Archive for the ‘Maths’ Category

Why learning tables off by heart is rubbish

October 23rd, 2011 liopge
rotelearning

As a conclusion to Maths Week 2011 and as an aside to this year’s Tables Tips Twitter Project, I am resurrecting an article about learning tables off by heart.  This post originally appeared on Anseo.net back in January 2008.  Almost 4 years later, I wonder if opinions about learning tables has changed.  Read on and see what you think!

A few days ago on Education Posts, I proposed that learning tables off by heart is rubbish. My proposal was met with general disagreement and in some cases, complete contempt! I decided to prove my point by asking teachers to learn a few sentences off by heart just like children are asked to learn number sentences off by heart.

They didn’t know the significance of them, exactly like a child doesn’t know the significance of learning tables when told to do so. So here’s the sentences. If you want to take part in the challenge, don’t read on after the sentences until you’ve tried learning them off by heart.

Fred Davidson lives in Aaron Zion Avenue
Greg Fredson lives in Aaron Clare Avenue
Isaac Davidson lives in Aaron Clare Avenue
Fred Davidson works in the Bill Davidson Building
Greg Fredson works in the David Bill Building
Isaac Davidson works in the Clare Fredericks Building

It was interesting to see how many people accepted the challenge.  There were 32 responses to my challenge but only 7 actually took it on.  The others used the thread to give their opinion on my hypothesis.  There was a strong sway of disagreement with me (57%) and only about 22% agreed.

The general views of those who disagreed were:

  • There is no other/better way to teach tables
  • The sentences have no relevance to tables
  • Learning tables off by heart did me no harm
  • Call me old fashioned but…
  • We shouldn’t spoonfeed children

Of those who attempted the challenge, only one out of seven considered it easy.  The rest who found it difficult gave the following responses:

  • I got muddled / scrambled / confused
  • My head got exhausted
  • I had no motivation / interest to learn them
  • I expected them to be easy to learn but they weren’t
  • They weren’t important to me
  • It was frustrating
  • I’m too busy

It was also interesting to see how some teachers recorded ways they tried to learn the sentences.  Some said they tried looking for patterns in the sentences. Others made a story up (e.g. similar surnames became part of a family) and finally others grouped similar names together to try and find patterns, etc.

Next, before the reveal, I’d like to compare how learning tables is very similar to having to learn those sentences off by heart.

  1. When you give children tables, many of them will experience feelings like those expressed by respondents, being muddled, confused, seeing no point, no motivation, etc.
  2. Perhaps the majority of teachers are good at learning things off by heart, due to the Leaving Cert relying heavily on this skill. To become a teacher, you have to score very high points in the exams. However, perhaps most people are not good at learning things off by heart.
  3. The six sentences below have complete relevance to learning tables. In fact, the sentences below represent 6 number facts. They seemed meaningless to many of you because they have appeared in a new way. A child comes across tables initially as something new too.
  4. Every word below is also a real word and you can explain what every word means. Putting them together, however, they don’t seem to have any pattern. Likewise, every number in a tables fact is also known to a child but put them together and they don’t seem to make sense or be very interesting. The reason some of you weren’t bothered learning the sentences off is the same reason a child wouldn’t either.

Here’s how the sentences below are, in fact, tables facts.

  • Any of the names represent a number. E.g. Aaron=1, Bill=2, Clare=3 and so on. Z words represent zero.
  • Lives = Plus
  • Works = Multiplied by
  • Avenue = Equals
  • Building = Is

So to translate:

Fred Davidson lives in Aaron Zion Avenue (6 + 4 = 10)
Greg Fredson lives in Aaron Clare Avenue (7 + 6 = 13)
Isaac Davidson lives in Aaron Clare Avenue (9 + 4 = 13)
Fred Davidson works in the Bill Davidson Building (6 x 4 = 24)
Greg Fredson works in the David Bill Building (7 x 6 = 42)
Isaac Davidson works in the Clare Fredericks Building (9 x 4 = 36)

As far as I would be concerned, all the responses in this discussion represented exactly how a child feels when given a list of tables to learn. Just because I represented my tables in words doesn’t mean they should have been any more difficult.

