The NZ Maths Curriculum states:
In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to: (then lists the Achievement Objectives)
What are Meaningful Contexts?
Is the usual Maths Text (work sheet) activities meaningful for most students? whether it be 4 + 5 = ? or use Pythagoras to find the length of the 3rd side of the triangle. Apart from doing as they are told or pleasing me the teacher, is there a real reason for the students to answer?
To me a meaningful context is something that appeals to students whether it be cultural, imaginative, a challenge, a real life situation, something that really engages the learning and they see a reason for finding the answer/solution.
In a previous post "TwentyOne" I suggest the "Challenge to beat me" was a meaningful contect for many of the students I worked with.
Over the years I have seen so much "busy work' in maths that no wonder we produce so many people who dislike or "cant do Maths"
At one mini conference I attended the Manager of a large printing company, welcomed us and said "I admire you people with your interest in maths and science... I just cant do maths!!" At morning tea I challenged him and asked why he said that. In the discussion he revealed that he was an accountant and did budgets and kept records etc. HE DID NOT EQUATE ACCOUNTANCY TO MATHEMATICS. I suggest that is because he did not explore maths through meaningful or real life contexts!!!
I suggest that many of the measurement activities that we get students to do are often not motivating for them and they therefore do not learn the required objectives we had hoped.
Is there a need for students to be measuring the netball court, when it is already marked out?
Why do we ask students to measure the height of the door to the classroom?
I suggest that we need to look for activities that have a “realness” to them so that the students can see a reason for investigating and completing the activity and hence developing the skills of measurement.
Perhaps we need to rephrase the netball court in a different way,
“There is a limited budget available for spending on court markings.
To help the BOT decide how much money is available we need to know how much it will cost to repaint the netball court lines”
We now have a situation where there is richness to the activity rather than just taking the trundle wheel and measuring the lengths of the lines.
As a School Committee member, before ‘Tomorrows Schools’ I was asked to estimate the cost of repainting the school swimming pool.
I felt that this was better done by the senior children as a ‘practical measuring activity.
We, the School Committee, told the students the cost of the paint was and what area a litre could be expected to cover.
The students had to work out the surface area and then decide on how many litres of paint and consequently the cost.
The important outcome was that their maths was used to assist the Committee and the students actual saw the pool being painted!!!!!
The challenge for each of us is to find, WITH THE HELP OF OUR STUDENTS, interesting tasks that will help them achieve the desired learning outcomes.
Have you considered reading books like "Counting on Frank" as a starting point for a maths investigation: (Is Frank's investigation about Ball Point Pens still true?)
How many students in the class have attached ear lobes? and how many have unattached?
In Gulliver's Travels Gulliver's thumb was measured and from this one measurement the Little People made him a coat. Would this be true today, How could we find out?
How much pocket money does each child get in the class?
What can you tell me about Kowhaiwhai Patterns?
In the Bible Goliath was described as being "6 cubits and a span". Would he fit through the classroom door without banging his head?
In the song "Twelve Days of Christmas" many things are mentioned. How much would all these things costs? check out this link 12 Days of Christmas
New class in 2020 (Year 4+) check in with me for an interesting "Trait" investigation with a Circular Graph
Monday, 16 December 2019
Saturday, 7 December 2019
We Learn Maths By DOING
I often wonder if we play lip service to the way children learn. If we spend time with pre-schoolers we see them constantly playing and through their play and questioning and interaction with the teachers the learn various skills and concepts.
Yet when we look at primary maths' learning there is often an emphasis on numbers and numerals. Yet these are quite abstract concepts. I am sure we would not just a put a book of words in front of students and expect them to learn.
As a Maths Adviser to schools/teachers and then as a private consultant I encouraged teachers to use the following:
When fluent with "Do" ask the students to explain what they were "Doing", When fluent with "Doing" then "Write" about what you said and did!! This then leads to mastery of the concepts.
A couple of years ago I saw a FB post in response to request for feedback about a Measurement Activity created by a Canadian Maths Teacher. The response:
"This is a great measurement activity but we are an iPad school so it would be great if it could be created for the iPad"
To suggest that I saw RED was a bit of an under statement ! How does a children learn what a metre a cm, a gram, a kilogram, a litre etc through manipulating abstract concepts of electronic screens. They need to measure length, estimate mass if they are going to understand the concepts of measurement.
