“…these children just don’t practice. How will they learn? I can’t do what they are supposed to do. Isn’t this how we learnt in school? said Veera as she walked into the staff room. She looked at me and I nodded my head in agreement.
It was not the first time I heard a teacher mention her agony about students not engaging in practice. So often had I felt the same. Being a mathematics teacher, I had spoken to my students about the need to practice to become better at algebra, fractions, and mensuration. There was nothing that practice could not help them achieve. Their parents reiterated the power of practice during the parent-teacher meetings and put some pressure on their child. The hapless student would nod his head and decide to plough into more practice to master the topics in the syllabus. It would not last though and in a few days, the results of not practicing would start showing again. Practice. What is the magic we expect it to do? What do we, as teachers, mean when we ask students to do it? What do students think of it? What would it help them become better at? How? Are all practice tasks alike? And finally, what does a good practice task look like? These are some of the questions I try to answer in this article.
This P word had been provoking my imagination for a while. So, I went about asking a few teachers* as to what are the kind of tasks they would term as practice and what would they expect it to achieve. This is what they said.
“I ask them (the students) orally…. questions are asked as per the board pattern after a small aspect of the topic is covered. When they orally answer I get to know their level of understanding and it gives them the confidence to write when the time comes.”
“Practice can be written…. question-answer practice that the students are asked to do at home. This is done so that they get practice to learn to write their final papers. They are asked to write it in the format as per the expectations of the Board (exams). They are asked to time the writing of their answers, within the time limit underline key words to try and reach perfection\quality in writing…”
“practice can also be running notes…as I speak in the classroom.”
“There is a homework task I recently gave them for practice….. I asked them to make a cabbage pH indicator using everyday materials found in their home.”
“In class or as homework I want them to practice the different sums given in the textbook at the end of the chapter…”
“When I did area and perimeter, they were given the task to research and find out where in life these topics are widely used. Similarly, with the topic on ratio and proportion…”
“It is about singing the same song… practicing it…”
Here, the meaning of practice seems to range from speaking and singing to writing (with reading and listening), it could be homework or class work, it could be to increase understanding or just to perform well in exams. The word, like a magic hat was throwing out rabbits of different colours depending on the subject and the teacher. Intrigued, I began to explore further and discovered a few things about practice.
Practice means different things
The meaning of practice differs according to the subject. In music, for example, practicing is usually singing/playing the same song or a part of it again and again till one reaches a state of mastery over (or should we say harmony with) its tempo, rhythm, pitch, lyrics, silences and so on. Most master singers take this practice or riyaaz as almost a holy grail to be done daily for hours.
In learning a new language, practicing usually involves memorizing new words, phrases and grammatical constructs and using them in different contexts to communicate and make our thinking audible or visible.
On the other hand, practice in mathematics commonly involves solving different questions from textbooks on the same topic. It is more like getting the technique woven into the fabric of the mind of the student, so that they are able to solve similar problems later on.
So, if you are in a staff room and overhear a conversation between a mathematics and an English teacher on giving students more “practice”, listen closely as they are likely to mean completely different tasks.
But certain things seem to be common among practice tasks
When I asked the teachers about practice tasks, they usually came up with tasks for homework. Although class time might touch upon it, it was rarely spent only on practice. Usually, a new concept is introduced in class by the teacher, followed by a question and answer session with a few examples and then the students would do most of the practice at home to get a deeper understanding. Also, the tasks were usually short – below 30 minutes. These details were in line with my own experience as a teacher.
But is this how the research and academia looked at practice tasks? Had they come to some kind of common definition?
My research did not show any common threads on this routinely used term. But some of the writers I came across were looking at the same landscape of practice from different perspectives.
Stephen Nachmanovitch, a violinist, author and educator says, “Our stereotypical formula, “Practice makes perfect” carries with it some subtle and serious problems. We think of practice as an activity done in a special context to prepare for the performance or the “real thing”.”
He explains that practice is as “real” as it gets. If we continue to judge what we are doing to reach some idealized “real thing”, we forget what we are creating. When a musician plays his piano, he is creating music as he is practicing. In search of a more “perfect note”, the very outpouring of the heart and the subtle change in tones is judged and censored. No wonder, practice looks boring, a hard struggle. The element of play has been lost!
