Teaching Algebra – it’s just like fractions!

I wrote a while ago about teaching basic algebra to children and taking away their fear. But what do you do when it becomes a little more complicated?

When working with a pupil recently we came across this problem:

algebra

and the child was unsure how to start. I reminded her of when we had looked at ordering fractions and asked how she did that.

“I can’t do 3/5, 8/10 and 12/15,” she said “because they are all different, so I have to make them the same. I know 10÷2 is 5 so I can do 8÷2 and turn 8/10 into 4/5, and I know that 15÷3 is 5, so I can 12÷3 and turn 12/15 into 4/5. Then I put them into order – 3/5, 4/5, 4/5 – and then I turn them back so 3/5 is the smallest and 8/10 and 12/15 are the same.”

I praised her for remembering so well and then told her this problem was just the same. It looked hard because k, m and n were all different, but maybe she could make them the same.

As soon as she started to think of the problem in that way she was able to see that m could be changed into 3n and k could be changed into 2n, so the problem was 2n + 3n + n = 1500 or 6n = 1500. Once she had worked out that this meant that n must be 250 she had no problem at all in converting 2n back to k and 3n back to m, giving the solution k=500, m=750 and n=250.

Algebra – it’s not too hard. It’s just like ordering fractions!

If you live in north Birmingham and would like to book my services as a private maths tutor, please get in touch.

Teaching sequencing and column addition to a child with dyscalculia

Recently a colleague asked me for some suggestions to help one of his pupils with her maths. She was having various problems, such as

  • Difficulty with sequencing numbers
  • Getting confused as to which way to move on a number-line to add or subtract single digits
  • Getting confused as to which way to move on a number-square to add or subtract multiples of 10
  • Not understanding whether an answer she got when performing a calculation was “reasonable” or way off
  • Confusing the Hundreds Tens and Units columns and so not always starting in the correct place when performing calculations.

He suspected that the child in question might have dyslexia and / or dyscalculia, and if that is the case then I can understand why they might have trouble with column addition/subtraction. They’ll be concentrating really hard on left to right, left to right for their writing, and then suddenly column calculations go right to left – no wonder they get confused!

My advice was to make the learning experience completely multi-sensory, even if it meant taking the learning outside.   These were some of my suggestions:

  • Make a physical number line on the floor/front driveway/back garden/anywhere with plenty of space. Place one object with the label “1”, then two objects labelled “2”, three objects and a label 3 and so on to help her equate the number 3 with the value 3
  • Chalk the numbers outside, and get her to walk along it counting forward, and then walk the other way counting backwards.  Get her to jump along it landing on every other number counting forwards in twos and then backwards in twos.
  • Move on to a number square in chalk so that they can change direction to add on/take away 10. This should also help with “reasonable” answers because, for example, she would come to understand that she had to walk further to add on 49 than to add on 12.
    • Always make the number square start with 1 at the bottom, rather than at the top like most number squares – then the higher numbers are at the top of the square and the lower numbers are at the bottom which also helps with understanding the value of numbers.
    • Use an abacus for additions/subtractions instead of written methods. I’ve done this with Y6 children who had no concept of place value and it made a huge difference!
  • When moving on to column addition and subtraction, colour-code the numbers in each column with a known sequence of colours (eg Red White and Blue so they do red units first, then white tens, then blue hundreds). Put the numbers either on coloured card – or even better use painted wooden numbers so she can pick them up and feel the shape of each number
  • People with dyslexia tend to think in pictures, so when finally moving onto pen and paper calculations, try putting pictures of Strictly Come Dancing / X-Factor judges at the top of each column. The judges always sit in the same order on the shows, so it’s easy to picture them sitting in a row – then you know that you always have to add Bruno Tonioli’s numbers first!

One final tip I picked up at a session on dyscalculia to help children with sequencing numbers was to give them something associated with each number so that they have something to relate that number to – seeing how the number matches the object and handling the objects while they count makes it more of a multi-sensory experience.  For example if you want them to count in 6s, rather than giving them something generic like pictures of 6 spots or sets of 6 cubes, give them egg-boxes.

Teaching Reading Comprehension

Of all the things I teach, I find reading comprehension the hardest. The retrieval type questions are OK, as are the technique ones, but teaching things like inference is quite tricky. I’ve found a workaround by teaching it from the opposite direction – giving the children a piece to read where the characters are shaking or crying and asking how they can tell the character is sad, scared, etc.