The people who looked for patterns, made relations and grouped similar sentences were all using strategies to help them learn. If you can give your pupils strategies like that, e.g. David Fredson also lives in Aaron Zion Avenue (commutative property), you’ll see the benefits straightaway.

My conclusion to this is that we need to accept that teaching strategies is far more beneficial than simply learning tables off by heart. No doubt, some will still disagree and I’d be interested to see your reasons, to which I’d be glad to respond.

Posted in Maths 1 Comment »

Maths Week Tables Tips

October 21st, 2011 Simon
vintage-table

Throughout Maths Week, teachers around the country were asked to give tips for learning multiplication tables using the hashtag, #tablestips, on Twitter.  We had a good response with 16 teachers tweeting lots of different tips.  The most popular tip involved the 9 times tables, with several good ideas to help learn them.  Other ideas included some interesting tips about patterns of 3 and 5 times tables.  I’ve compiled all the tips into a PDF document, which is free to download.  Thanks to @fboss for recommending Tweetdoc.org to compile the tweets.

Download the Document: Tables Tips.

Posted in Maths, News No Comments »

My take on the Maths Crisis

September 19th, 2011 Simon
math2

As with the Leaving Certificate results, the Junior Certificate’s showed that there is a worrying trend in maths results in this country.  Most of the focus is being centred on the estimation from the Teaching Council that over 30% of maths teachers hold no 3rd level qualifications in mathematics.

However, just because a teacher has a lot of knowledge about maths, does it make them a good teacher of the subject?  Isn’t a qualification in mathematics less important than a qualification in teaching mathematics?  Perhaps this is where the focus should be.

At primary level, the aims of the maths curriculum include fostering a love of mathematics, problem solving and making maths all about real life situations.  None of these skills (as far as I remember from my degree) are learned in a 3rd level mathematics degree.

From http://www.flickr.com/photos/cogdog/

Gone are the days in primary schools where children used “borrow and payback” and other random unconnected rules for solving problems.  Gone are the days of Maths books with pages of meaningless sums.  Gone are the days where a teacher gives an example of how to “do” a problem and children simply “do” more of the same problems with different numbers without really understanding why.

How primary school teachers help children learn these concepts has completely changed.  We encourage children to use concrete materials as much as possible before moving towards the more abstract areas of maths.  We build upon their previous knowledge so they don’t have to learn loads of rules off by heart.  We give children strategies and make them real to them. Do we see this at second level enough?

It’s often been said that teachers who struggled with Maths in their own school days make much better Maths teachers than those who got the abstract concepts easily.  There may be some truth in that.  Perhaps, these teachers can bring their students along the mathematical journey more effectively,  from figuring out that 1+1=2 towards differential equations and giving them the stepping stones along the way.

From http://www.flickr.com/photos/mikehamm/

Engineers Ireland claimed the “crisis in maths” at second level reflects a “total systemic failure in national education structures” and it’s difficult to argue against this.  The worrying thing, as usual, is that the blame is being centred at the wrong place.   While having a base level of knowledge and understanding of maths is important, the way it is taught is much more important. It’s good to see that Project Maths is being rolled out because it does focus on good methodologies and making maths more real to students.

Perhaps if Project Maths is given a chance, we may see increases in our mathematical scores.

 

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Missing the Point about Maths

July 10th, 2011 Simon
The "Scandal" in the Sunday Times

I don’t usually comment on second level education on Anseo.net.  However, I’m fairly open about the fact that I think the Leaving Cert is a dreadful idea.  Like many others, I feel that the Leaving Cert is simply a memory exercise, where one has to – in the words of Daithí Ó’Mhurchú – regurgitate the last 6 years of knowledge.  I did my Leaving Cert in 1997 and it hasn’t changed that much.  My main memory of the Leaving Certificate was choosing the bits of the curriculum that my teachers thought were going to come up in the exams.  Therefore, in English, we didn’t bother studying any of Shakespeare’s sonnets or Paradise Lost by Milton.  In Maths, I think I skipped learning a few theorems off by heart, which brings me to my point.

