Place value inquiry
Mathematical inquiry processes: Make connections; generate examples and problems. Conceptual field of inquiry: Place value in the decimal number system.
The prompt is a number line with a microscopic view of a section of the line. The teacher can run the inquiry as open or structured depending on the experience of the class with inquiry learning and on the pupils' levels of initiative (see Levels of Inquiry Maths).
The inquiry starts with students posing questions and speculating about the relationship between the two parts of the number line. Different pathways can emerge from the initial phase depending on the start number (positive or negative, whole number or decimal, tenths or hundredths, and so on) and on the intervals chosen. One point that often needs reinforcing is that the intervals on each line must be the same, but each interval on the blue line is a tenth of the interval on the black one. Pupils can decide to divide the blue line into more intervals, which might lead to the appearance of recurring decimals. This, in turn, can give rise to a discussion on degrees of accuracy (and rounding) and and on the advantage of fractions in this context.
In a structured inquiry, the teacher might pose questions and set tasks, perhaps after conducting a discussion around the meaning of the prompt. A planned lesson around place value (incorporating, for example, units, tenths and hundredths on the number lines) might be followed by tasks that involve filling in missing numbers when given two pieces of information about the black or blue line. The teacher might build up to posing challenging questions, such as: If the main number line started at 5 and finished at 19, what is the fourth number on the blue line? (An interesting discussion has arisen in a year 7 class about whether it is possible to complete the missing numbers when given one number on each of the lines). Once students have practised finding the interval on the black line by dividing the difference between the start and finish numbers by the number of intervals, then they could experiment with their own start and finish points.
Extending the inquiry
A pupil in a grade 5/6 class at the Fred Varley Public School (Ontario, Canada) extended the inquiry by introducing a third number line (see the picture posted here). This represents a valuable extension to the inquiry. If the top line represents units, other number lines could be added to introduce the concepts of tenths, hundredths, thousandths and so on.
The inquiry in year 7
Excitement generated by inquiry
Dr Dawn Jepson, a teacher of mathematics at Thurston Community College (Bury St Edmunds, UK), used the place value prompt with her year 7 mixed attainment class. In only her second inquiry, she was able to generate great excitement in the classroom as students reasoned about the meaning of the prompt (see the picture above). Dr Jepson described the lesson as amazing and remarked on how proud she was of her students. Her head of department added that inquiry, in producing high levels of enthusiasm in students and teacher alike, is infectious. Dr Jepson shared the outcomes of the students' inquiry in a departmental meeting.
Questions and observations
These are the questions and obseravtions of a year 7 mixed attainment class at Haverstock School (Camden, UK). In a structured inquiry, the teacher used the calculation that split 11 into ten equal steps to develop a mathematical approach to the resources on this page.
This document lists the questions and observations about the prompt from two other year 7 mixed attainment classes at the school. The teachers of the classes were Amy Flood and Ollie Rutherford. Ollie reports that, "There were so many amazing points from students and overall the inquiry went really well!"
Using students’ questions to structure inquiry
Matt Carvel used the second half of a one hour lesson to introduce the prompt to his year 7 mixed attainment class. Students asked questions and made observations:
Is it to do with measuring?
Are they centimetres and millimetres?
Is the top line the same but bigger?
What’s in between the gaps on the bottom line?
There are ten sections on top.
There are 10 little pieces in the line above.
Between 1 and 2, there is 1.1 to 1.9 points in between.
The units could be in between.
For the second lesson Matt chose to focus on the four statements in italics above. He started the statements on the board (see illustartion). Matt created a task that focused on the two numbers at the start and end of the blue line on the top number line. The six tasks involved an increasing level of mathematical complexity. (Students had previously been taught how to choose the task with an appropriate level of challenge.)
As the lesson developed, students began to make up their own questions. One chose 32.8 as the start number and 328 for the end. On the bottom line he went up in intervals of 32.8, but realised that he would not end at 328. When he presented his work to the class, another student, who had devised a method of dividing the difference between the start and end numbers by 10, offered a way of correcting the calculations. Matt called the second student up and the two discussed a solution in front of the class so all students would benefit. Matt reports that “it was the best moment of my teaching career so far.”
