My students' perception of understanding

In this essay, the first in a series, I shall share examples of what my students think understanding means in the context of their school subjects. As a majority of my teaching experiences over the past 5+ years have been in mathematics, the cases discussed are from that domain. However, I have attempted to present my points in a way that is accessible to a reader from any discipline by focusing more on underlying mindsets rather than going into the specifics of the subject matter.

Case 1

The image below lists three fundamental identities in trigonometry that students across the globe cover in grades 9 and 10. I have taught this concept to a wide spectrum of children across Indian and international contexts.

What follows is a typical conversation regarding these identities:

Me: Have you seen these identities before?
Student: Yes, these are in the section on important formulae at the end of the chapter.
Me: What would it mean for you to understand these identities?
Student: I need to remember the formula.
Me: Do you think it is important to understand how these identities were derived?
Student: No. That is not asked in the exam.
Me: Are you interested in understanding how these identities were derived?
Student: Maybe, but there is no time. I have to study many chapters in many subjects. It is more important that I remember the formula so that I can score marks in the exam.

Practically, guiding children through the derivation of these identities takes about 10-15 minutes. In fact, a majority of my students were able to derive these relations independently once I said, "Think about Pythagoras' Theorem!" - they marvelled at the elegant simplicity of the derivation and trigonometry became accessible to them rather than the property of an ancient Greek mathematician from a bygone era!

The conversation reveals a key mindset that I have observed in many children - understanding primarily means being able to recall and suitably reproduce material on an exam. All the textbooks that I consulted explain and derive these identities (means) but the child's focus is solely on the final results (ends). Anything that is not directly tested on the exam can be ignored. This mindset feeds into the way they approach all subjects - rote learning geographical facts, memorising poems, recalling chemical equations etc.  I believe that the essence of mathematics (or any subject for that matter) lies in the interconnections between topics - here, Pythagoras' Theorem and trigonometry. This essence is lost when compact formulae are directly given to students to consume and regurgitate on the exam.

Case 2

Here is another example from quadratic equations - a core topic in high school algebra. This formula helps in finding the solutions to a quadratic equation. The students that I have worked with are content with just plugging in numbers into the formula and arriving at the 'answer' without knowing how the formula was obtained or the graphical significance of the solutions.

Once again, to guide children through the derivation of the formula takes about 15-20 minutes. While the method used in the derivation is one that children might not be able to come up with on their own, the derivation can be easily comprehended (if brought out systematically) by them and integrates concepts from reasoning with algebraic expressions, algebraic identities and quadratic equations. Conceptually understanding the origins of this formula also helps in topics such as functions that are covered extensively in grades 11/12 and at the university level.

It is not that children are totally disinterested in understanding the "why" behind such formulae. They merely do not perceive the value of such understanding when they see it through the lens of the exams (as currently designed) that they give at the school level. The format and types of questions asked on an exam play a key role in deciding how the teacher teaches and the student learns. If a child is going to be tested largely on 'knowledge', a little on 'comprehension' and not much else (refer the image of Bloom's Taxonomy below), then he/she will end up learning only these lower order thinking skills.

Image source:

Case 3

To explore another facet to this point, I would like to take the example of statistics. The basics of statistics are covered in most schools in grades 9 and 10. Children are introduced to bar graphs, line graphs, histograms and ogives along with learning about the three common measures of central tendency viz. mean, median and mode. Recently, I tutored a grade 10 student who had completed the chapter in school but wanted to practice more questions. Here is a snippet of the dialogue we had when starting the section on median.

Me: How comfortable are you with the concept of median?

Student: I have understood it but want to practice more questions.
Me: Okay. What does the median of a data set tell you?
Student: I didn't understand your question.
Me: You know that the mean tells us the average of the data. You know that the mode tells us which data point occurs the most frequently. Right?
Student: Yeah, I know that.
Me: In the same way, I am asking what does the median of a data set tell you?
Student: I don't know that.
Me: Okay. But, when I asked you how comfortable you were with the concept, you told me that you understood it. Can you explain what you meant?
Student: I know the formula and how to calculate the median. 

