* Not from the examination point of view

These 7 seemingly innocuous words come together 12 times as starred footnotes in a popular class 10 mathematics textbook.

A textbook that is likely to have been consulted by around 21,50,000 students (perhaps more) during the 2020-21 academic year.

When do these notes appear? Is there a pattern to the excluded content? What is the subliminal message that it might send to children? Is the note subtly symptomatic of bigger issues that plague the Indian education system?

When does the note appear?

The note appears in conjunction with:

• 7 summative exercises
• 1 question within a regular exercise
• 3 solved examples
• 1 conceptual proof

Summative exercises are ones with questions that draw on from all concepts covered in the chapter; these come at the end of the chapter. By logical extension, regular exercises are ones that appear after a section (or chunk of a chapter) with questions that draw on the key ideas covered in that section. Solved examples are questions that are worked out in the textbook and used to demonstrate the application of a concept.

Evidently, majority of the content earmarked by the authors as being not (relevant) from the examination point of view pertains to summative exercises.

Let's go a bit deeper...

Is there a pattern to the excluded content?

I examined the content/questions in the summative exercises to try to understand what made them unsuitable from the examination point of view (as per the authors). How were they different, if at all, from the regular exercises?

I can sum up my findings in a single word - rigour.

All questions in the excluded summative exercises could be answered by applying the concepts covered in the chapter - this is similar to regular exercises. The only difference was the rigour of the questions - the questions in the summative exercises required children to evaluate and analyse the question (these are higher skills in the Bloom's taxonomy pyramid which I reference in this post). Arriving at the path to the solution wasn't straightforward and was likely to push their thinking.

On the other hand, the regular exercises, by and large, had questions that involved either direct recall of facts, the rote substitution of values into formulae to obtain solutions and applying the concepts to direct contexts and situations.

I would like to illustrate the above point with a few concrete examples from the textbook. You, the reader, might have studied high school mathematics years ago! 😅 Hence, I have chosen topics and tried to express myself in a way that will help you grasp the essence of my point.

The first example is from an algebra chapter - arithmetic progressions. In the column to the left, I have pasted (as is) a question from the regular exercises (RE) and, in the middle column, I have pasted (as is) a question from the summative exercises (SE). I have compared questions that cover the same concept and added comments to explain the difference in rigour in the column to the right.

Here are two examples from the chapter on surface area and volume.

A final example from the chapter on probability.

On studying the excluded content in the regular exercise, solved examples and conceptual proof, I concluded that all were excluded for the same reason - rigour. Grasping them required children to step outside their comfort zone but the concepts involved were very much within the scope of the syllabus.

These findings were disconcerting. Why were textbook authors and paper setters consciously excluding rigorous material from the exam?

(Rigour is a theme that I had discussed in my post on the strengths of the Finnish teacher education system.)

What is the message that the note may send to children? Is there a bigger issue here?

Children are perceptive to the underlying meaning behind such notes. I've taught students who used this textbook and were quick to point out that "This question won't come in the exam! Why are we solving it in class?" or "This problem is not in the syllabus!" Applying concepts to rigorous unfamiliar questions was deemed unnecessary once they realised that they wouldn't be 'tested' on it.

Another direct implication of the note is that there are types of questions that may be asked in the exam (from the regular exercises) and types of questions that will definitely not be asked (from the summative exercises) in the exam.

Speaking as a mathematics teacher/educator, children studying in education boards that use this textbook know the exact kinds of questions that will be posed to them. Is it any surprise that scoring in excess of 90% has become commonplace and that the country is seeing marks inflation like never before? Two articles that provide food for thought on this issue are here and here.

In the India Today piece, Board Examinations: Where grade inflation met quality deflation, the author writes that "examinations were reduced to answering the most simplistic and direct questions rather than testing the ability to solve more complex problems. The result is that, despite all warnings against rote learning, school education today is reduced to the task of making students better at memorising and recalling answers to simple questions." The author goes on to elaborate on how the marks awarded in board examinations are gradually losing credibility as a metric to gain admission to colleges and the stress that children (and parents) go through to score higher and higher. He concludes the piece by saying that "we (educators) are producing a barrage of "toppers" who are educated to live in a make-believe world where complexities are few and who will hit a hard wall of reality the moment they step outside their schooling system." 

In the Deccan Herald article, Marks inflation shows we need urgent education reforms, the author writes that "...exam questions are along the lines on which students have been trained. They are encouraged to write standard answers for which they have been coached... This does not test the ability of students to think critically, cogently and logically..." This lies at the core of the point I made earlier about children knowing how the questions will be asked.

I have observed a deep-rooted obsession among educators, parents and students with scoring 100%. I am all for giving one's best and aiming for high scores/grades but this fixation with turning in perfect answers to questions that have been asked, answered and revised many times over prior to the actual exam has led to a situation in which rigorous unfamiliar questions that deviate from what is expected have no place in the question paper. Can we imagine (and perhaps slowly move towards?) a reality in which scoring 75-80% (say) is exceptional because the quality of the questions asked are top-notch?

~ o ~ x ~ o ~

The crux of the matter is:

• Assessments influence the teaching-learning process in the classroom.
• Assessments are necessary to gauge whether children are learning concepts and skills that will equip them for different pathways in higher education.
• There are around 330 million children in the 6-18 age group in India; hence, written exams provide the quickest means to assess at scale (as opposed to, say, individual/group projects or independent research).
• Over the past couple of decades, the move towards multiple choice questions in exams across the country has further reduced/eliminated the effort of manual correction that was need for subjective questions.
• Doing well in these exams opens doors to opportunities to study in good colleges => land decent/good jobs => live a comfortable life.

Clearly, and for good reason, exams are given a lot of importance at every level of the education system and by all stakeholders (children, parents, teachers, principals, education department officials etc.). Improving the quality of questions asked on these exams can have a positive ripple effect throughout the system as we (educators) will be indirectly raising the bar and holding children, teachers and schools to higher standards. The challenge lies in radically changing our mindsets (again, applicable to all stakeholders) and moving away from a familiar status quo that has been in place for years now!

~ o ~ x ~ o ~

(I would like to thank my teammates in the math content team at Educational Initiatives as the seed for this post was planted during one of the research paper discussions that we had. 😀 As math content developers, we read and discuss material published in journals such as Journal for Research in Mathematics Education and Mathematics Teacher: Learning and Teaching PK-12 to deepen our understanding of how children interact with mathematics and try to feed it back into the work that we do.)