Monday, 12 June 2017
Unit 8: A summary
- Create a classroom environment where students are comfortable making mistakes and willing to persevere with problems.
- Make Math Talks an integral part of teaching, where students explain their thinking and listen to each other.
- Give students deep, broad, open-ended problems that they can't solve instantly.
- Encourage them to find out what they need to know to solve problems.
- Make maths fun. Help students think of the maths problems as puzzles.
- Encourage students to look for patterns and generalisations, and develop their skills in describing them.
- Encourage students to play with math. If they've found a solution, encourage them to change the problem in different ways to give themselves a challenge.
- Encourage students to find more than one way of solving a problem.
- Emphasise the process, not the answers.
- Show the relevance of math in everyday life
Unit 7: Algebra
Here's a summary of the key points in this unit:
Algebra provides a language for describing patterns, is a tool for solving problems and develops abstract thinking. There are now many jobs that need data analysts as the internet provides ever more data, and it's important our young people develop the skills needed for this. Algebra is one tool that can help make sense of data.
Traditionally students have been asked to 'Find the value of x' when learning algebra, where x is one number. This is known as procedural algebra. The course advocates that the focus should shift to students developing equations that generalise situations, where n can represent any number. This is known as structural algebra, and students often find it difficult to move from procedural algebra to structural algebra.
To help students make this shift, it's important to spent time in Kindergarten making, describing and predicting patterns. The emphasis is on describing the general form.
Helping students visualise what is happening helps them develop their own equations and understand why the equations work. Starting with patterns in KG is a good way of developing visualisation skills. As students become confident with playing with patterns, they are able to predict what the nth case will look like, and from there write an equation to generalise what is happening.
Students often have a misunderstanding of the equals sign. It's important to develop the idea that what is on either side of an equals sign must be equivalent.
Research also suggests that using the first letter of something to represent it in an equation often leads to misunderstandings. e.g. a=number of apples, not a=apples. To avoid this misunderstanding it's better to use a different letter. e.g. n=number of apples.
How a problem is presented to the students is important. If we want students to work out the pattern, we should not give them the materials in a form they can simply count them to find the answer. Briefly show a visual or restrict the quantity of materials available so students are forced to think in an abstract manner to generalise the pattern.
Maths talks are important in algebra. Allow students time to think about the problem, discuss it with peers and share their thinking with the class. Develop a growth mindset culture and use mistakes to highlight misunderstandings and as learning opportunities.
Always encourage students to visualise the problems and use tools to help them do this. The equations will grow out of their visualisation of the problem.
Making Patterns - an idea
I wonder if a progression from the ABAB type linear pattern would be to give students some blank templates for rotational symmetry, and create a pattern along one line. Students would then predict how many pattern shapes they would need to replicate the pattern along each of the lines indicated. The templates could be for various different orders e.g. 3, 4, 5, 6, 7 and 8 lines.
I think these have the potential to build up into fairly complex patterns, especially if differently coloured 1 cm cubes were used.
Algebra provides a language for describing patterns, is a tool for solving problems and develops abstract thinking. There are now many jobs that need data analysts as the internet provides ever more data, and it's important our young people develop the skills needed for this. Algebra is one tool that can help make sense of data.
Traditionally students have been asked to 'Find the value of x' when learning algebra, where x is one number. This is known as procedural algebra. The course advocates that the focus should shift to students developing equations that generalise situations, where n can represent any number. This is known as structural algebra, and students often find it difficult to move from procedural algebra to structural algebra.
To help students make this shift, it's important to spent time in Kindergarten making, describing and predicting patterns. The emphasis is on describing the general form.
Helping students visualise what is happening helps them develop their own equations and understand why the equations work. Starting with patterns in KG is a good way of developing visualisation skills. As students become confident with playing with patterns, they are able to predict what the nth case will look like, and from there write an equation to generalise what is happening.
Students often have a misunderstanding of the equals sign. It's important to develop the idea that what is on either side of an equals sign must be equivalent.
Research also suggests that using the first letter of something to represent it in an equation often leads to misunderstandings. e.g. a=number of apples, not a=apples. To avoid this misunderstanding it's better to use a different letter. e.g. n=number of apples.
How a problem is presented to the students is important. If we want students to work out the pattern, we should not give them the materials in a form they can simply count them to find the answer. Briefly show a visual or restrict the quantity of materials available so students are forced to think in an abstract manner to generalise the pattern.
Maths talks are important in algebra. Allow students time to think about the problem, discuss it with peers and share their thinking with the class. Develop a growth mindset culture and use mistakes to highlight misunderstandings and as learning opportunities.
Always encourage students to visualise the problems and use tools to help them do this. The equations will grow out of their visualisation of the problem.
Making Patterns - an idea
I wonder if a progression from the ABAB type linear pattern would be to give students some blank templates for rotational symmetry, and create a pattern along one line. Students would then predict how many pattern shapes they would need to replicate the pattern along each of the lines indicated. The templates could be for various different orders e.g. 3, 4, 5, 6, 7 and 8 lines.
I think these have the potential to build up into fairly complex patterns, especially if differently coloured 1 cm cubes were used.
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