Chapter 2 Jan de Lange’s Pyramid

In a balanced test, students should have the opportunity to show what they have learned and what they are able to do. Teachers may want to describe student understanding and capabilities. Also, teachers may want to describe students’ misconceptions, partial understanding, etc. They want answers to questions about students’ mathematical abilities, such as: does a student only master basic skills? is he able to give proper (mathematical) reasons? can he solve more complex problems?

Because a teacher wants to give the student feedback, (s)he needs to know which subjects are not mastered yet by the students and perhaps need more emphasis in class. It is also important to know which steps must be taken to help the least able students and give the more gifted ones enough challenging materials. To find answers for all of these questions teachers need a great variety of tasks and tests on different levels of competency. Moreover the mathematical content of the problem and the complexity of the task must be considered. As you see there are many factors to be taken into account! Help is needed to bring order to the chaos. We will show a model for the classification of problems known as ‘Jan de Lange’s pyramid’. This model was originally designed as a triangle, figure 1

figure 1 Triangle

Horizontally in the triangle of figure 1, you see the different mathematical strands. Vertically you see the different levels of competency a student needs to have in order to answer the question correctly. These are labeled 1evel 1 for the lowest level to level 3 for the highest one in the triangle. In this chapter is an explanation of this concept. In the next chapters it is elaborated further and many examples are shown to clarify the theoretical underpinning of the model.

Why is the model shown in figure 1 a triangle and not a rectangle? The triangle shape gives an indication of the number of problems and questions on the different levels in a balanced test. It also represents the division of score points over the different levels. As you can see in the model, the largest amount of time as well as the largest number of score points should be given for level 1 questions. They are at the base of the triangle. At the top of the triangle are the level 3 problems. They should appear in each balanced test but fewer in number than the other ones. Level 3 problems are harder to answer and take more time than questions where basic skills are used. That is another argument to have fewer level 2 and -3 questions in a test than level 1 questions.How a balanced test may be designed, using problems on all of the competency levels will be discussed in chapter ….

Using this model and discussing it with teachers, we found the triangle shape did not suffice. Some people thought that because a problem is a difficult one, it immediately should be labeled with a higher level of competence. But that is not necessarily true, so the triangle was reshaped into the pyramid of figure 2. The third dimension now available is the one that shows a distinction between simple and more difficult problems on the same level. A more detailed discussion of distinctions between easy and difficult problems on the same level occurs in the chapters that follow.

figure 2, Jan de Lange’s Pyramid

Level 1: Reproduction, procedures, concepts and definitions

Responses to a question labeled Level 1 often require knowledge of facts, definitions and routine procedures that have been memorized and have been practiced during previous lessons. Examples of questions that require knowledge of facts and definitions:

1What is an equilateral triangle?

Calculate 7 x 12

Draw a geometric figure with two parallel lines.

What is the name for a graph of a squared relation? Examples of questions that require the use of routine procedures:

2Draw the graph y = 3x – 2

Without using a calculator, compute 3674 : 26

Change 27,000,000 to scientific notation

Solve the equation x2 + 3x – 17 = 0

Compute

Examples of questions that require applying standard algorithms:

3Compute the area of a circle with radius r = 7

Use the Pythagoras’ Theorem to compute the length of the unknown side of this right triangle:

7

6

Solve this proportion:

A considerable part of a test will often consist of level 1 questions that are given to students to determine whether the basic facts or skills taught in a chapter of the book are mastered yet. Level 1 questions will often appear as separate questions. In larger context problems that contain more questions on the same context, level 1 questions are sometimes used as “entry”questions to give students the opportunity to explore the contents of the context. Posing a question on level 1 does not mean students do not have to show their work. As a teacher you want to determine if a student has mastered some routine procedure, however if it is wrong, it is important to know where and why. When there is no work to be shown, even

though the right answer is given, the student may have arrived at it in a totally wrong way as the next example shows:

4The tangent of the straight line 5x – 8y – 3 = 0 and the positive x-axis is:a. 5/8 b. – 5/8 c. 8/5 d. – 8/5

Answer: a

Age: 14, 15Level: 1Content: dependency and relationships, functions

Benno: The answer is a, because the height is always less than the distance and it is positive since it says positive x-axis.

In general, if a teacher does not know how a student got the answer, (s)he cannot give appropriate feedback. It may be a good idea to let the students use half of the test paper to make calculations, even if multiple choice questions were given. Sometimes important information is found there that could help to understand and interpret the student’s work afterwards.

Level 2: Connections and integration for problem solving.