The formula that the student was referring to is shown below. While the derivation is not within the scope of the 10th grade syllabus, what I found worrying was her blind acceptance of the formula without any question of its rationale or logic. Furthermore, she was firm in her belief that knowing the formula automatically meant that she had understood the concept of median and what it represented. A little probing showed that wasn't true.

Case 4

The last example I would like to illustrate pertains to arithmetic progressions (abbreviated as A.P.). This is a concept grounded in algebraic reasoning and patterns and is learned by students in grade 10.

An arithmetic progression is a sequence of numbers in which each term can be obtained by adding (or subtracting) a fixed amount to (or from) the preceding term. This fixed amount is called the common difference and is usually denoted by d.

For example, the sequence {2, 5, 8, 11, 14...} is an arithmetic progression with first term (denoted by a) as 2 and common difference d as 3 because each term can be obtained by adding 3 to the preceding term.

Similarly, the sequence {7, 3, -1, -5, -9...} is an arithmetic progression with first term a = 7 and common difference d = -4 because each term can be obtained by subtracting 4 from the preceding term.

How to obtain the total sum of the terms of an arithmetic progression is one of the key concepts in the topic. Conversely, being able to find the terms of an arithmetic progression given information about its sum is also a skill that children are taught. A snapshot from a widely used textbook on how to find the terms when given the sum is shown below.

I tutored two students (on two separate occasions) on arithmetic progressions. Both my students used the textbook from which the above snapshot is taken and had covered the chapter once in school.

Here is a conversation I had with one of them.

Me: How would you approach the following problem?

Student: We did similar questions in school. Our teacher told us to take the terms as (a - d), a and (a + d).
Me: Alright. Why do we take the terms as (a - d), a and (a + d)? Why not take them as a, (a + d) and (a + 2d) or maybe something else?
Student: I don't know. Our teacher said that it is better to take it as (a - d), a and (a + d).
Me: Okay. Can you please explain why it is better to take the terms as (a - d), a and (a + d)?
Student: I'm not sure.
Me: Alright. If you take the terms as(a - d), a and (a + d), do you know how to solve the question.
Student: Yes, I can.
Me: Okay, can you please solve it?
(3 minutes later)
Student: The terms are 5, 8 and 11.
Me: Right. Now, can you try explaining why taking the three terms as(a - d), a and (a + d) is better?
Student: I don't know. It is written like this in our textbook and the teacher told us we should do such questions like this only.
Me: So, am I allowed to take the terms as a, (a + d) and (a + 2d)?
Student: No, that would be wrong.
Me: Why?
Student: Because we are supposed to do such questions only in this way.

Mathematically speaking, the student's argument is wrong. Both methods can be used to arrive at the solution. The method outlined in the textbook (and followed by the student's teacher in the school) is algebraically easier to solve which is why it is a "better" method - it is more efficient in terms of time and computations involved. However, it is not the only method and the textbook fails to mention that or explain why this method is adopted. I followed up this dialogue by showing the student the alternate method that gives the same answer but requires more steps and algebraic manipulations for him to realise why the first method is considered "better".

This conversation illustrates that some students take the procedures in the textbook (or shown by the teacher) as incontrovertible to the extent that they believe it is the only way of approaching a problem. Maybe, they have lost the desire to question and find that accepting the procedure is the path of least resistance to scoring on exams. Maybe, they try asking their teacher but he/she doesn't explain the procedure due to time constraints or his/her own deficiencies in pedagogical content knowledge. Maybe, they are overwhelmed by the amount of material they need to learn across subjects and find it simpler to just use what is given to them rather than having a discussion on it. The net effect is an acceptance of the procedure without a sense of the "why" behind it.

To summarise:

  • Understanding means being able to answer the question types that appear on exams. As most questions on exams up to the grade 10 level at school test knowledge and comprehension, children's development is mostly restricted to these lower order thinking skills.
  • Exploring the deeper 'why' behind concepts is often sacrificed due to limitations of time, amount of syllabus, the aforementioned exam pattern and, in some cases, limited pedagogical expertise of the teacher. Perhaps, in some cases, students are not even aware of the superficiality of their understanding.
  • Recalling formulae without understanding their origins is accepted as understanding.
  • Carrying out procedures without understanding the reasoning behind them is also accepted as understanding.
In the next essay, I shall unpack my own evolving understanding of understanding - how it has changed over the years, factors that influenced those changes and my current views on understanding.


