Recently I had a conversation with some colleagues who teach in a university. They were very worried about something they had noticed about their undergraduate students - a fear of making mistakes. They had noticed that these students were very reluctant to hand in assignments in case they were 'wrong' and were often spending time very unproductively in checking and re-checking their answers. While it is, of course, important to encourage students to be careful about checking their work, and to help them to develop a repertoire of checking strategies, this conversation does seem to reflect a growing problem, that more and more students are becoming afraid to try new things in case they fail, and/or become depressed and question their own self-worth if they do make mistakes. Mathematics, with its emphasis on 'right' or 'wrong' answers can potentially reinforce these fears. On the other hand, however, the mathematics classroom can also be the perfect environment for sensitive teachers to help their pupils to face up to and overcome these fears - and, of course, the earlier in the child's school life that this support begins, the better.
The purpose of this article is to illustrate some ways in which mathematics teachers can help to create a secure, supportive classroom environment in which the pupils learn to not fear failure and to value making mistakes as an opportunity to learn and grow. Each section begins with a quotation from the Sathya Sai Education in Human Values programme, a world-wide, secular programme designed to support the integration of values education across the curriculum. The sources of these quotations have not specifically been acknowledge because they appear in similar form in many different places, but the quotations have been printed in italics. More details about these ideas are discussed in my book 'Education in Human Values Through Mathematics, Mathematics Through Education in Human Values'. For further information about how to order this book, please contact [email protected]
True education should make a person compassionate and humane.
It is likely that unwillingness to participate in the mathematics classroom arises from lack of understanding and compassion, which can often be unconscious, by teachers and other pupils. Consequently, we need to ask the question: how can we encourage more effective participation by any students not participating fully?
- Do not be angry if a child cannot understand something or makes a mistake, because this can lead to fear of failure.
- Show him/her how to recover from the mistake and try again.
- Tell them about famous people who were not afraid to make mistakes (see stories below), or about some of the mistakes you have made - but also encourage accuracy and patiently ask them to correct their careless errors. A useful source of ideas is a book called "Mistakes That Worked" by Charlotte Foltz Jones.
Students should not allow success or failure to ruffle their minds unduly. Courage and self-confidence must be instilled in the students.
- Use positive visual and body-language cues (nodding, smiling) and prompts (ah ha, hmm) to encourage them to arrive at appropriate answers.
- Be careful not to frown if a child makes a mistake, and don't allow other children to frown if a classmate makes a mistake either.
There is over emphasis on quick and easy gains rather than patience, fortitude and hard work.
Peter was a very clever eleven-year-old. In the final year of his primary schooling, there was only one test on which he scored less than 100%, and then he only lost half a mark. His classwork was always done quickly and correctly. When he knew that he could succeed, he was confident and willing to work hard. To challenge his thinking, Peter's teacher would give him some difficult problems. If Peter could not immediately see a way to solve a problem, he became a different child. He would sit, drawing on his notepad, or wander around the room. He would even ask his teacher if he could spend the time tidying the storeroom. Peter, who was normally so successful and confident, was afraid to tackle a difficult task because he was afraid that he might fail. So his solution was to quit, to make the fears go away. Fortunately, the story had a happy ending, because Peter and his teacher worked together to help him to develop more courage to tackle difficult problems rather than taking the easiest path of stopping.
Many writers have written about students such as Peter, who expect solutions to come to them quickly and easily and will give up rather than face negative emotions associated with trying the task. Another concern is that they often are not aware of when it is worthwhile to keep on exploring an idea and when it is appropriate to abandon it because it is leading in a wrong direction. They need to know when it is appropriate to use a particular approach to the task, and how to recover from making a wrong choice.
Clare, aged ten, was given the following problem to solve:
By changing six figures into zeros you can make this sum equal 1111.
Clare selected the strategy of changing numbers in all three columns simultaneously. She worked at the task with patience and fortitude for two hours. As she worked, she said to herself, "I know that this is going to work. All I need is time, to find the right combination." After she repeated the strategy 21 times, her teacher interrupted and suggested that it might be time to look for another way to solve the problem.
In Peter's case, it was not enough for his teacher to tell him that frustration, for example, is a normal part of problem solving, and to encourage him to spend more time working on the task. Clare, on the other hand, was "overpersevering", locked into persistently pursuing one approach when it may be more appropriate when stuck to use other strategies, even such as help-seeking. One of the responsibilities of a mathematics teacher is to help pupils to learn how to persevere when the problem-solving process becomes difficult. They also need to know how to make decisions about avoiding time being wasted on "overperseverance".
STRATEGIES FOR ENHANCING PERSEVERANCE
- Equip learners with a range of strategies/techniques for solving different types of problems.
- Encourage them to experience the full range of positive and negative emotions associated with problem solving.
- Promote the desire to persevere.
- Help them to make "managerial" decisions about whether to persevere with a possible solution path (when to keep trying, and when to stop).
- Encourage them to find more than one way to approach the problem.
One sequence of strategies which is used frequently by successful, persevering problem solvers is the following:
- Try an approach.
- Try it 2-3 times in case using different numbers or correcting errors might work.
- Try something different. (You might decide to come back to your old way later.)
One student used this sequence to persevere successfully with a problem.
Stories About Famous Mathematicians
When you are teaching the appropriate topic, take a minute to tell your pupils an anecdote about one of the famous mathematicians who contributed to this particular field of mathematics. It is important for pupils to be aware of the 'human' side of these famous people. "Using biographies of mathematicians can successfully bring the human story into the mathematics class. What struggles have these people undergone to be able to study mathematics?..." (Voolich, 1993, p.16)
Stories About Famous Mathematicians
Education should impart to students the capacity or grit to face the challenges of daily life.
