Tag Archives: problem solving

Away with the fairies

© 2014 Shelly ʕ•ᴥ•ʔ https://www.sketchport.com/drawing/6517152420986880/fairies. Licensed under CC-BY.

Are you a daydreamer? Were you accused of daydreaming at school? Many of us were. With minds that are easily distracted and work that is less than exciting, it is easy for thoughts to drift away into other realms. It can take anything, or nothing, and it is often difficult to back-track from where we find ourselves, along the path of thoughts to what initiated the journey. It can be no more tangible that the dream that escapes upon waking.

While daydreaming can be pleasant and good for relaxation and creativity, it is often frowned upon in students meant to be concentrating on what they are to learn. Children would probably find it easier to attend if the work was tailored to their needs, initiated by their interests, and involved them as participants rather than recipients. The fifteen minutes of play per hour that Finnish children enjoy would also help, I’m sure, in giving time for minds to be, not corralled into predetermined channels.

In this Conversation on Daydreaming with Jerome L. Singer in Scientific American by Scott Barry Kaufman on 10 December, 2013, Singer says, I think that teachers need to recognize that often, the daydreaming is because some of the kids are bored”.

Whether through boredom or not, daydreaming can sometimes lead to breakthroughs in solving problems, creativity and productivity as described in this CNN article by Brigid Schulte For a more productive life, daydream. Brigid lists a number of daydreamers; including:

  • J K Rowling
  • Mark Twain
  • Richard Feynman
  • Archimedes
  • Newton

Other famous daydreamers include:

  • Einstein
  • Edison
  • J. R. R. Tolkien
  • Boy George
  • Richard Branson

Here are a few other quotes about the importance of daydreaming:

Keith Richards is reported as saying that “Satisfaction”, the Rolling Stones’ most famous hit, came to him in a dream, and

Paul McCartney says the same thing about the Beatles’ hit “Yesterday”.

Neil Gaiman: “You get ideas from daydreaming. You get ideas from being bored. You get ideas all the time. The only difference between writers and other people is we notice when we’re doing it.”

George Lucas: “I’m not much of a math and science guy. I spent most of my time in school daydreaming and managed to turn it into a living.”

Professor Elizabeth Blackburn, the first Australian-born female Nobel Laureate, attributes her success as a molecular biologist, in part, to daydreaming.  She is reported by the Sydney Morning Herald to have said, ‘I think you need time to daydream, to let your imagination take you where it can … because I’ve noticed [that] among the creative, successful scientists who’ve really advanced things, that was a part of their life.’

While speaking to students at Questacon in Canberra after receiving her prize, she joked, ”Your parents and your teachers are going to kill me if they hear you say, ‘she told us just to daydream.’

So why is it, if the importance of daydreaming is recognised by successful creatives, thinkers, scientists, and business people, that it is still frowned upon in school? Why do we still insist that children sit at desks, repeating mundane tasks in order to pass tests that have little bearing on their future success or on the future of our species and the planet?

In a previous post I wrote about John Dewey’s dreamof the teacher as a guide helping children formulate questions and devise solutions. Dewey saw the pupil’s own experience, not information imparted by the teacher, as the critical path to understanding. Dewey also contended that democracy must be the main value in each school just as it is in any free society.” According to Pasi Sahlberg in Finnish Lessons, What can the world learn from educational change in Finland? schools in Finland have dreamed their own dream by building upon Dewey’s.

Of course, on a much smaller scale, I have my own dream of a better way of educating our children.

This week at the Carrot Ranch, Charli Mills dreamed a dream and challenged writers to In 99 words (no more, no less) write a story that involves a dream. This action could have happened while awake, such as daydreaming, or make up a dream when asleep. Go where the prompt leads as it could be a nightmare or just fond memories or ambition.

This is my response. I hope you enjoy it.

Off with the fairies

Each year the school reports told the same story:

He’s off with the fairies.

Poor concentration.

Needs to pay more attention.

Daydreamer.

Doesn’t listen in class.

Must try harder.

Needs a better grasp on reality.

Will never amount to anything.

Meanwhile, he filled oodles of notebooks with doodles and stories.

When school was done he closed the book on their chapter, and created his own reality with a best-selling fantasy series, making more from the movie rights than all his teachers combined.

Why couldn’t they see beneath the negativity of their comments to read the prediction in their words?

 

Of course, not all daydreamers become successful, and not all children have a negative schooling experience. For a much more appreciated and positive set of comments, read this post by Elizabeth on Autism Mom Saying Goodbye to Elementary School.

Thank you for reading. I appreciate your feedback. Please share your thoughts.

Is the ‘right way’ always the best way?

