How to promote real inquiry in mathematics classrooms
A colleague of mine has been struggling recently to understand how to develop inquiry in her mathematics classroom. Despite having secure content knowledge and a vast experience of teaching she finds it challenging to devise teaching activities that truly manifests as inquiry. After observing her a few times, it became clear to me that she confused Inquiry with discovery.
David Perkins, Professor at Harvard Graduate School of Education, proposes the idea of “playing the whole game” in his book Making Learning Whole. The central thrust being that learning should be provided in a whole form, rather than in topics as is more normal. This is particularly intriguing for mathematics instruction. His metaphor of a baseball game being a different experience compared to learning different skills in a training setting is curiously poignant.
So what is the difference? I can almost hear teachers going “yeah, what is the difference?” Or even “what’s wrong with discovery?”
For many teachers, this “inquiry method” is just another name for discovery method. But is it? I beg to differ and I can only support my view with experience and the tsunami of research out there.
Discovery teaching relies heavily on two things in my opinion:
- The learner is unaware of the learning intention
- A logical activity that is well planned leading to a predetermined outcome
The ignorance of the learner is critical to the success of discovery methods since if the learner is familiar with the content, the long process of proving sometimes disengages. How many times is the annoying learner encountered who chooses to blurt out the intended destination of the lesson? It is like watching a movie with someone who has seen it before. Since discovery teaching by its very nature is contrived, meaning the teacher is creating the impression that each “discovery” is new yet it is obvious by the directions that the outcome was predetermined.
Another pitfall of discovery teaching in Mathematics is the lack of transfer in weaker learners. They logically follow through the process, but struggle to connect it to the Mathematics on the board. We have all been there. A popular topic is the sum of interior angles in a triangle. The logical process of drawing a triangle, cutting out the ends, joining them and voila a straight line relies almost totally on the teacher guiding – telling perhaps – the learners. Despite the process being well supported by logic and the outcome undisputed many learners still do not remember the simple fact. Transfer is lacking. Or at least independent transfer.
Inquiry teaching takes another route to learning, as it is largely inductive and demands questions from the learners for it to thrive. Questions are promoted with further questions, one answer by a learner is followed by another question probing understanding, questions are focused on concept formation and the learners are leading the learning by virtue of their own curiosity.
As a teacher in an International School, I have found inquiry to be the safest and most effective method to introduce learning in Mathematics. My main reasons were:
- International students are normally second language English learners
- International students have covered different aspects of the syllabus before
- Variety of teaching experiences and methodologies seen by learners
- Differentiation required since most are largely non-selective
These peculiarities in International Education demand a holistic approach to ensure that all learners are included as well as the provision of a rich conceptual learning experience capable of being transferred to different education systems when the learner eventually returns home, or even migrates to another country.
Questions are the key and the versatility of the teacher to guide the learners, deeper into thinking, thus probing understanding and challenging certainties. I for one have been known to not provide answers, or as my students say, answer questions with questions. This is a great atmosphere to have in your classroom, it makes everyone equal and the pursuit of knowledge joins everyone together.
It means that the learning in your classroom becomes more deliberately focused on conceptual formation rather recognition of content validity. Proving that the angles of a triangle add up to 180, only adds another bit of content but it doesn’t explain why. Why does this happen? Why is there a limit to all triangles? Understanding the triangle explains all polygons including the circle. Is the circle a polygon? Ask your class to think about this, but with colleagues probe this deep thinking? Is the reason because it fits on a straight line? Surely there is something more conceptually profound that commands it to be so.
Inquiry teaching will challenge your thinking and understanding but it fosters a positive learning atmosphere in any classroom. I must admit that many teachers find the adjustment challenging, but in the modern world where knowledge is free, analysis and synthesis are invaluable skills. Deliberately pushing every interaction with your learners into a questioning atmosphere will develop your own inquiring skills.
Inquiry implies involvement that leads to understanding. Furthermore, involvement in learning implies possessing skills and attitudes that permit you to seek resolutions to questions and issues while you construct new knowledge.
Here are some pointers, to promote real inquiry in your MYP classrooms
- Write down the conceptual questions you want to cover in your lesson. This will help to keep you on track if you tend to lose it a bit in the questioning phase. Having a written record means that you would have thought about the activity and expects to discuss this as priority.
- Avoid answering learner questions. Develop a habit of not answering any question. Reward the questioner by saying good question and either follow up with another or pass it to someone else. This will promote good questioning and make the curiosity element in your class active. At first your learners will find this weird, but eventually will feel the benefits in their conceptual understanding.
- Ask at least two questions to the same responder. Begin with a factual question and follow up with a conceptual one. As a rule I try to ask at least two questions to probe understanding. Why did you say that? How do you know?
- Keep some question stems that promote deeper thinking. I like using “Can you think of another example?” “Is your answer always correct?” “Is there any value in knowing this?” You can see my point.
- Promote lateral thinking and transfer. Allow your learners’ questions to veer off as long as it naturally follows the topic being explored. Having a prepared list of questions will help to guide the conversation back to the learning. Thinking is deeply rewarding if freedom reigns in your learning space.
- Accept only written reflections. I always let my learners write their reflections down before sharing. As soon as the concept is covered, I ask them to write their reflections and then read them out loud. This forces deeper thinking and allows them to really reflect since writing is a more involved process. This works well if you ask them to “write three things they learnt today”. Do not ask for one! Three pushes them deeper.
- Focus the learning on concept rather than content. Keep the concept formation as the guide to the questions, be alert for opportunities to delve deeper in ideas proffered by learners. Concept acquisition is far more valuable than content.
- Video yourself and count the number of questions in your classroom. Check to see who ask the most questions. Try and improve the quality of your questions, watch for opportunities missed and learn to capitalize on them. An audio recording is equally useful if a video intimidates.
These are just some of the pointers I have garnered over the years. The list of suggestions, are by no means new and are not intended to solve every problem. However, the aim is to share some ideas as to how to convert your classroom into an inquiry one. As with most changes, it begins with you.
Megel Barker is a Google Certified Educator that has taught mathematics for 21 years. He’s currently Assistant Principal at an International School in Oman and has written two workbooks that support the Oman GED Exams. You can follow him on Twitter @mathter.