2-dimensional shapes inquiry
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:
Names of two-dimensional shapes;
Perimeters of polygons;
Methods to inscribe shapes in a circle;
The circumference of a circle;
Regular polygons (including interior and exterior angles); and
Areas of polygons.
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.
A new pathway for inquiry is suggested by the following questions from a year 7 student in Ollie Rutherford's class at Haverstock School (Camden, UK):
What is the ratio between the number of sides of a polygon and the area of the segments left over?
Is there a sequence to the ratios?
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.
Making thinking visible through inquiry
Amelia O'Brien, a PYP teacher at the Luanda International School (Angola), posted the picture (above) of her year 5 pupils' responses to the prompt on twitter. She reports that the pupils made interesting observations connected to pattern. There are a number of lines of inquiry suggested by the observations. In one line, for example, pupils tested the conjecture* that "the more sides that the inside shape has, the less space there is left over inside the circle" by working with the areas.
The pupils used their visible thinking cycle (claim-support-question) to develop the inquiry, which included an impromptu word inquiry into the suffix quad-. Overall, Amelia says, students were engaged and there was "so much excitement" in the classroom.
Structured and open inquiry
These are the questions and observations of Sally Pearson's year 7 mixed attainment class. 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 360 degrees 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.
Sally Pearson is a Lead Practitioner in the mathematics department at Brittons Academy (east London, UK).
Claim-support-question inquiry cycle
Amelia O'Brien's grade 6 PYP class at the Luanda International School (Angola) asked questions and made observations about the prompt to initiate an inquiry. The students used a cycle of claim-support-question 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).
Claim-evidence-reasoning inquiry cycle
Courtney Paull (a grade 7 math guide at the International School Manila) posted this picture 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."
After the claim that shapes with sides that are parallel to each other have an even number of sides was posted on twitter, one mathematics educator shared the picture below as a counter-example to the claim. To maintain the validity of the claim, the wording could be changed to specify 'regular shapes'.
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 below). 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 .
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.
Thus, 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.
Inquiry in the primary classroom
Adapting the prompt
Amanda Klahn adapted the prompt for an inquiry with her grade 4 IB Primary Years Programme class. The reduction in the number of inscribed shapes to three makes the inquiry potentially more open. Now the pupils can suggest moving to the next case with more confidence because the hexagon might be more familiar than the heptagon (the next case in the original prompt). 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.
At the time of the inquiry, Amanda was an IB PYP teacher at the Western Academy of Beijing (China).
Ju Bonilla Garcia, a teacher at the United World College Thailand in Phuket, used the 2-dimensional shapes prompt with her kindergarten class. She tweeted during the inquiry that, "Tons of knowledge [is] being revealed through discussions!" (1) and students are asking "interesting questions which will move our inquiry forward" (2). Ju 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 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.
Discussions about the prompt
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
There are three types of triangle and four types of quadrilateral.
This statement has 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 pathway for this inquiry might be to attempt to develop a classification for pentagons based on the length of sides and type of interior angle.
When you draw round the bases of different 3-d shapes, you will always get a different 2-d shape.