Giving children opportunities to question, to be creative, and to problem solve are high on my priorities. Children need to be given the time and opportunity to figure out things for themselves. While it is sometimes easier just to tell or show them what to do, or even do it for them, it is generally better for their development, to let them have a go at finding a method or solution. Please note: I am not talking about dangerous things here like playing with fire, testing to see how fierce that dog really is, or driving a car.

If children are constantly told there is a right way of doing things, they will stop exploring, discovering, and inventing their own or new ways of doing things. This is an issue because, if we always do what we’ve always done, we’ll never progress. There is generally no harm in, but much to learn from, each successive attempt.

Opportunities to explore, discover, and use intuition are also important to the development of mathematical thinking. When children are developing understanding of number, they often invent their own strategies for working with numbers. Sometimes, as attested in this paper by Heirdsfield, Cooper and Irons, the strategies used display more advanced thinking, and are more efficient, than those taught as ‘the’ correct way of solving a problem using pencil and paper.

I have noticed a change in the speed and agility with which my seven-year-old grandson works with numbers now that he has learned there are certain ways of; for example, adding two numbers. He tends to second-guess himself as he attempts to mentally calculate using the pencil and paper method he has been taught, rather than other more effective strategies he had previously invented and used. Perhaps you have noticed something similar.

Provocations, such as these 3 Fun Inquiry Maths Activities for the Last Week of School by Steph Groshell on Education Rickshaw,  are great to get children thinking about different ways of solving real problems.

Little Koala’s Party – a story for problem solving in the readilearn mathematics resources also encourages mathematical thinking and planning. Children help Little Koala organise a party for her family and friends, deciding who will be invited, the number of guests, and what’s on the menu. The suggestion is made that children plan a party of their own and they are asked to consider how they would go about it. The discussion and sharing of ideas, rather than the imposition of one ‘right’ way, is the important thing in developing mathematical thinking.

Now it might seem a stretch to tie this in with a piece of flash fiction, but I hope you’ll be able to follow my thinking through the mist and into the light.

This week at the Carrot Ranch, D. Avery took the reins from Charli Mills and challenged writers to in In 99 words (no more, no less) write a story that symbolically, mythically, mystically, or realistically involves dawn, as a noun or verb. Write about the dawn of time or the time of dawn, or the dawning of an idea. As always, go where the prompt leads.

The right way

Father and Son sat side by side. Father cracked his knuckles and sighed repeatedly while Son sharpened his pencils, each pencil, and arranged them meticulously according to undisclosed criteria.

“Come on. Just get it done. Then you can play.”

“I’m thinking.”

“Think faster.”

“I know it’s 96.”

“Well write it down.”

“Sir says I have to do the working out.”

“Then do it.”

“I don’t know how.”

“Like this. See.”

“That’s not how we do it. Sir says…”

“Then do what Sir says.”

Slowly it dawned on Dad: Sir’s way may not be the best way for all.

Thank you for reading. I appreciate your feedback. Please share your thoughts.

 

Easter holiday wishes – Readilearn

 

This week I have uploaded two new resources which are just as suitable for Easter holiday fun at home as they are for learning in the classroom.

Whose egg? A logic puzzle can be used with the whole class to introduce children to the steps involved in completing logic puzzles; or as an independent or buddy activity if children already know how to complete logic puzzles on their own.

Three friends, three eggs, and three baskets. But which friend has which egg and which basket?

Children read the story scenario and the clues, then use the information to deduce which friend bought which egg in which basket.

Great for reading comprehension and creative thinking; and for collaboration in a paired activity!

Continue reading: Easter holiday wishes – Readilearn

Delivery – just in time for Easter! – Readilearn

 

Many children around the world eagerly await the arrival of the Easter Bunny and his delivery of coloured, candy, or chocolate eggs or toys. The Easter Bunny has been delivering his gifts for more than three hundred years.

When Europeans arrived in Australia a little over two hundred years ago, they not only brought the Easter Bunny tradition, they brought real rabbits as a food source and for hunting. Cute little rabbits, you may say, but the rabbits were quick to breed. Without any natural predators, they soon became widespread, and created an enormous environmental problem. They contributed to the destruction of habitats and the loss of native animals and plants. They also became a serious problem for farmers.

One of the animals that suffered as a result of the introduced species is the bilby, a now vulnerable marsupial, native to the deserts of Central Australia. The cute bilby with its long rabbit-like ears and cute face is considered a possible native substitute for the Easter Bunny in Australia.  Chocolate makers and other organisations used the idea of an Easter Bilby to draw attention to its plight and to the Save the Bilby Fund, established to help its survival. (Check out the Save the Bilby Fund’s free education resources.)

This week I have uploaded some new Easter resources featuring bilbies. I hope you and your children enjoy them.

Continue reading: Delivery – just in time for Easter! – Readilearn

How do you know?

