Appreciating technology
Technology seems to be advancing at an exponential rate. Both hardware and software that were once top quality become quickly outdated or superseded. The history of mathematics can help engage students with some topics, even if it is only a brief mention. In a similar manner it might be interesting/worthwhile to see how far technology has come.
Where better than the curious website The Vintage Calculator Web Museum
Also check out the calculator time line which goes from 1600 to 1980
Interpreting Maths
In teaching, I feel it is inevitable that the teacher tries to imagine the understanding of his/her students and approaches a lesson with some estimate of both how much knowledge and how much understanding their students may have. However as you get deeper into specialisations I feel that this gap between teacher and student is blurred by specific language which is automated for the teacher and new or confusing for the student. Reading Mathematical Insight (Padula et al) I was reminded of the complex construction of english and the complex variety of ordinary English (OE) and mathematical English (ME) which is involved in maths teaching. The errors produced by language are many and varied, and are often quite logical. Large amounts of information are condensed into symbols while the structure of the language can trick or cause ‘the reversal error’. I agree with Padula’s focus on the need to practice deciphering the dense mathematical sentences and clearly identify the subtleties of these languages (OE and ME) in a mathematical classroom.
Brooker (Valuing language in Mathematics) delves more deeply in to the use of languge in mathematics by focusing on a foundation of clearly understanding numbers. This article is the most succinct expression of my belief in the idea that by of learning foundation ideas well, many future difficulties can be avioded. Brooker proves this with example; “how many millions are a billion”? Knowing the meaning behind the language of numbers solves the argument/confusion that people have with this idea. How ‘numbers’ are taught and expressed from the start gives clarity to the uses of maths and enables students to appreciate the beauty of maths.
Appreciation of Maths is a common outcome in the NSW syllabuses but if maths is portrayed as a bunch of procedures then no understanding of ‘Numbers’ becomes possible. Brooker argues that we rely on language and that care is needed in teaching it. Teaching procedures seems to me the most dangerous part, as so often it is easy to simplify a procedure into a ‘rule’ like “move the decimal point”, in which the meaning is lost and the language further confuses the process(the decimal point is always next to the 1’s).
Brooker suggests that the poor teaching leads to poor numeracy, which is in this day an age nearly the same as being illiterate. Not understanding numeracy leaves students at the mercy of other peoples interpretations of number (the news) and stops information being ‘problematic’.
Language implications for Numeracy: articles by R. Zevenbergen
Language is a part of all teaching and communication, but it seems to cause some problems and barriers in specialised subjects such as science and maths. Zevenbergen argues that these difficulties usually efffect already disadvantaged students. It effects their participation and legitamises their failure. Zevenbergen makes a strong case for the differences in home language, school language and math language effecting the equity of the classrooom. Explicit teaching seems the only solution to what seems a paradoxical situation. If exposure to language helps, but too much disadvantages then I feel it is the responsibility of the teacher to have an awareness and balance this difficult problem.
In another article Zevenbergen focuses on more explicit examples of language which might cause problems in the classrooms. Some excellent points such as the problems with spatial interpretations of ‘high’ or ‘big’ and the sublties of ‘off’ and ‘of’ in certain contexts. I agree very strongly with the point that a real world problem might not result in a mathematical answer and enjoyed the children’s answers to the bus problem (fit three kids on some seats). Zevenbergen does seem to labor on some weak points of language such as the confusion ‘ruler’ might create. Perhaps it is my secondary approach to the article, but this seems a weak point as so much of our language uses words in multiple contexts. I feel kids will know and understand both types of rulers from a young age, every classroom has a metre ruler. Maths isn’t the only place that words have contextualised meaning. The simple answer is perhaps that all teachers need to have be aware of their language use and teach explicitly.
Using Graphics calculators
Initially intimidated by the many differences to my scientific calculator, I feared that teaching with these calculators would be more trouble than teaching without them. However it doesn’t take long to see how simply designed they are and what a powerful math tool that they are. Unlike some of the computer software I have used they don’t require a new input language on top of already complicated mathematical language, and things such as ‘complex numbers’ can be entered as they read without ‘operators’ becoming an issue. I think they are simple to use, especially once you are familiar with some of the simple reset or ‘error’ procedures. They offer instant feedback, and simple opportunities to view and correct input. They are, I think, an excellent learning tool.
What is ‘numeracy’ in Australia.
The concept of ‘numeracy’ seems to have changed and been adjusted many times in the last 50 or so years. The various interpretations all seem to include understanding and applying maths in contexts which are mostly practical. Brian Doig shows the Australian Association Mathematics Teachers (AAMT) came to the conclusion that it is both “distinct from literacy” and “more than number sense”. From my readings of international views of both ‘numeracy’ and ‘Quantitative literacy’ all seem to point to something around a ‘year 10′ or ’school’ understanding of maths, but there seems to be no clear definition of ‘numeracy’ even while there are studies and politics that seem to relate to ‘numeracy’. Without a clear definition of ‘numeracy’ is there then no clear outcome for educators to aim for in schooling?
Technology in the Maths Classroom
It could be said that a crucial part of learning maths is to find some way to unite procedural practices with some connection to real world understanding . Certainly these two concepts offer educators different (though connected) challenges. With my limited experience of technology I can only imagine the benefits and problems involved with more technology in the classroom.
My first concern is that by increasing technology there will be a reduction of the amount of simple procedures students do. These simple procedures were the foundations of my own understanding, I was very quick at them and used a calculator only for complex equations. This, I believe, helps with estimation and having an understanding if an answer is of any value, and knowing if any error may have occurred. This would be important in the use of technology. I believe having a close link to the repetitive and simple procedures helps in this connected understanding.
On the other hand, the idea of being able to gauge the value of an answer (is it realistic?) can easily be made much clear with technology. Technology can surly help by limiting, for example, long and arduous calculations or inaccurate/wonky drawings of graphs and so offer clear and accurate support in a students understanding.
I believe it will be a fine line balancing the use of technology in the maths classroom. The possibilities technology creates are many and varied, but as long as students don’t loose their ability to construct meaning with numbers themselves also.
some further and interesting thoughts on this: http://gwegner.edublogs.org/2007/07/29/wild-goose-chase/
