Hopscotch Goats

I believe this activity, in this format, was originally published by NCTM. I have seen it with frogs on lily pads, and I have used the principle in the classroom with seats and children.

It is important in our math's classrooms that we incorporate more problems/investigations like this where students can not just say, "I know the answer!"  These types of activities encourage lateral thinking, investigation and blockages which students need to get used to.

I hope it proves successful in your math's class

For further information and extensions look at this site: https://nzmaths.co.nz/frogs-teachers-notes?fbclid=IwAR15UV2ujnpSmNqAT6eJ-5z3Faq_ExmhI-G7rq1C8rusHFc_SS-t5CszVOA

Why are we still fixated on teacher computation to the detriment of MATHEMATICS?

“Mathematics is the alphabet with which God has written the universe”  Galileo Galilei.

I have believed for a number of years that, the traditional approach to teaching has been incorrect.  It was based on the need for “Tally Clerks” during the Industrial Revolution and it has hardly changed since. 
Think about the way maths was taught to you, now how about your parents? and your grandparents?
My grandparents and father never flew on an aeroplane, none of them had a cellphone, tablet or a computer, how the world has changed in one short generation, because now my grandchildren have tablets, interactive TV etc
We are told that we have a “Child-Centred” Education system (Be damned we have not) We are told we are educating for the future (Be damned we are not)
Where is the creativity in Maths Teaching? The Thinking? The Cooperation? The open-ended questions? the child-centred investigations?
When was the last time you brought a flower, some fruit, into the classroom and asked the question “where is the maths? or, What can you see in the object?
When was the last time you asked the class, what do you want to know about: fractions, or, measurement, or, geometry, or, art and maths?
Len Cooper, retired maths adviser/consultant and teacher:
https://mathslen.blogspot.com/

Charlotte Wilkinson, “The Wilkie Way” (NZ) in her last newsletter asks similar questions so I share her thoughts and questions.

Maths and The Arts

Traditional school maths with endless practicing of calculating. The standard written algorithms are procedures that have been taught for endless years since the introduction of the base ten algorithmic number system (About 2000 years).
In making maths modern - mental strategies were the new way of solving - these quickly became more procedures for students to memorise (or not).
Why does school maths have such an unhealthy fixation with procedural knowledge?
The era of human computers has passed, so many of the arbitary skills promoted by school maths are also archaic - long multipication and long division, adding and subtracting fractions to name just a few.
We are presented with acronyms STEM and now STEAM which implies Maths is separate to Science, Technology and Engineering and now Art too.
While it is relatively easy to dismiss the stupidity of thinking mathematics is unrelated to science, technology and engineering let us now consider Mathematics and the Arts.
Bertrand Russell (Nobel Laureate, Philosopher, Mathematician) once said: “Mathematics, rightly viewed, possess not only truth, but supreme beauty - a beauty cold and austere...sublimely pure, and capable of stern perfection such as only the greatest art can show.”
GH Hardy Another famous mathematian in the earlier part of the 20th century known for being a harden purist advocating for rigour and abstraction, ultimately declared that: “I am interested in mathematics only as a creative art.”
If mathematics is an art form, what are mathematicians. Hardy called them “makers of patterns”. The late Maryam Mirzakhani, the only female recipient of the Fields Medal (the highest accolade in mathematics) was often mistaken by her daughter for an artist.
To mathematicians, school maths is something of a desecration of their subject. There is no doubt the rise of numeracy has crowded out more conventional arts subjects but it has also concealed the true nature of mathematics.
Paul Lockhart (American school maths teacher and research mathematician) wrote in his book A Mathematicians Lament - “No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial.”
Even where calculation has its place, the mathematican seeks rich representations that illuminate the procedures they call upon.
Do you see times tables as just a matter of memorising number facts (learn them by rote) or do you spend time exploring the myriad of patterns within a multiplication table and between the multiplication tables.
We spend alot of time talking about 21st century education - I see modern learning environments, I see school maths curriculums as completing workbooks. I see students working on laptops and maths apps but the mathematics I see them doing is still archaic, procedural and not the kind of mathematics our students need.
The kind of mathematics that our students need, that our world is increasingly dependent on, is much more aligned to its artistic tenent. I think this is why STEM has become STEAM. Skills like curiosity, persistence and resilience are not divorced from mathematics, they are the very traits that mathematicians through the ages have brought to their problems.
As with any art, there is a subjective element to deciding what should go into a mathematics curriculum, or how to assess these broader skills. However that is no excuse for persisting with an outdated brand of the subject that brings joy to so few and value to fewer still.
The history of mathematics is entwined with the history of technology. Humans have precedent for updating their ways of doing mathematics based on the tools available to them. (OK the complete change from using Roman numerals to the base 10 system took about 200 years). With the technologies now available at our fingertips, we can put calculation in its rightful place as the footnote to mathematical thinking power.
Calculation is simply the price we once paid to do mathematics.
When maths anxiety is a real issue for many people, and so many others show an indifference to a subject that promises such immense power and beauty, the price to continue with a fixation on procedural calculation as the backbone to school mathematics is too high.
Samantha, a year 5 student once said to me: “ I love doing sums (calculations) because you don’t have to think.”
How true - once a procedure is memorised no thinking is required.
While the content is the curriculum, how you choose to teach it to your students is dependent on your own creative flair and knowledge of the subject matter.
               My own daughter’s advice to Primary school teachers given 5 years ago was
                                                 “Calculate less, estimate more”
 
                                    Mathematics will never be replaced by the Arts.
                           Mathematics is an art, and it is time we embraced it as one.
©Copyright N C Wilkinsons Ltd 2019

Why are we still ability grouping???