  1. Enjoyed reading your post. I have witnessed and often found frustrating, this incorrect view of 'understanding.' I have also learnt that this view, developed during the school years, carries on in the corporate world. People are constantly in search of thumb rules and shortcuts to arrive at answers or take decisions. At times this is in the interest of time, but mostly because managers are just lazy to really understand 'why' or even think of 'first principles.' This often leads to undesirable outcomes in the long term as the context changes over time.
    Overall I feel that people (mostly students) don't see 'understanding' as an end goal. They look for practical applications in the world around them and feel that understanding is unnecessary if they can't find any. Practical applications are important, but in understanding, not only does your application become better, you given yourself the chance to come up with something that is your own! I guess it is incumbent upon the teacher to unearth the joy behind understanding the concept and possibly allow students to find their own method instead of a single solution.

    1. Rajiv,

      Your point about this spilling into one's professional life is excellent. In my opinion, it would be fine if one arrived at thumb rules with an understanding of the rationale behind them because that would then allow one to tweak them for different contexts. Blindly adopting a thumb rule or a shortcut can have repercussions in the long run that are not desirable and I appreciate you bringing that lens in your reading of the essay.

      Children are malleable during their school years. If they see their teachers (who are adults that they look up to) recommending shortcuts, formulae, processes etc. without an explanation, then they take it for granted that such 'understanding' is acceptable. This mindset, as you pointed out, permeates through other aspects of their life.

      Yes, most students do not see 'understanding' as an end goal worthy in itself. If that 'understanding' (in whichever form it be) is directly linked to practical opportunities (better colleges, better jobs etc.), then students see a value in it. Perhaps, something that can be inferred from your comment is that the curriculum needs to looked at more carefully (and maybe redesigned?) if children are not seeing any practical applications of understanding it?

      Your last sentence (" is incumbent upon the teacher to unearth the joy behind understanding the concept...") is unequivocally true. However, it requires time, planning (individual and collaborative within teacher teams) and materials - all of these are precious resources that are available in limited supply to teachers as they struggle to complete the syllabus, correct student work and perform administrative responsibilities. Can our schools give teachers time during school hours to afford them opportunities to plan such lessons?

  2. Loved reading all the 4 cases ( all the conversations that you've had with your students regarding study)
    I came across one main thought of your that we should get the deepest understand of what we are learning or what/why we are doing. Because if we get to know this then our concepts get cleared automatically. Understanding things to the depth excites us to get involved in the particular thing like to know a little extra about it. For instance if I give a life example like if we get to know the why behind our birth then we can visualise our goals and aims very well

    All the 4 cases allured me as it was a struggling case of a student and as I am a student even I've faced these situations wherein I just gave important to only remaining or learning what is already there and not the reason behind why does it exists

    1. Thank you so much for the comment. I am glad that you enjoyed reading the cases. :-)

      I appreciate you using the word "excites" in your comment because I agree that understanding something more deeply can make the topic genuinely interesting! A person will want to learn extra things about it and it makes learning a fun process rather than something that has to be done to pass examinations.

      It means a lot to me when a student comments on my post and I was happy to see that you could make connections between the post and your own experiences in school.

  3. Can you meet your students somewhere between their concept of understanding and the one you would like them to have? For instance, I've had students in calculus only memorize, say, the product rule for taking derivatives instead of the quotient rule. Should they need to apply / use the quotient rule, they have learned how to derive it from the product rule. Similarly, with your example of trigonometry identities, would students have understanding, or knowledge deeper than recall if they memorize sin^2 + cos^2 = 1, and use that to derive tan^2 + 1 = sec^2?

    1. Graham,

      Thanks for the comment. I guess that would be a good start. Perhaps, that approach will help them see a value in understanding the "why" behind formulae and relationships rather than just the end result. In your experience, did children show an interest in wanting to understand how the product rule was derived when they saw it being used to derive the quotient rule? Did it pique their interest in any way? So, if they see how connected sin^2(A) + cos^2(A) = 1 and tan^2(A) + 1 = sec^2(A) are, then do you think that they might be more inclined to explore how did the sin^2(A) + cos^2(A) = 1 come about in the first place?

