"Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten." (pp. 118-119, Adding it Up)
What is conceptual learning?
Though the above quote refers to mathematical ideas, it can be applied to conceptual learning in Science, and other disciplines too. Broadly speaking, we could say that learning consists of facts, procedures and concepts- all 3 are important. In fact, the three are generally interrelated; learning and practising procedures can help in the understanding of a concept and understanding a concept can help in the organization and retention of important facts.
Take an example of the topic ‘motion of falling objects’ typically covered at the grade 9 level. Students learn the following with respect to this topic-
Facts- All falling objects accelerate at the rate of ‘g’. The value of ‘g’ at most places on earth is 9.8 metres per second squared.
Procedures- Equations of motion such as v= u + at; using the equations together, substituting the value of one in another etc. (do note that the concepts related to the motion of falling objects are present in these equations also; the distinctions between facts, procedures and concepts is being made to illustrate a larger point)
Imagine now a student who has studied these facts and practised these procedures well enough to ace her Physics tests. How would such a student answer this question-
You are dropping a book (taped to prevent it from opening up) with a piece of paper placed on it. Do you expect the book and paper to reach the ground together? Or, do you expect the paper to ‘stay back’?
Source: Challenging Prior Mental Models, i Wonder Magazine, Azim Premji University-
Many people are unsure of what would happen in such a situation, including teachers and engineers. Given that the equations of motion do not have mass in them, this should not be a difficult question to answer, yet many of us are perplexed by this question. This is because we have not fully internalized the concept that “Objects from a given height will fall to the ground at the same time, irrespective of their masses’. We will discuss the reasons for why this is difficult to internalize, later in this post.
We can take many such examples to illustrate how deeper conceptual learning differs from acquiring facts and developing competence with procedures. Think about this for instance- you are in a completely dark room with a chair placed in front of you. Would you be able to see the mug or not? If you waited a few minutes, would you be able to see it then?
Source: Challenging Prior Mental Models, i Wonder Magazine
If you would like to know more about science misconceptions, you could watch the films ‘Minds of our Own’ and ‘A Private Universe’. If you are a science teacher, you should definitely get your school to acquire a copy of Rosalind Driver’s book ‘Making Sense of Secondary Science’.
You could also take a quiz to test you or your child’s/ student’s understanding of Science and Math concepts. The test is pitched at the grade 8/9 level. Click here to take the test if you are an adult and click here to take the test if you are a child.
Is conceptual learning important?
This is a valid question- we might argue that many successful adults have done quite well without being necessarily clear about many concepts. After all, most exams focus on testing facts and procedures. If the purpose is acing exams, conceptual learning may have limited value. To be clear, conceptual understanding can help many students get clearer about the topic and thus do better on tests too, but most tests are constructed in such a way that knowledge of facts and procedures is rewarded over conceptual understanding. There are many ‘test-wise’ students who can ace tests without deeper conceptual understanding, while non ‘test-wise’ students may suffer because they cannot make their peace with procedures that appear arbitrary to them (see the example of Liping Ma and ‘subtraction with regrouping’ below).
I argue that conceptual learning is important for 2 reasons-
1. It provides a strong foundation on which further learning can be built. When there is conceptual understanding, it is easier to see connections between different ideas and one has to ‘remember less’. For example, a student with a conceptual understanding of fractions, will work easily on decimals, percentages, ratios and proportions, seeing these as different representations of the same thing. In contrast, a student who has only a procedural understanding of these topics will feel the strain of having to recall different procedures- the seeming arbitrariness of the rules can be stressful.
“I don’t know what’s the matter with people: they don’t learn by understanding; they learn by some other way—by rote or something. Their knowledge is so fragile!”-
Thus, while conceptual learning can sometimes take more time, in the long run it is more effective as one can build on what one has learnt; unlearning wrong ideas later is more painful. So not only will one have to correct the basic ideas, one will have to go through the process of correcting all the wrong ideas built on that.
There is a striking example of the value of conceptual learning from a researcher, Liping Ma, who studied the different methods in teaching subtraction employed by US teachers vis-a-vis Chinese teachers. Take the sum
In the US, students are expected to ‘borrow’ 1 from 5, subtract 9 from 12 to get 3 in the units place, subtract 1 from 4 to get 3 in the tens place to arrive at the answer of 33. In contrast in China, students would perform the same subtraction by ‘regrouping’ the numbers in question. Thus a student may perform the subtraction by doing (40 + 12)- (10 + 9)= (40-10) + (12-9)= 30 + 3= 33.
Liping Ma found that Chinese students significantly outperformed American students on such questions on international tests like TIMSS. Her research shows that lacking an understanding of the procedure they were employing, American students tended to slip up. Students treated the procedure as arbitrary and some even felt anxious about it because of ‘borrowing 1 from 5 and not ‘returning it’.
2. Achieving conceptual understanding requires active engagement on the part of the learner, and this builds the ability to think from first principles. When the focus is on facts and procedures, the teacher’s role is to explain the facts and demonstrate the procedures clearly, and the student’s role is to pay attention and practise the procedures diligently, seeking clarifications where things are not clear. In contrast, when the focus is on conceptual understanding, both the teacher’s and the student’s role are quite different. The teacher must pose questions and tasks that challenge the student to build the concept, and the student has to go through the struggle of arriving at the concept.
