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.
Monday, 1 May 2017
Session 5: Number Sense
This was an important session in that it confirmed so much of what I consider best practice in mathematics education. The observations in the earlier section that low achievers over-rely on counting resonates with my experience. the over-arching idea here is flexibility in approach to calculation and that one size does not fit all. The multiple ways of approaching a calculation are important for students to see and give essential validation to individual students' methods. However, not all strategies are equal and I disagree with Jo's dismissal of the separation of conceptual and procedural. Whilst the separation may be artificial from a teaching point of view, from an assessment perspective, identifying errors that are conceptual or procedural is important in gauging the students' level of understanding.
I was interested in the William Thurston quote regarding compression and that without conceptual learning such compression was impossible! The examples of number talks shown in the video clips often gave examples of the teacher framing the calculation talks within deeper concepts, such as the rules of arithmetic and seeking to explain a problem algebraically.
For ISZL, much of this session should serve as a positive reinforcement of what we already do. I have recently seen examples of number talks in G1 and G4 that were very productive and pivotal in developing the flexibility we should aim for in approaches to calculation. Our students learn a repertoire of mental strategies and written methods but we should not be complacent. How consistent is our approach across grades for instance? My recent foray into G7 was interesting in that so many of the students showed evidence of the kind of compression that Thurston talks about; they had assimilated strategies, number sense, knowledge of operations etc and were applying them to a range of problems. What was significant to me was the variety of calculation methods successfully used.
A key feature of the examples of number talks in the session was the teacher modelling the spoken strategy using standard notation, and checking with the student that the recording was an accurate representation of their method.
The visual models employed were sometimes effective but at times I didn't think they accurately represented the strategy discussed, however, the message here is important: visual models are an essential component of helping students develop number sense...and we need to use a wide variety.
This was an important session in that it confirmed so much of what I consider best practice in mathematics education. The observations in the earlier section that low achievers over-rely on counting resonates with my experience. the over-arching idea here is flexibility in approach to calculation and that one size does not fit all. The multiple ways of approaching a calculation are important for students to see and give essential validation to individual students' methods. However, not all strategies are equal and I disagree with Jo's dismissal of the separation of conceptual and procedural. Whilst the separation may be artificial from a teaching point of view, from an assessment perspective, identifying errors that are conceptual or procedural is important in gauging the students' level of understanding.
I was interested in the William Thurston quote regarding compression and that without conceptual learning such compression was impossible! The examples of number talks shown in the video clips often gave examples of the teacher framing the calculation talks within deeper concepts, such as the rules of arithmetic and seeking to explain a problem algebraically.
For ISZL, much of this session should serve as a positive reinforcement of what we already do. I have recently seen examples of number talks in G1 and G4 that were very productive and pivotal in developing the flexibility we should aim for in approaches to calculation. Our students learn a repertoire of mental strategies and written methods but we should not be complacent. How consistent is our approach across grades for instance? My recent foray into G7 was interesting in that so many of the students showed evidence of the kind of compression that Thurston talks about; they had assimilated strategies, number sense, knowledge of operations etc and were applying them to a range of problems. What was significant to me was the variety of calculation methods successfully used.
A key feature of the examples of number talks in the session was the teacher modelling the spoken strategy using standard notation, and checking with the student that the recording was an accurate representation of their method.
The visual models employed were sometimes effective but at times I didn't think they accurately represented the strategy discussed, however, the message here is important: visual models are an essential component of helping students develop number sense...and we need to use a wide variety.
Number Talks - Session 5
Number talks have great potential in our primary maths
lessons. There are many benefits in doing number talks including: students work
in partners or groups of 3 to share their thinking and to problem solve, which
allows students to feel more comfortable in taking risks and sharing ideas; more students have
the opportunity to communicate their understanding; after students come
together and share with the class, they can see that there are many ways to
solve a problem; they get many opportunities to ‘play’ with numbers.
Students learn more about number sense. They can begin to see
maths as a flexible subject and to
start using and seeing numbers flexibly. They begin to realize and understand
that there is not only one way to find an answer. Making mistakes is okay and
encouraged – we can learn from our mistakes. The answer is not as important as thinking about and sharing the methods
that they used. I still have students that think ‘faster is smarter’. Doing
these number talks teaches students that there are more ways to show how to be
a good mathematician.
Number talks work well because they teach number fluency and
automaticity at the same time as
teaching a conceptual understanding of number. It was also emphasized in
session 5 how important it is for students to see visual representations as
often as possible when learning mathematical concepts.
Sunday, 23 April 2017
Session 5
This session has reall relevance for us in the primary school. I think the idea of the Number Talk has great potential for us. The number talks provide students with ways to share their thinking,
clarify their thinking and create a visual image to show their thinking.
All these come together to help students come to a deeper understanding
of number particularly and Maths generally. Focusing on just one problem allows for the depth of thinking required to enagage with a problem at a conceptual level rather than a procedural level.
I know many teachers already use something similar when sharing student work but maybe not in a deliberate way. Maybe having a name for this pedagogical strategy would be helpful. I'm not sure I have seen the strategy go as deeply as creating the visual to show student thinking and I think this is an area we could explore more. I really think there is potential here for challenging some of our more able students.
