Estimation or not?

This task is about identifying whether students are estimating or not.

Five students were asked to estimate the sum of these three numbers:

329 + 462 + 836

Some of them were estimating and some were not. For each person:

select either is or is not estimating and
explain why they are / are not estimating

Answers/responses:

For each part, students' responses followed this hierarchy of sophistication:

Students correctly identified whether a person was estimating or not, and identified the specific concept targeted by the question.
Students correctly identified whether a person was estimating or not, but only made a general comment about estimation. See table called Common correct responses, just before Next Steps, for further elaboration.
Students correctly identified whether a person was estimating or not, but did not give an acceptable explanation.
Students incorrectly identified whether a person was estimating or not.

		Y6 (11/2005)	Y8 (06/2006)
a)	Not estimating, and either: Josef has used a calculator to do an exact calculation. Other correct statement(s) about estimation other than the target one.	easy very easy	easy very easy
b)	Not estimating, and either: Gina has done an exact calculation or has an exact answer. Other correct statement(s) about estimation other than the target one.	moderate moderate	moderate moderate
c)	Estimating, and either: Describes or names an acceptable method of estimation (rounding). Other correct statement(s) about estimation other than the target one.	difficult moderate	very difficult moderate
d)	Not estimating, and either: Sam has done an exact calculation first, (and then rounded). Other correct statement(s) about estimation other than the target one.	difficult difficult	difficult difficult
e)	Estimating, and either: Describes an acceptable method of estimation (front-end). Other correct statement(s) about estimation other than the target one.	difficult moderate	very difficult moderate

Based on a representative sample of 173 Year 6 and 139 Year 8 students.

NOTE: Responses may be given orally to ensure that writing is not hindering the communication of ideas.

Teaching and learning:

Prior knowledge

A firm base of additive basic facts
Good place value concepts

Diagnostic and formative information:

	Common error	Likely misconception
b)	Estimating	Indicates that if numbers are rounded at any stage then it is estimation.
b) and d)	Estimating	Sees the vertical algorithm or an exact computation as a mental strategy.
b) and d)	Estimating	Thinks that estimation uses exact working or leads to an exact answer.
d)	Estimating	Sees rounding 1627 to 1600 as guessing
d)	Estimating	Believes it is estimation if the answer is close to the exact computation.
c) and e)	Not estimating	Believes estimation must perform a computation exactly or lead to an exact answer, (or comments that the answer is wrong).
c) and e)	Not estimating	Believes that no computation can be done in estimation, even an approximate one that can be done mentally.
e)	Not estimating	Does not see the front-end method as an acceptable way of estimating.

Common correct responses
Estimation is a quick mental way of getting an answer close to the exact one. It incorporates a wide range of different strategies that depend on the individual person making the estimate, and the particular problem. The following table of responses gives examples of general statements about estimation that are correct, but may restrict students' ability to explore and enhance their range of estimation strategies.

	General conception about estimation	Example
	Estimation is guessing.	"Gina is not estimating because she is not guessing." "Aroha is estimating because she is making an educated guess."
	Estimation uses rounding.	"Gina is not estimating because she is not using rounding."
	Estimation is doing it mentally.	"Josef is not estimating because he is not using his own brain." "Aroha is estimating because she did it in her mind."
	Estimation is when the answer is close to the real answer but it is not exact.	"Gina is not estimating because that's the right answer." "Peter is estimating because he is giving a rough answer."
	Estimation is a quick, easy way of way of getting an answer.	"Aroha is estimating because she did the sum in the shortest possible way." "Peter is estimating because his working out is simple."

Next steps:

To encourage students to understand what estimation is and what it is not. A class discussion based upon their responses to each of the examples, with students sharing why they think the student is estimating or not, is one strategy to challenge their ideas. Using the Think, Pair, Share strategy may be a good way of conducting a class discussion.
Discuss that estimation aims to eliminate exact computation, and is something that can easily be done with mental strategies. Emphasise that estimation is not just a pure guess, but is based on good number sense (i.e. affirm the values in the table of Common correct responses).
Extend the range of estimation strategies for students who can discriminate between estimation and exact computation (refer to Computational estimation information). This could be through a discussion of the ways different students in the class estimate using a variety of estimation problems.

Comparison between Year 6 and Year 8 students – Research findings
Year 6 and Year 8 students performed at roughly similar levels of achievement on most aspects of these questions, with the Year 6 students outperforming the Year 8 students in the second of the two aspects mentioned below. This is unusual, as generally you would expect Year 8 students to achieve higher success rates than Year 6 students would. It is unclear why this is the case, but it may indicate that little time is spent on aspects of estimation or that it is not looked at in more advanced ways with Year 8 students than it is with Year 6 students. In comparable questions in other areas of mathematics these Year 8 students outperformed the Year 6 students.

The two ways in which Year 6 and Year 8 students differed were:

In part a) students were equally likely to identify that estimation was not doing calculations exactly on a calculator. However, Year 6 students were significantly more likely to focus on the fact that the answer was exact and therefore it was not estimation.
In parts c) and e) Year 6 students were more likely to describe the method of estimation used, whereas Year 8 students were more likely to give a generic description about estimation such as "guessing", "doing it mentally", or "the method of estimating is not exact".

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