# The prompt

Mathematical inquiry processes: Explore and test cases; find rules; generalise. Conceptual field of inquiry: Measurement of angles; interior angles of polygons; inscribed shapes inside circles.

The prompt is adapted from one designed by Theo Giann (a PGCE student at the University of Brighton, UK, during 2011-12). In his original version, the fourth circle contained a question mark, rather than a hexagon. However, it is possible for students to interpret the question mark as a teacher's closed question and, hence, limit the inquiry to finding the next polygon in the sequence. In its current form, the prompt might lead to more open lines of inquiry, such as the ones below that have been generated by students' questions and comments:

Theo tried his prompt with a year 8 class as part of a Masters-level research project on encouraging autonomy. As this was Theo's first inquiry with an unfamiliar class, he decided not to use the regulatory cards and instead gave the students a limited choice of working in their books, making a poster or completing a worksheet.

### New line of inquiry

A new line of inquiry is suggested by the following questions from a year 7 student in Ollie Rutherford's class at Haverstock School (Camden, UK):

The answers connect multiple topics in the curriculum: ratio, sequences and expressions for the nth term, area of polygons and a circle, Pythagoras' Theorem, trigonometry (including the cosine rule) and properties of individual shapes (such as the centroid of a triangle) amongst others.

# Claim-support-question inquiry cycle

Amelia O'Brien, a PYP teacher at the Luanda International School (Angola), posted the pictures on twitter. The first one (left) shows the responses of her year 5 pupils to the prompt. Amelia reports that the pupils made interesting comments about mathematical pattern.

The pupils' questions and observations led to a number of lines of inquiry, including an impromptu word inquiry into the suffix quad-.

In another line, pupils calculated areas to test the conjecture that, The more sides that the inside shape has, the less space there is left over inside the circle. (The conjecture is similar to the inscribed shapes inquiry.)

The pupils used their visible thinking cycle (claim-support-question) to develop the inquiry further. Overall, Amelia says, students were engaged and there was "so much excitement" in the classroom.

June 2019

Amelia had used the claim-support-question cycle previously with a grade 6 PYP class at the Luanda International School. The pupils asked questions and made observations about the prompt to start the inquiry. They used the cycle to frame their thinking.

Amelia reports that, "The students were very engaged as they chose their own 'claim'." The cycle led to an advanced mathematical inquiry that included claims such as "irregular polygons cannot fit into circles" and angles in a heptagon sum to 820o (see pictures).

March 2017

# Structured and open inquiry

Sally Pearson, a Lead Practitioner in the mathematics department at Brittons Academy (east London, UK), used the prompt with her year 7 mixed attainment class. The picture shows the students' questions and observations.

During the inquiry, students were involved in structured or open inquiries depending on their levels of initiative and independence. Sally reports on the development of the inquiry:

It was the class's second inquiry so far. I was impressed with the variety of the responses and the way that students were eager not only to provide a new question or comment but also how they were listening so carefully to each other and building on or amending what others were saying.

Lots of answers to questions were provided by other students and they were all so desperate to speak we ended up spending about half of the lesson on debates about whether a square is a rectangle, how many lines of symmetry a rectangle has compared to a square, whether there are 360 degrees in a circle and whether a right angle is acute, obtuse or neither.

It was a great start to a unit on angles with a mixed attainment class as some naturally gravitated to exploring angles in triangles, some changed the prompt to see if the sum of the interior angles of any quadrilateral is 360o and some were intrigued by the interior angles in the pentagon.

For those that needed it, I closed the inquiry down fairly quickly as they wanted support investigating angles in triangles or polygons with direction from me, but I was interested in the number (about eight out of 30) that were happy to structure their own further inquiry based on changing the prompt with quadrilaterals or extending the pattern they saw to hexagons and beyond.

March 2017

# Claim-evidence-reasoning inquiry cycle

Courtney Paull (a grade 7 math guide at the International School Manila) posted the picture above on twitter. Students created their own claim-evidence-reasoning cycles from the prompt.

The picture shows one claim: Shapes with sides that are parallel to each other have an even number of sides. Courtney said that she was not sure where the inquiry was going, but the children made "some really awesome claims."

June 2018

Counter-example

After seeing the claim, one mathematics educator shared a counter-example.

To maintain the validity of the claim, the wording could be changed to specify regular shapes only.

# Co-regulating inquiry

Co-regulation refers to a process in which the teacher and students direct an inquiry together. In the account that follows, the teacher offered a year 7 mixed attainment class the opportunity to decide what to do within a restricted set of options.

At the start of the inquiry, the class posed questions and made observations about the prompt (see the picture). The students focused on the names of the shapes and the interior angles of the polygons, and on the fact that the polygons are inscribed inside circles.

