Maths is Creative Thinking: Measuring Time

Maths involves creative thinking.

When kids are given opportunities to discover this and why creative thinking is important to mathematical thinking, even those who might harbour negative feelings, become much more open-minded towards maths. They start to rethink and understand that maths is not about getting an answer.  It helps them to understand that mathematical thinking is about coming up with your own ways to try to solve problems. It is about creating your own strategies and then reflecting on their effectiveness. It involves analysing strategies to find their pros and cons. 

Often, in traditional maths learning classrooms, children are spoon-fed strategies to replicate and then repeat. That robs them of rich opportunities to create their own strategies and to really delve into the concepts being explored. When we give children opportunities to creatively come up with their own strategies (and allow those strategies to be mad or ineffective) we help tear down those misconceptions that there is only one way to solve problems. We also help tear down feelings of apprehension some children harbour fearing they are not doing the 'right' thing. 

When we ask children to be creative in maths, we are giving them the much needed freedom and opportunity to explore the way their mind works. 

That's important.

We all think differently and we should be encouraging children to develop their own thinking processes, not programming their brains to think a certain way.


To begin our investigation into ways we can measure elapsed time, we used a Padlet and shared our initial understandings by answering the question: 

How and why is creative thinking important in maths?



Padlet is a wonderful way to give a voice to everyone in the class, not just those courageous enough to share their thinking.

After ideas were shared, we were asked to find someone else's thought that made is think about it differently then we had. 

I like this strategy for a few reasons:

1. It encourages children to actually read and think about others' thinking.
2. It helps validate the thinking of class members and helps them feel their thinking is appreciated.
3. It helps generate deeper levels of discussion. When a student shares another students' idea, we all become more involved in analysing it.
4. It is a more active way of sharing thinking. When we discuss orally, some children miss out of the initial and then struggle to catch up with what is being discussed. Using the Padlet, they can read and see what thought is being discussed and this helps them to think about it more.


We then used another Padlet to share and discuss our understandings of:

How does visualising help us with mathematical thinking?



These thinking and discussions helped 'set the scene' for the sort of thinking and investigation were about to do.


We then looked at the following problem:





Using the phrase 'How many different strategies.........' instantly makes the enquiry open-ended. There are now many possibilities and that is what we are valuing.

To help us understand what we are valuing, I explained how the strategies we create do not have to be the most effective. In fact, we should try to challenge our thinking by also creating mad, ineffective strategies because they will help us with our thinking. We can create strategies that will take a long time to help save. That's alright.  Let's just explore different possibilities.

We used the think-pair-share routine.

Children had 10 minutes creating different strategies on the paper.

After 10 minutes, we discussed what we thought about creative thinking.

Some very in depth and interesting reflections emerged such as:

° Before I thought creative thinking was just about doing something in a fun way. Now I understand that you actually need to think really hard and deeply about something. 

Everyone agreed that creative thinking does require 'hard' and deep thinking.

° I think it's interesting that creative thinking is a balance between hard thinking and fun thinking. 

° I get different answers with the different strategies I created and I'm trying to find out why.

This last comment is exactly where we want children to be: without being prompted, find reasons why a strategy is working or not. We all agreed this was a great thing she was doing.




I then explained we were going to do a 'silent gallery walk' to see what others were experimenting with and exploring as possible strategies.

Why do we think we are going to do this?

- So we can see how creative we can be?

- We could be inspired by other people's ideas and use them?

That's interesting.  Do we think it is alright to use other people's ideas?

- Yes, because we might see an idea and be able to change it to make it even better.

- I don't think we should just copy someone else's idea though. It's the same as stealing their thoughts and pretending its yours.

- But we can be inspired by someone else's idea and see if we can be even more creative with it.

-Its similar to what we did last week with creating logos. We learnt that designers often collaborate and share all their ideas to make a great idea together. If we see someone's strategy we like, we aren't speaking it. I think we are becoming a team with that person. Even if they don't know they are part of the team.


With those ideas we agreed that the purpose is to be inspired. If we find an idea interesting, we should try to experiment and play with it.

Some of our creative strategies:

So many unique and interesting thoughts can emerge when we give children the freedom......

We then silently walked around examine the strategies being created. When children found something inspiring, the returned to their seat and continued creating different strategies to solve the problem.  

After another 10 minutes had passed, table partners then shared their strategies with each other. Some rich discussions took place as they analysed the strategies and the reasoning behind them. The children naturally evaluated the effectiveness and visual creativity without being encouraged to.


To expand our creative thinking further, we then looked at our next task:



The new strategy created needed to have an element from each person. Some initially felt this was too tricky, but after time we all found possible solutions to do this. 


We then shared our new strategies on the data screen.  We explained each person's strategy and then how they were combined.  Their classmates were encouraged to ask them questions about the thinking process rather than the product and that generated some interesting perceptions on what creative thinking entails.




