Teachers who have to comply with a state curriculum are often under pressure to reach a pre-determined learning outcome, and this pressure militates against running a truly open inquiry. In channelling a lesson down a set path, however, the opportunity for students' creativity to flourish is lost and so is the potential for students to make connections between mathematical concepts. This is exemplified in the search for a prompt to 'cover' the angle properties of parallel lines.
Mark Greenaway - an Advanced Skills Teacher in the UK - used this prompt (right), which looks like two pairs of parallel lines. To me, this immediately suggested an inquiry about the angle properties of those lines. However, students used at least five areas of maths to attempt to interpret the prompt (illustration below):
- parallel and perpendicular lines;
- coordinates on a graph, gradients and equations of straight lines, and equations of circles;
- areas of shapes;
- problem-solving involving the radius of a circle (requiring the use of Pythagoras' Theorem in one way to solve the problem); and
Indeed, if this prompt had been designed to 'target' angles, then it is clear the inquiry teacher has failed by making the prompt open to multiple interpretations. Interestingly, Mark called this a "straight lines inquiry" and, in providing such an ambiguous prompt, ran it as an open inquiry. (For a description of the levels of openness in inquiry maths lessons, click here).
One alternative is to use a guided inquiry, which might start with the prompt on the left. Marking an angle with a red dot and the parallel lines with arrows has, in the past, steered students towards posing questions and making comments about angle properties. Often, they speculate about the size of other angles by measuring or 'by eye'. The teacher can intervene further by guiding students towards a consideration of how many independent angles are required to find all the angles in the diagram. The answer in this case is two (see diagram below). The students would, as normal, decide upon the direction of the inquiry from there.
Another alternative is a structured inquiry, which is similar to an investigation in mathematics classrooms. (For a comparison of inquiries and investigations, click here). The teacher structures the inquiry by posing the question: How many independent angles would you need to find all the angles in the diagram? Students would be directed to draw diagrams with straight lines of which none, some or all will form pairs of parallel lines. They would then work out the number of independent angles required and tabulate the results. The final step would be for each student to induce (or 'discover') an answer - in this case, the formula giving the relationship between the number of lines (l), the number of pairs of parallel lines (p), and the number of independent angles (i).
You can follow Mark Greenaway on twitter @suffolkmaths. His website contains wealth of lesson resources and teaching ideas.