To go back to the example of the falling objects, the teacher may need to pose questions such as the following, in addition to teaching the relevant equations of motion-
Do objects of different masses fall at the same rate? If they do not, share examples (a student may share the example of a feather and a ball falling at different rates)
Why does the feather fall slower than the ball? Is it because it has a lower mass? How can we find the answer to this question?
What does the equation of motion say about the time it takes for a falling object to reh the ground?
Active engagement with questions such as this is crucial to develop conceptual understanding, and this has great value because the student is developing the ability to think on his own from first principles.
Elon Musk, a trailblazer in many areas, including building rockets at a tenth of prevalent costs, has spoken about the importance of thinking from first principles. He explains how thinking from first principles is needed to innovate; he uses the analogy of a chef vs a cook to make his point.
“The chef is a trailblazer, the person who invents recipes. He knows the raw ingredients and how to combine them. The cook, who reasons by analogy, uses a recipe. He creates something, perhaps with slight variations, that’s already been created. If the cook lost the recipe, he’d not know how to cook the dish. The chef, on the other hand, understands the flavor profiles and combinations at such a fundamental level that he doesn’t even use a recipe. He has real knowledge as opposed to know-how.”
Coming back to the question of ‘many successful adults who have done quite well without being necessarily clear about many concepts’, I would say that-
Many of them may have switched disciplines and would be clear about concepts in their current domain (Statistics from the US indicate that only 40% of STEM students continue in STEM).
Some of them would have good know-how (like the cook) but not the real knowledge needed to be a chef.
How can we help students develop conceptual understanding?
Achieving conceptual understanding can be hard for a number of reasons, like-
The use of the same standard examples time and again may give students a wrong idea about the concept. For example, squares are rarely drawn in an orientation other than having a horizontal base. This might lead students to conclude that the shape is not a square if it is turned.
Important ideas may not be dealt with sufficiently in the curriculum; some important steps in understanding a topic may even be skipped. For example, understanding the particulate nature of matter (kinetic theory) well, can help students to understand many physical phenomena well like melting, freezing, dissolving, relationships between pressure, temperature and volume and so on. However this is often not dealt with sufficiently.
Students not being encouraged to make connections between different things they learn (even in the same subject). Taking the example of the falling objects again, if students say that the sheet of paper will fall slower than the book, are they asked to look at the equations of motion (that do not have mass in them) and reconcile their answers with the implications of the equations?
The lack of opportunities to visualize and observe phenomena- whether through hands-on experiments or virtual simulations. For example, the concept of electricity can be hard to understand, given that we cannot see it. However an approach where students can visualize the phenomenon as well as be challenged to think about it (‘minds-on work’ as emphasized in point 3) can be effective.
The teacher’s belief (often unconscious) that students’ minds are a ‘blank slate’. When teachers teach without recognizing students’ prior mental models, students may continue to hold on to their existing unscientific beliefs while providing the ‘correct answers’ in exams. Going back to the example of falling objects, a student may retain his prior belief that ‘heavier objects fall faster’ even though the equations of motion learnt in school contradict that belief. In fact the student is often not aware that he has these contradictory beliefs.
While all the factors mentioned above need to be considered in ‘teaching for conceptual understanding’, point no. 5 is particularly important because it is less commonly understood. All of us have prior mental models about the world, based on our observations, and these mental models are valuable. For example, it is of value to our survival to believe ‘heavier objects fall faster’- we definitely need to move away instinctively and instantaneously from the path of a falling ball than a leaf falling to the ground. However this belief is not scientifically correct. See this striking video of objects falling in a vacuum.
It is therefore an important task of the science teacher to recognize prior mental models children possess, surface these mental models, and create a ‘cognitive conflict’ between different ideas (in this case, between feathers falling slower than balls, and the equations of motion suggesting time of fall doesn't depend on mass). Working through this cognitive conflict with the help of the teacher results in deeper conceptual understanding (Students figure out the existence of air resistance or drag in this process and reconcile the different ideas).
Watch this film ‘Khan Academy & the Effectiveness of Science Videos’ to understand how research shows that cognitive conflict and confusion for a period of time can actually be productive in enhancing conceptual understanding.
The research of Phil Sadler (Harvard) and his colleagues shows that when teachers know the concepts themselves, and are aware of common student misconceptions, their students perform better on tests of conceptual understanding. In other words, it is not enough if the teacher himself knows the correct answers, the teachers should also have an idea about how students think.
In conclusion, I would say that if the purpose of learning is ‘for life’ and for excellence, aiming for conceptual understanding is important. If one wants to be an excellent researcher or practitioner in any discipline, this is possible only if one is constantly striving to improve one’s understanding of concepts already learnt and relate new concepts to concepts one already understands. In fact, if one gets used to cracking tests and exams based on memorization and ‘figuring out patterns’, one has a much harder time unlearning and relearning when work and life situations require deeper understanding. So why not make conceptual learning a goal in itself? Even though it can be challenging, it is fun and enjoyable, and developing a taste for conceptual understanding, is good preparation for life.