Partner Sample:

Partner A's:

Partner B's:

                         

New Combined Strategy:

Another Pair showing each's strategy and then combining it: 

We then wrote a quick reflection using the following prompt:


Here are some excerpts from our reflections:

° I feel like this is the first time I've ever really understood what creative thinking is REALLY about. I used to think it was just fun thinking, but I know now that it involves hard and complex thinking too. 

° ...we can even take the easiest word problem and make it super complex when finding ways to solve it. 

°......visualising helps me make sense of what I am trying to work out in my mind.

° This helped me discover that I can create many different ideas and I feel more successful as a mathematician. 

°......helps me make more sense of what I'm doing.

° .....When I kept hitting road blocks with the strategies I was trying to make, it made me think of ways I could solve them. Actually, so much thinking went on in my mind doing this. I was amazed that my mind could handle them all so well. 

°I like how there isn't any pressure when we are creating different strategies. Its fun and challenging at the same time.

Behold! A New Year is Born!

How the Year 1 B.D was created.......

I love how creative children can be when given opportunities!

We've been looking at whether the year zero exists or not for a few weeks now.  Today the concept of
1 BC and 1 AD without a year zero came up again when we were finding ways to measure elapsed time from 2105 to 2637BC   (a student had found out that was the year a lunar calendar was developed).



We discussed whether we need to add a year between the year 1BC and the year 1 AD.
We had lots of different theories about what we should do when suddenly a student had a creative solution to our dilemma.

Let's combine 1 BC and 1 AD to make the year 1B.D!!!

 B.D?

= Before Domini.   So the second before Jesus was born is B.C and the second he was was born it becomes A.D (Anno Domini- Latin for 'In the year of our Lord')

So if your birthday was that year, you could say, "I was born in 1 B.D"

Plus, we just solved our confusion of that gap of time between 1 BC and 1 AD. It's now the same.

We wondered if we should contact the maths department at Oxford University to share with them our newly created measurement of time idea.

Just then, someone laughed and showed us how B.D could indeed stand for Before Domini, but it could also stand for Birthday = Jesus' birthday. We felt for sure we had created something really great!

We then thought we should also let Oxford Uni know about our other unit of measurement of time we created last week.

How the unit of measurement called 'Twenty-oney-nighty' was formed.....

Some students figured that since we have:

° a week       =   1 week
° a fortnight =   2 weeks
° a month     =  4 weeks

......we need a measurement of time for 3 weeks.  They thought of how fortnight comes from '14 nights' then a 3 week period should be called a 'twenty-oney-nighty'.

We all loved the idea and also appreciated the need for a 3 week unit of time so it was officially approved in our classroom and we use it quite often in discussions such as:

- When are our next holidays?
- In twenty-oney-nighty  :)

Kinesthetic Time Travelling.......

At the weekend, I read this interesting article Why kids need to move to learn and thought about how I could get my kids to use their bodies to measure elapsed time. So, I came with a game which I hoped would serve the following purposes:

° provide an opportunity to mentally measure elapsed time away from visualising on paper

° give kids a chance to learn kinesthetically

° take learning outside our classroom

° have some fun whilst learning maths

After explaining the body movements for the game (see pic), I explained we are in the year 2015 and in a time machine.  We are about to travel back in time and you need to try to work out what year we have travelled to.  

We made silly time machine noises and I told them to travel 1 millennium back in time.  They all made a huge leap.  In their minds, they were asked to think what year they are in now.  Next, they needed to travel 3 centuries back in time. At each jump they should think what year they are at.  Then, they took 2 heel-to-toe steps back to show 2 x decades. Think: what year are you in?  Then, they tip-toed 6 years.

Giving everyone time to think, we then each shared what year we thought we had travelled back to.

Since we had travelled back 1 326 years, we were in the year 689 A.D.

We tried this a few times till I felt they understood the body movements and the thinking needed occasionally taking them back into B.C.

Then, it was time for them to take control of their own learning.  Partner A had to think of a year back in time. They told Partner B how many leaps, jumps, steps or tip-toes they needed to take, after, of course, making a silly time machine noise! When finished, they asked Partner B what year they thought they were in.  They both then discussed whether both of them had been successful and if not, what had happened. 

It became an interesting observational assessment as I noticed which students were confident in heading way back into B.C, which preferred to travel back only a few hundred years and which were confident in taking the numbers below the thousand or hundred mark easily.  

After a regroup to share strategies or difficulties we might have faced, I asked them to come up with a way we could play the same game but to help us practise measuring elapsed hours and minutes. 
They came up with the body movements (see pic).

This time though, we moved forward in time. 
We practises a few games together till I felt they were confident in doing it with their partners.

