You as a learner of Mathematics: a reflection upon a personal journey ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ***Introduction*** My earliest recollection of a Mathematics lesson stretches back to when I was aged just four years. I was at Infant School, playing with ... tens
and units blocks. I remember the whole class around me were having fun
as they learned. This, I feel, is how Mathematics should be portrayed to
children and adults alike, as a subject which is not dominated by
sitting in silence and slaving over textbooks, but one which can be
enjoyable to learn, spark an interest in the subject and most
importantly develop both a child's knowledge and confidence.
This is a reflection upon my own learning of Mathematics,
recalling events and emotions throughout my educational journey and
examining how my personal learning and development has been impacted by
a variety of teaching methods. Finally, I discuss what I believe to be
the central issues regarding the connection between the teaching and
learning of Mathematics.
Any other memories I have of Infant School Mathematics are at best
extremely vague, although this is not to say that this period did not
have a strong impact upon my Mathematical learning and development. I am
sure that it did. Throughout my education Mathematics has always been my
favourite subject and I feel that the encouragement and support which I
have received throughout my studies has played a great part in my
interest, enjoyment and hard work.
A key factor, of course, is that I have a talent for the subject. I
imagine many people would agree, that it is far easier to take enjoyment
from a subject at which you excel than one with which you struggle. I
hasten to make clear that I am not saying children can only enjoy
subjects for which they have a talent, far from it, for during my
educational journey I have seen examples of both students whose talents
have been neglected and resigned by poor teaching methods, but also of
students who maybe did not have the same interest or academic capacity
for the subject, but whose interest has been engaged by a teacher who
was talented and resourceful enough to connect with that child.
At Junior School I did sometimes find Maths lessons hard, but not for
the same reason as most other pupils as I tended to find that the lesson
moved too slowly and I was eager to progress to more difficult questions
whilst the teacher was still explaining previous ones to other pupils. I
do think that this was a problem throughout much of my primary
education, as I feel my knowledge could have developed both further and
faster had I been able to work at my own pace.
I can only remember one criticism of my Maths work being made at this
stage of my education. This criticism succeeded in improving my
Mathematical skills substantially. It came from my Year Four Maths
teacher at a Parents Evening. He explained that he couldn't fault my
work or mathematical ability and that my answers to questions were
always correct, however I never wrote down any method or working out. He
said it was clear that I found the work easy and the answers came to me
almost instantaneously, but as he knew it would not always be as such,
he had a way to correct it. At my next Maths lesson I was given the same
work to do as everybody else, however instead of giving the answers, I
had to write out the calculations I performed, the way in which they
were done and why they were necessary, as step by step instructions,
finishing my work on the line before the final answer. At the time I
thought this to be highly unnecessary as I already knew the answers but
as time progressed and work became more complicated I found that this
method of showing me to do workings out on paper had set me in good
stead for the rest of my education.
Two years later, in my final year of primary education, our teacher
began organising a times tables 'race' every Friday. The competition,
which involved completing tables and doing sums, was timed, with a prize
for the first person to finish. For the first four weeks I came first
with a clear lead each time. This made me feel clever and proud of my
achievements, but on the fifth week I was told I was no longer allowed
to participate and my teacher asked me to help her check people's
answers. At first I felt proud that I was thought too good to need to
join in, but soon that changed to feeling that I had been unfairly
separated from my friends when I just wanted to join in. I think that a
more sensitive approach would have been simply to provide a prize for
second and third places also, as children do not only feel they have
been segregated or victimised for being bad at something.
When I joined Secondary School I found the Maths Department's teaching
strategy to be far different to that which I had been accustomed in
previous years. I found being taught in ability groups a definite
advantage to my learning as I could work at a faster speed alongside
people of a similar ability. As a result I found my work improved at a
much faster rate. We were also encouraged to think more for ourselves,
to consider problems alone and in groups instead of getting instant
feedback from the teacher. I reacted extremely well to this style of
teaching, I found that I could get deeper into the subject as well as
the individual problem and as importantly, I found that areas I had
conquered myself were much easier to remember and recreate in exam
One teacher who I dearly remember from Secondary School began each
lesson with a game, usually bingo, which he managed to make relevant to
each individual lesson. This was an excellent idea, not just for myself
but for the class as a whole, as this, at an age when people can become
disinterested in school, brought a little light into the classroom.