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Being able to verbalise the steps taken to arrive at an answer or a conclusion is a valuable skill. It is particularly important with mathematical thinking, even at beginning stages. Sometimes it feels as if the answer appears without the need for thought. Of course we aim for this with automaticity of number facts and times tables. However, I am referring to problems that may involve two or more different calculations.

maths problem

Unless one understands the processes involved in solving one problem, it can be difficult to apply the same strategy in other situations. Sometimes it may be difficult for a teacher who finds the answer effortlessly to understand a learner’s confusion leading to mis-steps and miscalculations.

I generally collect my six-year-old grandson (G1) from school once or twice a week.  Now that he’s a big year one boy he thinks he doesn’t need to hold my hand to cross the road. However, there is a bit of traffic around his school and I prefer him to do so, telling him it’s for my safety too.

holding hands to cross road

One day I, erroneously, told him he needed to hold an adult’s hand until he was ten. (Ten is closer to the age for riding a bike unsupervised.) While this post is about mathematical thinking rather than traffic safety, if you wish to do so, you may find some information about traffic safety with young children here.

Always on the lookout for teachable moments that encourage thinking, I then asked if he knew how many more years it would be until he was ten. It was a bit of a redundant question because, of course he did. He is an able mathematician, just like his dad who constantly extends his ability to compute and think mathematically.

When G1 didn’t answer immediately, I assumed that he was either ignoring my question as it was too easy for him and not worth answering; or that he was thinking of the implications of my initial statement about holding an adult’s hand until he was ten.  Then again, perhaps teacher-type questions from GM are sometimes better ignored, particularly after a day in school. I was happy to accept his silence as we continued across the road and didn’t press him for an answer.

Once safely on the footpath he said, “It’s four.”

“Know how I know?” he continued, pre-empting my question (he knows his grandmother well).

He held up both hands. “Because 5 and 5 are 10.” (He put down 4 fingers on one hand.) “And that’s 6. And 4 more makes 10.”

6 + 4 = 10

Although I hadn’t asked how he knew (on this occasion), I was pleased he was able to tell me, even though, in reality, he “knew” without having to work it out. On previous occasions when I had asked him how he worked it out, or how he knew, he hadn’t always provided an explanation. He may have shrugged, said “I don’t know” or simply ignored my question. Being able to explain his thinking demonstrates his growing mathematical knowledge and metacognition.

The ability to think through and verbalise steps is important to understanding. How many of you talk yourself through steps of a procedure you are following? I certainly do at times. While knowing that six plus four is ten is an automatic response for most of us. It wasn’t always so and we needed a strategy to help us understand the concept and recall the “fact”.

Opportunities for mathematical conversations occur frequently in everyday situations but are often overlooked. Recently I described some ways the children and I discussed different ways of combining five of us on an outing to show that 3 + 2 = 5.

maths in the car

Recently when two grandparents and two grandchildren were travelling together in the car four-year old G2, without any prompting from me, began describing how we could be combined to show that 2 + 2 = 4, for example

2 adults in the front, 2 children in the back, that makes 4. There were many different ways that we could be combined; and just as many that would group three together and leave one out, for example

1 driver and 3 passengers.

We had other mathematical discussions during that car trip. I had cut an apple for each child into a different number of pieces. Each was to guess how many pieces there were. G1 went first. I’d introduced him to prime number just once previously when I’d cut his apple into seventeen pieces so I didn’t expect him to be proficient with them. He certainly wouldn’t have been introduced to them at school at this stage.

These are the clues I gave him, the guesses he made and my responses supporting his growing understanding. I thought he did very well. He requests a guessing game each time I cut apple for him now. It’s sometimes a challenge for me to think of new clues.

pieces of apple

 

I had cut 15 pieces for G2. G1 helped her work it out when I told her that she needed to count all of her fingers and the toes on one foot.

For an additional challenge I asked G1 if he knew how many fingers and toes there were in the car all together. He thought for a moment before giving the answer. Then proceeded to tell me how he knew as I started to ask. He explained that he had counted in twos because each of us had twenty, and counting twenties was just like counting twos but they’re tens. A quicker and more effective way that adding on twenty each time which I may have suggested he do.

Asking how do you know or how did you work it out helps children think about their own thinking. Listening to their responses helps adults understand where they are in developing mathematical concepts. Asking questions about their thinking can challenge and extend them further, but it is important to not expect too much and to support their developing understanding.

What maths did you engage in today? Did you even realise or was it automatic?

Thank you

Thank you for reading. I appreciate your feedback. Please share your thoughts.

I’ll just look that up: lazy or smart thinking?

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I often lament that I was born too soon. I love what can be done with information technology and am constantly learning more of its uses. I am in awe of what can be achieved with the aid of digital tools. Unfortunately, some things, because they can be done, are vastly overdone.