Structure and independence
These are the questions and observations of a year 7 mixed attainment class at Haverstock School in Camden (London, UK). Each pair of students went on to consider the six regulatory cards before sharing their choice with the rest of the class. As a whole, the class selected the following four cards:
Decide on the aim of the inquiry: The majority of students had a clear understanding of the number lines. However, those that picked this card required a further discussion on what they could do with the number line to inquire further.
Change the prompt: Students who selected this card wanted to experiment by using different increments on the top line to see how this changes the bottom line. They proceeded independently from the teacher.
Practise a procedure: The teacher structured the inquiry by handing out one of two tasks depending on the prior attainment of each student.
Inquire with another student: Students who chose this card, it turned out, were expressing a desire to be given direction. The teacher gave them one of the same two tasks as above.
The teacher responded flexibly to the choice of cards to structure the inquiry for those students who required guidance, while allowing others to develop their inquiry independently. At the end of the lesson, students fed back on their reasoning during the tasks or on their own findings.
Nichola Sowinska, a teacher of mathematics in Peterborough (UK), contacted Inquiry Maths after using the place value prompt with her year 7 class: "Some really great thinking came out of it." The students posed the following questions (see the photograph for a sample):
If the line was between two and three, what would the intervals represent?
Is the black line showing units or tenths? This will affect the blue line.
Is it positive or negative?
If it was between five and six, what would the blue 'bits' be?
If either line was longer, would it affect any of the other numbers?
How many decimal places could the blue line show?
Is there a specific number of blue lines (intervals) between two black lines?
If there were more numbers on the black line, would it affect the blue line?
What would the decimals be as fractions or percentages?
Does there have to be a specific number of blue lines between black lines?
Is there a formula to work out the next term?
This set of rich questions shows a class in the process of trying to construct an understanding of the number line based on their existing knowledge of place value. Multiple inquiry pathways could develop around the relationship between the black and blue lines. Once the ends of the black line are defined numerically, then the intervals in both lines are fixed. In their questioning, students have begun to speculate about what section of the number line is represented in the diagram (for example, between two and three or five and six). How would intervals on the blue line change if the intervals on the black line represented tenths or negative numbers?
As the class was new to inquiry, Nichola decided to keep the students together: "We went on to look at how many different ways we could split up the line." The class finished the inquiry by presenting diagrams supporting their ideas on mini-whiteboards. In the space of one lesson, the students had been involved in questioning, discussing, reasoning and presenting their ideas.
Structuring the inquiry
Caitriona Martin also tried out the prompt with a year 7 class. She invited students to pose a question or make a comment, but then she structured the rest of the inquiry: "It’s the first time we’ve done an inquiry lesson properly together so I kept it fairly closed and made the students all look into the same thing.
"Our key question was 'What if it didn’t go up in ones?' They started trying out different intervals on the scale. Some made it go up in twos, some in fives, some in tens, some in tenths. It was interesting to see how they tackled working out the 10 sub-divisions in the lower scale from there."
In the second lesson, the class continued with the structured inquiry. Caitriona evaluates the learning that occurred through the inquiry: "In the style that we did it, it was a useful activity to see how the students made sense of 10 smaller steps being equal to one big step. I think it honed their reasoning and logic skills." She also discussed the structure of the inquiry: "As an inquiry, it would be interesting to see where other classes take it, whether they could be more creative or abstract with it as it felt rather closed, but that would definitely have been influenced by the style with which I conducted the inquiry."
At the time of the inquiry, Caitriona was second-in-charge of the maths department at St. Andrew's School, Leatherhead (UK).