You probably are thinking, what brought this on Len? This post on FB originally from Jo Boaler
It could be a great poster in every maths class and teachers staffroom
Yet when we look at primary maths' learning there is often an emphasis on numbers and numerals. Yet these are quite abstract concepts. I am sure we would not just a put a book of words in front of students and expect them to learn.
As a Maths Adviser to schools/teachers and then as a private consultant I encouraged teachers to use the following:
Do
Say
Write
When fluent with "Do" ask the students to explain what they were "Doing", When fluent with "Doing" then "Write" about what you said and did!! This then leads to mastery of the concepts.
A couple of years ago I saw a FB post in response to request for feedback about a Measurement Activity created by a Canadian Maths Teacher. The response:
"This is a great measurement activity but we are an iPad school so it would be great if it could be created for the iPad"
To suggest that I saw RED was a bit of an under statement ! How does a children learn what a metre a cm, a gram, a kilogram, a litre etc through manipulating abstract concepts of electronic screens. They need to measure length, estimate mass if they are going to understand the concepts of measurement.
You probably are thinking, what brought this on Len? This post on FB originally from Jo Boaler
It could be a great poster in every maths class and teachers staffroom
Some activities to get you started.
These could be arranged as Stations around the room for students to explore and complete.
Thursday, 28 November 2019
Twenty One
Another Nim activity.
As an adviser/consultant I had this activity at hand(in my head as it does not need counters or pens and pencils)Objective: To be the last person to take away either 3 or 2 or 1 to leave zero
Players: Two or two teams. (when introducing the activity I would always play the "Class" or Group, so I was one team and the class was the other. This of course made it easier for me to win as a class would try numbers at random, with no clear strategy)
How: First player starts by verbalising ("I will take away 1 and leave 20")
Second player takes away 1, 2, or 3 ("I will take away 2 leaving 18)
First player takes away 1, 2, or 3 verbalising and leaving the remainder
Continue until one player takes away 1, 2, or 3 to leave zero
Encourage: The students(and yourself to find a strategy that will allow you to always win.
Does the first player always win?
Does the second player always win?
What is the last number that you must leave to always win?
Adaptation: Start at Zero and be the first to reach 21
Take away a different set of numbers, e.g. 1, 2, 3, or 4?
Please Dont: Share your strategy, encourage the students to come up with their own. This is the basis of Creativity, Investigations, Becoming Stuck and moving through, Problem Solving.
As soon as two players know a strategy then whoever starts will control the game this is whn you use an adaptation.
When I was working in classes and schools as an adviser/consultant/teacher I would often use this activity as by doing it mentally as explained above helps mental agility with students.
At any break time of the next visit there would be queues of students wanting to play against me so that they "could win" It often took a number of days before I came a cross a student who knew the strategy.
Yes: We could show the working on the board, but this takes away the mental aspects and the strategy is much easier to see!
We could use a pile of 21 counters
Sunday, 24 November 2019
Encourage Creativity and Investigations: STOP USING WALTS etc
This photo is self explanatory. Whoever thought we should write WALT's, Specific Objectives on the board, screen, before students working, must have been more interested in Test results than student's real learning.
See below for another post about Writing Objectives before a lesson:
Thanks to the person who shared this on FaceBook.
Writing Objectives A waste of time
See below for another post about Writing Objectives before a lesson:
Thanks to the person who shared this on FaceBook.
Writing Objectives A waste of time
Friday, 8 November 2019
Struggle/Being Stuck with Maths
I can remember seeing in a lecture room, at Auckland College of Education some years back, a wall poster along the lines of: STUCK, congratulations. I just wish I could remember the rest but it nwas along the lines of "this is when the learning starts"
When I read the article below, by Jo Boaler, I thought I should share, because often as teachers we often try and make everything so easy we break math's learning into small steps, each step taking us towards the big goal. Is this really the way children learn best? (Most Text Books and Programs do just this!)
In the real world, outside the classroom, most of us learn best when we become STUCK, figure a way through and achieve what we wanted.
I would like to encourage all teachers to have activities available for all students to become STUCK and perhaps FRUSTRATED, so that REAL Learning can take place, and the student can feel real achievement in what they have done.
Feel free to share this article and perhaps join Jo Boaler's You Cubed email list.
