The Weekly Column

Article 9, April 2000

HOW TO CATER FOR LOGICAL-MATHEMATICAL INTELLIGENCE

Logical-mathematical intelligence involves the capacity to analyze problems logically, carry out mathematical operations, and investigate issues scientifically.

The brain responds best in conditions of high challenge with low stress, the state Lozanov refers to as relaxed alertness. Multi-path, individualised and thematic learning with the mental work of engaged problem-solving enriches the brain. So does novel challenge and real-life experience. All the activities in this article present the learners with problems to solve and so cater for the logical-mathematical Intelligence type. As the problem-solving frequently entails spotting the mistakes, the starting point is the identification of what is meant by error.

Although the word error carries negative connotations, it is an intrinsic part of the learning process and can be seen in a positive light. As Michael Lewis says: "Language mistakes are not sins, they are creative experiments" Instead of concentrating on right or wrong, our main concern should be successful communication. Unlike other subjects, there is a wide range of correctness in a language and even dictionaries disagree about some things. An important factor in learning a new language is that of hypothesis forming and there will be no room for this to take place if the students are inhibited by the correction process.

What is a mistake? A mistake is caused by learners not putting into practice something they have learned whereas an error is caused by the learners trying out something completely new. In practice, it is nearly impossible to distinguish between the two so it is probably more helpful to talk about slips and errors. With a slip, the learner needs the least help that is sufficient to activate his existing criteria for correctness. With an error, the learner has to be given something new as he cannot derive the solution from his existing knowledge.

According to Michael Lewis, three main sources of error can be identified - interference, lexical deficiency or partial mastery. He recommends reformulation rather than correction as it attempts to imitate the way in which real-life correction happens. Krashen suggests that if a student makes a mistake, the teacher should try to expose the student to language just above the student's current level - language which they are ripe to add to their own language reserve. The advantage of this approach is that it offers information on how an efficient speaker would have said the same thing. The disadvantage could be the negative message that is conveyed - "I need to act as your interpreter for the rest of the class as you are unable to communicate effectively" It also fails to take into account the importance of emphasising the positive aspects of the learner's output. What about praise for the examples of effective language produced? Another problem could well be the amount of Teacher Talking Time involved in the process. That is why if you are going to adopt this approach, you need to do so with caution.

Over-correction results in teacher-dominance, the stiffling of creativity, the inhibition of independent thought, undue caution on the part of the learner and possibly stress. Learners unders stress resort to the "fight or flight" response and learning then becomes biologically impossible. Under-correction can result in complaints from students and loss of confidence in the teacher. Getting the balance right between the two extremes is a bit like walking a tightrope! We need to help the careless to become more careful and the careful to become more careless.

The art in correcting is in knowing when to intervene. Interrupting the free flow will damage the learners confidence. This is what Adrian Underhill has to say on the subject: "I will intervene when the learners need something from me and not when they don't. This reduces the impulse to keep approving the learners right actions, often by showing transparently artificial pleasure" and so acting in an inauthentic way. A common sense attitude would be to say that if the students make a mistake when working on the target structure of the lesson, it should be dealt with on the spot. However, if the mistake is unrelated to the target structure, it can be dealt with on another occasion.

A distinction needs to be made between correcting spoken and written work. When speaking, mistakes often do not matter if people can understand what you mean. Unfortunately this is not the case when you are writing. Your message must be clear and understandable first time. Outside the classroom, written work with mistakes tends to be frowned upon and also in exams. Research shows that up to 20% of written mistakes can be identified and corrected by the writers themselves. A basic re-viewing check list can be produced for each student on an individual basis which can then be used for reference. Students need to be actively involved in the process of dealing with these problems as this will engage them intellectually, induce a more co-operative atmosphere and promote learner independence.

How can error be dealt with and learner independence promoted at the same time? By providing the students with an opportunity to self-correct on an organised basis and by encouraging them to direct you to the problems they would like help with in their written work.

Too often students assume it is their task to write and the teacher's to evaluate but this is a disempowering process. Reading students work involves responding to the text as a reader and not simply as a marker. Conferencing is a procedure in which the teacher and writer work together, motivated by a concern with clarifying the writer's intentions. This can be combined with various forms of self-evaluation, in which students annotate their own texts at points where they would like the teacher's advice. A code can even be developed for the process.

The approach recommended for correcting spoken work is to make a note of the errors that crop up during an activity on an OHT. This can then be flashed up on the board/screen at the end of the session and the learners can be given an opportunity to self correct, to point out what is wrong and also why. Examples of effective language can also be noted and praised instead of simply focusing on the negative aspects during this error correction stage.

It has been suggested that when students reach an Intermediate stage they come to what is known as the 'plateau' level. This is characterised by their frustration at their apparent lack of progress which is often less noticeable than it is at the lower stages.

A change of approach on the part of the teacher can often help them to overcome this hurdle and to start climbing the mountain again. However, it can be argued that there is a plateau to be overcome at every level, in particular with post-Cambridge First Certificate level learners.

Learner involvement is frequently at its best when the student is perplexed and confused, but not yet frustrated. "Positive dissatisfaction" can be used to engage learners at peak levels of motivation and understanding. You can purposely put participants into a state of controlled frustration in order to develop better quality thinking, patience and mental toughness.