The Sunday Times reported today that there is a bit of a scandal about Paper One of the Maths Leaving Cert.  I read the first words of the argument:

Paper 1 was the most difficult yet, featuring questions that…

I expected the end of the sentence to be something like “were not on the curriculum” or “a higher standard than previous years” but no.  The problem was that these were questions “that pupils could not have anticipated”.

In other words, they were questions that pupils guessed would not be on the paper so didn’t learn them.  There’s one of two things wrong here.  The first is that perhaps the curriculum for second level subjects is too broad and thus there’s too much to learn.  The second possibility is that students are not learning the full curriculum relying on predictions and rote learning based on previous years.

However, the biggest thing that is wrong is the general acceptance that not being able to anticipate questions is an issue.  The Maths paper, I believe, should test Maths ability not rote learning.  However, unfortunately, I anticipate that nothing will change.

Posted in Maths Tags: No Comments »

The Targetboard Project

July 9th, 2011 Simon
targetboard

In our May edition of Anseo.net, I wrote about how I thought one could use Google Apps to support a collaborative Mental Maths project using Targetboards.   I launched Targetboard.net and tried to drum up as much support for the project as possible before we began on June 7th.

Thanks to the support of mailing lists such as CESI and DICTAT and from the many retweets of my Twitter friends, over 50 schools signed up for the project from all parts of the country.  I had no idea whether Google Docs could support that many people collaborating at once but I lived in hope.

On June 7th at about 9:30am, I logged on to my Google account and started up the slideshow where the Targetboard would appear at 10am.  I was heartened to see two schools already logged in.  By the time it reached the magic hour, there were 7 schools frantically typing solutions to reach 20 in the five minute session.  I intermittently sent words of encouragement to the participants and the five minutes came and went incredibly quickly.

Even more schools logged in on Wednesday, Thursday and Friday.  It was brilliant to see the number of solutions gushing down the screen.  Some schools took on personal challenges of using all 9 numbers in the grid to solve the target number.  Some schools learned some tricks to get their answers in more quickly – copy and pasting a list of answers proved quite popular!  However, overall the schools seemed to get a great buzz from mental maths.  The tiny competitive streak involved in the exercise gave huge motivation.  And the prize?  Nothing. The only thing that was up for grabs was the glory of being the school to have the most unique solutions to the targetboard.

I was lucky enough to sit in a classroom where the Targetboard was going on in my school.  The classroom teacher was working with her pupils and the atmosphere in the room from everybody was electrifying.  All sorts of emotions were on show – excitement, panic and pride.  The conversations were super and the collaboration between everyone in the room was brilliant.

The teacher commented upon the democratisation the project had on her class.  Children all felt equal. They were able to work at their own level and they were able to succeed in some way at the board. Some children became pseudo-heroes for part of the week such was their talent for spotting solutions.

On my part, moderating the project proved a little more difficult that first anticipated.  While watching the project and giving words of encouragement during the project was easy, collating the results each day was time consuming and not as easy as I’d hoped.

The main thing I didn’t realise was that Google App’s sidebar for discussions is written in Flash, which means I couldn’t copy and paste the answers into a separate document to delete duplicate answers and sort them into an easy way to read them.  Unfortunately, I had to manually go through each answer and check that no one else had done the same.  Thankfully the number of participants was low enough to do this within an hour.  However, obviously this would not be something I could do everyday.

To end the project, all participants were sent web badges to thank them for their participation.  An email went out to thank them for their support and a wish for the project to happen again, perhaps during Maths Week 2012.