As the students read out their questions and observations, the teacher took the opportunity to introduce formal vocabulary, such as ‘segment’, ‘inscribed’ and ‘interior' angle. The teacher then used a protractor on the interactive whiteboard to measure the angles in the triangle and quadrilateral, confirming that they were all, respectively, 60o and 90o

Regulatory cards

The teacher introduced the six regulatory cards he had chosen for the class. In previous inquiries, students had started to choose particular cards automatically instead of thinking afresh about how to regulate their activity. Those cards described how to inquire (for example, Work with another student), so now all the cards were about what to do in the inquiry.

However, before asking students to select a card, the class discussed what each card might mean in the context of the current inquiry. Under the guidance of the teacher, the students proposed the following:

All the students went on to select Use a worksheet or Look for a pattern. The teacher showed those who chose the latter how to split a polygon into triangles so they were able to calculate the sum of interior angles in multiples of 180o before working out the value of one angle in a regular polygon.

Co-regulation in this case involved the teacher in designing the regulatory cards and deciding when instruction was appropriate, while students developed the meaning of the cards in the specific context and selected a card that enabled them to learn.

April 2015

Amanda Klahn, a PYP teacher at the Western Academy of Beijing (China), adapted the prompt for an inquiry with her grade 4 class.

Reducing the number of shapes makes the prompt more accessible as pupils have less to consider in the question, notice, wonder phase.

As Amanda says, the pupils' questions and observations provide enough material to keep the class inquiring for weeks - from area and perimeter to constructions using a pair of compasses.

March 2014

# Kindergarten inquiry

Ju Bonilla Garcia, a teacher at the United World College Thailand in Phuket, used the 2-dimensional shapes prompt with her kindergarten class.

During the inquiry Ju posted that tons of knowledge [is] being revealed through discussions! and students are asking interesting questions which will move our inquiry forward. She reports on the development of the inquiry:

As part of our transdisciplinary inquiry into shape and space, kindergarten students showed an interest and curiosity in a series of structures that were built by Grade 4 in the playground at UWC Thailand. We looked for shapes, lines and angles within the structures, which led us to the Inquiry Maths prompt.

We started by viewing the prompt as a whole class and student observations were sorted through the concepts of form, function and connection. This initial exploration of the prompt did not result in any student questions.

However, students demonstrated great enthusiasm when asked, 'What would happen if we continued to play with the lines?'

Each student was then provided with a shape from the inquiry prompt to play with individually. This encouraged experimentation with lines within the shapes which, in turn, led to student questions.

Interestingly, students drew additional lines within the shapes which formed various sized triangles. We then looked at student examples and many learners made conjectures about how many triangles were present within each shape (as drawn by their peers) and the characteristics of the triangles.

Through this inquiry, many gained the understanding that shapes can be partitioned to form other shapes, just like numbers. I have used IM prompts with older grade levels (upper primary) and was curious to see how this approach to learning maths would work with kindergarten.

The key differences included the extent to which I needed to scribe for students (obviously!), the time frame and how questions were elicited. It was certainly just as valuable, engaging and rich as when facilitating this approach with older learners.

December 2019

### Inquiry and misconceptions

In his Masters-level essay, Theo Giann considered the impact of inquiry on one student and raised an important question about misconceptions during inquiry:

One boy, who usually was unmotivated in my lessons, produced a poster with many examples of different inscribed polygons and clear mathematical labelling of each one. He went on to eloquently explain his work to the rest of the class; however, he made an error in his work – he deduced that because he had drawn an irregular inscribed pentagon, all irregular pentagons could be inscribed. I asked him to investigate this statement but he did not see his error and I ran out of time to explain where he had gone wrong.

On the one hand, the student clearly developed his own line of inquiry, explored relevant aspects of the problem and quite frankly worked harder than I had ever seen or ever saw afterwards. The freedom of being able to decide his own workload, his own methods and his own definition of success inspired him to have a really fantastic lesson. On the other hand, his work contained errors which, due to their complex and unique (to him in that lesson) nature, could not be fully addressed – I cannot guarantee that this has not led to misconceptions being formed or even supported.

As a mathematics teacher, I was now faced with a seemingly impossible question – to what extent can mathematical accuracy be discarded for the sake of enthusiasm? No doubt, there is ready at hand an army of purists who would yell from the hilltops “NONE! EVER!” – I myself verge very closely towards that camp – but can we really cast out of hand a method that promotes a child’s enjoyment of mathematics simply because it can lead to unresolved misconceptions? Surely no system can be completely free of that unfortunate phenomenon.

### Planning and evaluating inquiry

Maria Esteban (a trainee teacher at London Metropolitan University) used the prompt on her second school placement. She discusses how she planned and evaluated the inquiry in her project report.

### Functioning with geometry and fractions

In this article from the ATM's Mathematics Teaching 207 (March 2008), Derek Ball and Barbara Ball describe the types of students' thinking that arose when considering the prompt above.

# Alternative prompt

This statement seems plausible to pupils. Even when it is clear that the statement is false, it has encouraged classes to seek a pattern in the number of different types of polygons.

At its most open, the inquiry has led to students asking why pentagons are not classified in the same detail as triangles and quadrilaterals. One possible line of inquiry might be to attempt to develop a classification for pentagons based on the length of sides and type of interior angle.