Why Struggle Is Essential for the Brain — and Our Lives
As parents and teachers, we do just about everything we can to make sure that children don’t struggle. It turns out we are making a terrible mistake. Research shows that struggling is absolutely critical to mastery and that the highest achieving people in the world are those who have struggled the most. The more I communicate this message to parents and teachers the more stories I hear of complete personal transformation.
Neuroscientists have found that mistakes are helpful for brain growth and connectivity and if we are not struggling, we are not learning. Not only is struggle good for our brains but people who know about the value of struggle improve their learning potential. This knowledge would not be earth shattering if it was not for the fact that we in the Western world are trained to jump in and prevent learners from experiencing struggle.
“When students look at me and say: “This is hard,” I say, “That is fantastic.”
An international study of mathematics teaching found that teachers in Japan put their students in places of struggle 44 percent of the time in classrooms—they saw this less than 1 percent of the time in U.S. classrooms. What do we parents and teachers do instead? We jump in and show the way, offering steps to a solution to help save our students from struggle. This is in large part because this new science is not widely available and we are culturally trained to feel bad, and to rush in and help, when this is probably the last thing we should do.
The research on the impact of struggle turns out to help adults too—in all sorts of jobs. I interviewed sixty-two people for my new book, “Limitless Mind.” Many of them shared similar accounts of how they used to go into meetings afraid they would be found out for not knowing something. After learning about the importance of not knowing and of engaging in struggle they now proudly show up and say “I don’t know, but I will find out.” They display a mindset of discovery and curiosity, which has helped their lives in many ways.
Once we stop the charade of knowing everything, and embrace knowing less, with a willingness to sit with uncertainty, unexpected things happen.
When I was teaching middle schoolers in a research math camp a few years ago one girl stood out to me; she was nearly always wrong in her thinking, but she was always engaged, arguing her case, pushing to understand better. An observer of the class would have described her as a low achiever, but she improved more than any other of the 84 students we taught that summer. Her standardized test score in mathematics improved by 450 percent after 18 struggle filled lessons. Our messages to the students—that struggle would be valued and mistakes are productive—had helped her feel good about struggle and embrace it.
When I tell young learners that struggle and mistakes are the best times for our brains it is freeing. Students no longer give up on problems when they find them hard—they push through the struggle to the wonderful places on the other side. When students look at me with a puppy dog face and say: “This is hard,” I say, “That is fantastic. That feeling of ‘hard’ is the feeling of your brain developing, strengthening and growing”.
“We cannot achieve anything creative without being comfortable with mistakes and struggle”
In 2016, two young computer scientists rocked the world of mathematics by solving a previously unsolved math problem, an event that many described as audacious. The two young men reflected that it was knowing less that allowed them to solve the difficult problem. It freed their mind to think in better ways.
I am not arguing that knowledge is bad or knowing answers is not helpful. What I am saying is that knowledge is less important than a mindset of discovery and curiosity. We cannot achieve anything creative without being comfortable with mistakes and struggle—and we should all embrace times of struggle, knowing they are helping our brains. When we adopt a limitless perspective, approaching different jobs and conversations with a comfort with uncertainty and struggle, with a willingness to learn from others and with a flexible approach to problems, outcomes improve—in learning and in life.
Millions of students start the school year each year excited for all they will learn, but as soon as they struggle or see someone solve a problem with ease, they start to doubt themselves and mentally shut down. This starts a less productive learning pathway for them. Instead they should value the time of struggle and know that they are on their way to being better, wiser and equipped with a stronger brain. Getting answers right is OK, being stuck and finding them hard is fantastic.
Jo Boaler is the Nomellini-Olivier Professor of Education at Stanford, co-founder of youcubed.org and author of the new book “Limitless Mind: Learn, Lead & Live without Barriers.”
When I read the article below, by Jo Boaler, I thought I should share, because often as teachers we often try and make everything so easy we break math's learning into small steps, each step taking us towards the big goal. Is this really the way children learn best? (Most Text Books and Programs do just this!)
In the real world, outside the classroom, most of us learn best when we become STUCK, figure a way through and achieve what we wanted.
I would like to encourage all teachers to have activities available for all students to become STUCK and perhaps FRUSTRATED, so that REAL Learning can take place, and the student can feel real achievement in what they have done.
Feel free to share this article and perhaps join Jo Boaler's You Cubed email list.
