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.)
Thursday, 23 March 2017
Jo Boaler's Latest Ted Talk
Link below. She discussed things we have not covered yet in our course. Two big take-aways for me were the impact of showing students you believe in their abilities and good ways to phrase math questions that are engaging & visual.
https://www.youcubed.org/oxford-tedx-talk/
https://www.youcubed.org/oxford-tedx-talk/
Wednesday, 15 March 2017
Mistakes Are Our Friends!
“Mistakes are our friends!” Is such an important message to share with our learners. Every year, I get on my soap box and share the importance of mistakes in our classroom. It is always nice to hear educators and researchers share information in regards to the brain. Knowing now that not just mistakes , but awareness of mistakes activates and grows the brain is fascinating. It reminds me to not only celebrate when a mistake is made, but also prepare and provide opportunities for students to make mistakes in the classroom.
I also appreciated the importance Carol Dweck and Jo Boaler put on the word, yet. So many students and adults think if something is easy for them, then they are smart. But really they should feel cheated from an experience to grow and learn. One of my students came in the other morning and asked,”Ms. Waring, do you have any challenges for me today?” I would never have done that as a student. I was so worried about getting the right answer and scoring well on my math facts tests. So I do not think we need to educate our young learners about growth mindset. I hope the adults in schools and at home can change their mindsets from how they grew up. Parents and teachers should focus on the hard work and effort put into student work rather than the perfection and score they get.
Sessions 3 and 4 reminded me of the importance of providing opportunities for students to make mistakes, get messy, and grow as learners. We’ll continue to make “challenge the new comfort zone."
Mistakes and the Growth Mindset
Here are some of my favourite bits and pieces from these two modules.
-When your brain is struggling that's the best time for growth.
-We want students to be struggling and getting it wrong at times.
-"I want being challenged to become the new comfort zone, not this is easy."
-"If students are afraid of mistakes, then they're afraid of trying something new, of being creative, of thinking in a different way."
-Mathematical Practice 1 - Make sense of problems and persevere in solving them.
-Encourage students to be sense makers.
-Reward students for experimentation and trying things out, not for correct answers, but for having ideas and being willing to test them out.
-Effort is the secret of life.
-In this class mistakes are expected, inspected and respected.
-highlight mistakes positively in feedback and then get them to correct their own work.
-Don't teach to tiny curriculum standards but to big ideas in maths.
-Covering everything doesn't mean students learn everything.
-When students finish work have them think deeper, ask them to think of other problems similar to those they have been working on and work on those.
-When your brain is struggling that's the best time for growth.
-We want students to be struggling and getting it wrong at times.
-"I want being challenged to become the new comfort zone, not this is easy."
-"If students are afraid of mistakes, then they're afraid of trying something new, of being creative, of thinking in a different way."
-Mathematical Practice 1 - Make sense of problems and persevere in solving them.
-Encourage students to be sense makers.
-Reward students for experimentation and trying things out, not for correct answers, but for having ideas and being willing to test them out.
-Effort is the secret of life.
-In this class mistakes are expected, inspected and respected.
-highlight mistakes positively in feedback and then get them to correct their own work.
-Don't teach to tiny curriculum standards but to big ideas in maths.
-Covering everything doesn't mean students learn everything.
-When students finish work have them think deeper, ask them to think of other problems similar to those they have been working on and work on those.
Tuesday, 14 March 2017
Reflection
"Making mistakes is the most useful thing to be
doing" reaffirms to me the
importance of the children showing their thinking when they are solving a
problem and not just focusing on the answer.
Being able to describe the strategies that they are using confidently to their peers/teachers is an
important skill for children to learn.
Being able to work through their strategy and see where they went wrong
is also very important. Continually
encouraging the children to understand that getting a question wrong doesn't
matter so long as they are confident with the strategy they are using and are
able to see where they made the mistake.
It has made me think about the terminology that I use with
the children. "Not quite yet" is an interesting term that can be used
with the children to encourage them to keep trying and is very positive. The need for me to give children time to correct their work is very important.
Knowing that by developing children's ability to be
confident and persistent problem solvers will really help then in later
life. These are skills that they will
then go on and use to make them successful in their careers.
