(With apologies to Shakespeare and his Julius Caesar)
A few things caught my educational eye this week.
Last weekend there was an article in one of the papers that mentioned that David Hicks had not been permitted to accept a package of books to read. One of these books was Harper Lee’s To Kill a Mockingbird. Books contain ideas and ideas are dangerous things. Reading gives us a way of reflecting on our own lives and belief systems as we continually interpret, analyse and compare and contrast our experiences with the ideas formed by the words on the page.
Education should be about ideas but sometimes it is not. Sometimes it’s more about ‘knowing stuff’. Teaching for understanding involves thinking – deep, sustained thinking that leads to students being able to construct an abstraction in their minds that actually makes sense of many distinct pieces of knowledge. If students understand something then they can provide evidence of this by being able to transfer knowledge from one situation to another when presented with a different context. For example: we can all teach students the steps to solve a simple quadratic equation. Obtain a zero on one side, factorise the other, use the Null Factor Law. But the real mathematical understanding we should be emphasising isn’t the method itself, it’s the Null Factor Law. This is the idea that can be transferred to many other contexts of solving other equations involving expressions that can be factorised; it’s the basis of the factor and remainder theorems. Do our students understand why they have to get a zero on one side? Do they know why a different approach to solving quadratic equations is needed to that used when solving linear equations? Has the question ever been asked of them?
Unfortunately, in my opinion, these sorts of ‘connecting ideas’ questions are not regularly asked in mathematics classes, because they aren’t in the textbooks. In my opinion, too many teachers in schools use the prescribed text as the decider and director of their curriculum. It is very important that we think carefully about what the key mathematical understandings are that we want our students to grasp then design a curriculum that teaches to these understandings. The text should just be another resource we have at our disposal.
I have also attended a number of professional development activities this week. At one of these, the attendees were given a list of characteristics of strong understanding performances – the seven Rs. That is, a list of things we should endeavour to have in tasks that we use to assess students’ understandings. These are:
(1) Rigour: task should embody and afford a high level of understanding. Does the task have ‘stretch’ to it? Is there room to extend thinking if students want or need this?
(2) Revealing: task should uncover a level of understanding and identify misconceptions.
(3) Rewarding: task should be intrinsically motivating, not just ‘busy’ work
(4) Require a level of independence: task should be self-directed and have an element of choice
(5) Real: task should have an authentic quality to the discipline. ie. does it involve working as a mathematician would?
(6) Rich in thinking: task should have identifiable parts that develop thought. Were connections sought? Cause and effect? What thinking is being targeted by the task?
(7) Reflective: task should involve some element of written reflection that enhances student learning.
At the same session, we were told that a reflective and questioning classroom had students:
(1) wondering about things
(2) looking closely at ideas
Thinking back over the classes we’ve had so far this year, in how many can we truthfully say all of the above behaviours were evident?
Are these ideas for the teaching of mathematics just pipe dreams? I hope not. I hope that teachers don’t hear these ideas and say, as Shakespeare’s character in Julius Caesar does, “[S]he is a dreamer; let us leave [her]”