Why Struggle Is Essential for the Brain — and Our Lives
As parents and teachers, we do just about everything we can to make sure that children don’t struggle. It turns out we are making a terrible mistake. Research shows that struggling is absolutely critical to mastery and that the highest achieving people in the world are those who have struggled the most. The more I communicate this message to parents and teachers the more stories I hear of complete personal transformation.
Neuroscientists have found that mistakes are helpful for brain growth and connectivity and if we are not struggling, we are not learning. Not only is struggle good for our brains but people who know about the value of struggle improve their learning potential. This knowledge would not be earth shattering if it was not for the fact that we in the Western world are trained to jump in and prevent learners from experiencing struggle.
“When students look at me and say: “This is hard,” I say, “That is fantastic.”
An international study of mathematics teaching found that teachers in Japan put their students in places of struggle 44 percent of the time in classrooms—they saw this less than 1 percent of the time in U.S. classrooms. What do we parents and teachers do instead? We jump in and show the way, offering steps to a solution to help save our students from struggle. This is in large part because this new science is not widely available and we are culturally trained to feel bad, and to rush in and help, when this is probably the last thing we should do.
The research on the impact of struggle turns out to help adults too—in all sorts of jobs. I interviewed sixty-two people for my new book, “Limitless Mind.” Many of them shared similar accounts of how they used to go into meetings afraid they would be found out for not knowing something. After learning about the importance of not knowing and of engaging in struggle they now proudly show up and say “I don’t know, but I will find out.” They display a mindset of discovery and curiosity, which has helped their lives in many ways.
Once we stop the charade of knowing everything, and embrace knowing less, with a willingness to sit with uncertainty, unexpected things happen.
When I was teaching middle schoolers in a research math camp a few years ago one girl stood out to me; she was nearly always wrong in her thinking, but she was always engaged, arguing her case, pushing to understand better. An observer of the class would have described her as a low achiever, but she improved more than any other of the 84 students we taught that summer. Her standardized test score in mathematics improved by 450 percent after 18 struggle filled lessons. Our messages to the students—that struggle would be valued and mistakes are productive—had helped her feel good about struggle and embrace it.
When I tell young learners that struggle and mistakes are the best times for our brains it is freeing. Students no longer give up on problems when they find them hard—they push through the struggle to the wonderful places on the other side. When students look at me with a puppy dog face and say: “This is hard,” I say, “That is fantastic. That feeling of ‘hard’ is the feeling of your brain developing, strengthening and growing”.
“We cannot achieve anything creative without being comfortable with mistakes and struggle”
In 2016, two young computer scientists rocked the world of mathematics by solving a previously unsolved math problem, an event that many described as audacious. The two young men reflected that it was knowing less that allowed them to solve the difficult problem. It freed their mind to think in better ways.
I am not arguing that knowledge is bad or knowing answers is not helpful. What I am saying is that knowledge is less important than a mindset of discovery and curiosity. We cannot achieve anything creative without being comfortable with mistakes and struggle—and we should all embrace times of struggle, knowing they are helping our brains. When we adopt a limitless perspective, approaching different jobs and conversations with a comfort with uncertainty and struggle, with a willingness to learn from others and with a flexible approach to problems, outcomes improve—in learning and in life.
Millions of students start the school year each year excited for all they will learn, but as soon as they struggle or see someone solve a problem with ease, they start to doubt themselves and mentally shut down. This starts a less productive learning pathway for them. Instead they should value the time of struggle and know that they are on their way to being better, wiser and equipped with a stronger brain. Getting answers right is OK, being stuck and finding them hard is fantastic.
Jo Boaler is the Nomellini-Olivier Professor of Education at Stanford, co-founder of youcubed.org and author of the new book “Limitless Mind: Learn, Lead & Live without Barriers.”
Tuesday, 22 October 2019
Homework is damaging our children
Admittedly this is from the USA but similar feelings and understandings can be found in NZ. As a father, Grandfather and working in Education for 54 years I believe we are not allowing kids to be kids and setting homework is often not justifiable, as it can be:
I wonder what has happened to all the Family Math books schools and teachers bought? Along with the monthly activity newsletters the Trust published?
“There is no evidence that any amount of homework improves the academic performance of elementary students.”
This statement, by homework research guru Harris Cooper, of Duke University, is startling to hear, no matter which side of the homework debate you’re on. Can it be true that the hours of lost playtime, power struggles and tears are all for naught? That millions of families go through a nightly ritual that doesn’t help? Homework is such an accepted practice, it’s hard for most adults to even question its value.