As I stayed on at my school's Sixth Form to take my A' Levels, my Maths
lessons continued in a similar fashion to the previous few years. There
was, however, one teacher who very much sculpted the pattern which my
higher education was to take. My A' Level Maths teacher knew my
abilities very well and he spoke to our small class as a peer rather
than a superior. At the time when I was choosing a University course
this teacher put considerable effort into trying to dissuade me from
embarking upon a Maths degree, so far as to say that he didn't think I
would be able to do the work. I, however, was determined to prove him
I took up a Maths degree at Lancaster University and although I haven't
found it easy, it is a challenge which I fully enjoy. Once again the
teaching style has changed, being much more focused on individual study,
which at the same time provides the essential skills needed for future
life. Recently I saw my A' Level teacher again, he was extremely pleased
with my progress at Lancaster and it seems he knew me well enough to see
exactly how to push me to do my best here.
Overall I have found my journey through Mathematical education to have
been a positive one. I feel I have flourished both in knowledge and
confidence, thanks to caring and inventive teachers who have encouraged
me along my way. I think that there are many intertwined issues
regarding teaching in Mathematics, issues concerning funding and teacher
shortages and at classroom level issues such as ensuring lessons are fun
to attend and appealing to your target audience as well as ensuring that
teachers build a child's confidence as opposed to damaging it. I think
it is important that teachers can interact with the children as it
matters not how well you know the subject if you cannot adequately
convey that knowledge.
I believe that the most central issue as regards teaching and learning
in Mathematics is making the connection between the teacher and the
student. I think that the best teachers are those who recognise that
each pupil is different, they have different learning abilities and
approaches to problems and each responds to a method of teaching in a
slightly different way. A first-rate teacher forms a connection with
each individual and it is this connection which can be the main
influence on Mathematical learning. I use as an example; that the way in
which my A' Level teacher pushed me to do a Mathematics degree could
just as easily have pushed someone else away.
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Home Schooling versus Conventional Schools
Education. Without it, opportunities are limited, with all that implies both for the future both of an individual as well as the economy of where they live. I do not think that there are very many people who disagree with the notion that a standard of education should be reached, with at least basic literacy, numeracy, and scientific ... understanding of how the world works being an absolute must. So important to the well being of both an individual as well as for the health of society that governements themselves typically have a body to oversee that educational opportunities are being given and that certain standards are at least aimed for. The main way this is met is through the setting up of institutions for the masses, known as schools, where the masses can go and have trained teachers share the knowledge.
This of course, is the easiest method of delivery, and to ensure that the basic standards are being reached for adequately, the state oversees these schools at a local level, checking that the standard of teaching is high enough that the knowledge is imparted adequately. That is the ideal of course, but therein lays a problem. It is the problem of "one size fits all". The truth is "one size will fit the most number of people more or less" is much closer to the truth. This of course means that while most will do well, some will find it too limiting, others too difficult. This is where private institutions and even state ran special schools often come into play. Those with certain disabilities or needs can go to the special schools if need be, and private schools can be paid for to challenge the child who needs it. But again, these are not options available to all, for special schools are far fewer, because the overall percentage of students who need them are less, and so, some get left out, or have to travel far away, and so on. And if they can get to one, all of the students with "special needs" are there, each with a different need; so again, a sort of one size type of thing comes into play, with many of the problems as before. As for fee based schools, this is fine should one be able to afford the fees, but this leaves many children out of reach, even with scholarships. Again, given they are for large groups of students of differing abilities and gifts, the one size issue still rears its head.
A longstanding alternative to institution based education has been home education. For centuries, families around the world employed tutors, governesses, and the like, or, if well educated themselves, often taught their own children. In the past, free state schools were admittedly either non existent or few and far between, or the child was being prepared to go to an elite prep school that was for older children, or even to gain the special skills needed to take over a family business empire or the reins of a country. This was not always the case, however, as many families continued to choose this method in order to tailor the education to their children's personal educational requirements, whether the child was needing extra help, or needed more advanced studies for his age compared to other children. Many modern families choose this option for these very reasons, as well as a host of other reasons.
