This post provides a few practical tips on how to develop fundamental abilities (i.e. the first level of the Maths ability pyramid), thus helping students to become more confident by increasing their awareness and fluency with the mental manipulation of objects and processes such as order, numbers, causes and consequences.
There is a double benefit in working on this development: not only does it help teenagers to focus and develop mental resources, but it does so by involving them in a series of lively exercises that look very much like collective games with relatively little Maths involved. In other words, developing fundamental abilities is both low-cost and high-benefit.
To emphasize this point, one should remember that the stumbling block for many Maths students is the how-to, or the strategy. When faced with a typical Maths activity, they must first understand what to do, and then know how to do it. Students that are intimidated by these stumbling blocks will thrive in fundamental activities because these activities are repetitive and immediate.
The exercises I describe below are among the easiest to introduce to a group of students, but needless to say that teachers can imagine new exercises depending on the needs they identify in the class: there is virtually no limit.
Some of these exercises are oral, some written. If oral, you usually use cold calling, which adds pace and thrill. If written, students usually use mini-whiteboards, or exceptionally paper if you want to collect and assess results. They are best done at the beginning of a class, or at the end, or to gather attention before a whole-class moment. They are usually done with students seated, but some teachers prefer a standing position, this is up to you.
These exercises are divided into 4 categories:
- Attention / instant memory
- Mental maths, quick calculations
- Mental images
Attention / instant memory :
For many students, working on the development of fundamental abilities means first and foremost settling down. This is really the first stepping stone to reach for, the one stage that conditions much further development. That’s why it is so important to start with this first category of games.
- Write a number on a card, show the card, then hide it, ask students to write the number. Do this a second time, then ask students to write the first number, etc.
- Give students a sequence of numbers. Then (orally or in writing) ask students to repeat/write the sequence with specific instructions, for example: forward, backward, every two numbers, only the last three numbers, only even numbers, only primes, order the numbers in ascending / descending order, etc.
- Developing collective concentration : counting Blobs and Glups
- Instruction: I will ask you a question and you will answer with a number between 1 and 9 (or 1 and 20, or an odd number between 10 and 29, or an integer between -10 and 10, etc.)
- To Student 1: how many Blobs ?
- To Student 2: how many Glups ?
- Same to Students 3 and 4, so you get 2 numbers of Blobs and 2 numbers of Glups
- To all students : add ! (or subtract)
Mental maths, quick calculations:
For the importance of regular Mental Maths practice in the development of a number foundation, see my earlier post: why is it so important for students to have a sound number foundation ?
- Times tables, but more interestingly multiplication/division facts (yes, for one multiplication fact, you get 4 multiplication/division facts, don’t you ? so all these facts have to be practised inside out)
- Times table chain: this is probably the best way to learn/revise multiplication facts, as it also develops collective concentration. It works like this:
- Divide your class by rows or columns. Say, for example, that you have 5 rows of desks that represent the x3, x4, x5, x6 and x7 tables
- Start with a multiplication, for example 6×9, ask answer to first student of row 4; result is 54
- The next student to answer (which can be determined either sequentially, or through cold call, or any other order) has to take the unit digit (4) and multiply it with his own times table, etc.
- Mental Maths strategies (bridging, compensating, etc.)
- Countdown, from simple to complex
- Collective ‘Guess my number’: 3 students give 3 different clues, from which all students find the possible solutions. For example:
- I’m a multiple of 3
- I’m a factor of 48
- I’m higher than 20
Getting students to work with mental images is important, but should be introduced very gradually. Using the same activities (instant memory, mental maths), three levels are possible.
- Basic level: the idea is to do it ‘all in your head’, delaying writing until the very last moment. However, not all students are confident with working mentally, so there may be a feeling of getting lost, that is why the second level is often necessary.
- Awareness level: there again, students do it all ‘in their head’, but they receive explicit instructions to form mental images of the numbers they are playing with. This is useful for students as the game operates a bit more slowly and it makes them more aware of what is happening in their mental space.
- Deep level: this consists in consciously relating mental images to specific parts, sides or movements of the body. This approach should be used with caution, but the potential benefits are enormous, particularly for SEN students.
This category of exercises aims at getting students to practise basic deduction based on clues and hypotheses.
- Quick logic, using braintraining puzzles
- Causes and consequences: give a sequence of 4-5 words, ask students to write them in a logical order
- More advanced logical puzzles, involving combinations of hypotheses
Referring again to the Maths ability pyramid, please note the following additional benefit of these activities: implicit learning.
Implicit learning is an intermediate learning level located in between the first layer (fundamental abilities) and the second layer (skills) of the pyramid. As indicated by the word ‘implicit’, it consists in learning something totally new without being aware that you’re actually learning. It’s a way to remove stumbling blocks, for example as a preliminary to algebra.
There are two examples of this implicit learning in the activities described above:
- Counting Blobs and Glups is an implicit introduction to using variables
- With number chains, students implicit learn to process functions (input, output)