National Education Policy 2020 lists down several initiatives to improve learning levels among students, especially reforms in assessments. The Central Board of Secondary Education (CBSE), one of the largest boards in India, has announced a series of initiatives to implement the education policy and initiate the reforms. The increased weightage of multiple-choice questions (MCQs), and inclusion of innovative question types like case-based questions are excellent steps in this direction. To take forward the reforms and introduce them to the schools and students in India, we, as the teacher community, need to step-up and upskill ourselves in making good MCQs to enable the shift from assessing rote learning to assessing competencies and mathematical thinking. Some of the innovative question types like context-dependent or case-based MCQs and assertion-reasoning type MCQs are in line and appropriate for the intended shift. **The assessments dictate what is expected of students and hence play a major role in accelerating the shift.** Students need to shift their learning process from, let us say, solving every textbook question multiple times and solving board exam papers of last 5 years to demonstrating proficiency in applying the understanding in an unfamiliar situation/problem largely.

Let us now try to understand the reality. Take a pause and try to answer the following question:

Which of these statements is/are true?

Though contradictory, both the statements are true. The article, “The PISA test – why do Indian students struggle in tests like these?” by Sridhar Rajagopalan beautifully explains the reasons for this bizarre dichotomy. The limitation of having only ‘textbookish’ examples in important exams like board exams and their familiar pattern make them predictable. This also makes it easier to score higher with rote learning without developing any critical thinking. In the pre-independence era under the British rule, the education system may be designed to produce people for clerical jobs which is no longer the current need. On the contrary, students are expected to be critical thinkers solving real world problems and be innovators in their adult life. How will the current form of education prepare them to be critical thinkers? One of the means is to give them greater exposure to real world problems to solve and help develop problem solving ability. PISA questions are context-based or case-based questions; often the context is based on real-life application of maths.

In this article, I argue how good questions, particularly MCQs and innovative question types like context-dependent or case-based and assertion-reason will help create the shift towards assessments that promote critical thinking. I discuss with the examples from maths but the ideas can be easily generalised to sciences. I would like to give some ideas on how teachers can go about creating such questions. A recent survey by Educational Initiatives (Ei) with responses of 785 Maths teachers across the nation demonstrates the need among teachers to upskill themselves on making case-based questions along with other important skills in creating good MCQs like developing strong plausible distractors.

### Why should we make good questions?

If beauty can lie in the eye of the beholder, a good question too can be different for different people. One of the qualities of a good question is how much more information it provides about students’ level of understanding so that appropriate actions can be taken and pin-pointed feedback to learners shared.

Which of the two questions would you use to gain maximum insights about students’ proficiency in finding area of a trapezium? Which one would you pick for a competency-based assessment?

One may argue question 2 is not good as it has many side-lengths mentioned that are not required to find the area. This would confuse a test taker. Is it unfair to expect the students to know that the formula for the area of trapezium requires side lengths of **parallel sides** (not non-parallel sides) and altitude on them? Also, students should be able to find the area of any trapezium given its dimensions, irrespective of its orientations. In a real-world problem, knowing what information is required to solve the problem at hand is also a skill required, right? Often, students also struggle to understand that the altitude on one of the sides can lie outside the shape. Hence the 2^{nd} one is my preferred example. Only about 33% of 7809 grade 9 students of English-medium private schools taking Ei ASSET[1] were able to answer the question correctly and about 29% students took lengths of non-parallel sides and altitude on them in the formula, only to find the area incorrectly.

### What are context-dependent or case-based questions?

These are the questions based on a context/case, usually a real-life example.

Can you visualise case-based questions for grade 10 involving Ferris wheel? What maths concepts/skills can be assessed with it for grade 10 students? Now compare with the example below.

**An example:**

*[Common text (group text) based on which questions are asked:] *

**Ride on a giant Ferris wheel:** The London eye is a giant Ferris wheel in London. Passengers are made to sit in sealed and air-conditioned passenger capsules as shown in the figure.

It takes approximately 40 minutes to complete one revolution. The diameter of the Ferris wheel is 120 m and its lowest point is 10 m above the ground.

Dimpy enjoys a ride sitting in a passenger capsule on the London eye for 3 revolutions.

Answer questions based on this information.

*[Questions based on the given common text:]*

What good qualities do you see in the above questions?

### What are the qualities of good context-dependent or case-based questions?

- The context on which the group text/common text is based should be
**authentic**and not contrived. - The context should be
**relatable**so that students can understand the situation/context/real-life situation. - Questions should require common text on answer and should be based on key mathematical ideas involved in the common text.
- Involves a
**situation not found in a textbook**so that questions based on them are not textbook questions. (These questions differ from the word problems in textbooks.) - Questions assess important mathematical concepts/skills/competencies.
- Each question assesses different mathematical concepts/skills.
- Questions are of
**varying difficulty assessing different cognitive levels**mentioned in Bloom’s taxonomy. - Information in the questions and common text is authentic as far as possible. At least should not be contrived like saying the giant Ferris wheel is 500 m or 1 km high.