It is interesting to note that after much research into ability groups/class streaming(setting) and the danger they can cause to a child's learning, it is surprising that it is still happening and often encourage by the powers above us. 

Recently a retired colleague(like myself retired) was sharing how her grandchild stopped reading because she was put in the "cant read/bottom" group. We believe her thinking was, "they have me in the bottom group with the children who cant read so I will show them by not reading" It took intervention by the grandmother and a retired (Reading Reading Recovery Teacher) to snap the child out of this thinking and get on with reading like she know how to.

30 years ago when working with my "bottom group" year 7/8 I was using materials to help them understand various number concepts. I looked up and the rest of the class had stopped what they were doing and wanted to join in. From there on it was open ended problems and investigations with groups of 2, groups of and a real cooperative learning approach.

Check out what Jo Boaler's experience is with open ended problems and students taking control of their learning.

The following is from this website: NCET "Not working in ability groups has been a revelation"NCETM "Not working with Ability groups has been a revelation

One of the first things Year 4 teacher Tracey Baruah tried when her school joined the Mastery Readiness Programme was abandoning the practice of putting children in different groups according to their perceived ability.
But Tracey, maths lead at Spring Bank Primary School in Leeds, with over 25 years' teaching behind her, quickly saw the benefits:
Not working in ability groups has been quite a revelation, she enthuses.
Watching Tracey lead a class discussion with Year 4, it is striking that all the children are involved, and that the discussion she’s leading is only possible because the children have all been working on the same, open-ended question.

…and that speaks volumes about how children rise to the challenge. Regardless of what their ability is deemed to be, if you give them an equal opportunity at something, they rise to it. They will have a go.
Sarah explains that they now often use the format of question we have seen in the Year 4 lesson: asking children what they know, and what they can find out. It has really helped pupil confidence because it removes the feeling of ‘I can’t do it’.
Spring Bank Primary School is a one-form entry school in the Headingley area of Leeds. Following an Ofsted judgement of Requires Improvement (RI), the school joined the Mastery Readiness Programme with the West Yorkshire Maths Hub. They took the decision to remove ‘ability’ groups early on, hoping to increase all children’s success in maths.
Pupil confidence is one of the things that has been improved by removing ability-groupings, says Tracey:
Some of my children came into Year 3, last year, thinking ‘That work’s not for me’. And now they would never say that.
Sarah agrees:
Children will still say ‘Yes, I find maths tricky, but I know what to do to help me, I know who I can talk to, I know what I can use’. I think historically, when we did do that three-way, five-way differentiation, we were labelling. I really do feel it labels children. And children aren’t daft – they know when they’re not getting the hard work.
So, how does giving the whole class the same problem help build all children’s confidence? Sarah explains that those children who might previously have been given work for ‘lower ability’ pupils would never get the opportunity to see or hear the most difficult maths being done in the class and to engage with it. She says, ‘Now all children are seeing the best possible (mathematical) outcomes (from the problems set)’. Not only that, Tracey points out, but their contributions are being valued by others in the class:
They are also getting someone saying to them, ‘Oh right, actually that’s a really good way, I’d not thought about doing that,’ or, ‘Show me how you did it, oh right, that’s a really good way to do it’.

Furthermore, Sarah points out that ability-grouping was crude and didn’t allow for times when children found a topic easier or harder than usual:
We’ve also talked about how one size doesn’t fit all. All children have different skills, different areas of expertise, different understanding. You can have a child that really struggles with calculations and problem-solving but is a complete whizz when it comes to time or money. They bring those skills to the table.
And what about those children that might previously have been on the ‘top table’ or doing ‘harder’ work? Tracey says that children can still be challenged, but by deepening their understanding rather than racing on to the next topic with only a procedural understanding of the previous one:
At the other end of the spectrum, the ‘more able’ children, they’re still engaged in what you want them to do. And they’re able to share their expertise. There are lots of very subtle ways that you can extend them. It doesn’t have to be a different activity. It can be through your questioning. When children are busy, there’s lots of scope to go and ask individual questions.
Sarah adds:
It’s not just about ‘they are very good at that, let’s see what’s harder, or more challenging or a bigger number’ – the old style of looking at challenging maths. The staff collectively really do understand that going deeper is about the reasoning, the explanation, the ‘how can you show this?’ I think the children have really embraced that as well – they understand that it’s not just about harder maths and bigger numbers, it’s about truly understanding.

Removing ability-grouping is just one aspect of how things have changed at Spring Bank Primary School. To hear the whole story of how the school has thrived as part of the NCETM Mastery Readiness Programme, supported by West Yorkshire Maths Hub, listen to Tracey and Sarah in this NCETM podcast. It’s an inspiring story.