So what next for Targetboard?  I do hope that we can do the project again some time.  If it were to be done on a larger scale, I guess we would have to build a more automated program, which could give instant feedback at the end without the need for manually compiling the solutions.  In the meantime, seeing as I bought the domain, I might as well get some use out of it!  I’m going to expand its use from just Maths Targetboards to other grid-type mental starters in subjects such as English and Irish.

As I stated near the beginning of the article, projects like these rely completely on the goodwill of friends, colleagues and various supportive organisations.  Projects like these have no commercial value but have potentially huge educational value and thus have no budget.  I’d like to once again thank anyone who tweeted about the project, the schools who linked to the web site from their own, the CESI and DICTAT mailing lists for allowing me to publicise the event and of course all the children who gave up their time everyday for a week to ready, steady, think.

Posted in Google Apps, Maths No Comments »

Mental Maths Project

May 26th, 2011 Simon
targetboard

Last month on Anseo.net, I reported on an idea I had to use Google Apps to help with Mental Maths.  Thus, Targetboard.net was born.  I decided to buy the domain and encourage as many schools to take part in an event where we can find as many solutions to a targetboard as possible.  It’s all going to happen on the morning of June 7th just after the bank holiday.

So what does the Targetboard project involve?

On the morning of 7th June, schools will be invited to log on to http://www.targetboard.net.  They will be presented with a 4×4 grid with numbers in it and a target number below.  Schools will then have the opportunity to type in ways to reach this target number using the numbers in the grid.  Many schools do this as a warm up to maths lessons anyway but with lots of schools doing it the same time, we could find many more solutions together.

Essentially the aim of the project is to showcase another way that ICT can be integrated into the curriculum in a real and effective way.  It also showcases a simple way that schools around Ireland can join together in education.

The main thing over the next two weeks will be to spread the word as much as possible.  While this is not a commercial venture, it is very difficult to get word out to schools.  I would urge anyone who is interested to send posts to education fora, mailing lists and blogs.  The more schools on board, the more interesting the project will be.

Each day (from Tuesday to Friday) I will publish the number of schools that took part, the number of unique solutions we gathered and the name of the school that offered the most solutions.

Over the coming days, a practice targetboard will be published on the web site each school day.  So, if you can, please spread the word and let’s see if we can make mental maths social!

Posted in Featured, Maths No Comments »

Google Apps: Sharing Maths Across Schools

April 30th, 2011 Simon
shareGoogle

Mental Maths is one of the key challenges facing teachers today.  If I were to ask you to tell me what 1287 dived by 29 was, most adults would wish they had a pen and paper in front of them (or a calculator!)  What doesn’t come into our heads first is trying to mentally calculate a rough idea of the answer.  We should probably be seeing that 29 is quite close to 30 and go from there.  We need to give children these abilities, particularly in a world where rote learning is becoming less relevant.

Many teachers have come up with good Mental Maths starters in their classrooms. (I don’t mean an infamous book called “Mental Maths” which is anything but!)  One such concept is called the Target Board.  Essentially, this is a grid with numbers in each box.  The teacher sets a target number to reach and children must find as many ways of reaching that number through any numerical operation.  An example is below.

For example, children could go with 20 x 5 or 5 x 5 x 4.  The great thing about this exercise is its open-ended nature.  It can be as easy or as difficult as one wishes.

Other examples of mental maths starters in the classroom are: 24 and Countdown.

Countdown is the equivalent of the numbers’ game in the quiz show of the same name.  The teacher displays a target number and 6 smaller numbers to get that answer.

In the above example, to get 10 points, one might try (100 x 2) + (8 x 7) + 1.  Depending on how close the student is to the answer, the points lessen.

To play 24, the students are given 4 numbers and they must use each number once with any operation to make 24.  For example, if the teacher listed 4, 6, 6, 8 – a valid answer could be (6 x 8 ) – (6 x 4) = 48 – 24 = 24.