On this level of competence, students have to choose their own strategies and choose their own mathematical tools. Problems on this level more often can be solved in several correct ways. Sometimes a student solves a problem in an informal way, where he already has been taught more formal ways. This provides important information about the level of students thinking.

Moreover, at this level, we start making connections between the different domains in mathematics. Information is integrated in order to solve relatively simple problems. It is also possible there is redundant information given and the student will have to decide which information is relevant for solving the problem. On this competency level, students are expected to handle different representations according to a specific situation and purpose e.g. text, diagrams, formulas, tables and so on. An example to demonstrate where students can use different strategies to solve a problem is shown:

5On a copying machine you can enlarge or reduce the size of a drawing. Nathan used the machine to make a copy of a drawing, reducing the size to 60%. He was not satisfied with the result so he took the copy and made a new one at 140%. He assumed that now he would have the original measuremennts again. Was Nathan right? Explain your answer.

Answer: No, Nathan is not right.Note: If no explanation is given, no score points should be awarded for the correct conclusion. “Never take ‘no’ for an answer!”.Possible student work: I calculated 140% of 60%; 1.40 x 0.60 = 0.84. The second copy is 84% of the

original. I assume the length of the original drawing was 20 centimeters.

60% of 20 = 12; 140% of 12 = 16.8 The second copy is smaller than the original. Student made drawings to support her reasoning.

Age: 13, 14Level: 2Content: Number

The connections aspect requires students to distinguish and relate different statements, like definitions, examples, assumptions and proof.They will need to know the difference between a realistic situation and the mathematical model and be able to translate one into the other. At this level, they are not expected to model the situation themselves. We will address this aspect later.

Level 3:Mathematization, mathematical thinking and reasoning, generalization and insight.

At this level, students will have to:a. mathematize situations;

6Uncle Peter brought a large bag of marbles for the three of us. “Share fairly”, he said. How would you do that?

Note: The answer could be given orally!Possible student work: We give a marble to each child, one at the time. If one or two marbles are left, we put

those back in the bag. We make three rows of marbles. If these rows are equal in length, we shared fairly. We count the marbles, five at a time. Each child gets five marbles until all are gone.

Age: 6, 7Level: 3Content: Number, smart counting

b. recognize and extract the mathematics embedded in the situation,

7Which is the best bargain: 3 pieces for $10 or $4 a piece and a 15% discount?

c. choose mathematical tools to solve more complicated problems;

8Explain why you can be sure the white part of this drawing is larger than the shaded part.

Possible student solutions: using reallotment (cut and paste) computing areas using fractions using percent

Age: 11,12Level: 3Content: ratios

d. be able to compare the mathematical content with that in other context problems and generalize. For example, the situations in the next problems, as well as in many others, make use of the same mathematical content where linear functions play a role. The students may have learned what linear functions look like and how you can recognize them if a table with data on the variables involved is shown or how to recognize the graph that represents a linear relationship. During the lessons they should have seen many different contexts on linear relationships.

9Peter works as a handy man. He charges $10 for call-charge and $75 per hour. His friend Nolan charges $25 for call-charge and $70 per hour. How many hours does a job take where they charge the same amount of money?

10The height of a stack of coffee cups can be calculated using this (word)formula: height (in cm) = 7.5 + (number of cups – 1) x 3.5The kitchen shelf in a restaurant where these cups will be stored has a maximum height of 30 centimeters. How many cups can be stored in one stack?

11As a general rule you can use this (word)formula to know whether you have the right weight: length (in cm) = weight (in kg) – 100 Give an example of a group of people for whom this formula does not work.

In a test, which may be given much later in the school year, a completely different context is shown that the student has not seen before. Now he must be capable of generalizing this type of situation and think, “This is this type of relationship where you start with a fixed amount of something and with each next step you add (or subtract) the same number.” The competencies on this level include a critical component and reflection on the process. Students should not only be able to solve problems but also to pose questions and communicate. Correct mathematical reasoning is required and the students must be able to criticize a mathematical model and remodel if necessary. They can make a mathematical model of a realistic situation, try to solve the problem, remodel, solve the problem, make a transition back to the realistic situation and decide whether the solution is useful within this situation.

12The walls in the cellar will be painted. The total area is 22 square meters. How many cans of paint do you need? Each can covers for three square meters.

As is to be seen in the triangle, at the highest level of competence the difference between the mathematical strands disappear, sometimes the same problem can be solved both geometrically or algebraically! Once again, the student will choose his or her own strategy or even invent new strategies.