I would like to see teachers use number talks more in classes and hope that through discussion, collaboration and sharing of ideas this can happen- maybe sharing one of the videos in a grade level meeting would be helpful?
The session once again makes me consider the role of the teacher in Maths teaching- putting forward a problem and teasing out the different ways of looking at it from different students rather than teaching a strategy and practicing it on differnt problems.
As Joy noted in her post-I think we do have to think about the anount of sitting and listening students do and this will be the challenge for the teacher-engaging all students a all times. I think in Primary the number talks would benefit from more 'turn and talk' time or working together to create the visual representation of their thinking.
I like the idea of bar-modelling for thinking about problems- this is an inetersting blog post with more details: https://thisismyclassroom.wordpress.com/2014/03/31/throwing-out-that-old-rucsac/
I know many teachers already use something similar when sharing student work but maybe not in a deliberate way. Maybe having a name for this pedagogical strategy would be helpful. I'm not sure I have seen the strategy go as deeply as creating the visual to show student thinking and I think this is an area we could explore more. I really think there is potential here for challenging some of our more able students.
I would like to see teachers use number talks more in classes and hope that through discussion, collaboration and sharing of ideas this can happen- maybe sharing one of the videos in a grade level meeting would be helpful?
The session once again makes me consider the role of the teacher in Maths teaching- putting forward a problem and teasing out the different ways of looking at it from different students rather than teaching a strategy and practicing it on differnt problems.
As Joy noted in her post-I think we do have to think about the anount of sitting and listening students do and this will be the challenge for the teacher-engaging all students a all times. I think in Primary the number talks would benefit from more 'turn and talk' time or working together to create the visual representation of their thinking.
I like the idea of bar-modelling for thinking about problems- this is an inetersting blog post with more details: https://thisismyclassroom.wordpress.com/2014/03/31/throwing-out-that-old-rucsac/
Sunday, 26 March 2017
Unit 6: Relationships and process
I think two big ideas are highlighted during the course came out strongly in the interviews in this unit. One is the confidence to tackle complex open-ended problems. The other is developing a culture where students are focused on sharing thinking. They are comfortable learning through mistakes.
I think simply giving a problem that students don't know how to solve to a class that is not used to problem-solving leads to frustrated students and teachers. You have to first develop a culture where students become as interested in discussing the maths as in getting the correct answer. As I mentioned in my last post, I've started using dot-cards as a way to develop confidence and discussion, and help students build that all important positive relationship with maths, which is a focus in unit 6.
The unit also talks about using a mathematical process, and making this process explicit to students. This fits in with many other areas - the Writing Process, the Inquiry Cycle and Design Thinking (which is a process designed to develop innovation and problem-solving skills). It would be great if students are able to make connections and comparisons between processes in different subject areas as they grow through the school. I think the exact wording of such processes can vary, but the understanding that deep learning happens, problems are solved and new ideas are generated when we spend time immersed in the work and approach it in a variety of ways as we move through different stages is fundamental. It fits in with the problem-solving 'way of life' and a growth mindset culture.
Unit 5: Dot cards and maths talks
The unit encourages teachers to create an environment where students can find their own ways to solve problems and then share their strategies with others. A good starting point for this in Grade 1 is using dot-cards, where students explain how they know how many dots are on a card without counting them individually. This involves them seeing combinations of dots. Students share different ways they combined the dots to find the total. Some ready-made print outs of these dot cards taken from John Van de Walle's book can be found at http://www.mathcoachscorner.com/2013/07/using-dot-cards-to-build-number-sense/
Currently we're enjoying discussing one dot-card a day in 1T, and students are getting better at using different strategies such as seeing doubles e.g. 7+7 and finding ways to make ten e.g. 5+6 = 5+5+1 = 10+1 =11
Going beyond the dot cards, students can be given numerical problems to solve and then discuss their solutions and how they arrived at them.
The teacher's role is to draw out student thinking and make it visible to others, both algebrically and visually. This may involve 'rewinding' several times until they have understood the student's thinking. Teachers should focus particularly on incorrrect answers, allowing students the opportunity to talk through their work in the hope they will see their mistake. They should also encourage students to see similaries between the different strategies.
One question I'm wondering about is the optimal time ratio between students working alone to solve problems versus time spent listening to class mates share solutions, which falls into the 'sitting and listening' category (apart from when they are sharing their own ideas.)
Currently we're enjoying discussing one dot-card a day in 1T, and students are getting better at using different strategies such as seeing doubles e.g. 7+7 and finding ways to make ten e.g. 5+6 = 5+5+1 = 10+1 =11
Going beyond the dot cards, students can be given numerical problems to solve and then discuss their solutions and how they arrived at them.
The teacher's role is to draw out student thinking and make it visible to others, both algebrically and visually. This may involve 'rewinding' several times until they have understood the student's thinking. Teachers should focus particularly on incorrrect answers, allowing students the opportunity to talk through their work in the hope they will see their mistake. They should also encourage students to see similaries between the different strategies.
One question I'm wondering about is the optimal time ratio between students working alone to solve problems versus time spent listening to class mates share solutions, which falls into the 'sitting and listening' category (apart from when they are sharing their own ideas.)
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