For example, some find that their children are school refusers. The cause can be many; these children may simply reject the rigidity of the school day schedule and wish to express a sense of nonconformity, are bullied, suffer from depression, or simply feel frustration due to poor classroom environments. Other families may cite religious or philosophical reasons, such as feeling that their way of practicing their beliefs means that they should ensure that their God's Creation be taught across all of the curriculum, whether it is to learn the Biblical verses that apply when studying astronomical concepts, or the infinite wonder of Creation through maths, or devotional passages for literature, such as when studying the works of Coleridge. Religion is a very personal thing, so that how one feels they are called personally upon to practice it will differ from individual to individual even from within a certain faith or denomination, and from faith to faith. There is also the opposite side of the coin: people who do not wish religious ideals of any sort taught to their children, preferring to study religions as a comparative subject sociologically. Many state schools in the UK are run by churches, and while religious classes may be abstained from, it does not mean that assembly does not have a sermon, or prayer, that school plays and activities are not somehow religiously tied, and so on... Also, staff and pupils from these schools are often closely allied to the church that runs them, so even ordinary conversation may be at times religiously oriented in a manner that influences how the child's mind forms opinions.
Still other families choose home education because of family circumstances. It is not convenient to have a child at a school when the family must move about for work, or if one parent works "unsociable" hours. True, one could use a boarding school if one wished to and could afford it, but many families prefer to remain together, and choose to home educate in order to fit education around work hours, travel, and other obstacles. Indeed, we ourselves home educate, and this is part of the catalogue of reasons we do, as well as the philosophical aspect towards religions, and the fact that my two children need to be stretched a lot further educationally than most, suffering extreme boredom if not allowed to move on ahead once they master a concept or skill.
My husband works away all week, and is home only from the Friday night to the Sunday about lunchtime. Quite often he arrives home at a time where the children are preparing for bed, or if we are lucky, sitting down for the evening meal. He does have a mobile phone, and when he is not otherwise occupied, he will ring home several times a day for short conversations to find out how we are doing, what our day has been like, and to just connect with us and anything we may feel a need to talk about. This is important for children; knowing that Daddy is there to listen to their troubles, and share in their triumphs. Due to his work commitment, most of the free time he has to ring is during the first half of the day. This means that after typical school hours, bar a short call before bed, there would be no contact at all between the children and their father. Add in the fact that in his industry, it is not viable to offer holiday periods to all or most employees at the same time due to the nature of the work, and definitely not during public or school holidays for similar reasons, and you can see that a rigid time table spent elsewhere is not really conducive to our family maintaining a close relationship with each other with any kind of deep emotional and mental bond.
As for the need to be stretched educationally, this is an absolute must with my two. By this I do not mean they need to be pushed or hot housed. No, what I mean is, once they master a concept or skill, they need to be able to move ahead. My daughter actually did attend preschool and at age 4, went into reception. She developed severe eczema and developed severe stomach troubles due to extreme stress. The cause was extreme, deeply rooted frustration. Explain a concept once, and she got it. She went in already knowing how to read, and she read VERY well. While the other children were mastering Jolly Phonics and proud if they managed to read the latest Biff and Chip offering, she was bored to tears as back home, she was reading simple chapter books already, such as the Bobbsey Twin series. Telling the time was another source of frustration, she felt like banging her head on the table each time they went over the o'clocks, half past, and quarter pasts AGAIN and AGAIN off and on for three whole months. Science? Well, let me put it this way, at four, when studying the seasons, the teacher told the children that spring was when baby animals were usually born and leaves began to reappear on trees. She asked if anyone else could tell her anything about spring, and of course, children raised their hands and said things like, it gets warmer, and we get rain for the flowers, and so on. My child however, decided to share about how the earth travels around the sun, and how the earth is "divided in two" (hemispheres) and how that determines seasons. She also thoughtfully mentioned global warming and how it has affected how we experience seasons. Thank you Discovery Channel!
My son is the same. He suffered a stroke prenatally, and a lot of work was needed, so he had to be able to take his time when being introduced to things such as pencils and what not due to issues with grip, and also he tired quite easily. However, there was far from anything wrong with him cognitively, as his preschool teacher found out, during the single term he went before we moved house. "Why is the sky blue?" one child asked during a walk. "Because God made it that way" was the answer given. My son however told her it was actually because there is air and light. That air is made up of gases we cannot see, and that also there is water in the air that will come down later as rain. The sun shines through the gases and reflects off the water, making us see blue if it is nice, or grey if it is bad, and that at night, because the gases are otherwise transparent, and the sun is not there overhead to reflect its light, we can see the blackness of outer space.