The inquiry in primary classrooms
Making connections through inquiry
Matthew Bernstein, a teacher of a grade 5/6 class at the Fred Varley Public School (Markham, Ontario), posted the pictures above on twitter. He describes how the inquiry enabled students to make connections:
When starting this task, students immediately made connections to measurement and the 'step' or base 10 nature of metric units. The prompt is a useful way for students to understand this concept as it is a natural connection to place value.
Students who initially struggled to represent their ideas independently on the number line needed some scaffolding, which I was able to provide by asking them to count by tens on the top number line and then fill in the bottom one. Once they saw the subdivisions it was easy for them to make connections down to ones and tenths, but could also see what a number line above might look like. Students quickly saw the connection to place value and then this was reinforced the following day when we examined a 'zoomable' number line.
Making meaning through inquiry
Barbara Paraskevakos, a grade 3 PYP teacher at the Frankfurt International School, posted pictures of the inquiry . The inquiry started with Barbara posing the question: "What questions come to mind when you look at this image?"
The pupils then attempted to give the prompt meaning by drawing on their prior knowledge. Some thought it was a tool for measuring; others said it was a number line. The class went on to discuss why pupils had chosen to count in units, tens and hundreds.
Elizabeth Anderton, a newly qualified teacher at Springfield Primary School (Tilehurst, UK) reports on using the place value prompt:
"I used the place value prompt with a Year 3 maths class! The class loves sharing their ideas and I wanted to take advantage of their enthusiasm and use it to develop their curiosity about maths. I introduced the whole class to the idea of Inquiry Maths using this prompt. I gave each group of five children a large piece of sugar paper with the prompt stuck in the middle. I also printed off smaller versions and left them on the tables. They took them and used them, without any need to explain what they were for, to try out their ideas.
"I wanted to keep the inquiry as open as possible as they had been learning about measure and scale, place value, number patterns and sequences, so they had several ways in which to approach this from and I didn't want to close down any of their options. I explained that there were no right or wrong answers and that this was their chance to write and share all those ideas and questions. I learnt a lot about where the children were in terms of their understanding and the children had to draw on all their learning to try and make sense of it. I was impressed by the range of questions they asked and the way they ran with their ideas. I would highly recommend this approach for younger children."
Dewey on curiosity
In Democracy and Education (1916), Dewey defines child development as “the direction of power into special channels: the formation of habits involving executive skill, definiteness of interest, and specific objects of observation and thought” (p. 59). These habits permit the adult to solve complex problems. However, Dewey characterises the process as potentially dialectical with the adult also developing qualities of the child: “With respect to sympathetic curiosity, unbiased responsiveness, and openness of mind, we may say that the adult should be growing in childlikeness” (p. 59).
Later in the book, Dewey discusses the observation that children’s levels of curiosity are lower in school than in everyday life. The key to closing that gap is to provide a school environment in which children can participate in self-directed inquiry: “There must be more actual material, more stuff, more appliances, and more opportunities for doing things, before the gap can be overcome. And where children are engaged in doing things and in discussing what arises in the course of their doing, it is found ... that children's inquiries are spontaneous and numerous, and the proposals of solution advanced, varied, and ingenious” (p. 183).
Dewey reminds teachers that they can develop the their pupils' curiosity by creating meaningful activities that are rich in intrigue and connections: "Curiosity is not an accidental isolated possession; it is a necessary consequence of the fact that an experience is a moving, changing thing, involving all kinds of connections with other things. Curiosity is but the tendency to make these connections perceptible. It is the business of educators to supply an environment so that this reaching out of an experience may be fruitfully rewarded and kept continuously active" (p. 245).
An alternative prompt
Billy Adamson, head of mathematics at Thurston Community College (Bury St Edmunds, UK), used the prompt with his year 7 class, tweeting "Loved my inquiry lesson with year 7!" He supplemented the original prompt with one of his own:
0.7 < 0.76 < 0.764
0.7 > 0.67 > 0.467
Billy's prompt has the potential for students to engage in exploration and reasoning. Why, for example, does the inequality sign not change when more digits are placed at the end of the decimal number on the top line, but it could change when another digit is introduced on the bottom line?