What are the effects on the learning levels of students with such good quality case-based questions? It can kindle interest towards Mathematics, right? Will these not make them more competent for good assessments like JEE, college-admission tests like SAT, GRE etc.?

These questions are not only good sources of assessments but also stimuli to discover new mathematical ideas and thus *hooks* to introduce new concepts/skills and develop mathematical thinking.

### How does one develop good context-based or case-based questions?

Equip yourself with knowledge and draw inspiration from good quality questions from other sources.

- PISA released items are a rich pool of such questions and a highly recommended resource. Similarly, relevant released items of the Trends in International Mathematics and Science Study (TIMSS) can also be resourceful.
- Ei ASSET questions based on real-life applications are also another rich source.
- Assessments of good international boards of countries performing well in TIMSS and PISA like Singapore, Finland, the UK etc. can also be very resourceful.

Look for good sources for case-based questions. Whenever we see maths around us in day-to-day life, we must remember that they are good potential sources for case-based questions. Refer to a blog article written by this author on the same for case-based questions in lower grades.

Look for similar sources. One can have different sets of questions based on them. For an example, different sets of questions can be asked on a general Ferris wheel in a fair or famous one in India or abroad. (High roller in Las Vegas is the tallest Ferris wheel!) Look for sources which involve uniform circular motion. Simple harmonic motion can also be described with uniform circular motion. E.g. a ride on a giant swing!

Questions on period, amplitude (highest height during the swing from the ground), nature of the graph showing variation of height of the swing from the ground during a ride etc. One can also ask if the swing is getting faster, how the graph changes.

Some of the other sources:

- Applications of Maths in other Sciences especially Physics (Watch out Eklavya 3030 webinar series by CBSE!)
- Applications of Maths in sports
- Market transactions, best buys, best scheme, discounts/tax calculations, EMIs, investments etc.
- Advertisements and survey results in magazines, newspapers, articles on web etc. (These are rich sources of graphs to frame questions assessing data interpretation skills.)
- Floor plans of a building, maps etc.
- Different types of coordinate systems being used
- Board games/card games

Recommend readers to watch out this YouTube video which has a section on how to make good case-based questions along with best practices in developing good MCQs.

### What are reason-assertion type MCQs? How can one develop them in Mathematics?

Study this example based on a case of London eye discussed earlier.

**Assertion (A)**: The shape of the graph of the height of Dimpy’s head from the imaginary horizontal line passing through the centre of the Ferris wheel remains the same as that from the ground with graph shifting downwards.

**Reason (R)**: The imaginary horizontal line through the centre of the Ferris wheel (new reference point) is at a constant distance from the ground (old reference point).

- Both A and R are true. R is the correct explanation of A. (correct answer option or key)
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false.

There are two statements A and R in a reason-assertion type MCQs. A student is expected to see if assertion statement A is correct and R is the correct explanation or reason for the statement A being true. There are 5 choices, one of them having the correct answer. To develop such MCQs, one makes the claim as the assertion statement and the reason statement which may or may not be the correct explanation/reason/justification for the claim/assertion statement. The 5 choices are standard choices for any such MCQs. One may remove one of them if developing a 4-choice MCQ. The assertion can be an observation or hypothesis. The reason statement can be a theorem, a result or a property which maybe the justification for the assertion statement to be true.

Such types of MCQs are also good to diagnose misconceptions. Study the example below:

**Example**:

**Assertion (A):** The surface area of the cuboid made of unit cubes decreases if the shaded unit cube is removed.

**Reason (R):** As volume of any solid decreases, its surface area also decreases.

- Both A and R are true. R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are false. (key)

It assesses the misconception that the surface area of a solid always decrease (increase) if its volume decreases (increases).

MCQs in general and reason-assertion type MCQs in particular are helpful in diagnosing misconceptions and common errors. So, misconceptions/common errors are one of the good sources of question ideas for such MCQs.

### One of the other forms of innovative MCQs

**Example**:

Circle the statements which is/are correct.

- The opposite angles of a parallelogram are equal.
- The lengths of the opposite sides of a parallelogram are equal.
- There is a type of a parallelogram with all its angles as right angles.
- There is a type of a parallelogram which has 4 sides of equal length.
- Some parallelograms have more than two lines of symmetry

The above example is helpful to assess properties of a parallelogram and that a rhombus, square or a rectangle is also a parallelogram.

In general, not only such questions and types of questions described in the article can stimulate students thinking but also makes the assessments and learning of Maths engaging and relatable. Assessments are **for** learning and not necessarily **of** learning. Such innovative question-types are good for high quality assessments and for developing mathematical thinking.

[1] Ei ASSET is a skill-based test that measures students’ conceptual understanding and benchmarks the school’s performance at international, national & regional levels with actionable insights through easy-to-understand reports.

### Maulik Shah

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