Maths Teaching

There is an idea that maths is hard and unnatural and that is often considered the reason why people struggle with maths and yet maths is a very practical subject and is in fact highly visible in nature. In my opinion most people (if not all) can enjoy and progress well at maths if they have a good teacher.

In a talk that Jesse posted on his BLOG Dylan Williams talks about teaching and how many teachers use a louder and slower approach to teaching when a student says that they didn’t understand. This will only work for the student who simply wasn’t listening, those who were and are complaining of not understanding clearly need the concept explaining differently, but many teachers do not take this obvious approach choosing instead to do it ‘their way’ and only their way. The main problem is that only learning a concept one way is not enough, with Maths there are multiple ways to solve problems and doing things differently helps you to learn a process.  In the 25 principles is the principle of desirable difficulties and this fits really nicely here.

The reason that desirable difficulties and learning how to do something multiple ways works can be explained by Craik and Lockhart in terms of levels of processing, the deeper you process the information (so doing instead of repeating for example) the stronger the synapses become in your brain. Also if you learn something in multiple ways you create multiple pathways, making it easier to remember as forgetting from long term memory is a retrieval problem and so the more accessible the information (due to multiple pathways) the less likely it is that blocking will occur. I have tried to find one specific piece of research on this but I have found a wikipedia article on synaptic plasticity which fits everything together reasonably well.  This is a good refresher article on long term memory It is long though.

When looking at Maths teaching I looked at other principles that can relate to failures, The negative suggestion effect has a big significance in maths. The effect is basically that if someone is doing something wrong they could learn the wrong way. In maths students are learning the process and so it is ok for them to have the answer and an example to work through the first few times they try to solve a new kind of problem. It most definitely is not cheating and helps the student learn the right way. This relates well to the concept of Metacognition and getting students to demonstrate whet they think they know to see if they truly know it.

The concept of the testing effect and the generation effect don’t relate as well to maths learning as you would think, because though you spend a lot of time doing questions and testing yourself Jacoby 1978 points out that solving a problem makes the solution memorable but not the process, and so whilst answering lots of questions is helpful and helps make the process innate it may be better to learn the steps individually and then fit them together. Also going back to desirable difficulties, multiple ways of coming at the same problem are useful, despite it giving the student more to learn.

A problem with Maths is that it appears very abstract and yet it has a lot of grounding in reality and therefore teachers need to get across how all of the things they teach relate to the real world. This brings in the concept of multiple and varied examples and this relates to creating multiple pathways, as having real world examples will help encode the information better in memory, especially as it will then be dual coded.

There is a myth that Boys are better at maths than girls, and this is definetly a myth, Rosalind Barnette has shown this. There is however a cultural difference when boys are pushed more towards maths than girls.

One last point is that there is a learning disability called Dyscalculia which is like dyslexia with numbers, however it tends not to be picked up on as well as dyslexia is which is bad because just like with dyslexia, if dyscalculia is noticed in a student then that student can usually with help perform well at maths.

Thank you for reading, My take home message is basically that maths need not be impossible and when teaching mix things up and use desirable difficulties and dual coding as the synapses will be stronger and therefore memory and understanding will be better!!

A little thought for you:

Do you think that higher level maths exams would be better as open book exams where a student can show understanding but not neccesarily have to remember everything?

Or perhaps that the higher level exams should ask a question, and give the answer but not the way to the answer, so what the student has to demonstrate is there ability to obtain the given answer?

By the way I know that some teachers are very good and I personally experienced some very good maths teaching, but I have heard from a lot of people that their experiences with Maths were not so good, which prompted my topic.

The less I do the less I want to do … is this generalisable

I have always found that the less I do the less I want to do, therefore while I am at university I have a job as well to keep myself in check so to speak. I tried to research the idea that if someone has too much free time they want to do less. I wanted to look at the problem of how much free time we have at university and whether this was detrimental, I cannot find research however, So is it just me that has this problem?

I have found things saying that we have too much time to get drunk at university, but I am more thinking of waking up in the morning and reading academia, or the news or going for walks etc instead of just staying in bed or watching the television.

 

NM

Education in the news

Just a few stories I have been noticing since we started this class:

Large class sizes are beneficial

Girls learn better without boys

Should under fives be taught to read and write?

The free schools movement

The end of GCSE’s?

The benefits of Grammar schools being left to get on with it

The pros and cons of private education

Cuts to music budgets

There will be a lot of changes to education in the next four years with the current government and I believe it is important to keep on top of the changes. feel free to comment your thoughts on these articles

 

NM

Distance learning, home schooling and the internet

I listened to Emily’s talk on Learning through digital mediums and second life and it got me thinking about Distance learning.

