Extending an inquiry into the second lesson and beyond
   
Teachers have used Inquiry Maths prompts as stimuli for tasks that last one lesson. Through the prompt, they generate one-off deep mathematical discussion or encourage students to connect concepts during a period of exploration. Such approaches go far beyond the majority of maths lessons, which, lamentably, remain limited to learning disconnected 'knowledge' and focus on procedural fluency. However, there is the potential in the Inquiry Maths model for students to extend an inquiry over more than one lesson.
    
Extending an inquiry has the advantage of developing students' independence and initiative. They have the space to devise and direct their own lines of inquiry. Pursuing inquiry over a longer period also requires perseverance and resilience. These are important attributes, especially when students are required to learn in non-formal settings outside school or with less support in higher education.       

We were reminded this week that extending the inquiry can be difficult. A teacher contacted Inquiry Maths to tell us about his experiences during an inquiry with his year 10 class. The head teacher who was carrying out a learning walk focused on 'stretch and challenge' observed part of the the first lesson. The feedback was extremely positive. The head teacher remarked on the high levels of independence, continuing: "The students were very engaged and commented that they enjoyed this method of teaching because 'it helped them to discuss their opinions and figure out the answer together'. There was certainly a very high degree of challenge for all learners in evidence that was very pleasing to see." However, the teacher was dissatisfied because he could not sustain his students' engagement and motivation into the second lesson. Students had lost sight of the overall aim of the inquiry and the pathways they had decided to pursue at the end of the first lesson did not seem as intriguing second time round. 
 
Below, we give advice on how to develop an inquiry over a series of lessons when students have yet to develop the ability to direct their own inquiry from one lesson to the next.
    
Just above
It is important to set a prompt just above the understanding of the class in order to promote levels of curiosity and questioning that sustain prolonged inquiry. If, on the one hand, the prompt is presented to a class simply as a means of applying recently acquired knowledge, then its potential will be exhausted quickly once the knowledge has been applied. If, on the other hand, the prompt contains properties that are intriguing and unfamiliar, then students will require new concepts to understand it fully - and the new concepts can be built into the course of the inquiry.
   
'Hanging' lessons on questions
When the prompt is set at the right level, students will often ask a varied set of questions. These normally include questions about how to carry out a procedure or the meaning of a concept. The teacher can use the questions to design a structured inquiry over a series of lessons by 'hanging' lessons on a question or collection of questions. This proves to be far more meaningful for students than simply teaching unconnected lessons. Now each lesson has a context (the inquiry) and a meaningful purpose (answering a student's question). In order to give the teacher time to plan the first lesson of a series, students could be invited to pose questions at the end of the lesson immediately before the one in which the inquiry is due to start.
  
This presentation shows how the teacher has selected specific
questions (highlighted) on which to 'hang' lessons.
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One problem about this approach, it might be argued, arises when the class does not ask the 'right' questions to address the mandatory curriculum content. The teacher can avoid this issue by selecting or designing a prompt that creates a 'conceptual field' linked to a particular area of the curriculum. Then students' questions will be on the terrain established by the teacher. Another problem arises when there are insufficient questions to develop a series of lessons (perhaps because the students are new to inquiry). In this case, the the teacher can take it upon herself to structure lines of inquiry related to the properties of the prompt.
    
Direction and purpose
Whether the teacher structures the inquiry or opts for a guided or open inquiry (see Levels of inquiry), a student should be clear about the direction and purpose of their inquiry at each stage. When students are exploring or testing cases, they are often concentrating exclusively on mathematical procedures. At the end of such 'search' activity, they will often have to re-orientate themselves towards the original aim in order to appreciate how their results link to the direction of the inquiry. The teacher can achieve this reflection (individual or group) by facilitating regular discussions based on the choice of a regulatory card. When the physical cards, which can stand alone or be sequenced into a series of steps, are laid out on the table, they act as a reminder of the direction and purpose of the inquiry. At the end of each lesson, the teacher can summarise how lines of inquiry have developed, particularly by calling on students to present their work in progress. Referring to these presentations at the beginning of the next lesson is an effective way of bridging between lessons.
   
