Claire Perry Subject: Maths
'Pure Brain Power'
AIM: Develop pupil logic & thinking – by improving problem solving skills and enabling pupils to explain the process and teach others.
OUTCOME: improved motivation and understanding of problem solving processes (metacognition); improved communication skills in maths; improved maths scores.
Typically, students have great difficulty accessing geometry problems. Instead of a “one algorithm fits all” solution as in solving equations, or number problems, in geometrical problems it is not enough to substitute the given data into some formula – students must apply and combine the known theorems. Typically, there are several stages to any geometry question at GCSE and students are often reluctant to progress through a problem where the result is not immediate.
In addition, proof is increasingly rearing its head at Maths GCSE Higher level, but within a variety of different contexts, making the skills of proof a priority over the individual subject knowledge.
Both geometrical reasoning and proof rely heavily on logic- and yet typically we assume that the pupils are logical - without ever directly teaching it. This project, I hoped, would enable students to share and practise logical reasoning to unravel a complex weave of information and deduce truth. Ultimately, I hoped that these skills would transfer quite directly into students approaches to geometrical problems.
As a year 9 group, just finished with their SATS, I also hoped this project would provide year 9’s with challenge and intellectual success within a fun and exciting environment, yet simultaneously providing them with valuable higher order critical thinking skills required at GCSE.
The umbrella of this
project was “Logic Problems” - A logical puzzle is one that can
be solved by pure brain power –without any external knowledge.
Over a 6-lesson period, students were:
-Introduced to two different types of logic puzzles: Griddlers and Logic Problems
-Given time to familiarise themselves with the puzzles individually and independently
-Invited to work in groups of 3 to convince one another of ultimate truths and untruths – giving them an opportunity to share reasoning and logical progression.
-Challenged to write instructions for their individual puzzle, for students not yet introduced to this particular logic problem
- Invited to evaluate the instructions of one another and suggest improvements
Moving on from this mini-topic on logic, the group has since looked into Circle Theorems, a GCSE topic on geometry, typically not taught until year 11, and have progressed to proof of these theorems at an A-grade level.
I hoped students
would find the challenge of these complex problems, requiring a continual derivation
and cross-referencing of truths and untruths, exciting – and upon experiencing
success with these types of problems, would recognise the value in taking small
steps to achieve the bigger picture. Indeed, the students were enthralled by
the puzzles, finding them:
“Challenging and team building. Gripping” – Lewis Osbourne
“Interesting” Edward Lucas
“Challenging and you have to think about it” – Joe Butler
“A bit hard” Alfie Kelly
“Puzzling and yet interesting. Its exciting” Dilir Habibi
“Its fun and challenging. You have to finish it” Lukas Kulaitis
“its cool, its cool” Eric Wong
“Baffling” Martin Wilson, Mohammed Sayed
“Challenging & hard” Hussein Omar
Whilst the students enjoyed working individually on these puzzles, their reasoning to one another developed throughout the project – enabling them by the end to seamlessly explain their logical progression to establish a truth or untruth.
Prior to the project,
students had completed a geometry problem, requiring application of several
angle facts. One student had achieved the full 4 marks, but otherwise the class
had performed poorly – over half the class had not even attempted the
question, whilst those that did failed to show their understanding or reasoning.
As a class, the total mark scored: 20/92
On completion of the logic project, but with no additional work on angles (and in fact at least a month on from the initial baseline test), the class trebled their result and scored 60/92
Progressing on from the logic topic, the students have approached the topic of Circle Theorems with excitement and vigour. Their reasoning has been thorough throughout. The project has finished with each student working on the proof of a theorem and the majority of the boys have produced animated power-points to demonstrate the progression of their proof from first principles – only required from students at A grade or above on the Higher GCSE paper.
The timing of the project enabled myself and the students to break away from the constraints of the typical curriculum, providing greater variety for pupils. It enabled the teaching and learning of skills critical to problem solving, geometry and proof within a fun and challenging environment.
Students are more comfortable approaching a larger problem where an answer is not immediate and confidently communicate their progress towards an answer – a skill increasingly required of students at GCSE level.
The intellectual success experienced by most students has renewed the enthusiasm for maths after the stress and hard work of SATS
The project enabled me to teach a skill as opposed to a specific subject and allowed me to recognise the importance of talking to students about their approach to more complex problems.
Outcomes - Pupil Learning
The principle that any logic problem must be solved by logical argument alone has enabled the students to recognise their ’leaps of faith and avoid jumping to conclusions and making unwarranted assumptions.
“When I looked
at this before, I assumed b (the angle) was 45° because it looked like half
of the 90° rectangle angle, and then because x and b looked the same, I
said x was 45.
But this time, I tried to think more logically. I worked it out bit by bit. I didn’t even try to think about it more logically, I just noticed afterwards that my mind was working automatically” Lewis Osbourne
“I have learnt how to focus better on my work” Mustafa Hassan
“To carefully assess instructions”, Imran Badshah
“To get my point across to my team and get them to understand not just myself”, Lewis Osborne
“To think carefully and look at all sides of a puzzle…It has also been challenging and I have learnt to think more strategically upon starting problems and puzzles” Dilir Habibi
“I learnt that you need patience to success and it has improved my team communication skills”, Amir Shahin
“How to cross reference” Joe Butler
“To break a problem into smaller parts ….When I completed the challenge I got a strong sense of achievement which makes me feel proud of myself” Thomas Spurway
“To assess all the information before doing anything” Lukas Kuleitas
“It boosted my communication skills” Alfie Kelly
“I was able to discover new ways of approaching problems” Majid Noor
Outcomes - Adult Learning
Throughout this topic I was given insight in how group work can develop students powers of reasoning – by explaining the progression of a problem to one another, they had to clearly formulate, order and articulate their ideas.
It was interesting for me to plan a topic around a thinking skill (logic) as opposed to a topic on a particular area of subject knowledge. Talking to students about their approach to complex problems highlighted that students are often comfortable making assumptions where this is absolutely against the principles of maths and ultimate truth.
I have been amazed at how the development of the students thinking has given them access to higher level subject specific work, simply because they are more comfortable working on a complex problem via a series of small deductions. Through this work on logic, students have realised a sense of achievement from hard and often convoluted work to achieve an ultimate truth.
As a department,
we have introduced similar logic puzzles (Suduko) to several of our classes
– enabling them to practise logical thinking in a fun environment.
To further this, I would like to encourage the department to consider turning these ‘one-off’ lessons into more whole topics – to highlight the importance of:
· Logical progression
· Persuasion of logical truth and untruth
· Problems with assumptions
· Ability to achieve full focus
· Ability to work in small steps to achieve the bigger picture
The idea of using
group work to formulate, order and articulate ideas and logical progression
could be transferred to several topics:
· DT – writing up product instructions
· Food Technology – food timelines
· English & History – providing evidence to substantiate claims
In Conclusion – a personal response to the whole project –
I started this project thinking time would be a constraint – concerned that my energies would be stretched. However, it has been delightful working with such a cohesive group of students and seeing such excitement in them as well intellectual progression in critical thinking. The project has reaped rewards: Personal – students are still engaging their brains in these puzzles; Intellectual – their progress in proof and geometry has been astounding).
I feel that my energies have been well spent, and well rewarded.
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