For this reason the use of shock tactics to start off with can be helpful. Instead of attempting a conventional presentation, you might like to consider making use of a 'How much do you know about...?' type questionnaire, like the examples presented below. The students' embarrassment at their poor results will act as a spur to further study and help them to overcome their initial complacency. Moreover, as they entail problem-solving, they provide an ideal way of catering for the logical-mathematical Intelligence type:

HOW MUCH DO YOU KNOW ABOUT IDIOMS? Decide whether the following statements are true or false and give reasons for the choices you make:

1. All phrasal verbs are idioms and all idioms are phrasal verbs.
2. Idioms are a separate part of the language you can choose either to use or omit.
3. Idioms can be defined as colloquial expressions.
4. Idioms are fixed expressions that cannnot be changed.
5. Words with grammatical functions can be used idiomatically.
6. All proverbs are idioms.
7. All proverbs are sayings.
8. You are more likely to find idioms in quality newspapers than in the tabloids or popular press.
9. All idioms, by definition, are cliches.
10.One of the problems for language learners is that idioms tend to be culturally bound.

All the answers are false except for numbers five, seven and ten. To find out the reasons, take a look at the explanations below:

1. False. All phrasal verbs are idioms but not all idioms are phrasal verbs. Certain idioms contain no verbs. You appeared "out of the blue" and caught me by surprise, for example.
2. False. They form an essential part of the general vocabulary of English and "I hope you get the point and see what I mean" to give you two examples!
3. False. They can appear in formal style and in slang, in poetry or in the language of Shakespeare and the Bible.
4. False. Not all idioms are fixed. Sometimes the tense of the verb can be changed: "I'm going to have forty winks" or "I had forty winks" And sometimes the adjective can be varied: "You'd better keep a careful/close/sharp/watchful eye on her"
5. True. An example is the use of SHALL to offer help or to make a suggestion: "Shall I carry the bag for you?" or "Shall we have an early night for a change?"
6. False. All proverbs can be used idiomatically but a number of proverbs are also statements of fact and easily understood in their literal meaning. For example: "If at first you don't succeed, try, try, try again"
7. True. In the "Advanced Learner's Dictionary" a proverb is defined as "a short well-known sentence that states a general truth about life or gives advice" and a saying is defined as "a well-known phrase, expression or proverb"
8. False. You are far more likely to find idioms in the tabloids.
9. False. According to the "Advanced Learner's Dictionary" a cliche is a phrase "which is used so often that it is no longer interesting, effective or relevant" If this definition could be applied to all idioms, there would be little point in teaching them!
10. True. A good example of this is "as cool as a cucumber" The connection between cool and cucumber is far from obvious. However, every native speaker will naturally put the two words together. In English we say "as stubborn as a mule" whereas in Turkish the association is "as stubborn as a pig" In English we say "as strong as an ox" whereas in Dutch the association is "as strong as a bear"

HOW MUCH DO YOU KNOW ABOUT THE GERUND AND THE INFINITIVE?!?!?!

1. TO is always followed by the infinitive of a verb.
2. MAKE is the only verb that is followed by an object and the plain infinitive.
3. Modal verbs are always followed by the plain infinitive.
4. The verb TO START can be followed by the gerund or the TO infinitive but with a change of meaning.
5. The gerund can be used as the subject of a sentence but not the TO infinitive.
6. Verbs expressing likes and dislikes can be followed by the gerund or the TO infinitive without any change of meaning.
7. The following sentence is gramatically incorrect: "I suggest that she buy an English-English dictionary."
8. The following sentences both have the same meaning: "He helped me carry the luggage" and "He helped me to carry the luggage."
9. The verb LET has no passive form.
10. Both the gerund and the infinitive are defective because they have no perfect forms to talk about actions earlier in time.

All the statements are false except for number eight. To find out the reasons, take a look at the explanations below:

1. False. "I'm looking forward to meeting you" and "He confessed to murdering his mother-in-law" are two of many examples when TO is followed by the gerund.
2. False. LET is also followed by an object and the plain infinitive: My parents never used to let me smoke.
3. False. OUGHT is followed by the TO infinitive: You ought to buy yourself an English-English dictionary.
4. False. The verb TO START can be followed by the gerund or the TO infinitive and the two forms are interchangeable: When did you start studying/to study English?
5. False. The gerund is more commonly used as the subject of a sentence but the TO infinitve can also be used: To be or not to be, that is the question.
6. False. Verbs expressing likes and dislikes are followed by the gerund when talking in general and followed by the TO infinitive to refer to particular events: Do you like dancing? Would you like to dance with me?
7. False. The verb TO SUGGEST can be followed by the gerund or a THAT clause: The guide suggested visiting the Tower of London. The guide suggested (that) the tourist (should) visit the Tower of London.
8. True. HELP can be followed by an object and the plain or the TO infinitive without any change of meaning.
9. False. "The prisoners were let out of their cells twice a day" is an example of the verb used in the passive. However, when LET SOMEONE DO is used to express permission there is no passive form of the verb: My boss let me go home early.
10. False."Joan of Arc is said to have been a witch" and "The suspect confessed to having committed the crime" are examples of the perfect forms.