Eg, It was 2:45 and they took 3 leaps forward.  Think in your mind what time it is now.  Next, take 3 jumps.  Think: what time is it now? Now make 4 heel-to-toe steps. Think: what time is it now? Finally, move 4 tip-toes.

Everyone shared what time we thought it was and when we agreed we practised another game. 

After they played the game with the partner, we reflected on strategies we used in our minds.  A few children shared that they felt the number line practise we have been doing in class recently has really helped them to visualise when measuring time and so it is becoming a lot easier for them.

As we were finishing off, one student suggested we try an even bigger challenge by combining the years and the hours!! 

You can't ignore excited cheers, so we got back in our 'time machines' and travelled back 2 453 years and 2 hr 45 min.  We did it slowly and we all arrived at the same time!  

I wondered if we had done this on paper, whether we would have had an equal 100% success rate at such a complicated problem to solve.  I don't think we would have.  There must be a lot of truth in how engaging and helpful to learning moving our bodies is especially with maths. 

I also reflected on how I doubt I would have ever thought of combining the two games: years and hours, but when you constantly give children the control over their own learning, they can take themselves to really interesting and creative avenues to challenge their own thinking.  

Measuring Elapsed Time- Student Created Formative

Student-Created Formative Assessment: 

 Measuring Elapsed Time In My Life

Lead In:

We began together discussing the following:

° Do you think you spend more of your awake time 
with people from school or with your family in an 
average week?

° Do you think you spend more time reading printed 
materials like books or screen reading like on our laptops 
or ipads in an average week?

° How many hours a week do you think you spend at a 
table eating meals?

We shared our predictions and then possible strategies we could use to find out. 

We then brainstormed other possible questions we could investigate and came up with the following:

 ° Compare the amount of time you are awake to the amount  
of time you are asleep in an average week?

° What is the amount of elapsed time from when you leave 
home in the morning till when you return home on a usual day?

° Compare the difference in time between
a) The time you finish breakfast and begin lunch
b)   The time you finish lunch and start supper/dinner

° Estimate the amount of time you spend reading in a 
usual week.  (Include your reading time at school and at home)

° Do you spend more time at school or at home during a work week (Monday to Friday)?

With these ideas, we were ready to investigate...........

Investigation:

We had a choice of which to investigate and needed to present our mathematical process clearly so someone could easily understand what we did.

Most of us chose to use the 'mountain-hill-rock' strategy to show the measured elapsed time and to make it clear for the reader, most chose to include a key. 

Reflection:

We shared with a partner our investigation and asked them for feedback over how we approached the investigation.  

Formative Assessment Feedback:

I like to use Learner Profile attributes to assess and provide feedback to my students maths.  I like to think it takes away the 'tick / cross' mentality maths is usually given and instead give the thinking and the creativity involved more merit than whether a question is answered correctly or not.

A sample rubric and short feedback comment below:

Measuring Elapsed Time Investigation                  Mathematician: XXX

	Emerging	Developing	Consolidating	Expanding
Communicator: Ability to visually show mathematical thinking in a clear manner
Communicator: Ability to clearly explain thinking process (and neatly so it is easily communicated to the reader)
Risk-taker: Willingness to select an investigation that will challenge according to own level of understanding

XXX, you showed you able a good ability in investigation a situation in a logical manner.  Taking time to look back at your question and double-check you have answered it correctly is an important step when we are thinking mathematically.  Thinking about using a number line to visually show the maths could have been a helpful strategy, though I do understand that you prefer to work things out in your mind because that is easy for you. Sometimes we need to try to be creative in how to present our thinking visually.  

Experimenting with Strategies to Measure Elapsed Years......

 Lead In:

We began by brainstorming some    

of the key inventions we have learnt about that have helped shape our measuring of time system we have today. (see pic)

We then jumped onto our laptops and found out when 5 or 6 of these time measuring devices were invented and to challenge our thinking by including a few that were invented in B.C.

We then used placed these on a time line and thought about how they need to be spaced out.  A few of us continue to wonder about how B.C works on a number line and another offered a wonderful memory hook:  Think of B.C and A.D like a mirror reflection! 

Another helped by explaining how our measurement of years time line is symmetrical- the numbers on for infinity in both directions! Indeed they do. :)

Student-Created Investigations:

When we were satisfied that our time lines were well spaced out (quite a few us decided they wanted to redo them at home), we then did the following:

1. Choose any event on your timeline.

2. Create a number line and show a mental strategy you can use to find out how many years ago it was.

3. With the same year, repeat but use a different strategy.  can you find a different friendly number to experiment with?  Moving back from 2015 or moving forward from the given year?