Maintaining momentum by learning a new concept
One advantage of setting the prompt 'just above' the level of the class is that students will need new concepts to understand all of its properties. For example, in the inquiry The areas of a rectangle, a triangle and a circle are equal, students will often request instruction on working out the area of a circle. Learning a new concept during inquiry acts to boost the students' momentum and provides the impetus to explore new pathways. (See the box below for a description of a series of lessons with the introduction of a new concept as a central part of inquiry.)
  
Structured inquiry
Andrew Blair reports on a inquiry he developed with a year 8 mixed attainment class. The nature of the class led him to run a structured inquiry, in which he designed activities that addressed the students' questions and comments (above). Levels of motivation remained high during the inquiry because students could relate their learning to the starting points they themselves had created.

Lesson 1 
While the students had carried out inquiries before, the class had a reputation for being 'challenging' with some students having a poor attitude to learning. Nevertheless, in the initial phase of the inquiry, they all listened attentively as each pair posed a question about the prompt or responded to a peer's comment. Before the lesson, I had decided to restrict the regulatory cards I offered the class to these five.
However, when the time came, I judged the students required an immediate focus and handed out a sheet for them to discuss and work on (see two examples below). Students commented on the connection between the areas using the large squares as a unit of measure and the areas using the small squares. At the end of the lesson, the students had found the areas of the rectangle and triangle and had started to make suggestions for changing their dimensions in order to make the areas equal. We also had estimates for the area of the circle (using the large squares) of between 30.5 and 33.5. Students reasoned that the small squares would give a more accurate estimation because there were more whole squares to count.


Lesson 2
I based the second lesson on the questions about whether it is possible to work out the area of a circle and, if it were, how to do so. We started with a discussion of the diagrams below that link the area of the square on the radius to the area of the circle. 
Students realised that the area of a circle must fall in the range 2r2 < A < 4r2The class seemed to have settled on 3r2 before one girl tried to justify 
“slightly more than 3” because “the circle bends towards the outside.” I then introduced the idea of π as a mathematical constant, which we went on to use accurate to three decimal places. The students practised using the procedure of drawing the square on the radius of a circle and multiplying its area by π
  
Two students who had independently researched the formula for the area of a circle after the first lesson then presented the formula A = πr2. They modelled how to calculate the area of the circle on the worksheet from lesson one by substituting the length of the radius (3.25) into the formula. The area (33.2 accurate to one decimal place) was towards the top end of the estimates from lesson 1, which led to a short discussion about why that might be. As lesson 2 drew to a close, another student presented her dimensions for a rectangle, a triangle and a circle that have the same area (taking π accurate to three decimal places): 
RectangleTriangle Circle
length 15.71, width 20 base 31.42, height 20 
radius 10

Lesson 3
The final lesson of the inquiry started by addressing the two remaining points from the initial questions and comments. The first related to the question about whether other shapes could have the same area. Students selected one of three tasks:
(1) Draw a rhombus, parallelogram and regular trapezium;
(2) Draw the three shapes with equal areas (by counting squares); or
(3) After completing task 2, make one cut to the three shapes and rearrange the pieces to make rectangles with equal areas.
  
The second point involved the perimeters of the shapes. After I explained why C = πd, students either practised finding the the lengths of circumferences or tried to establish if the shapes with the same areas (introduced at the end of lesson 2) had the same perimeters. The class decided that if the areas of a rectangle, a triangle and a circle are equal, it would be unlikely they would also have the same perimeter. Many in the class wanted to go further and say it was impossible, but no student could establish a solid reason why this might be so. The contention remained at the level of intuition.
 
The Inquiry Maths model gives teachers the potential to develop an inquiry over a series of lessons. This can be achieved by setting a prompt just above the level of the class, 'hanging' lessons on students' questions, regularly reviewing and reflecting upon the direction and purpose of the inquiry and maintaining momentum by introducing a new concept that supports new lines of inquiry.

Andrew Blair
August 2018