Distance learners do so for many reasons, they may have other commitments to dependant children, sick relatives or jobs that they cannot afford to leave to go to university properly. These things should not leave these people at a disadvantage and the use of second life to create a ‘presence’ at lectures has been used by the open university in distance learning. There is a lot of information on distance learning and I found this article interesting from a teacher of distance learners. The main gist of it is that you cannot teach distance learners the same way students are taught in person.

Second life classes have also been used for english language learning and people have many suggestions for ways in which second life could best be used.

It is not just second life through which people conduct distance learning, there are also video link ups and you tube as well as black board.

In terms of home schooling the internet is very useful to help learners and their parent as a teacher to have a diverse amount to learn. I have seen a few blogs on home schooling this week so I won’t go into it too much but there are many children in rural communities, especially in countries like Africa, or the very rural areas of Australia or America, who simply cannot get to a school. For these children it is very important to look at how education through the internet can work. This is a very long article about a system of distance education called Northnet What I got from it was that they had to change the system of teaching so that it wasn’t just a repeat of classroom teaching but on the web, and they found that by their innovative approach the results were better than if these children had learned in a regular classroom. The other thing I got fro  it was that they say that what worked for them and the students they taught wont necessarily work elsewhere and that lessons should be tailored.

 

This  form of learning has the opportunity to use the 25 principles especially the multimedia ones. Feedback and tests can easily be conducted through the internet as well.

One more thought I had was that in the talk that followed Emily’s, Athanasia talked about teaching children with ADHD and she mentioned that they learnt well through video games. I wondered therefore would they improve through general online learning, and I found a report to suggest that they would.

 

There is loads and loads of research out there on this topic and I could look at and talk about it all day, but I think the best thing now is to hear other peoples thoughts.

Discovery Learning and Learning Through Play

I have been looking at a lot of Blogs about this and reading a lot on the net and I am getting confused about where I stand on the issue, I think that it is good for children to play and that people do learn by ‘doing’ but I think that schools miss the point when they implement these things. The learning through play scheme in Wales for example does not have enough staff or funding to employ the scheme effectively. There is limited structure to children’s days and very little emphasis appears to be put on literacy, essentially the child can sit at the writing table if they wish, but what child really wants to when there are shiny toys in the vicinity and all their friends are doing other things. This overview from 2008 makes a good case but the scheme had only just been launched so the children in the anecdotes have had the old form of schooling. The recent reports on welsh schools are further evidence that the scheme is not really doing a lot of good.

There is nothing wrong with teaching actively, children learn things like counting and other basic maths from playing shops etc, they just seem to have taken the scheme to far. And as people have been saying discovery learning only works if there is a foundation of knowledge.

Tests tests tests tests test

This week I paid particular attention to the testing effect, I listened to an interesting talk on it (but i can’t find the blog of the person who spoke) and read about it on Blogs and in the New York Times. The idea of the testing effect is that regular tests help students learn and the nature of these tests can make a difference too, as the generation effect shows, if a student generates the answer rather than chooses one from a multiple choice they will remember better. another principle related to this is spaced effects which is principle 5 and says that the tests should not occur too soon after learning as it gives an illusion of knowing, which relates to problems in metacognition. A further principle related to the testing effect is principle 6 which talks about exam expectations and that ‘students benefit more from repeated testing when they expect a final exam’.

Reading all of this makes you wonder why there is often outcry that students in the uk are tested too much and as a class it is easy to think that really students should be tested more. However the testing referred to by the outcriers are standardised tests. Students are often just taught to prepare for a final exam  in order for schools to achieve on league tables. This is especially the case with SATs in primary school many schools teach very specifically for the SATs but offer no wider curriculum, leaving students at a loss.

In my quest to look at how tests are used in schools I learnt about the Finnish school system. They have the best school system in Europe according to OECDs PISA survey and yet they don’t do standardised tests until aged 17. This does not mean they do not get tested, the tests they do are internal and teachers and schools are trusted to do well without interference from the government in the form of standardised tests.

I believe that testing is great as a learning tool but the pressure of standardised tests is not, I think that it infringes too much on curriculum and that though ‘exam expectations’ appear to be important the expected exams should not take up as much of the curriculum as they do.