When you look at the facts, however, here’s what you find: Homework has benefits, but its benefits are age dependent.
For elementary-aged children, research suggests that studying in class gets superior learning results, while extra schoolwork at home is just . . . extra work. Even in middle school, the relationship between homework and academic success is minimal at best. By the time kids reach high school, homework provides academic benefit, but only in moderation. More than two hours per night is the limit. After that amount, the benefits taper off. “The research is very clear,” agrees Etta Kralovec, education professor at the University of Arizona. “There’s no benefit at the elementary school level.”
Before going further, let’s dispel the myth that these research results are due to a handful of poorly constructed studies. In fact, it’s the opposite. Cooper compiled 120 studies in 1989 and another 60 studies in 2006. This comprehensive analysis of multiple research studies found no evidence of academic benefit at the elementary level. It did, however, find a negative impact on children’s attitudes toward school.
This is what’s worrying. Homework does have an impact on young students, but it’s not a good one. A child just beginning school deserves the chance to develop a love of learning. Instead, homework at a young age causes many kids to turn against school, future homework and academic learning. And it’s a long road. A child in kindergarten is facing 13 years of homework ahead of her.
Then there’s the damage to personal relationships. In thousands of homes across the country, families battle over homework nightly. Parents nag and cajole. Overtired children protest and cry. Instead of connecting and supporting each other at the end of the day, too many families find themselves locked in the “did you do your homework?” cycle.
When homework comes prematurely, it’s hard for children to cope with assignments independently—they need adult help to remember assignments and figure out how to do the work. Kids slide into the habit of relying on adults to help with homework or, in many cases, do their homework. Parents often assume the role of Homework Patrol Cop. Being chief nag is a nasty, unwanted job, but this role frequently lingers through the high school years. Besides the constant conflict, having a Homework Patrol Cop in the house undermines one of the purported purposes of homework: responsibility.
Homework supporters say homework teaches responsibility, reinforces lessons taught in school, and creates a home-school link with parents. However, involved parents can see what’s coming home in a child’s backpack and initiate sharing about school work--they don’t need to monitor their child’s progress with assigned homework. Responsibility is taught daily in multiple ways; that’s what pets and chores are for. It takes responsibility for a 6-year-old to remember to bring her hat and lunchbox home. It takes responsibility for an 8-year-old to get dressed, make his bed and get out the door every morning. As for reinforcement, that’s an important factor, but it’s only one factor in learning. Non-academic priorities (good sleep, family relationships and active playtime) are vital for balance and well-being. They also directly impact a child’s memory, focus, behavior and learning potential. Elementary lessons are reinforced every day in school. After-school time is precious for the rest of the child.
What works better than traditional homework at the elementary level is simply reading at home. This can mean parents reading aloud to children as well as children reading. The key is to make sure it’s joyous. If a child doesn’t want to practice her reading skills after a long school day, let her listen instead. Any other projects that come home should be optional and occasional. If the assignment does not promote greater love of school and interest in learning, then it has no place in an elementary school-aged child’s day.
Elementary school kids deserve a ban on homework. This can be achieved at the family, classroom or school level. Families can opt out, teachers can set a culture of no homework (or rare, optional homework), and schools can take time to read the research and rekindle joy in learning.
Homework has no place in a young child’s life. With no academic benefit, there are simply better uses for after-school hours.
Heather Shumaker’s new book It’s OK to Go Up the Slide (Tarcher/Penguin Random House) was published on March 8, 2016.
- Just busy work (As we have to set homework)
- Not practise of already learnt ideas
- Set as punishment for behaviour or other reasons
I wonder what has happened to all the Family Math books schools and teachers bought? Along with the monthly activity newsletters the Trust published?
Homework is wrecking our kids: The research is clear, let's ban elementary homework
Heather Shumaker March 6, 2016 4:00AM (UTC)
This statement, by homework research guru Harris Cooper, of Duke University, is startling to hear, no matter which side of the homework debate you’re on. Can it be true that the hours of lost playtime, power struggles and tears are all for naught? That millions of families go through a nightly ritual that doesn’t help? Homework is such an accepted practice, it’s hard for most adults to even question its value.