These have great opportunities in the classroom and the concept of competing with their classmates adds an extra dimension of fun to the whole affair.  However, wouldn’t it be great if they could compete with other classrooms around the world?

Google have recently added a new concept to their Google Docs, which I believe has very interesting educational value.  The concept allows authors of a document to have discussions about that document to the side of the screen.  Google’s own blog outlines how this works really well:  http://googledocs.blogspot.com/2011/03/introducing-discussions-in-google-docs.html

This new concept works in all of Google Docs applications and I’ve thought of an idea, which I’d love to try out with other schools.  Taking our Mental Maths approach, if I designed a slide with a targetboard, a countdown numbers game or a 24 game and then shared it with everyone who wanted to join in, it opens the whole thing to a much wider audience.  It could be used as a tool to get schools to compete with each other in Mental Maths.  What child in Cork wouldn’t be motivated to kick the ass of a child in Kerry in some sport – be it football or mental maths!

To see how this could work, let’s take a look at an example slide created using Google Docs:

Right now, this targetboard can only be viewed by me.  If I make it public, anyone with the link can access it.  I do this by clicking on the link, which, at present says: “Private to only me”.  A box will pop up and I decide to change this to “Public on the Web”.

 

Once this is done, I’m now free to advertise a public Targetboard challenge.  The link to the slide is given, (in this case click here to go to the slide), and you will see the Targetboard.  So far, so good – but how do you get the schools to compete with each other?  The secret is in the “View Together” link at the bottom right of the presentation.

Once you click on this, a side panel appears on the right hand side with all the names of all the users logged on to this slide.  Now each school can type in their answers to the target board.  At the end of an agreed time, the number of answers for each school are calculated and a winner is determined.

For the Countdown game, the first school to correctly type in the formula for the target number wins;  likewise for the 24 game.

Let’s say 10 schools agreed that each day at 10am, they would log on to the agreed link and compete with each other.  A league or tournament could be set up.  In fact the possibilities for fun ways to learn maths in an open-ended way are enormous.  For example, a classroom of children with their own laptops or iPod Touches could all compete.

To me, this all sounds very possible and requires very little to set up.  The only thing to do now is see if it can actually happen in real life.  If there’s any schools that are interested in trying this out, please comment below or send me an email and we’ll see if we can try it out for one week.  I’ll report our findings here and see what happens after that.  If anyone wants to organise it amongst themselves, let me know how you get on too.

Google Apps always surprises me with their tools that can be used for almost anything.  Let’s see if we can grab another educational opportunity!

Posted in Featured, Google Apps, Maths 12 Comments »

Maths Week 2010 – 9th – 16th October

September 13th, 2010 Simon
mathsweek

Maths Week is on from the 9th to the 16th October this year.  Read about it by clicking on the link below.

Maths Week 2010 – 9th – 16th October.

Our partner site, Mash.ie has a number of excellent maths related themes to play around with so have a look there and maybe add your own links.

Posted in Events, Maths Tags: No Comments »

Turning the Tables on Multiplication (Part 8)

June 8th, 2010 Simon
tables11

Over the last couple of months, we’ve seen how we don’t need to simply learn any of the tables by rote.  We’ve reduced the 121 facts to learn down to just 4: 3×3, 3×6, 3×7 and 3×8.

Now there is a strategy for all 3 times tables, which most maths teachers introduce after the four times tables rule.

Double then add the number again

It works like this: For example, with 3×7, you take the 7 and double it.  That’s 14.  Now you add 7 again and you get 21.  This is a decent rule and it works really well.  However, I found that my class got confused by it and always seemed to add 3 instead of the other multiple.  So 3×7 became 17 (14+3) a lot of the time.

We decided to do some visuals for these last 4 tables.

3×3 is the baseball team table.  There are 9 baseball players on a team, as far as we know so we drew 3×3 grid of baseball players with bats.

3×6 is our horror film.  The kids I teach know that most horror films have an 18s certificate.  So we created a movie called “Attack of the 3 sixes”  The cover of the DVD shows 3 sixes with an assortment of weaponry.