Jo has really made me think about the type of problems and
investigations that I should be doing with the children. I think it is really important to teach them
the skills they need for good problem solving: Team Work, Communication,
Flexibility and Persistence. Nrich is a great resource for investigations for the children to work on to develop their skills of problem solving.
Finally SLOW DOWN.
Finally SLOW DOWN.
https://nrich.maths.org/
Feeling positive about changes that I am going to make to the way I teach maths in grade 4.
Making Mistakes - Pushing Our Thinking
As Carol Dweck said, math is something you learn. It’s a set of skills. Everyone can get better at math. I’ve always encouraged students to value their mistakes and to understand that this is part of the learning process. I didn’t realize how mistakes can actually make the brain grow. After participating in this session, I would like to go further with my students about how we can look at the mistakes we make - I could share more literature with the children such as, 40 Mistakes that Worked or 11 Science Experiments that Failed to prompt discussion. I would also like to give time for students to share their responses to how they feel when they make mistakes. We can further the discussion by talking about the attitudes that develop.
I’d also like to make sure I give plenty of opportunities for everyone in the class to engage in rich, challenging open tasks to push their thinking and allow them to experience the ‘complexity and messiness’ of real world mathematics. Also - I will reevaluate ‘timed’ fact quizzes or tasks as ‘faster isn’t smarter’ and I do not want students to associate feeling nervous or anxious with completing math.
Mistakes: The Struggle Is Real
Mistakes are powerful. We know this already. Anyone who has ever failed at something in their life would probably agree with this statement. Unfortunately, mistakes also have a way of hanging around for awhile. Depending on the mistake, an individual could feel slightly embarrassed or ashamed, which makes the mistake incredibly hard to forget or overcome.
Furthermore, the impact a mistake has on an individual is directly linked to their mindset and the way they perceive certain events. A person with a fixed mindset may view a mistake as a roadblock.... a warning sign reading 'PROCEED WITH CAUTION'. On the flip side, a person with a growth mindset might view that same mistake as a simple detour; an opportunity to learn something new.
Jo's mention of the 'didactic contract'- the feeling that you want to spoon feed math to students who struggle- was also really interesting to read about. It takes time for a student to work through a challenging problem, and it takes time for them to develop their own understanding of the concept at hand. However, teachers know that time is precious and in order to 'get through' what needs to be done, sometimes it is easier just to show them how, or do it for them.
On a different note.... Jo mentioned that challenging, problem- based mathematics helps to create future entrepreneurs and innovators. This statement made me think about a group of students who I am currently teaching. All very strong mathematicians, and all very interesting in inquiring into industry, innovation, and infrastructure! An interesting personal connection to the session... and I look forward to sharing this information with my students.
Furthermore, the impact a mistake has on an individual is directly linked to their mindset and the way they perceive certain events. A person with a fixed mindset may view a mistake as a roadblock.... a warning sign reading 'PROCEED WITH CAUTION'. On the flip side, a person with a growth mindset might view that same mistake as a simple detour; an opportunity to learn something new.
Jo's mention of the 'didactic contract'- the feeling that you want to spoon feed math to students who struggle- was also really interesting to read about. It takes time for a student to work through a challenging problem, and it takes time for them to develop their own understanding of the concept at hand. However, teachers know that time is precious and in order to 'get through' what needs to be done, sometimes it is easier just to show them how, or do it for them.
On a different note.... Jo mentioned that challenging, problem- based mathematics helps to create future entrepreneurs and innovators. This statement made me think about a group of students who I am currently teaching. All very strong mathematicians, and all very interesting in inquiring into industry, innovation, and infrastructure! An interesting personal connection to the session... and I look forward to sharing this information with my students.
Reflection
I feel that the evidence about brain growth and growth mindset is a very powerful message since it allows children the freedom of making mistakes and encourages them to learn from it. I discussed this in class and they loved the idea of ‘ Maths neutrons’. We always talked about ‘stretch mistakes’ in our class room but I feel that the growth mindset is a more positive idea on learning as it gives all children a level playing field since they can shift their focus from a perfect final product to process-oriented learning. Luckily, we are at a school where this is part of the learning culture. It was encouraging to get confirmation that when Maths methods are discussed in the context of a problem, it leads to students developing persistance and confidence in later life. They loved hearing the story about how Laurent Schwartz, felt stupid because he was one of the slowest thinkers!