When you look at the facts, however, here’s what you find: Homework has benefits, but its benefits are age dependent.
For elementary-aged children, research suggests that studying in class gets superior learning results, while extra schoolwork at home is just . . . extra work. Even in middle school, the relationship between homework and academic success is minimal at best. By the time kids reach high school, homework provides academic benefit, but only in moderation. More than two hours per night is the limit. After that amount, the benefits taper off. “The research is very clear,” agrees Etta Kralovec, education professor at the University of Arizona. “There’s no benefit at the elementary school level.”
Before going further, let’s dispel the myth that these research results are due to a handful of poorly constructed studies. In fact, it’s the opposite. Cooper compiled 120 studies in 1989 and another 60 studies in 2006. This comprehensive analysis of multiple research studies found no evidence of academic benefit at the elementary level. It did, however, find a negative impact on children’s attitudes toward school.
This is what’s worrying. Homework does have an impact on young students, but it’s not a good one. A child just beginning school deserves the chance to develop a love of learning. Instead, homework at a young age causes many kids to turn against school, future homework and academic learning. And it’s a long road. A child in kindergarten is facing 13 years of homework ahead of her.
Then there’s the damage to personal relationships. In thousands of homes across the country, families battle over homework nightly. Parents nag and cajole. Overtired children protest and cry. Instead of connecting and supporting each other at the end of the day, too many families find themselves locked in the “did you do your homework?” cycle.
When homework comes prematurely, it’s hard for children to cope with assignments independently—they need adult help to remember assignments and figure out how to do the work. Kids slide into the habit of relying on adults to help with homework or, in many cases, do their homework. Parents often assume the role of Homework Patrol Cop. Being chief nag is a nasty, unwanted job, but this role frequently lingers through the high school years. Besides the constant conflict, having a Homework Patrol Cop in the house undermines one of the purported purposes of homework: responsibility.
Homework supporters say homework teaches responsibility, reinforces lessons taught in school, and creates a home-school link with parents. However, involved parents can see what’s coming home in a child’s backpack and initiate sharing about school work--they don’t need to monitor their child’s progress with assigned homework. Responsibility is taught daily in multiple ways; that’s what pets and chores are for. It takes responsibility for a 6-year-old to remember to bring her hat and lunchbox home. It takes responsibility for an 8-year-old to get dressed, make his bed and get out the door every morning. As for reinforcement, that’s an important factor, but it’s only one factor in learning. Non-academic priorities (good sleep, family relationships and active playtime) are vital for balance and well-being. They also directly impact a child’s memory, focus, behavior and learning potential. Elementary lessons are reinforced every day in school. After-school time is precious for the rest of the child.
What works better than traditional homework at the elementary level is simply reading at home. This can mean parents reading aloud to children as well as children reading. The key is to make sure it’s joyous. If a child doesn’t want to practice her reading skills after a long school day, let her listen instead. Any other projects that come home should be optional and occasional. If the assignment does not promote greater love of school and interest in learning, then it has no place in an elementary school-aged child’s day.
Elementary school kids deserve a ban on homework. This can be achieved at the family, classroom or school level. Families can opt out, teachers can set a culture of no homework (or rare, optional homework), and schools can take time to read the research and rekindle joy in learning.
Homework has no place in a young child’s life. With no academic benefit, there are simply better uses for after-school hours.
Heather Shumaker’s new book It’s OK to Go Up the Slide (Tarcher/Penguin Random House) was published on March 8, 2016.
Circular Nim
Here is another NIM activity that is different from the usual type. It encourages players to think about spatial(geometry) issues rather than just number as in the traditional.
The activity could be approached, through a class/group demonstration and then assigned as an independent activity.
Once students have found a strategy for winning encourage them to tweak the rules to create a new/similar activity.
As usual I do not share answers or strategies, as I believe that this takes away the investigation, thinking and creativity of learning. If we as teachers know the answers we can subconsciously encourage students to look for the answer, strategy we have when there could be others as well.
Enjoy the exploring
The activity could be approached, through a class/group demonstration and then assigned as an independent activity.
Once students have found a strategy for winning encourage them to tweak the rules to create a new/similar activity.
As usual I do not share answers or strategies, as I believe that this takes away the investigation, thinking and creativity of learning. If we as teachers know the answers we can subconsciously encourage students to look for the answer, strategy we have when there could be others as well.
Enjoy the exploring
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