A big part of this first year subject was on the emphasis that it placed on mental calculations and algorithms. Being faced with the prospect of solving mentally things such as multiplication of more than two digits was daunting until it was explained how to break it down and make it easier to figure out. At first I was alarmed thinking things such as ‘oh no it’s taking me too long to do this’ and ‘surely it would just be simpler to be able to write it down on paper as an algorithm?’. Thus we come to one of the most important things that I learnt in that first week, how to recognise and face the misconceptions that I had about mathematics and working mathematically. Sure I would never be a professional mathematician but it didn’t matter if I couldn’t work everything out super fast in my head. In fact a very common misconception is that ‘Smart people do maths fast, in their head, and in one sitting’ and that it in fact requires ‘slow, careful thinking and ‘pacing’’ (Frankenstein, 1989, p.19). In fact reading the paper by Marilyn Frankenstein (1989, p.17-26) I realised allot of things about the misconceptions surrounding maths and allot of them applied to me or I could understand how people could have those views.
According to Frankenstein (1989, p.17-26) these are the other misconceptions that people commonly have about maths: that there is one correct way to solve and answer a maths problem, they feel stupid if they make a mistake or ask a question in class, that they are the only one that didn’t learn some aspect of maths when they should have and that the teacher is the only one who can tell them the answers. Looking at all these misconceptions I will expand on how they have each affected me in some way. I always thought that if you were able to understand it maths would be simple because there was only one way to do and a set answer unlike English were you are required to analyse and interpret things and in lots of cases there are no wrong or right answers.
As a student I was also was hesitant to ask questions in class, most of all in front of my peers while we were in the middle of a group explanation because I didn’t like the feeling that they knew that I didn’t understand and didn’t want them to think less of me because of this. It was marginally better to go up to the teacher when the group lesson was over and ask for assistance then because all though other people could see that you were getting help it was more private. It was always better though if you felt confident talking about not understanding to a friend, someone who you felt confident sharing things with and they could help you or if they themselves did not understand because then you knew that you were not alone. I think in some ways once I was in high school in the higher year levels it was easier to say that I did not understand particular aspects of maths. Allot of other people were doing maths as part of a sense of obligation the feeling that they should do maths because it would help them in the future and it was another subject to fill up their timetable. So the feeling of not understand something was something that I knew was shared by plenty of others. In year 11 I chose to do maths methods, the highest level of maths possible at that stage partly because I thought that I could do it, partly because I wanted to prove that I was able to do it to myself and that I was not stupid- not that I actually thought I was stupid. I was able to do it and pass satisfactorily but it was allot of work, allot of needing to ask for help because I did not understand. I did learn some things from it though, apart from learning maths that is. The first is that perseverance is gold. I eventually decided to go to the after school help sessions that the school ran which were supervised by the two year 11 maths methods teachers. By doing so I realised that sometimes you just need something explained a different way and that not all teachers are able to explain it in a way that will make sense to you.
My faith in my maths ability was not always and is not always bad. I always loved trigonometry but I hate the complexities of the unit circle, which gave me allot of grief in year 11 trying to understand it. I liked and was good at Matrices and probability and geometry was always easy. However financial maths was a killer as was allot of things were I had to graph something. I was in the top maths group in year 6 and doing work ahead of the rest of the class that the teacher said was high school level, I thought that it was so cool to be looking at the circumference of a circle and learning about things above the expected level. However there was things that I did not learn ‘on time’ time tables being probably the most prevalent one- even at this stage, top maths group in grade six I still had troubles with times tables, those troubles being that I lacked the ability of instant recall which seemed so important. I was able to do them I just needed a little figuring out time that most other students didn’t seem to need; this made me hesitant when playing games like maths tiggy as the majority of the questions were multiplications. Now I think that not being able to recall all my times tables instantly was probably not as bad as I thought and I know that although having that skill would have allowed me to finish maths problems faster it did not make me any less smart.
