………
Age: 15, 16
Content: space and shapes
Level: 3
Context is relevant, situation: daily life
Classroom Assessment, Truus Dekker, Nanda Querelle, Corine van den Boer
2000
6
Summing up our findings in this chapter:
1. Level 3 problems are hard to solve and hard to design.
2. The mathematical strand or content of the problem is not always clear.
3. More often than not there are more than one correct answers.
4. Students have to give explanations, mathematical reasoning or proof.
5. Level 3 problems are also possible for younger students.
Classroom Assessment, Truus Dekker, Nanda Querelle, Corine van den Boer
2000
7