I did the activity that was mentioned in this session where children took a piece of paper and crumple it up with the feelings that they have when they make a mistake, and then throw it at the board, expressing those feelings. Then we took the paper back, unfolded it,and colored in all the lines that have formed to represent the brain growth they had when they made errors. Children loved this activity!!
I also showed the 2 videos from youcube in class and the discussions that followed were great!
I agree that often children are the “victims of excellence” and the pressure is not just from the teaching community but parents who have inflexible notions of excellence and push their children to achieve it.
I love workshops where you can take the ideas straight back to the classroom!
Monday, 13 March 2017
These sessions absorbed a lot of my thinking time. It took me a long time! I too, am a slow reader. I am not particularly quick as a mathematical problem solver either. But I enjoy them both equally.
Sessions 3 and 4 raise important questions for me concerning attitudes to mathematics amongst students and teachers. I meet many teachers and parents who admit to a residual anxiety about maths, mostly born out of their school experience. The testing culture discussed in this session is an obvious contender for the source of this maths anxiety. Similarly, the association of speed with recall tests or other mathematical tasks is another cause of stress in the learner. The session highlights the negative effects of a testing culture, in particular, the learner's association of mathematics with testing. To the learner, mathematics and testing have become synonymous. In test-driven environments, students become preoccupied with test scores/grades and the net result of this is often raised anxiety.
In our school we are not constrained by such a testing culture and have worked hard to develop positive attitudes to mathematics with our students, however, negativity concerning errors still persists amongst some children and finding ways of changing this is important. The references to the brain research were very thought-provoking; the realisation that your brain is actually growing when confused! (I need to read more about this). An environment with a growth mindset accepts that mistakes are inevitable, especially when confronting difficult problems, and they can pave the way to conceptual leaps.
We need to move away from maths problems that are always resolved in one lesson, or a set time, and lead students towards greater complexity and challenge. Persistence was a theme here. I was reminded of 'Thinking things through', by Leone Burton, written in 1984 and still a very relevant book on, in my view, dismantling the didactic contract. The problem solving work of mathematicians working in Silicon Valley referenced in the session had to deal with 'messiness' and complexity, persist and be resilient ... We need to continue to engage students in rich problem solving that resonates with the real world.
Sessions 3 and 4 raise important questions for me concerning attitudes to mathematics amongst students and teachers. I meet many teachers and parents who admit to a residual anxiety about maths, mostly born out of their school experience. The testing culture discussed in this session is an obvious contender for the source of this maths anxiety. Similarly, the association of speed with recall tests or other mathematical tasks is another cause of stress in the learner. The session highlights the negative effects of a testing culture, in particular, the learner's association of mathematics with testing. To the learner, mathematics and testing have become synonymous. In test-driven environments, students become preoccupied with test scores/grades and the net result of this is often raised anxiety.
In our school we are not constrained by such a testing culture and have worked hard to develop positive attitudes to mathematics with our students, however, negativity concerning errors still persists amongst some children and finding ways of changing this is important. The references to the brain research were very thought-provoking; the realisation that your brain is actually growing when confused! (I need to read more about this). An environment with a growth mindset accepts that mistakes are inevitable, especially when confronting difficult problems, and they can pave the way to conceptual leaps.
We need to move away from maths problems that are always resolved in one lesson, or a set time, and lead students towards greater complexity and challenge. Persistence was a theme here. I was reminded of 'Thinking things through', by Leone Burton, written in 1984 and still a very relevant book on, in my view, dismantling the didactic contract. The problem solving work of mathematicians working in Silicon Valley referenced in the session had to deal with 'messiness' and complexity, persist and be resilient ... We need to continue to engage students in rich problem solving that resonates with the real world.
Thoughts on Sessions 3 & 4
Many interesting and often new (for me) ideas were raised in these sessions. First session 3...
I liked what was said about the need for greater resilience and persistence. It was important to note that successful entrepreneurs made MORE mistakes than others. We shouldn't be designing work so that students get most things correct-make it more challenging. Unsurprisingly, there are many benefits to problem based learning approach. We need to show students how to use the feedback they are given to get better. Loved the idea of using "yet" in you don't know this yet. There is an over emphasis on speed.