4. Look at the two strategies and then evaluate which you find easier to do in your mind by circling it.

5. Repeat for other events on your timeline.

By doing this, I was hoping they would explore how we can create different strategies and that for some, perhaps they might discover a more effective mental strategy than what they usually use.  It was also hopeful that they would take risks by experimenting with different friendly numbers. A lot of the kids did try this and would sometimes remark how they had made a great discovery with a particular friendly number such as 1950, or how silly their strategy had become by trialling a particular year as a friendly number.  When that happened, we discussed how when we try something and deem it to fail, it is actually a success because we then see even more benefits in the other more effective strategy we could use.

Most of the children created timelines that showed many different events such as the invention of sundials, when the Mayan calendar was first used, the introduction of the pendulum clock in Europe, the first known water clock in Ancient China etc

This student (pic) decided to find out key events only in the history of the Ancient Egyptian calendar. Such an interesting approach! She explained that she is fascinated by Ancient Egypt and was happy she could link that personal passion in with her maths learning. :)

After investigating these strategies for how many years ago their events occurred, we then measured the distance in time between events on our timelines.  

Eg, How much time had passed between when the lunar calendar was invented in Scotland till when the candle clock was invented? etc

We repeated using number lines to show how we could mentally measure the difference in time and again explored two strategies for each question and then circled the strategy we found the easiest to do in our minds. 

Reflection:

We paired up and shared our thinking, explaining why we felt the strategies we circled were easiest for us to do mentally and also explained some of our experiments with trying to identify friendly numbers that might have turned out unexpectedly unfriendly and why that was so.

Assessment Feedback Using Learner Profile Attributes:

Measuring Elapsed Time Investigation                                  Mathematician: Patrick

	Emerging	Developing	Consolidating	Expanding
Thinker: Able to identify multiple strategies for measuring elapsed time in years, decades, centuries and millenniums
Communicator: Ability to visually show mathematical thinking in a clear, neat and well-spaced out manner

Patrick, you showed some good thought in spacing out your timeline, though you should have rethought how close the year 1656 is to year 1 compared to how close you had placed 600 BC.  However, your decision to spend extra time redoing your timeline so it was better spaced out shows your wonderful attitude in wanting to achieve your best in your learning. It’s a great attitude you have! You show you have a strength in determining how to use different mental strategies and analysing which you personally find more effective.  It was interesting to see you taking risks by experimenting with different 'friendly' numbers like you did with 1850 and 340 B.C even though, like you said they sometimes became 'unfriendly' in your mind, it is important to experiment and it is helping you strengthen your ability to measure elapsed years more confidently. A very good investigation and an excellent effort in presenting your thinking neatly. J

Which Strategy is Better? - Measuring Elapsed Years

Lead In:

Suppose you're reading a website and it explains that WWII began in 1939.  

What different strategies could we mentally use to work out how many years ago that was?

Children volunteered to come out and show us the strategy they would do in their minds. (see pic)

We found out most of us have a preference for starting at the year in history and measure 'upwards' to the present year.  A few of us though prefered to start at 2015 and subtract backwards.  When asked why we thought that is less popular, some suggested it might be because we are stronger at adding mentally than subtracting.  We also looked at the strategies shared and talked a bit about finding 'friendly' numbers to make life easier.  1950 was a a good example of a 'friendly' number to jump too.

Investigation: Is there a better strategy to use mentally?

Some years were displayed on the data screen in two columns: some a bit easier and some a bit more challenging.

                        Easier:   How long ago was:

° 1935 ° 1965 ° 1924   ° 1895 ° 1845      ° 1942

In how many years will it be: ° 2060 ° 2100

More Challenging: How long ago was: 

° 1933 ° 1977 ° 1865 ° 1845 ° 1845             ° 1755

In what year will you be ° 50 years old ° 85 years olds

Using mini-whiteboards and in pairs, they chose a year to measure how long ago it was.


Partner A explained to Partner B the strategy they would use mentally to work out. Partner B needed to listen and write down what Partner A was explaining to them. Then, on their other whiteboard, Partner B explained to Partner A how to measure how long ago that same year was, but using a different strategy. The idea behind this was they would explore two different mental strategies to solve the same question. After they explored the two strategies, they then determined which they felt was easier and more effective for that particular year. Then, they repeated for another year displayed.

This pair explored whether it is easier to start back and move to the present OR start at 2015 and move back.

This pair also explored whether it is easier to move forwards or backwards along the number line and they experimented with using 1940 as a 'friendly' number to help solve mentally.

This pair were curious about an earlier strategy shared at the beginning of our inquiry and thought they would give it a try.


Afterwards, they discussed their difference in preference. One girl explained that she doesn't like mentally seeing numbers visually on a number line and prefers to see them calculated like we do sometimes on pen and paper. Interesting......

Reflecting:

Afterwards, we regrouped together as a class and discussed:

° Did we discover any new strategies that we might find useful?

° Did we discover any strategies that might not be so useful when mentally working them out?

° Did anyone's initial preference change regarding starting at the year or starting at 2015?