Amelia O'Brien, a year 5 teacher at Caulfield Grammar School in Melbourne (Australia), used the place value prompt after launching a unit with philosophical dialogue. 

Pupils started their inquiry through a Socratic circle discussion in which they were invited to agree or disagree with the statement 'There is more future than past.'

They then discussed the prompt, posing questions and speculating on the meaning of the diagram. Some wondered if there is a link with past and future, while others showed how the diagram could represent the decimal system.

Pupils recorded their a-ha! moments, which included:

• "A hundred small intervals fit in the bigger line on top."

• "Any number could fit in the intervals."

• "There are infinite possibilities of patterns."

A new line of inquiry develops from one pupil's suggestion to use different bases (see below).

Amelia reports that the prompt is "a great way to gauge prior knowledge, generate questions, and create excitement and interest."

May 2024

A new line of inquiry: different bases

During the a-ha! phase of the inquiry, one pupil linked the prompt to other bases - binary (base 2) and ternary (base 3). 

The pupil's diagram shows four intervals between 0 and 1 in base 2. Following the logic of the prompt, the numbers would be presented on three levels. The top number line would show 0, 1, 10, 11, 100, and so on. In the level below, 0.1 would appear between 0 and 1.

The use of other bases is an impressive innovation.  As a line of inquiry, it could be used as a challenging extension to the main inquiry. 

Ben Grundy, a primary inquiry teacher from Melbourne (Victoria), posted the pictures below on twitter. He used the place value prompt at the start of a unit from the scheme of learning on fractions, decimals and percentages. 

Ben reports that his year 6 students "launched the unit with some great thinking and questions", adding that the class also discovered some misconceptions.

An interesting dilemma relates to the equivalence between decimals and percentages. In the final picture, is 3.5 equivalent to 350%, 35% or 3.5%? 

Ben differentiated the task sheets by using two or three zoomable number lines and requiring students to consider numbers in different positions on the lines.

September 2022

• 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.

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.

September 2019

Dewey on curiosity

In Democracy and Education (1916), John 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).