This is a teaching resource I developed to demonstrate exponential growth. Have a go a solving the problem on the first page before moving onto the solution lilies-in-the-pond3
The following is an explanation of the Lilies in the pond problem
Teacher’s resource: Exponential growth: Lilies in the pond
Detail of outcome: 2 unit mathematics HSC Course: Series. To give an understanding of important mathematical ideas, such as variable, function, limit etc. and to introduce students to mathematical techniques which are relevant to the real world (Board of Studies NSW, 1982).
Where the resource will be used: The resource will be introduced after the students are working on their first geometric series problems. The resource requires a computer, an OHP and Microsoft PowerPoint
Ideas for its use: Display the first page and ask for a volunteer to solve the problem. The idea is to get the students to dive into a mathematical solution without thinking about the problem. Students typically will develop an equation of the form A(t) = k2x then they will finally get 2x = 64 which will be solved using logs on their calculator. The context of the problem is lost. A logical problem is analysed as a mathematical problem (Janvier, 1990)
The equation for the model and with substitution is Tn = arn-1and Tn = 4*2n-1
Instructions for others: Most students will go straight into a mathematical solution and the formula are provided to guide them along that path. The solution is that they need to read the question carefully and use modeling and logic to work out that half the area is one day back from day 7, the last day. Most students work forward as well and this is another trap for them to fall in. Slides 2 and 3 show models to use. This resource demonstrates that reading and understanding the context of the problem is important. Also concepts such as sustainability of natural resources can be linked into the topic.
References
Board of Studies NSW. (1982). Mathematics 2/3 unit: Years 11-12: Syllabus. Sydney NSW: Board of Studies NSW.
Janvier, C. (1990). Contextualization and mathematics for all. In T. J. Cooney & C. R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s. Reston VA USA: National Council of Teachers of Mathematics Inc.
A very interesting demonstration of the use or misuse of mathematical algorithms. I am not sure how the writers found an incorrect algorith for multiplication that worked but provided the wrong answer.
Referring to Kal’s post. Kal you are too positive with regard to this article. From experience with my son, if I did not get him away from his multi-media centre in his bedroom I think he would turn into a techno-hermit. I feel a bit freaked out with a society comminicating solely with their digital devices.
Online presentation by the author – showing how earnest he is in condemning the way digital technology has made the current generation mentally deficient. He feels that teenagers are no longer engaged in subjects such as history, litaracy and civics and that they are more focussed on the ME rather than as members of society.
A lot of the responses from younger members are very critical of his ideas and the fact that he is using digital medium as amethod of communication.
Reading Thao’s post and the article about suicide of two teenage girls is alarming, as if the internet is the cause of these types of incidents. I’m pretty sure this was happening way before the internet eg Romeo & Juliet.
This a very interesting article recently published in The Australian newspaper that argues that the technological revolution associated with e-learning dumbs down the learning process of students. The author is a professor in liberal arts at Emory university and is writing from experience. The title of his current promoted book is The Dumbest Generation: How the Digital Age Stupefies Young Americans and jeopardizes Our future: Or, Don’t Trust Anyone Under 30.
The author cites research showing that young users of computer scan the screen using an “F” pattern. The information learned is only transient and is broken up into chunks. No depth of learning occurs just a cut and paste of the data. If the topic to be learned does not fit with in this format the student will just disconnect.
My practicum recently was in the area of Computer Studies at a public secondary school. The sentiment of in-service teachers was generally in agreement with Bauelein. The arguments he present are diametrically opposed to the subject Digital Leaning Generation that I am currently studying at the UTS.
My personal view is that the Digital Generation are using the technology just as they see fit outside of an educational framework. As soon as you talk about the application of e-learning my 18 year old son and his friends looks at me a bit strangely. I’d say let them get on with it without interruption from the so called digital immigrants.
The paper was part of a presentation about the use of computers in education. The author discusses contemporary teaching pedagogy such as learner centred environments and the use of new technology in delivering these new teaching methods. As part of a reform for effective learning learner-centered, knowledge-centred, assessment-centred and community-centred environments are described and promoted. The central role that effective teachers play in promoting these environments is emphasised.
The paper concludes with the analysis of two fictional schools in the near future, the first an high achieving private school, Highville that is oppressive in it’s use of technology (a bit like 1984) is concerned primarily with their students getting the highest university entry scores. The role and the place of the teacher in this environment is minimal.
In contrast the second school, Plainville that is favoured by the author uses the teacher and the technology in a facilitative manner where everyone is learning and happy. The author states that both of theses extremes are idealised and that the truth may be in between.
These web pages are series of speeches that Papert gave to a Japanese audiences in the 1980s. The author talks about a student learning mathematics in particular using the programming language LOGO. The talks might have been interesting 20 years ago but I think it has lost a lot of the impact. Programming languages are not that interesting to the new generation of learners as demonstrated by the decline in student numbers in ITC subject enrollments particularly for girls McDougall, B (2008).
To explain what the author is talking about he has embedded video files in the web pages but alas these links are Real Audio Metafile (RAM). format that requires a proprietary player. On installing the player the links on the web page were found to broken so I was not able to view the videos compounding the frustration.
Paper part 4: Advanced Math and LOGO was particularly difficult to understand. I am studying to become a maths/IT secondary school teacher and have some idea of turtles due to my age. (Turtles are using a programming language to move a graphical representation of a turtle around the screen – big deal). The use of mathematics and acronyms would both inappropriate and a bit of a turn off to current learners. I agree at lot with what Papert is trying to say in that student learning by self discovery using technology is the best way for the current generation but the message got lost because the web page is out of date and requires freshening up.
Both these articles are upbeat in their nature in what they say about letting children be children and the idiom of learning as play. The theory is based on work by Dewey, Montessori and Piaget leading Parpert and a method called “constuctionism” which looks a lot like constructivism.
The author is quite enthusiastic about the way children engage in developing their own software using Logo. The learning program appears to be set in the 1990s but she demonstrates the applicability to today’s internet and web page development can be learnt in exactly the same way.
The author emphasises the importance of exposition of software built by the students. The use of exposition has been a consistent theme in digital or e-learning. The target for these articles is the teacher of students especially infants. The author constantly promotes their company called mamamedia.com.
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