Imagination takes different forms for different people. In our minds, we can form and manipulate images of past, future or purely abstract objects and occurrences. In The Mind Teaches the Brain, Gattegno describes the importance of imagination:
Work with the imagination may be one of the most important aspects of mathematics education. But how often are students allowed time to imagine? If a student is sitting looking into space, it is often assumed they are doing nothing. I have often heard teachers advise seemingly 'inactive' or 'stuck' students to 'do something', meaning to write something down. But this might be counter-productive when solving problems. In his book Creative Thinking, JG Bennett suggests putting pen to paper may inhibit creativity, the ability to think of something new:
Bearing this in mind (!), try to solve the problem below without putting ever pen to paper, even if you are very tempted! [If possible, you may also find it interesting to notice how you form and manipulate images in your mind] Three fifths or four sevenths? Continue the pattern... could this help you decide? Is this approach always possible? In Being Peace, Thich Nhat Hanh asks:
Before I arrived, the river was undifferentiated energy, like myself. I come near to the river and dwell. In Conversation on a Country Path, Heidegger describes thinking as a ‘letting-oneself-into-nearness’.I watch the water flow along its path, I listen to the gentle bubbling sound. In this act of attention, there is differentiation: between the river and its surroundings, and between the river and me. But there is also an attraction, a sense of wholeness. In The Principles of Psychology, William James states:…each of us literally chooses, by his ways of attending to things, what sort of a universe he shall appear to himself to inhabit. A river is never the same, but we consider it the same if it follows a recognisable path. With recognition comes identity. We may gain our identities in a similar way, as described by James: A man has as many social selves as there are individuals who recognise him. Can we be a river and experience the hopes and fears of a river? Does the river have hopes and fears, a sense of self? From where do we derive our sense of self?William James says ‘the stream of our thought is like a river’, and describes consciousness as:
Thoughts flow past, but sometimes we bring our attention to the flow itself. This may be regarded as stepping out of the flow of time and becoming aware of time itself. We cannot directly experience the past or future, only the present. But even the present is elusive, as James describes:
He suggests that the present moment is an abstraction, and that we experience the present not as a moment, but as an extended period of time which he calls the ‘ specious present’:
The brain-processes that create this feeling of succession are those that give our sense of time, and ultimately our sense of a continuing self:
Assuming that the pattern continues 'as shown', and that there must be at least one red in a line, find the minimum and maximum possible fractions of white and red.Same question for this pattern: Explore similar patterns of your own. Can you predict the minimum and maximum fractions for any pattern? In my excitement about this upcoming ATM book, I've been playing with Cuisenaire rods.
Each diagram below shows three terms taken from a sequence (click on the image to see more). The question is: How would *you* extend the sequence (in both directions) in each case?[Do any problems arise with your way of extending each sequence?] May I invite you to watch this video, which shows that 13 can be written as 2^2 + 3^2, which is the sum of two consecutive Fibonacci numbers squared ('Fibonacci squares'). Q1. Which other terms of the Fibonacci sequence can be written as the sum of two consecutive Fibonacci squares? [why?] Q2. Show that the rest of the Fibonacci sequence can be written as the difference of two Fibonacci squares... [and deduce that these terms are also multiples of some Fibonacci number (apart from 1)].Q3. Show that difference of two consecutive Fibonacci squares can be written as the product of the two Fibonacci numbers on either side (of the consecutive ones).For example, 8^2 - 5^2 = 3 x 13. Q4. Here's an interesting pattern I discovered after playing around with Cuisenaire rods: 8^2 + 2^2 = 2.(3^2 + 5^2) 13^2 + 3^2 = 2.(5^2 + 8^2) ... Can you find any other interesting patterns like this? |