I remember being at an education conference in the early 80s, at the time when computers were becoming more common in classrooms and in homes. A presenter at the conference excited us about the wonders of digital technology and its ability to ease our work load. He predicted that computers would be used to do so many of our menial tasks that by the year 2000 we would have so much spare time we wouldn’t know what to do with ourselves.

His prediction, in my opinion, was way off the mark. While the use of computers has greatly enhanced our capabilities and our knowledge, it has also driven us to constantly do more. I notice this particularly with data collection in schools. If it can be quantified, then it must be quantified, analysed and compared. Whether what is being measured is of any value is of no consequence. If it can be done, it will be done.

1985 was the year I first used a computer at school and bought my first computer for home. They were Apple IIe (huge) desktop computers. I can remember how excited I was to be able to program little games using BASIC coding language. There were some good problem solving software programs on floppy disks I used with children at school; and my son played his way through the series of Ultima role playing games.

Back then, the thought of having a computer that could be carried in a pocket was as absurd, to the general population, as having a phone that wasn’t attached to a wall. Fortunately, there were others with imagination and vision who were able to make these things a reality.

I recently read a post on Daniel Willingham’s blog Science and education entitled The brain in your pocket. The brain that he is talking about I carry in my handbag. In fact, I often have two with me, though I’m not convinced that this supports the theory that two brains are better than one.

In the post Willingham shares some observations about our current use of technology to replace thinking. He suggests that, as thinking requires effort, people don’t like to engage it and find ways to avoid it. He referred to this as “miserly thinking”, a term coined by Robyn Dawes in the 1970s.

Two strategies for avoiding thinking include use of:

  • Memory – repeating earlier actions or doing things the way you’ve always done them; such as ordering the same item from a menu or always travelling the same route to work
  • Associations – using strategies that worked in other similar situations.

Willingham then shared this question as a test of miserly thinking:

bat and ball question

He suggested that if you, like many others, answered 10 cents you were using miserly thinking triggered by the word “more”, immediately thinking that subtraction was required and not checking to see if the answer was correct.

He used this to demonstrate a similar effect that use of the internet has had upon thinking in recent years, when answers to many questions are just a motion (click, tap or swipe) away. In more recent years of course nearly everyone is carrying a smartphone in a pocket or handbag, that is, if it’s not grasped firmly in hand. This easy access, he suggests, makes people less reliant on their own memories as they look to the internet for a quick answer.

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Research cited by Willingham suggests that a higher use of the internet for answers reduces the ability to solve problems like the bat and ball question shown above. He says thatpeople who are more cognitively miserly are more likely to search information out on their smartphone.” He adds that “The reason is not clear. It may be that low-cognitive-ability people seek information—look up a word meaning, calculate a tip—that high-ability people have in their heads.

He asks the questions:

daniel willingham questions

What do you think?

I love having the “external memory” that I can use to find out what I want to know when I want to know it. Previously I would have had to remember to look it up at some other time. If I didn’t know the right question to ask, or the appropriate term to look under, I may have been left in the dark forever.

I love being able to spell check my work or check a dictionary or thesaurus to confirm that I have used a word correctly. These things enhance my knowledge and improve my skills. I am a bit peeved at the idea that it is those with low cognitive ability who look things up. I thought it was a smart thing to do!

Do these actions make me a miserly thinker and decrease my cognitive skills? I really don’t want to think about that, but I sincerely hope not!

I first became familiar with Willingham’s work through his book Why Don’t Students Like School?: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom. I thought I had mentioned it in a previous post, but if I have, my search feature, upon which I am reliant, couldn’t find it. I’ll have to remember to do so in the future!

Thank you

Thank you for reading. I appreciate your feedback. Please share your thoughts.

 

@cesarharada, Encouraging innovation and problem solving through science

Cesar Harada

I found this TED talk by Cesar Harada totally engaging. Cesar, who describes himself as half Japanese half French, teaches science and invention to students from aged 6 to 15 at the Harbour School in Hong Kong.

Cesar opens his talk by explaining that, when a child, he was allowed to make a mess, but only if he cleaned up after himself. As he grew up he realised that he had been lied to: adults make messes too but they are not very good at cleaning up after themselves.

He closes his talk by suggesting that children should not be lied to. He says,

“We can no longer afford to shield the kids from the ugly truth because we need their imagination to invent the solutions.”

He then adds,

” we must prepare the next generation that cares about the environment and people, and that can actually do something about it.”

In between he describes some scientific thinking and inventions made by his students to solve local problems initially, then problems that affected other kids remotely, and finally problems that have a global impact.

I’m sure that you too can only be impressed by the learning and the positive actions that are being undertaken by these innovative students and their inspirational teacher. When I hear (true) stories like this, it certainly gives me hope for a better future.

I hope you enjoy Cesar’s talk as much as I did.

You can also find Cesar on Twitter.

Thank you

Thank you for reading. I appreciate your feedback. Please share your thoughts.