It’s really hard to find good resources to help though. There are books with lots of practise questions, but if you don’t know how to answer them then no matter how many questions you attempt, you still won’t be able to.

comprehension booksAt last I have found a solution. It’s a series of books called Teaching Comprehension Strategies from Prim-Ed. They take the various types of questions: summarising, predicting, concluding etc and explain step by step how to answer each type. Each question type is split into three stages. On the first page are some multiple choices with an explanation for each choice as to why that answer is good, unlikely, perfect or impossible. Next up are a few questions with hints on where to look and how to work out the answers. To finish are questions to answer independently with no clues.

As a bonus, the books aimed at younger readers are not at all babyish, so I can use them with my struggling readers without them feeling demotivated at reading things aimed at “babies”.

I’ve found them really useful, and the feedback I’ve had from the children I’ve used them with has been really positive, both in terms of usefulness and enjoyment.

Related post: Beast Quest Comprehension

The Difference Between Teaching Children and Adults

I was once asked in an interview what I thought the difference was between teaching adults and teaching children.

There is of course the obvious factor that adults are in a classroom because they want to be there and because, for the most part, they have chosen to study the subject you are teaching. Children on the other hand are in the classroom because the law states that they have to be. You would think therefore that the adults would be more motivated.

However, children have learnt to learn, whereas adults have forgotten how. Children come to school each day expecting to learn, and they know that they will be required to put in some effort and take responsibility for their own learning. Adults arrive at their evening class tired after a full day at work and think that sitting in a lesson and just listening is the same as learning. Children are prepared to practise a new skill for a longer period of time because they know they need to. Adults tend to try one or two examples and decide that’s enough, so they don’t complete the embedding process. I’m not criticising. I’m often guilty of this myself.

So much for the difference in learning between adults and children. What about teaching methods? When I first started out I thought that teaching adults and children would be very different. I imagined that teaching adults would be a lot more serious, but this turned out not to be the case. I discovered that the more games I introduced, the more the adults engaged with the lesson and the better they learnt. Songs and video clips proved equally popular. After talking to other teachers of adults I have come to the conclusion that there really is no difference in methods that work.

So, back to the original question: What is the difference between teaching adults and children? I think the main difference is the content and context rather than style. Depending on the subject adults may require content to be more in-depth than children, or they may wish to focus on a smaller area such as handling money and paying household bills. I wouldn’t teach children how to order a beer in a foreign language, whereas this is a favourite for adults! Context for children will focus on their limited life experiences, and relate to school, playground games, holidays. Adults will be less interested in school and playtime, but will relate to the context of work, home and holidays.

What do you think the main differences are?

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What’s the Point of Homework?

Before I begin, I feel I should point out that I agree with homework. I quite often set homework at the end of a tutoring session: sometimes it’s because I can see they understand what we have been working on but I feel that they need more practice for the method to become embedded. Sometimes it’s because I want to see whether they have the confidence to work through a problem without me sitting next to them saying, “Yes” every time they ask, “Is this what I do next?” And sometimes it’s because I want to see how they do with a particular topic before I decide how, or if, I need to teach it.

Regardless of the reason, we’ll go through the homework together afterwards. If it’s maths, we’ll address any areas where they went wrong; if it’s English, we’ll look at what was good about what they wrote, and identify any areas where they could improve. But their independent work is always used as a basis for “where next?”

When I go to a child’s house to tutor, they or their parents will often ask me to help with homework. I’m happy to do that – helping children to understand what they do in school is my job. Obviously I never do the homework for them, but I make sure they understand how to approach it. Increasingly however the homework in question is some sort of puzzle such as sudoku or a wordsearch, which leaves me wondering, “What is the point?” What are the children going to learn from Sudoku? How is it embedding or improving their maths skills?

And wordsearches? So many times I have had children tell me they are stuck with their homework and saying they can’t find some of the words. I do give wordsearches sometimes –  I use them to help children focus on vocabulary or spellings, but I tell them why they have a wordsearch and what to do if they can’t find the words. Where is the learning in not being able to find the words in a wordsearch?

As I said at the beginning – I do believe homework is a useful idea….as long as it is meaningful. I can’t help but feel sometimes that children are given homework for the sake of homework.

What do you think? Do you agree with giving homework and how do you decide what to set?