For students who have tried but are still having difficulties, McDonough (1984) advised that the teacher:
- ask the pupils to restate the problem in their own words and if this indicates that they have mis-read or mis-interpreted the card, ask them to read the instructions again,
- to help with the understanding of the written instructions question the pupils carefully to find out if they know the meanings of particular words and phrases (i.e. mathematical terminology),
- have the pupils show the teacher what they have done, compare this to what is asked in the instructions, and question the pupils to see if they could think of another method, for example, "Could you have done this another way?" or, "Have you ever done a task like this before?"
- if necessary, give the children a small hint but only after questioning them carefully to find out what stage they have reached.
If the teacher follows procedures such as those described above, the pupils will be encouraged to be more thoughtful and self-reliant. If pupils are panicking or unable to think what to do, introduce them to the valuable technique of silent sitting - that is, sitting for a few minutes in a state of complete outer and inner silence.
You can tell them about famous mathematicians who have solved problems by using this technique. For example, Sir Isaac Newton.
By example and precept, in the classroom and the playground, the excellence of intelligent co-operation, of sacrifice for the team, of sympathy for the less gifted, of help...has to be emphasised.
Some teachers' comments:
I was concerned about two things. One was the way I could use praise to develop self esteem. The other thing was the way in which I was involved in my pupils' activities. I chose these issues because I had got into the habit of teaching from the front of the room and responding to the students' answers with comments such as "Okay", "Good", "Sensible". I was also concerned that the girls were outnumbered by boys in the class and there was an underlying assumption that the boys were better than the girls, made particularly evident by a vocal group of boys. I consciously placed myself with different pupils in the classroom and moved to groups when asking or answering questions. I deliberately targeted the quieter children to encourage them to participate in group/class discussions. I developed a repertoire of responses to students' answers, including, "Good thinking strategy," or "Can you clarify that response?" I allowed more response time, focused on permitting girls to respond following incorrect answers and followed their answers immediately by further questions. Although I only had two weeks in which to implement these initiatives, I felt sufficiently positive about the change in quality of the students' responses to warrant continuing this approach. (Primary School Teacher)
four components of language skills: speaking, listening, reading and writing. Interactions are indeed the heartbeat of the mathematics classroom. Mathematics is learned best when students are actively participating in that learning. One method of active participation is to interact with the teacher and peers about mathematics. (Primary School Teacher)
I chose to work with a group of children about whom I felt I knew very little. I realised that these children could have ability which was not being shown, so I decided to make a more concentrated effort to provide a variety of experiences and activities, to allow some 'non-performing' children to demonstrate their skills. I also recognised the need to discourage a group of 'noisy' boys from putting down the girls and their contributions. A colleague undertook a similar exercise with an older class. She was surprised that she knew the boys better as being more confident and responsive. She intends to investigate this further by asking a colleague to observe her teach to find out whether her suspicions are true that she is responding more to the boys than to the girls. (Secondary School Teacher)
Education must award self-confidence, the courage to depend on one's own strength.
- Some of us may believe that it is acceptable to be untruthful if it is to avoid hurting somebody else's feelings. On the other hand, some people can also be cruelly truthful and blunt if they do not like something about another person. We need to realise that neither of these behaviours is really appropriate.
- If we are patient and consistent in our approach and give criticism with compassion, we will have a more significant influence on the child's subconscious levels of thinking than we realise.
- This does not mean that you have to be blunt or to hurt somebody else's feelings by telling them something unkind. For example, when correcting students you could say, "I don't like the way you answered that question. I like it better when you give me a sensible answer and I know that you have put thought into it." Or you could say, "I don't really like the way you have done this piece of work. I prefer it when you do it more slowly and make fewer mistakes". This means that you are making it very clear to the other person why you are not happy and how you would prefer her to behave.
The teachers who wrote the comments above were asked to recommend ideas which they could try in their classrooms to encourage more understanding of those students who may not feel safe to participate as fully as they should or could be. Recommendations included:
- give continuous encouragement, mainly verbally. Value everybody's responses and have firm rules about interruptions and 'put downs',
- encourage a balance between co-operative and competitive teaching and learning styles,
- demonstrate an 'expectation' for students to participate,
- encourage group work and peer tutoring, particularly on activity-based and problem-solving tasks,
- allow students sufficient time to complete their work,
- encourage different strategies for approaching and solving problems,
- talk to the non-participators about their reasons for lack of participation - perhaps our perceptions are invalid.
Bell, E.T.(1937). Men of Mathematics (Touchstone Edition, 1986). New York: Simon & Schuster. ISBN 0-671-62818-6 PBK
Lovitt, C. & Clarke, D. (1992). The Mathematics Curriculum and Teaching Program, Professional Development Package, Activity Bank Volume 2. Victoria, Australia: Curriculum Corporation. ISBN 0 642 53279 6.
McDonough, A. (1984). 'Fun maths in boxes', in A.Maurer (Ed.). Conflict in Mathematics Education, (pp.60-70). Melbourne, Australia: Mathematical Association of Victoria.
Perl, T. (1993) Women and Numbers. San Carlos, California: Wide World Publishing/Tetra. ISBN: 0-933174-87-X
Voolich, E. (1993). 'Using biographies to 'humanize' the mathematics class'. Arithmetic Teacher, 41(1),16-19.