After Teachers’ Day, ICTS organised an in-house teachers’ training program.
BY DEBDUTTA PAUL
Professor Joseph Samuel welcomed 42 science and mathematics educators teaching classes 8th to 10th selected for the one-day, in-person workshop at ICTS on 6th September 2024. Then, he handed over the floor to Sita Ayyar, who demonstrated various ideas on reflective mathematics teaching.
What is reflective about teaching? Sita explained that reflective teaching is the practice of reflecting on three questions about the teaching process: What? Why? How?
Sita argued that the process of teaching can be improved by reflecting on past and present practices and thinking about various aspects of the teaching process. Some of the reflection, she said, can be done while teaching itself.
Mathematics teachers know that mathematics is fun! But, if they say that to students, does that communicate to the students that mathematics is fun? Taking this example, Sita introduced the concept of showing rather than telling and using the power of demonstrations. As a corollary, she said, mathematics teachers need to acknowledge students are scared about the subject, no matter how much some may enjoy it. Then, Sita put the onus of making mathematics fun on its teachers. She said that since mathematics is all around us, including in nature, teachers can give various examples in classroom scenarios to demonstrate mathematical principles.
Next, Sita explained different levels of learning using Bloom’s Taxonomy: a hierarchical model of learning consisting of six steps in ascending order of cognitive value: [1] remember; [2] understand; [3] apply; [4] analyse; [5] evaluate; [6] create.
Collaboration, Sita said, is important in climbing this ladder, as it helps students reinforce principles of mathematics through dialogue. She suggested the teachers break down the learning process into small, achievable steps. Sita demonstrated, using examples, how teachers can initiate dialogue among students and guide them through these steps progressively. To aid the process, “ask them why” at every step, she said.
In this context, Sita also mentioned that teachers should not practise behaviour against or for specific students. She said that all humans have subconscious biases, and teachers, too, have subconscious biases based on the gender, social, and economic standings of students. She strongly encouraged teachers to recognise their subconscious biases and work against them actively. “Each child is a plant with enormous potential,” Sita said.
Next, Sita did a role-play exercise with two groups of five teachers each. In each group, one teacher played the role of a teacher while the rest played students. The two teachers taught the same topic: opposite angles are equal. In the first group, the teacher followed the standard textbook conventions and carried out the lesson entirely using the blackboard, while in the second, the teacher did not use the blackboard at all. Instead, he could make use of five plastic rulers.
After the role-play was over, Sita asked everyone which class was more engaging. All the teachers agreed that it was the second. Through this exercise, Sita demonstrated the power of using simple examples from everyday life to demonstrate ideas — instead of sticking to the blackboard only.
After a short break, the session resumed.
Sita Ayyar explained how, with the leap in technology, digital platforms could be used to make geometric thinking fun. It helps students learn by exploring, making mistakes, and collaborating. Most of all, games make students happy. “Happy students are good students,” Sita said.
In her demonstration, she used the game Police Quad developed by the Tata Institute of Social Sciences. While the teachers attempted it and got immersed in re-learning geometry, Sita said: “We are also children.”
Using examples from the game and otherwise, Sita demonstrated how teachers can encourage different ways of thinking about the same concepts. Students can arrive at the same conclusion by thinking along different lines, especially in geometry, she stressed, adding that teachers have to help students explore them.
In conclusion, she emphasised on passing on the values of respecting diversity among the students. As students collaborate, teachers can encourage them to listen to others and reason in the face of apparent conflicts respectfully.
After a lunch break, the workshop resumed with all the participants. H.R. Madhusudan delved into teaching processes using examples from optics. The session opened up with no specific agenda but with questions from teachers who have taught optics in classrooms.
Primarily, two questions came up:
- How can teachers motivate solving numerical problems in and outside the classroom?
- How can different scenarios of specific examples, like the lens formula, be explained clearly?