      This actually reminded me of something that teachers do that is related to this but not mathematical - moving children from extrinsic motivating factors (like rewards and points) to intrinsic motivating factors (learning something because it is interesting and has value in itself). This shift is gradual and occurs towards the end of primary and start of middle school. Maybe this transition from extrinsic to intrinsic, while slow, is a way in which teachers meet students at their developmental stage?

    2. Interesting connections, as always.

      *In your experience, did children show an interest in wanting to understand how the product rule was derived when they saw it being used to derive the quotient rule?

      I suppose my motivation for this example as a middle ground between knowing the what (and not understanding the why) is that I could motivate students as to the simplicity / relative elegance of the product rule compared to the quotient rule. This is ultimately in service of memorization / test preparation, as I don't think there's anything inherently more elegant about f'g + g'f than (f'g - g'f) / g^2, rather that the former is briefer and more concise for memorization. I'm not sure it piqued their interest in why it works any more than the explanation I offered in deriving the equation initially.

      *Intrinsic / Extrinsic

      Certainly there's some movement in this stage, but one can perhaps be in service of the other. If I offer an extrinsic motivation like the AP test, and students develop intrinsic motivation to be derivers, not memorizers, I see that as another small win. Is it true that derivers are intrinsically motivated, and memorizers must be extrinsically? Or am I making a mistake / overgeneralization in bringing these two together?

    3. Thanks for the response, Graham.

      Alright, I got what you meant by the 'middle ground' now... In a sense, the ends are still the test that they need to do well in (whether this is a positive or a negative is another discussion!) but the means are now more grounded in the 'why' than in simply the 'what' and 'how'.

      I think derivers are more intrinsically motivated that memorisers (and, correspondingly, memorisers are more extrinsically motivated than derivers) - talking in absolutes does not seem ideal here. Maybe, getting children onto the path to deriving will require some memorising and external motivators but, once they get onto the path, they might see an intrinsic value in it?

  4. I find all four cases that you describe to be typical for students in the United States. There is a huge focus on procedural fluency in math classrooms, where we "teach" students the formulas or procedures so that they can perform on our assessments but we do not explain why the formulas or procedures work because it is not tested. I believe, instead as teachers, we should be teaching the conceptual understanding of formulas and procedures before expecting students to be procedurally fluent. I have found that some routines, like Number Talks, can help build a students conceptual understanding. I believe if we spent the time to build conceptual understanding of the mathematics at all grade levels, we will see an increase in procedural fluency and ultimately standardized test scores.

  5. Kerry,

    Thanks for the comment. It is interesting to note the parallels in math instruction in India and the U.S.A. with regards to their focus on formulae and procedures.

    Your comment made me think about the comment made by another friend of mine, Rajiv, on this post. If you scroll up, you'll see that that is the first comment on the post. He wrote about how such kind of teaching can spill into other aspects of life as adults - specifically, how people are "constantly in search of thumb rules and shortcuts to arrive at answers or take decisions". So, by giving the children the formulae and procedures, are we subconsciously communicating a message to them that using the final result is acceptable even if they have no idea of its origins?

    Another thought that comes to mind is how powerful the assessment is in dictating the way the teacher teaches and the student learns (I mentioned this in Case 2 of the post). Do you believe that assessments that test and drive a conceptual understanding will help? Or, is it not practical to design and evaluate such assessments at scale in large classrooms which is why we resort to pen-paper assessments that test procedural fluency?

    On your final point, I agree that giving "time to build conceptual understanding of the mathematics at all grade levels" is likely to positively impact procedural fluency and test scores. However, and this is something we discussed even during our master's program, are current systems and curricula equipped to give teachers and schools the time to explore concepts more deeply? For example, in India, students in grade 10 need to cover a vast array of topics in the span of about 8-9 months - this includes holidays, time for school tests and activities etc. Teachers and students mention the severe time crunch that they are under to simply "cover" the content, let alone go into it with the goal of conceptual understanding. Do teachers and students in the U.S.A. also find the content to be vast in high school? Or, are there a manageable number o topics/concepts that then allow them to go deeper?


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