Identifying Conceptual Difficulties

Formative Assessment in Science and Mathematics Education

Identifying conceptual difficulties

Introduction

This unit is intended to help you reflect on the nature and causes of students’ mistakes and to consider ways in which we might use them constructively to promote learning.

Outline

In this PD unit, four discussion activities are suggested:

Assessing students’ responses to diagnostic questions.
This involves examining some genuine student responses and trying to identify the thinking that is behind the errors.
Diagnosing the causes of errors
This involves thinking about various mistakes and the local generalisations that often cause them.
Responding to errors and misconceptions.
This involves considering research advice on the best way to respond to mistakes and errors. This issue is taken much further in later units.

D. Looking at classroom activities
This involves the close examination of two sample lessons and considering how effectively these lessons handle mistakes and misconceptions. The choice of the lesson should be related to the topic chosen for activity A.

Time needed

Approximately 2 hours.

Activity A: Assessing students’ responses to diagnostic questions

Look at the samples of (genuine) students' work shown on Handout 1.

Write a few lines summarising:

· The nature of the errors that have been made.

· The thinking that may have led to these errors.

Afterwards, compare your answers with those of a colleague.

Activity B: Diagnosing the causes of errors

Draw out the different possible causes of mistakes. These may be due to due to lapses in concentration, hasty reasoning, memory overload or a failure to notice important features of a problem. Other mistakes, however, may be symptoms of alternative ways of reasoning. Such ‘misconceptions’ should not be dismissed as 'wrong thinking' as they may be necessary stages of conceptual development.

Consider the errors and underlying generalisations shown on Handout 2: Generalisations made by students. Ask participants to contribute their own examples to the list.

Such statements are valid in many contexts that occur in early education. They work in limited domains that do not generalise. (For example, when children deal solely with natural numbers they infer that ‘when you multiply by ten you just add a nought’. Later on, this leads to errors such as 3.4x10 = 3.40). Many 'misconceptions' in students’ work may be attributed to the use of such local generalisations.

Discuss the following questions together:

Can you think of other generalisations that are only true for limited domains?

For what domains do the following generalisations work?
When do they become invalid?

· If I subtract something from 12, the answer will be smaller than 12.

· The square root of a number is smaller than the number.

· All numbers may be written as proper or improper fractions.

· The order in which you multiply does not matter.

Activity C: Responding to errors and misconceptions.

There are two common ways of reacting to pupils' errors and misconceptions:

· Avoid them whenever possible:
"If I warn pupils about the misconceptions as I teach, they are less likely to happen. Prevention is better than cure."

· Use them as learning opportunities:
"I actively encourage students to make mistakes and learn from them.”

Discuss the views of the group.

Now issue Handout 3: Some principles to discuss.
This describes the advice given in the research. How do you feel about this advice?

Activity D: Looking at classroom activities

Now look at Handout 4: Looking at Classroom activities.
Choose one of the lessons on Handout 5.

Work through of the activities a colleague.

As you do so, try to write notes under the following headings:

· What major mathematical concepts are involved in the activity?

· What common mistakes and misconceptions will be revealed by the activity?

· How does the activity:
- encourage a variety of viewpoints and interpretations to emerge?
- create tensions or 'conflicts' that need resolving?
- provide meaningful feedback?
- provide opportunities for developing new ideas?