InquiriesWith minimal guided instruction coming under fire in the Explicit Instruction section, you may not think there is a place for inquiries in mathematics. But if done carefully, I believe there still is, with potential benefits in terms of motivation and problem solving.
Research Paper Title: Inquiry Teaching
Author(s): Andrew Blair
Inquiry teaching is often misunderstood as "learning by discovery". This article provides a good summary of the key distinction, together with the perceived benefits of inquiry teaching. The amount of structure or guidance offered by the teacher is determined by students' ability to think independently and critically. A key feature of inquiry lessons is "negotiation" between the teacher and the students over the structure and direction of the lesson. It is worth noting, however, that proponents of the direct instruction approach will likely balk at statements such as "Students not only meet concepts and skills in a meaningful pursuit of answering their own or peers’ questions, they also participate in a debate about how to learn and why to learn in a certain way" and "(the teacher must) redistribute authority to students". But I think that if the inquiries are well structured, and students only participate in them when they are ready (i.e. they are not used to teach concepts, just to apply them), then they have a valid place in the maths classroom.
My favourite quote:
In teacher directed strategy lessons, there is little guarantee that a method permits students to reach an ‘ideal’ understanding. In the investigative classroom, the learning experience is rigidly structured, leaving students the unenviable responsibility to discover a conceptual relationship. It is only the inquiry classroom that offers students a mechanism to harmonise conceptual learning with the method of learning.
Research Paper Title: Extending example spaces as a learning/teaching strategy in mathematics
Author(s): Anne Watson and John Mason
This is a fascinating paper which looks at the importance of examples for the understanding of mathematics. We have seen in the Cognitive Load Theory section the importance of worked examples for early skill acquisition under a model of Explicit Instruction, but here we look at something different - the use of student generated examples. The paper cites several studies in which student generated examples have aided concept development. Asking students to "find an equation of a straight line that has two intersection points with the parabola y = x2+ 4x+ 5." gives them opportunities to test their core coordinate geometry skills, but also opens the door for further investigation and generalisation - why do some lines cross a parabola twice, whereas others do not cross it at all, for example? Hence, we have examples leading to inquiry. Having read the Cognitive Load Theory section, there is the obvious danger that such questions are too much for students' fragile working memories to deal with during early skill acquisition. In the absence of well formed schema, novice learners may engage in inefficient problem-solving search via means-end analysis, and hence could actually be thinking hard but not learning anything. However, the authors point to a study involving bottom-set year 9 students explaining the definition of a prime number by generating their own counter-examples, suggesting that student generated examples may in fact aid the important encoding process. For me, there is little doubt that framing a question in terms of asking students to generate their own examples is a great way to promote inquiry, foster motivation and to provide opportunities to make deep connections. Without wishing to sound like a broken record, I feel we just need to be careful to use these strategies when students' understanding of the basic concepts are relatively secure.
My favourite quote:
We have illustrated how seeing teaching/learning mathematics as creation and extension of personal example spaces can inform the construction of tasks in which students can work directly on their own mathematical structures and relationships. Achieving competence in mathematics can be seen as the development of complex, interconnected, accessible example spaces.