Learning Mathematical concept

 

I have been running a tuition business now for over 4 years. We have children ranging from 5 years to adults needing support with their Maths, and other subjects.  

When teaching maths, I have found that our students are reliant on using methods that they are learning in school, with no understanding of how or why the method works. This frustrates me because my job and every maths teacher's job is to develop student's understanding of mathematical concepts.  My approach is to encourage students to investigate and question around using numbers. 

For many years there have been study after study that advocate the use of concrete materials for helping children to make the connections between what they already know and new concepts. Unfortunately I have rarely seen this in mainstream classrooms beyond year one. I see lots of worksheets with pictures of concrete materials, like Base 10 and Numicon, rather than actually having the children use these to learn the maths. There is an over emphasis on making sure there are lots of lovely looking exercise books with written work in them.  Research shows that when these materials, or resources you can manipulate, are used they are not used appropriately. Is this due to how teachers are trained? Most research talks about the need for these materials to be used in primary school, but as a teacher, who has taught in primary, secondary and further education I say they should be used with any student who would benefit from them. 

I have used Cuisenaire rods to investigate how many different ways there are to make the numbers one to ten. I explained and demonstrated with one and two. I then asked a child, aged 10, to carry on. He made the graphemes for each number rather than the quantities. When he started to work with the rods, the way I wanted, he was amazed that 1+2+1, 1+1+2, 2+1+1, 2+2 and 1+1+1+1 all made 4 and then wanted to see the combinations for the other numbers. 

In another lesson, I used some dominoes and asked the student to group them so there were eight dots in each group. She needed to keep counting the dots because she didn’t recognise the dot patterns or the combinations of dots that made totals. The task did not contain any domino tiles with a total above 6. The actual task was to arrange the 10 domino tiles into a square with a total of eight dots on each side. We didn’t get this far. 

A 7 year old who can real off his times tables and tell me equivalent fractions struggled to make square number sequences. I had him make them out of counters. I decided to do the same task with a 14 year student. Her confidence with maths was very low and she did not have a fluency with any calculations yet. However she loved the task today and was able to verbalise the concepts she was learning and talk about connections to previous school lessons, she had had but not understood. The notations on the table were her thought processes when finding cubed numbers. 

The benefits of using these resources are:

• Student have processing time while carrying out the task, 
• they ask questions or talk about their thinking while working, 
• They have a memorable lesson 
• They make new connections 
• and they stay focused and on task. 

Games are also a wonderful tool we use during tuition. See my next blog. 

Bibliography

Black J, 2019, Manipulatives in the primary classroom, Nrich.math.org

Szandrei J, Concrete Materials in the classroom, International Handbook of Mathematics

                   Education Pg411-434

Importance of Environment

I read this last week on another blog "Remember it's never the child's fault. it's their environment that is affecting their attitude". I have always been a firm believer that all behaviour is communication and, as the professionals and adults, we should focus on working out what that communication is trying to tell us. 

All of my private tuition has moved online due to the current restrictions in the UK. I have been very aware, over the last few weeks, of how much attention I pay to my student's behaviour when I am teaching. I am constantly reading the small changes that tell me if I am getting the balance right between challenge and enjoyment. This is far trickier when you cannot see the whole child. Spotting that tapping foot or fiddly fingers are out of site. 

When teaching face to face, I would always point out the behaviour I was seeing and then asking the question. One such student was a 10 year old boy with ADHD. very bright but his self esteem at school was being undermined by the environment he was expected to work in. When he worked with me he needed to fiddle, he needed tasks that involved kinetic tasks. I could always tell when the challenge was becoming overwhelming. He would stand up and need to change his balance from one leg to the other. The first time I told him what he was doing he had no idea that this action was a physical reaction to how he was feeling. He also did not realise he had stood up. He said this was always getting him in to trouble in class because he was meant to stay in his seat. With me we often built in moving around when dealing with a new concept, but I worry this will never be put in place when he is in school. As an adult he will be able to find a job where this is ok, I would image seeing him competing as a professional golfer in a few years. 

Online is still an environment and I am working hard to include all the things I know work in my teaching room with my online teaching. 

Education with intention

Education with intention, the connectedness of the classroom to the real world, and teacher assessing students on how they used their knowledge, not how well they remembered the facts!, is the only way we should see education. Gavin McCormack, LinkedIn 

I saw this on a post today and it really resonated with me. As a teacher I have found one of my best tools has been to ask students ‘how did they....’ and often the student is unable to tell me or will say they guessed an answer. This leads on to me helping them make connections with the skills, knowledge and experience they know, but sometimes do not realise they know, and how they used this to “guess” the answer. 

On Tuesday I was working with a 7 year old who is dyslexic, via zoom.  She had cards with the numbers one to ten written on them and was asked to find me the two cards that had a sum of 16 and difference of 2. Within seconds she showed me 9 and 7 but looked unsure of herself. When we delved into her choice she realised the earlier task of finding 3 cards with a sum of 14 and her base ten knowledge had enabled her to pick out the correct cards. Her confidence and self-esteem are blossoming. Teaching is a joy.