I am not going to lie to you and say that given matters, my husband and I were not willing to afford our children the opportunity to advance to the best of their abilities, and given our then current status of not having any quality family togetherness time, we decided to change over to home education. Once we did, our daughter's stress levels decreased dramatically, and both she and our son happily forged ahead feeding his thirst for knowledge. We are, as with other UK home educators, overseen by a dedicated visiting representative from the county's educational department, and he has said that given our circumstances he feels we made the right choice. That is to say, the state has given official approval as to the correctness of our choice as compared to our other available options. Should circumstances change, the correct choice may also change, so that is something we keep in review as time passes.
Of course, with children being "at home", one may be concerned about social issues and learning to function in a group setting, or fitting into society. This would certainly be true if we lived in an isolated place, but even then there are ways around it. For example, home education via the radio is not an uncommon facet of life for children in the far reaches of the Australian Outback, and it has been so for many decades. These children grow up just fine, fitting in where they need to. The reason is actually quite simple. Even someone living in an isolated place has contact with others, for supplies, trade, etc, and the children form a friendly relationship with these people and if possible, their families. People like us who live in a town or village have it easier. There are many opportunities to join clubs, attend workshops (most are free or nearly so), visit local activity centres and pools, and also interact with neighbours and people in our greater neighbourhoods. My children do Scouts and Guides, so each week they go off with friends to attend the meeting, go to camps, and so on. Likewise, they know the children in our neighbourhood, and play together on a regular basis. They also are on friendly terms with many of the local shop staff, the postmistresses, the chemist, and even the elderly lady we always seem to meet who is always walking her Jack Russell! In fact, they count many people of different ages as being their friends, so whenever we go to a gathering, or to the park, they are not the children standing about thinking, "I am 8, I don't see anyone who looks 8, so I can't really play." No, they smile at the toddler on the swing and offer to give them a push, they ask the older child if they can join in on the monkey bars, or go ask that elderly lady at the Christmas party if she would like them to fetch another cup of tea for her, and then stand and have a nice chat. This is actually pretty typical of even the "shyest" home educated child.
In fact, Dr Paula Rothermel of Durham University conducted three years of research devoted to the study of children who were school educated, and those who were home educated. For three years, she followed 1,000 children, and what she found surprised many people. What she found was that while in schooled children, lower socio-economic status indicated a general decline in educational attainments, particularly in regards to literacy; with home educated children this was not true. Even children from households with few parental formal academic credentials and a lower socio-economic background but who were home educated, typically outperformed their schooled peers in literacy and maths, from as young as the age of 4 and continuing the trend right up to 11 (the age cap of her study sample). Also, she found the home educated children typically were more socially adept, in that in actual social settings, they were more at ease with a wider range of people such as one encounters outside of a school's closed environment, and so better able to function in any given situation requiring social skills.
This research echoed the findings of Alan Thomas, who conducted research on home educators living in Australia and in London. In addition, he noted something interesting about home educated children's personal character skills that relate to social behaviour: home educated children typically were able to admit errors, mistakes, and wrongs without resorting to defence mechanisms in order to save personal face. This is a skill that is directly linked to a healthy sense of personal self esteem, and is a good indicator of general well being, so it is an important benchmark to consider. It is also a skill that spills over into academics, as not feeling a loss of face also means that that the home educated children typically would ask for clarification or expansion on a subject if they did not understand it completely or to question an idea they disagreed with, without being aggressive or dismissive. Many other scholarly research studies and dissertations have been done on this subject, both here in the UK and abroad, and all say the same things: home education is a viable alternative to school, and produces children who are not only academically accomplished to the best of their personal abilities, but socially adept.
This is not to say that I think every family who wishes to home educate should. I personally think that there is a small minority out there who wish to do it for ideological reasons that are best avoided. These are admittedly rare, but I do feel that using schooling or education as a platform to pass on ideologies without any chance of an opposing viewpoint is wrong, whether it be a school or a home education environment. Germany and Holland's stance against home education is to prevent such cases, and it is admirable, but I do think a blanket ban is a bit of overkill. For example, take the recent case of the German family who fled to Tennessee, citing religious persecution. They were evangelicals, and the authorities refused to allow them to home educate solely for religious reasons, citing the fact that a leading evangelical happily sends his children to the state schools. Now, granted, there are many, many different evangelical churches, so what one person's faith might say may differ to another. Likewise how each person within those feels their personal relationship with said God calls for them to act. Because of this, many of the churches have their own evangelical schools, both actual buildings as well as correspondence schools. It is not said whether or not the family had wished to use a correspondence school method or not, but I suspect not. Even so, many of the materials these use can be ideologically and academically suspect.
