Parallel lines inquiry
Mathematical inquiry processes: Interpret; explore; generate other examples; find rules. Conceptual field of inquiry: Angle properties of straight and parallel lines; independent and dependent angles.
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 the prompt above. Appearing to show two pairs of parallel lines, it suggests an inquiry about the angle properties of those lines. However, students used at least five areas of mathematics 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 mathematics lessons, read Levels of Inquiry Maths).
One alternative is to use a guided inquiry, which might start with either of the prompts above. Marking an angle with a red dot or adding a statement (and indicating 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.
Two independent angles are required to find all the angles in the diagram. Given one red angle, all the other red angles are known. Similarly, given any green angle afterwards, all the other angles are either known or could be calculated.
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).
Number of independent angles (i) = l - p - 1
Developing independent and self-aware learners
Language acquisition as the first phase of inquiry
Sonya terBorg blogged about using the prompt with her primary class in Idaho (US). The post describes how the class carried out a preliminary inquiry into concepts and language related to angles. The pupils then conducted an open and collaborative inquiry by applying the language acquired in the earlier phase.
Content coverage through open inquiry
Aoife Lynch (a mathematics teacher in Ireland) used the prompts on this page to 'cover' the angle properties of parallel lines with two classes. Her experiences show that it is sometimes difficult balancing the need to cover content and the desire to keep the inquiry open.
When Aoife first tried the inquiry she combined the second diagram (with the dot) and the question for investigation ("How many independent angles would you need to find all the angles?"). Students were very reluctant to pose questions or make comments, perhaps because they were unsure of how to answer the question and did not want to take a risk in front of the class. The inquiry very soon became a teacher-directed investigation, with students following along.
On the second occasion, Aoife used the question again, but did not refer to it. When the focus was mainly on the diagram, the class was far more willing to question and comment. This gave Aoife a much clearer picture of students' understanding of the topic. The inquiry took on a more co-constructed nature because students set the starting point of the lesson through their responses to the prompt. In this case, running a more open inquiry with less emphasis on the teacher's question encouraged students to engage with the concepts and properties underlying the diagram.
Read more about how curriculum content is incorporated into open inquiry in Inquiry and Curriculum.