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Conjecturing and Generalising is part of our Developing Mathematical Thinking Primary and Secondary collections. This page accompanies the Conjecturing and Generalising Primary and Secondary resources.
This follows on from Exploring and Noticing, and Working Systematically.
You may wish to watch the recording of the webinars, which draw on the resources below to discuss how teachers can offer students opportunities to conjecture and generalise.
Mathematicians are not only interested in answering questions, they also pose questions, make conjectures and test ideas.
Assuming students have been given the opportunity to explore a context, have noticed patterns and possible structures, and have worked systematically to generate more examples, they may then be in a position to conjecture, expressing ideas about what they think will happen more generally:
"I think ... will always happen."
"I think this method will always work."
"I think this is not just a special case."
"I think, if I change ...., then .... will happen."
Often, students go from question to question to question, rather than going from question to question to general. Moving from the particular to the general is a key mathematical process, and if we are to offer students opportunities to use specific examples as a springboard to generalising, then we may need to think about the following:
“The aim is to reduce the pressure on ‘being correct’... when a learner offers a response to a question, try to catch yourself before you declare whether it is right or wrong; praise it as a conjecture, and invite others to consider whether they agree with it, or whether they would like to suggest a modification or a counter-example. In this way you can be responsible for the process of thinking, while the learners work together to decide correctness." (John Mason)
The teacher models the questions and comments a mathematician might make and, over time, encourages students to ask these questions themselves ('scaffolding and fading'):
We're assuming students have found all the Magic Vs using the numbers 1-5. If you haven't had a go, you may like to try to do so now.
There are only three solutions:
Let's call the total of the three numbers in one arm the 'magic total'.
The magic totals of the Vs above are (from left to right) 8, 9 and 10.
This may lead to these initial conjectures about magic Vs:
I think...
Further testing may lead to these modified conjectures:
We're assuming that students have spent time exploring ways of expressing the numbers 9 to 15 as the sum of consecutive numbers. If you haven't had a go, you may like to try to do so now.
9 = 2 + 3 + 4
10 = 1 + 2 + 3 + 4
11 = 5 + 6
12 = 3 + 4+ 5
13 = 6 + 7
14 = 2 + 3 + 4 + 5
15 = 4 + 5 + 6
This may lead to these initial conjectures:
I think...
Further testing may lead to these modified conjectures:
Further exploration could lead to conjectures about multiples of 5, 6, 7...
We're assuming students have played Got It or Have You Got It? and have worked out a strategy for the default setting (target of 23, using the numbers 1 to 4). If you haven't already played, you may like to do so now.
This may lead to these initial conjectures:
I think...
Further testing may lead to these modified conjectures:
Assume that students have been exploring Keep it Simple, and in particular, how to express $\frac{1}{6}$, $\frac{1}{7}$, and $\frac{1}{8}$, as the sum of two different unit fractions (fractions which have numerators of 1). If you haven't had a go, you may like to try to do so now.
$\frac{1}{6} = \frac{1}{9} + \frac{1}{18}$
$\frac{1}{7} = \frac{1}{8} + \frac{1}{56}$
$\frac{1}{8} = \frac{1}{10} + \frac{1}{40}$
This may lead to these initial conjectures:
I think...
Further testing may lead to these modified conjectures:
We're assuming students working on Tilted Squares can find the areas of squares drawn in their 'usual' orientation, and have drawn and found the areas of squares with a 'tilt' of 1. If you haven't worked out the areas of squares with a 'tilt' of 1, you may like to do so now. The Getting Started section offers some advice.
These results may lead to this initial conjecture about the areas of squares with a 'tilt' of 1:
If the base of the square goes:
1 along and 1 up, area = 2
2 along and 1 up, area = 5
3 along and 1 up, area = 10
4 along and 1 up, area = 17
So, I think if the base of the square goes
5 along and 1 up, area = 25 + 1 = 26
6 along and 1 up, area = 36 + 1 = 37
x along and 1 up, area = x$^2$ + 1
Students may draw a few more examples to convince themselves that their conjecture is valid.
This may then lead to a follow-up conjecture about the areas of squares with a 'tilt' of 2:
I think if the base of the square goes x along and 2 up, area = x$^2$ + 2
If you haven't worked out the areas of squares with a 'tilt' of 2, you may like to do so now.
These results may lead to a modified conjecture:
If the base of the square goes:
1 along and 2 up, area = 5
2 along and 2 up, area = 8
3 along and 2 up, area = 13
4 along and 2 up, area = 20
So, I think if the base of the square goes
5 along and 2 up, area = 25 + 4 = 29
6 along and 2 up, area = 36 + 4 = 40
x along and 2 up, area = x$^2$ + 2$^2$
Further exploration can lead to Pythagoras' Theorem!
We're assuming that students have explored Wipeout using numbers 1 to 6, and then different sets of six consecutive numbers, such as 2 to 7, 3 to 8, 15 to 20... If you haven't had a go, you may like to try to do so now.
Starting with 1 to 6, and wiping out either 1 or 6, leaves five numbers with a whole number average
Starting with 2 to 7, and wiping out either 2 or 7, leaves five numbers with a whole number average
Starting with 3 to 8, and wiping out either 3 or 8, leaves five numbers with a whole number average
Starting with 15 to 20, and wiping out either 15 or 20, leaves five numbers with a whole number average
This may lead to these initial conjectures:
I think...
Further testing may lead to these modified conjectures:
We're assuming that students have had a chance to explore Searching for Mean(ing), and in particular, how to combine different weights so that they average 4kg. For example, combining 3kg and 8kg weights, or 3kg and 9kg weights, or 3kg and 10kg weights. If you haven't had a go, you may like to try to do so now.
3kg and 8kg weights
3, 3, 3, 3, 8 average of 4kg
3kg and 9kg weights
3, 3, 3, 3, 3, 9 average of 4kg
3kg and 10kg weights
3, 3, 3, 3, 3, 3, 10 average of 4kg
This may lead to these initial conjectures:
I think...
Further testing may lead to these modified conjectures:
The tasks above are a small sample from our full collection of Primary and Secondary Conjecturing and Generalising resources.
Thinking Mathematically by John Mason with Leone Burton and Kaye Stacey
Learning and Doing Mathematics by John Mason
Developing a Need for Algebra by Alf Coles (for more information about 'common boards')
Podcast and transcript of Dylan Wiliam discussing effective questioning in the classroom - the conversation includes reference to a 'no hands up' approach and creating a community of learners
Thinkers by Chris Bills, Liz Bills, John Mason and Anne Watson