I speak from the standpoint of someone who actually attended two such schools, which used two different evangelical curriculums. I was in high school, and I can tell you that in both cases, we were all, 1st grade to 12th, in a single room, desks facing the wall, in cubicles so we could neither gaze upon nor speak to another student at all except as permitted during break. Each subject was Bible based. One curriculum had this corker for science: dinosaurs are pretend and God put bones in the ground as a joke. The other curriculum had this to say: The earth is only 2000 years old. Dinosaurs were real but only around a short while and existed at the same time as people. Indeed, in the USA, there is even a "science museum" that explains this very "fact" and has dioramas to share the "scientific truth as revealed by God" to the masses. Given that we were sat in a closed environment with only these materials, and a constant running dialogue about how the minions of Satan are out there to lie to us and try to disprove God, I am going to come out and say it. This was brainwashing. The academic subjects were being taught alright, but in such a manner as to support one group's ideology. This is wrong, whether it is a school, or a family wishing to sequester children and indoctrinate them. I don't care if the indoctrination is religious, such as the German family and those schools apparently wished to pursue, or political, such as families who are members of the BNP or KKK. Such things should not be permitted to masquerade as academic truth, full stop.
I do believe, however, that such a small minority should not be confused with the greater population. Just as most religious based schools do not apply such methods, neither do the vast majority of home educators, regardless of where in the world they live, despite the large press given to evangelical homeschoolers. Most are simply families wishing to exercise a choice in order to provide an educational opportunity that suits their child and the needs of the family. I believe that access to a suitable education is a basic human right, as is the right to family life, and as such, the choice should be open.
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Home Schooling versus Conventional Schools
I am a parent (ok no duuh to that one really) but what some people don't know is that we have 5 children, and only 3 are in main stream public schools. Our option to Home educate came out of necessity. The summer we were due to move home our 2 youngest children were in the transition from infant primary to junior primary, and ... as is the way with our local authority, all junior school places were assigned by mid June - and no one but SERCO knew if they had placements for the children that september.
Now, we were left with a tough decision. One of the middle 2 children has educational and behavioural issues, and as such a swift transition into a new school was a absolute must. However, there was no guarantees at all we could get placements for the youngest 2 at the same school. The next nearest junior school was 3 miles away with our new house in-between the 2 schools!
We were also faced with the fact placements may not have been found till at least December, forcing us into a state of 'home education' for at least 3 months.
Well I am a firm believer in if a jobs worth doing, its worth doing well. So after some extensive research and questioning of some well rated Home Educators we chose to deregister the youngest 2 a month prior to our home move.
Now unlike a lot of home educators, we did have a view of doing home education for at least 1 year, and each year we review the situation and how we feel the girls are progressing. We have a firm grounding in the core - literacy and numeracy - basically covered with workbooks and a whole spiral course divided up into American style 'semester' grade system. The course has been validated and approved by many countries.
Our Home Education officer visits once a year but quite often we talk on a monthly basis. WE go to cubs, scouts, HE meets, tours, educational facilities....
I've even had the nerve to challenge the schools on many occasions on their handling of my other children's education. And my teaching doesn't stop at 4 o'clock when my 'schoolies' come home. I then have to for 2 hours a night re do what they did at school, explaining not just how things are done, but WHY things are done. While my little He'd ones are happily playing monopoly on the dinner table I have 3 children having a whole extra school day packed into 2 hours just so they can understand in context what they are supposed to be doing!
Add on to the fact I have been on 20 "school trips" in the last 3 months with my HE'd to the total cost (including travel and food) coming to less than £200, I told the school about free cinema tickets they did not even look at, I have had to shell out £90 per schooled child in 2 trips since September.
My children relish the 1 on 2 time we spend on their education. Yes occasionally they say they would like school, but that's because one of them specifically thinks it will get them out of 10 minutes daily hand writing exercises!
HE relies on many things. It relies on the child's thirst for knowledge and the parents ability to involve themselves.
With the current fall in standards in exams and schools drive for SAT scores, pushy teachers expecting parents to collect tokens from higher priced foods & goods to pay for equipment that should have been covered with funding took out in tax from every working parents pocket, a de-christianised school system which still pushes Christianity.
Personally, before you think of abolishing HE, maybe you should look at shaking up the British school system, maybe even align it with our American or even Japanese counterparts. They may have their issues, but right now they are heads and shoulders and a huge ladder above our schools.
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Discussion / University of Newcastle and University of Northumbria at Newcastle student radio station
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