Looking at this from an LS lens I can attest to the stress caused to students when they feel under pressure to quickly produce an answer. The didactic contract also resonated with me as students often try this approach. PBL sounds great in theory but hard to do with younger kids. Good stuff in this session!
Session 4 head some good points as well. It is useful to have our students make sense of maths. In the video we saw many students thought the answer was six but how can the quotient be bigger than the dividend? that didn't make sense so was a clue to students to try something else. There is value in explaining what someone else said-it is a mark of understanding. It is important to select tasks that promote engagement-not just the search for an answer. Formative assessment is much more useful than summative--students need to know where they are when they are able to do something about it! The self-esteem movement was wrong-it has diminished children's growth mindsets. The Growth Mindset Task Framework consists of: openness; different ways of seeing; multiple entry points; multiple paths / strategies; and clear learning goals and opportunities for feedback. Tracking sends a message that damages students (high and low), impacts teacher expectations, defines the type of work students are given and results in teaching to the middle.
I liked what was said about the need for greater resilience and persistence. It was important to note that successful entrepreneurs made MORE mistakes than others. We shouldn't be designing work so that students get most things correct-make it more challenging. Unsurprisingly, there are many benefits to problem based learning approach. We need to show students how to use the feedback they are given to get better. Loved the idea of using "yet" in you don't know this yet. There is an over emphasis on speed.
Looking at this from an LS lens I can attest to the stress caused to students when they feel under pressure to quickly produce an answer. The didactic contract also resonated with me as students often try this approach. PBL sounds great in theory but hard to do with younger kids. Good stuff in this session!
Session 4 head some good points as well. It is useful to have our students make sense of maths. In the video we saw many students thought the answer was six but how can the quotient be bigger than the dividend? that didn't make sense so was a clue to students to try something else. There is value in explaining what someone else said-it is a mark of understanding. It is important to select tasks that promote engagement-not just the search for an answer. Formative assessment is much more useful than summative--students need to know where they are when they are able to do something about it! The self-esteem movement was wrong-it has diminished children's growth mindsets. The Growth Mindset Task Framework consists of: openness; different ways of seeing; multiple entry points; multiple paths / strategies; and clear learning goals and opportunities for feedback. Tracking sends a message that damages students (high and low), impacts teacher expectations, defines the type of work students are given and results in teaching to the middle.
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Thursday, 23 February 2017
Reflections on Session 3,4,5
Reflection 3
The main ideas for me this session were...
Mistakes can grow your brain. The need to change the
classroom culture. We should keep track of the progress students make and share
it with them. We should give specific feedback a student can use. Give a
hopeful message-you've not learned this 'YET' We should have children asking
for 'harder' work.' Challenge' is the new comfort zone. Effort is worthwhile,
enjoyable and productive. Take away from evaluating ability into opportunities
to learn. We need to keep students at the edge of their understanding. Speed is the enemy of a mistake friendly
culture. Do we have too much 'content'? What are the 'big ideas' in maths?
Reflection 4
The main ideas for me this session were…
Giving time for group discussion
Finding students with an answer and letting them convince
their group first
Keeping away from the rules
Letting students present their answers
Asking students to challenge one another
Sharing multiple paths to the same answer
Slowing down the lesson
My action is going to be to design a 'growth mindset' poster
for the class reinforcing the values of effort to be worthwhile, enjoyable and
productive. Something along the lines of "You can grow your brain when
you...make mistakes...when you don't know the answer straight away..when you
convince a friend..when you are challenged...
I like the idea of a 'low floor...high ceiling task'.
Reflection 5
I would like to try number talks with my class incorporating
some of the key 'teacher moves' as demonstrated by Cathy Humphries.
Cathy's introduction was "let's get ready to
think" alerting the students to participate.
She gave a lengthy 'wait time' after asking a question and
slowed quick students down by saying, "try to see it in a different
way."
She gave a very open invitation to share, "Is anybody
willing to share what the answer might be?"
She used a 'mistake' by one of the students to learn from
his methodology.
She asked for other solutions and didn't close the question
down.
She checked for understanding, "I think I heard you
say.."
She illustrated the student's solutions.
She gave the students time to discuss with one another.
She called on a student who hadn't participated yet.
I also see the importance of students visually representing
their thinking.
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