Madhusudan demonstrated how both these can be answered by repeatedly making the students go back to fundamental principles. That is, whether it comes to explaining various branches of the same scenario or solving numerical problems, if teachers can use the classroom time to help students create a way of thinking from the core physics principles, then the motivation to apply them logically follows. He used various examples from textbooks to demonstrate that certain statements or ideas get indoctrinated among students as well as teachers, making the process of learning and teaching both difficult and dry.
Discussions on different questions in optics followed. Madhusudan broke down each question into multiple questions and answered them from fundamental principles. The correct answer, ultimately, is the one we observe in nature, and he showed that those answers could be arrived at by thinking through the fundamental principles in each logical step.
For example, to answer questions on spherical lenses, Madhusudan framed the equivalent questions for plane mirrors. While answering, various concepts that teachers take for granted became clearer. He used this way of thinking to motivate the equivalent answers in the case of the spherical lens.
Madhusudan used the case of lateral inversion of mirrors to drive home his points. Plane mirrors do not exhibit lateral inversion, he showed with the help of a plane mirror, unlike what is written in most mathematics textbooks across the world. Then, he explained how our brains trick us into thinking about inversion and why it does so. “The same brain creates illusions and explains them,” Madhusudan said, adding that no matter how much we explain the illusions, we still see them.
Going a step ahead, Madhusudan questioned how textbooks and teachers sometimes create logical inconsistencies by using different principles in specific contexts only, even when experimental scenarios demand apparently distinct principles be clubbed together. He gave examples to show that in classroom scenarios, teachers make wrong statements by failing to acknowledge different physical processes at play.
Madhusudan demonstrated that once the teacher establishes a language of reasoning, connecting the dots becomes easier. Moreover, he stressed that teachers should actively use existing knowledge about pedagogical practices documented in various papers and books to reflect on their teaching practices.
When teachers asked Madhusudan how to tackle the practical scenario that they need to teach students the textbook content and give students marks in exams, he acknowledged that teaching is located within the social limitations of hyper-competition. For the sake of examinations, Madhusudan recalled that, in his days of teaching, he would first demonstrate the correct usage of general principles. Then, he said that in specific cases where textbooks countered the practical experience, he would tell the students that only for examinations, they need to give particular answers while the reality was different.
When a particular teacher asked why even simple textbook examples of experiments were complicated and what alternatives could be devised to demonstrate that light travels in straight lines, Madhusudan said, “Light should travel in straight lines no matter what you do.” He pointed out ways of demonstrating the principle.
After a short break, he demonstrated how principles of light can be used to demonstrate not only how reflection phenomena occur but also how refraction phenomena occur. Following the train of thought, Madhusudan explained dispersion. The active discussion veered towards specific examples, giving rise to acknowledging complex phenomena in nature, like how dispersion by water droplets causes rainbows. Unlike textbooks, the discussions tackled the full complexity of the problem.
Madhusudan warned that over-simplification can lead to errors in understanding. He distinguished between mistakes and errors. While mistakes can be rectified by reasoning through and doing experiments, errors can be long-lasting and fatal. So, he argued, teachers need to think about the extent to which simplifications are valid. He used examples from dispersion to demonstrate limitations in analogies.
During the discussion, it became apparent to everyone that light travels not only in straight lines but also as waves. That is, to explain various phenomena, the wave nature of light becomes important to consider.
Faced with the inconsistencies of specific analogies given in textbooks, teachers asked Madhusudan for alternate examples. He said the same examples could be used in addition to explaining to the students the nuances of why the analogies are valid in those specific contexts. The session arrived at a general conclusion: the exercise of teaching is to help students create mental models of the world.
“If these analogies are wrong,” a teacher asked, “why are we teaching them?” Mr Madhusudan argued that students can be taught principles instead and for specific cases and examples — their limitations. Ultimately, not every analogy that teachers can use to teach principles may be useful, but they help the students create a mental model of the world. The open discussions led to conversations on constraints on society put by society. Everyone agreed that how exactly teachers can navigate these constraints is subjective.