A little more on Metacognition

The bjork paper talks extensively on Metacognition though not in the most obvious manner, on page 196 he talk about the learner using ‘one index to predict another. he talks about the learner being unable to tell that they have or have not reached the level of learning required and explains that the learner attempts to gauge their level of knowledge through cues that are not demonstrative of their knowledge level. For example a person may just go with what they recognise because it ‘rings a bell’ in their mind even though what they have chosen as the answer is completely wrong they will feel confident, as demonstrated by Kelley and Lindsay (1993) and Reder (1987, 1988). Many feeling of knowing judgements and Judgements of learning studies have been conducted and in they indicate that people are more accurate of how much they truly know if they have a gap between testing and learning, this is because people tend to use the index of what they just read to predict how much they know. For example if you read a passage of a book then reread it moments later you will think you know it but if you re read a few hours or days later only some of the information will be familiar and this will tell you that you do not know it.

A lot of research has been conducted on this phenomenon for example Thiede, Anderson and Therriault (2003) got participants to read texts and then immediately write down keywords or write the keywords later. The immediate keyword group had more positive feelings of knowing than the delayed recall group but it was found with a later test that those who had a delay were more accurate about their knowledge.

An explanation for all of this comes from Wixted (2004) who says that recently formed memories are vulnerable because their consolidation can easily be interfered with and so though you know something immediately after learning, interference may easily occur and you may remember a lot less later on.

In terms of teaching metacognitive monitoring the motivation of the student is important as for them to improve their metacognitive skills they need to take an active role in learning and they need to set themselves goals and test themselves in ways that demonstrate to them how much they know and how much they thought they knew (Cotterall & Murray)

Thank you to Emily who commented on my previous post providing me with the Cotterall and Murray paper

Cotterall, S. & Murray, G. (2009). Enhancing metacognitive knowledge: structure, affordances and self. System, 37, 34-45. doi:10.1016/j.system.2008.08.003

Kelley, C. M., & Lindsay, D. S. (1993). Remembering mistaken as knowing: Ease of generation as a basis for confidence in answers to general knowledge questions. Journal of Memory and Language, 32, 1-24.

Reder, L.M. (1988). Strategic control of retrieval strategies. In G. Bower (Ed.),The Psychology of Learning and Motivation, Vol. 22, New York: Academic Press, 227-259.

Reder, L.M., (1987). Strategy selection in question answering. Cognitive Psychology, 19(1), 90-138.

(Reder’s list of publications, many with pdf files  http://memory.psy.cmu.edu/publications.php)

Thiede, K. W., Anderson, M. C. M., & Therriault, D. (2003). Accuracy of metacognitive monitoring affects learning of texts. Journal of Educational Psychology, 95, 66-73. doi: 10.1037/0022-0663.95.1.66

Wixted, J. T. (2004). The psychology and neuroscience of forgetting. Annual Revview of Psychology, 55, 235-269. doi: 10.1146/annurev.psych.55.090902.141555

Metacognition

I am in Jesse Martin’s project group and for my project I am looking at Metacognitive monitoring. Metacognitive monitoring is knowing what you know and so we used a sample of students from Jesse’s first year statistics class and had them complete supplementary weekly tests telling us how confident they were on each answer, thus testing their metacognitive skills. Some of them had metacognitive training, others did not. We then looked at their in lecture weekly test scores to see how they compared to students who did not participate in our research. Sadly there was no difference. Luckily all is not lost as we did find that our participants who showed the highest metacognitive skills also seemed to perform the best in the tests and so we can at least say that metacognitive monitoring is a very useful skill, we just need a way in which to train it.

With the principles of education that Bjork talks about the problem is that people are unwilling to try them in a classroom context. This is due in the most part to Bjork’s initial point about teachers being conditioned to only teach in ways that gain them positive feedback. In the long run however that is detrimental to the pupil. Therefore our testing of metacognitive monitoring was done without much precedent. We had a paper by Nietfeld et al to work from where the principle seemed to work in a classroom. However the issue with his research was that despite it being conducted in a realistic classroom he was working with educational psychologists who were already well versed in the field. This may have been one of the reasons our results didn’t reflect his.

Despite the negative results we still believe that metacognitive monitoring is an important skill we just need to know how it can be trained, I am researching this in terms of my project and will keep the blog updated on this but any ideas from others out there would be fantastic, for a basic overview of metacognition visit : http://gse.buffalo.edu/fas/shuell/cep564/metacog.htm

 

Reference for the Nietfeld paper:  Nietfeld, L. J., Cao, L. & Osborne, J. W. (2006). The effect of distributed monitoring exercises and feedback on performance, monitoring accuracy, and self-efficacy. Metacognition Learning, 1,159-179. doi: 10.1007/s10409-006- 9595-6)

Hello world!

Welcome to WordPress.com. This is your first post. Edit or delete it and start blogging!