3×7 is our slot machine.  If you get three “Lucky Sevens” in the slot machine you win €21.

3×8 is our 3 egg timers representing morning, afternoon and night-a full day.  The digit, 8, kind of looks like an egg timer so this helps.  And how many hours in a full day?  24.

So, that’s it.  We’ve managed to learn lots of different ways to remember our tables.  At this stage, your students will be able to play any consolidation games for tables and there are plenty of them on the web.  I use http://www.playkidsgames.com/games/mathfact/mathFact.htm which simply gives the children a load of tables in a certain amount of time.  We then graph their results on a board.

There are loads of other strategies out there to learn tables and hopefully people will add their own ideas to the comments below.  I hope you have enjoyed the series of lessons and I had better leave with an image – the fully greyed-out tables grid!

Posted in Maths Tags: , 2 Comments »

Turning the tables on multiplication (Part 7)

June 1st, 2010 Simon
tables10

With only 10 tables left to learn, perhaps it’s time to call it a day on the strategies.  Surely, 10 tables aren’t that difficult to simply learn off by heart?  In my opinion, I think even one table to learn by rote is too many.  I feel there needs to be a strategy for every table.

You may have noticed that I’ve been calling the nine times table strategy, “the finger trick (palms down)”.  This next strategy is called “the finger trick (palms up)”.  This trick will help you know every one of these tables:

6×6, 6×7, 6×8, 6×9, 6×10, 7×7, 7×8, 7×9, 7×10, 8×8, 8×9, 8×10, 9×9, 9×10

Now we already have a number of these covered through other strategies but there’s no harm having another one, just in case.

YouTube shows a great video of this strategy.

Multiplication strategy Play Now | Play in Popup | Direct Link

You can download the flv file here if you have problems with NCTE’s filtering.

I think this video explains it better than I could.  However, for those of you who learn better by reading:

Step 1: Put your palms out and in your head label your thumbs as 6, the pointing fingers 7, the middle fingers 8, the ring fingers 9 and the little fingers 10.

Step 2: Using an example, 6×8, the left hand will represent 6 and the right hand will represent 8.  Put down the thumb on your left hand (that’s 6).  On the right hand, put down all fingers to 8, (that’s the 6, 7 and 8 fingers).  You should have your thumb down on the left hand and your thumb, pointing finger and middle finger down on your right hand.

Step 3: The fingers that you have down represent the tens.  You should have 4 fingers down altogether.  So your answer is 40 something.

Step 4: The fingers left up will form your units.  The left hand should have 4 up and the right hand should have 2 up.  Multiply these numbers together (4×2).  This makes 8 units.

Step 5: Add the 2 numbers together (40 + 8 = 48) so 6×8=48

This may seem very confusing at first and it does have a good number of steps.  However, children get used to it very quickly – give them plenty of practice and they should begin to get it easily after about 15 minutes.  The only hard ones are 6×6 and 6×7 because you’ll be adding 20+16 and 30+12 respectively.

This knocks off another 6 tables from our chart.

We’re now left with 4 tables – all of them 3 times tables.  Before I finish these tables, there’s a couple of extra visual strategies that might help if children find the “finger trick (palms up)” strategy really hard.

8×7=56 can be remembered with 5, 6, 7, 8 as in 56=7×8

8×8=64 can be remembered with the number of squares on a chess board

6×6=36 can be remembered as 6×6 ends in 6. Count the 6′s to get the tens.

7×7=49 is my American footballer from the San Francisco 49ers.

6×7=42 is the meaning of life according to Douglas Adams’ Hitchiker’s Guide to the Galaxy

The only one we couldn’t get a picture for was 6×8=48.  We eventually said it would take two days to get a picture for this one – 2 days = 48 hours?  Clever?

The final piece of the jigsaw will be revealed in the next session, with a number of images and strategies to get rid of those last 4 tables.  See you then.

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