Aside from the misconceptions that I realised that I had the content that was presented through the lectures and tutorials was really informative and made me realise and think about things that hadn’t crossed my mind before. I am of course referring once again to algorithms. In an article by Emilia Mardjetko and Julie Macpherson (Mardjetko, E & Macpherson J, 2007, p.4-9) they put emphasis on teaching and developing students’ abilities to do calculations mentally. In their article they refer to research done by Mackinlay who found through a study of grade 3-5 classes who were tested on things for which they had not been taught algorithms that there were two major strategies used by the students. The less successful strategy was attempting to use an algorithm and the most successful was being able to ‘manufacture legitimate and effective strategies for mental computation’. The article also talks about how algorithms take away focus from the development of number sense and place value and that using algorithms leads to high error rates as ‘the carry operation can be quite problematic for mental imaging’. These problems with using algorithms had never occurred to me before undertaking this subject all I knew was this is how we were taught to do it so that was how it had to be done. But putting more emphasis on mental calculations is a good thing because students who can use mental calculations flexibility have proved to have a higher understanding of underlying mathematical concepts (Mardjetko, E & Macpherson J, 2007, p.7)
It all comes down to how maths is taught, teachers need to know their students and what they know and understand. In the end it is the ability to understand rather than the ability to achieve a correct answer that is important. Students need to know the basic maths; one thing that can confuse students is the equals sign. 8+4=__+5 will confuse allot of students and they will think that either 12 or 17 belongs in the space (Baroudi, 2006, p.28). Baroudi (2006. P.29) says that ‘recent research suggests that students need to be helped, from an early age, to construct valid generalisations of the arithmetic operations’. The article which discusses algebra says that students need to be ‘exposed to arithmetic in a variety on contents’ and that they need to be able to explore and discuss ideas. I think that this is one of the most important things that I have learnt in this subject is that not all students will learn the same way and that you need to present problems differently so that you can be sure that they understand the concept.
How the students view maths is important and how well they understand it, but so is the teacher’s viewpoint and understanding. Beswick (2006) looked into the different types of maths teachers. Some of the beliefs that were seen as most important by the teachers Beswick interviews were that: ‘Mathematics is about connecting ideas and sense making’, ‘mathematics is fun’, ‘all students can learn mathematics’, ‘the teacher has a responsibility to actively facilitate and guide students’ construction of mathematical knowledge’. To teach maths though the teacher need to be able to understand it. John McAdam (2007, p.289) says that ‘many prospective teachers may lack the basic conceptual understandings that are promoted by the current reform, constructivist learning theory, and as “experienced learners”’. He also says that teachers are a product of the classroom that they were in as student, ‘teaching by telling’.
I have been greatly affected by what I have learnt in this subject this semester and so I am determined to make sure that in my future classroom to teach strategies to help with mental computation over the traditional algorithms. I am also determined that I will not allow a student to not understand something by demonstrating how to do something multiple ways and encouraging the students to problem solve and not just give them the answer but encourage them to find it on their own in whatever way they feel comfortable. I also want to avoid students being made to feel like they are stupid like I have been sometimes made to feel by a teacher because I didn’t understand something in maths. I want to promote an environment where my students can work together and help each other to learn and where maths is fun and not something to be apprehensive of.
Baroudi, Z (2006) Easing Studentts’ transition to algebra. Australian Mathematics teacher v62 n2 p28-33 2006
Besswick, K (2006) The importance of mathematics teachers beliefs. Australian mathematics teacher. 62 (4) 2006
Frankenstein, M (1989) mathematics anxiety: misconceptions about learning mathematics. Relearning mathematics: a different third R-Radical maths, Free Assoc. Press
Mardjetke, E & Macpherson, J (2007) Is your classroom mental? Australian Primary mathematics Classroom. 12 (2) 2007
McAdam, J. (2007) Prospective teachers’ use of concrete representations to construct an understanding of addition and subtraction algorithms. In W. Martin. M Strutchens & P Elliot (eds.) The learning of mathematics. Sixty-ninth yearbook of the National Council of teachers of Mathemactics.