I had experienced the same numbness in students when they were preparing for mathematics exams. When we reinforce one right method to reach one right answer, no play is possible. When we switch to standard answers to questions in reading comprehension, the same thing happens. The students hate it.
But practice is not just about being “playful” and turning out whatever one feels like. There is a distinct method to it and it does lead to mastery.
Donald Schön, an influential thinker and author, looks at practice as critical to achieving mastery in a discipline. He takes the example of design studios in architecture and how students become master professionals through practice. He visualizes practice as a continuous reflection-in-action as the student creates a plan for, say a building. It is a moment-to-moment engagement with what is happening in front of her on the drawing board. She brings all her knowledge and technique to traverse the landscape of design and imagine how the completed structure could look like.
Later, she and the teacher go through the design together and reflect on what was happening when the design was being created and what the other ways are in which this design could be improved. The teacher would talk through the work done, ask specific questions to highlight the assumptions of the student. So practice involves both: reflection-in-action and reflection-on-action1.
Both these authors have taken something that most teachers (including me) had considered to be boring, dull, repetitive and made it something to wonder at and essentially creative.
I began to search for examples of practice tasks in classroom and see how they would compare to such perspectives. Both these authors claimed that practice does lead to a richer understanding of the subject. And given that our notion of practice tasks was something different, would students achieve mastery over topics in schools?
Practice – to what kind of mastery?
If a student solves 10 questions at the end of a chapter in a textbook on finding the area of a shape, what does she really know after attempting those questions? What has she become better at? I had often given students different questions and somehow trusted that they would learn the concept I was trying to teach with that practice. For instance, for practice tasks, I often began with questions like:
Q 1) If the length, breadth, and height of a cuboid are 5cm, 10cm, and 20cm, what is its volume?
then moved to seemingly more complex questions like:
Q 2) Your geometry box needs to be wrapped in a shiny red paper. The box is 5 inches long, 1 inch wide and 1/2 inch high.
a) What is the minimum amount of paper needed to cover the box?
b) How many cubic inches of sand are needed to fill the box?
A close look at the two questions shows that although the second question sounds complex, it uses the same principle of Volume = Length X Breadth X Height to find the answer. Students are often extremely smart to know that if the question is about volume and there are three numbers given, then they have to multiply all three to find the answer.
So, if the instructional aim of such practice is to make students “memorize” the formula, then this practice task will work. But if the aim is to help them understand volume or encourage the students to think and reflect on to arrive at volume, such tasks miss the mark. Worse, these kinds of questions reinforce the notion that a mathematics question is usually closed, solvable in 5-7 minutes, has one correct method/formula, one correct answer. It does not encourage mathematical thinking2. So much for what Nachmanovitch would like practice to be like!
But where does that lead us? How could we restructure practice to make it really help the student understand and become better at conceptualizing what volume really means. As I thought through, I came up with some characteristics of practice tasks which would be useful and definitely far removed from boring, unmindful repetitions.
a) Practice task provides for play and experimentation. It does not need to be many different types of questions that a student must do. It can be the same task done repeatedly (think of riyaaz) and each time revealing something more to the student. By repeatedly, I do not mean the same way. Each time the task is performed, it could be slightly different from the way it was done the previous time – which leads to a sense of wonder and discovery. It also leads to an appreciation of the nuances of the concept and a richer understanding of it.
For instance, students could be asked to find multiple ways to solve the same mathematical problem. Or they could be asked to write different stories with the same title. The focus of the teacher is not to see the efforts as something which will become a “good” story but rather to see each story as complete and good in itself. The practice is like a work-in-progress where the stone is getting cut to reveal the sculpture. Each stroke that the student makes is like hammer and chisel cutting away at the stone.
b) The task provides feedback without always being too dependent on the teacher. The required presence of the teacher for feedback is usually not practical for the classrooms of today. The task must allow the student to get a sense of whether she is on the right track. In programming, for instance, when a program is run it will show the output immediately and often highlight the bugs. In mathematics, a magic square or a sudoku can give feedback to the student if she is getting it right. Students could be paired in a language learning class and asked to have conversations wherein they could correct each other as they go along.
c) The practice task is authentic. It takes the students closer to what a practitioner would go through in the world outside the classroom. Mathematicians don’t go about finding area and perimeter like it is usually shown in the textbook. But they do think of whether the number of tiles needed to cover a floor would always be odd or even. Or whether the middle number in a 0-9 magic square would always be 5. In a language class, a letter to the municipal commissioner on how to make parks more child-friendly or how potholes could cause accidents would be closer to what people think and write about.
d) Practice tasks encourage reflection in action. The students need to pay close attention to their thought process, a kind of meta-awareness, as they engage with the task. This reflection brings them more keenly to the present and be ready to learn any nuances which the task offers. For instance, they could write comments by the side as they solve mathematical problems. Or they could think aloud as they write answers to a reading comprehension task.
A useful way to do this is to perform a ritual before beginning practice. A simple folding of hands for a few seconds often brings us into the space of meditation and prayer. Musicians often touch their instruments with both their hands as if they are seeking blessings from it before they start practicing. In the same way, a simple gesture or activity can set the tone and remind the student that she is about to do something important and full awareness is required. It could be a simple closing of eyes for a minute, a methodical cleaning up of the desk or re-arranging the items before she begins to solve mathematical equations or write letters as practice. This is likely to make practice a respectful activity and encourage the student to reflect. Perhaps it is no coincidence that in Hindi, practice is also called as sadhana – a devotional experience.
e) The practice task needs to be followed with reflection later. Often, we quickly move from one topic in class to another. As teachers, the pressure of covering the curriculum is high. But this often creates problems later on because students never really grapple long enough with a topic to develop mastery over it and we have to re-teach the same things later on. This reflection-on-action is extremely critical for the teacher as well as the student as they both try to figure out different paths taken to complete the practice tasks.
If the class size is too large, students can discuss and compare solutions with other students. The teacher can provide clarifications, suggestions and ultimately nudge them to reflect on their own thinking when they were trying to find the solution. Also, for the teacher, it would reveal common inappropriate conceptions of a topic.
Continuing the discussion: towards a finer understanding
To highlight a few points, I believe that practice is a complex term, interpreted and used in different ways. However, it can be interpreted and designed through tasks that are not boring or unmindful or lead to hating the subject domain. Done sincerely and playfully, practice tasks can be extremely intense, fulfilling, fun and a powerful learning experiences. It is a devotion, a sadhana My aim in writing this article was to highlight some of the complexity and richness underlying this term and suggest ways we could restructure our current practice tasks. I hope the article serves as a starting point for discussion in classrooms and staff rooms where teachers argue what has been proposed here and develop a finer understanding of this notion we so often use.
A few practice tasks in mathematics
So, how does this really get concretized. Let me share a few examples of practice tasks from mathematics:
- Ratios: Using rice/dal/pebbles fill different vessels in your home to the ratio 1:2. How do you know your solution is correct?
- Volume: The city department has been asked to build a water tank for the residents of a new colony – each house using roughly the same amount of water every day as your own house. What shape (cylindrical, cube, cuboid, pyramidical) should this water tank be? Give dimensions.
- Logical reasoning, Patterns: Fill a 3X3 empty square using numbers 0-8 such that each row, column and diagonal adds up to the same number. Compare your solution with other classmates’. What did you find? What about a 4X4 magic square? Form and check your hypothesis about what should be the central number.
Area: You have a length of rope which you can use to fence any area of unclaimed land and mark it for your family’s use. If you wanted to have the largest area of land, what shape of land would you choose? Why?
*I would like to thank the teachers of Shishuvan, Mumbai for sharing their views on practice.
1. Schön, D. (1983). The reflective practitioner. Basic Books: New York.
2. To explore this further, I suggest reading:
(a) Schoenfeld, A.H. (1992). Learning to Think Mathematically: Problem Solving, metacognition, and Sense-Making in Mathematics. Handbook for Research on Mathematics. 334-370.
(b) Hersch, R. (2006). 18 Unconventional Essays on the Nature of Mathematics. Springer.
The author works on strengthening teacher education and educational leadership at Tata Institute of Social Sciences, Mumbai. He can be reached at [email protected].