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What (if any) pedagogical benifits do you believe the interactive whiteboard offers and why?

Teaching mathematicics is involves communicating ideas and concepts. Some of these can be difficult with wordy explanations s or even blackboard diagrams. The interactive whiteboard can give understanding to ideas through easy manipulations on the whiteboard which would be difficult to do any other way. Also students seem to be more interested in the lesson simply because of the technology. These things have can increase learning of the students.

“Chords of a circle” This lesson would be suitable for the NSW syllabus for arc and chords and to see the relationship between chord lengths and angles. The benefit of using the Geometer’s sketchpad software in this lesson is that resizing of the circle, chords and angles are all done with the movement of the mouse. It is an easy way for students to visualise the relationships without having to draw a whole heap of diagrams. A little lenghy to originally set up but it wouldn’t be a problem if student were familiar with the program. Would be easier if students used a printed version to follow to set up the sketch. A big tick for visual learners. 

How could the lesson be improved?

Are the instructions for the lesson clear?

The webquest “Buying your First House” http://questgarden.com/45/79/1/070120175316/ is about performing calculations to determine cost of buying a home with various options such as with or without a deposit and includes budgets for running a household. This is one of a collection of webquests at http://webquest.org/search/index.php  

Efficiency Expert :This project is short and to the point and has clear outcomes  

XXX The Affiliator : A single person could do this. It doesn’t require discussion or collaboration.  

The Altitudinist : High level thinking is required as an analysis is required by students to determine the best options   

The Technophile : While the font and colouring are not good, it does have useful links which all work ok. 

 Lee has some innovative ways of engaging students regarding language use in mathematics. He believes that talking is essential for a learning classroom. Classrooms need to be physically set up to enhance discussions so that students can see and hear one another to allow increased understanding of concepts. Teachers benefit from students talking about the math ideas so that they themselves can know what the student’s level of understanding is. Teachers can communicate to students that getting a wrong answer is fine because it helps the teacher to assess what further learning is required. If the activities are engaging then students can support one another in their learning. To help students’ confidence to articulate their understanding, they can team up and work in pairs for an assigned time to think about their ideas, as time to think and reflect is essential for students to learn. An interesting method he has used is to have groups of students describe a Fibonacci sequence, Pythagoras and quadrilaterals. The class compares each groups definition and decides which one is best and they invariably choose the most general case with sufficient detail. This aids students ability to “talk mathematically”. Also they discussed percentages amongst themselves to make the relevant connections with decimals and fractions. Further they described and named their own functions such as the “Brochi Number Pattern”.  

I would expect that Lee is an experienced teacher to do this successfully. It would take quite a lot of skill to ensure the discourse stayed on track to achieve the desired outcomes of the class. These methods, when done well, should enhance the student ability to communicate and think in a more mathematical way, that is, should increase students “Literacy” and “Numeracy”. 

Maybe a good dialogue would be to describe in mathematical terms why this clip is not so  http://www.youtube.com/watch?v=sDerqFGmwNI 

Most of these wesites are of the American country.They are real-time information. I’m sure there would be similar websites for Australian conditions which would hold the interest of Australian students better. There are two sections to the table 1) Real-time Data and 2) Data Sets

REAL TIME DATA

There are two http://k12science.stevens-tech.edu/ that cannot be found, either on Explorer or Firefox.

http://airnow.gov/ This website shows the current wind direction and speed across the American country.

 http://earthquake.usgs.gov/eqcenter/recenteqsww/ This website shows all earthquakes in the world in the last seven days. You can move the mouse over each one to see its magnitude.

 http://iwin.nws.noaa.gov/ this website is the national weather service. It has links to all the different weather warnings, but they are pretty boring as they are all text based.

http://noaa.gov/satellites.html This website took quite a while to download. It has the differnt states with a satellite image showing colours to identify levels of ”Base Reflectivity” in DBZ’s, whatever that is?

oceanweather.com/data took ages to seemingly download, but then didnt work.

DATA SETS

http://mathforum.org/~pdaley/datalibrary/ This website has lots of interesting facts that can be discussed or turned into lessons not only fo rmaths but other subjects too. These include world population growth, the way water swirls down a drain and the prices of a Big Mac across the world.

 Wow, what a great article by Jamison; far and away the best so far. He has such an easy way of describing the problems that math teachers need to consider, and how he himself deals with them.  

Overall the purpose of teaching mathematics is to teach reasoning and analytical thinking through intellectual discipline. To get there, students must go through two stages of mathematical development. Firstly the school-based “plug and chug” where students just plug in numbers to a formulas to get correct answers, and secondly, the quantum leap in thinking to a proof and argument level of thinking and expression. Unfortunately students typically do not develop to this higher level of expression.

 A great insight was the fact that students look for hidden assumptions in math problems, however there are none due to the exacting nature of mathematics. Students are so conditioned with everyday English with deciphering its ambiguities that this is transferred to their reading of math problems, and they create their own misconceptions just by looking for hidden meaning which does not exist. The clarity and precision of maths becomes a stumbling block to them.  “One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts”         – Albert Einstein   

Teaching and learning maths requires language to be used as a tool which requires just as much precision as the written expression. Students need to learn the “language of maths” with its rules of interpretation just as if it is another language and so many students fail to grasp this. Everyone loves Einstien but even Albert Einstein was a notorious underachiever as a child. At the age of 16 he failed the entrance examination which would have allowed him to study for a diploma as an electrical engineer. Quoted as saying “Do not worry about your difficulties in mathematics, I assure you that mine are greater.” So even the greats have struggled. It is encouraging that in 1900 Einstein eventually graduated as a teacher of mathematics and physics.

Week 3 – Summary of Readings.  

FIRST READING – Padula and Schmidtke

Mathematical language uses terms familiar to other areas of English, but have different meanings associated with them. Teachers need to take care to ensure student understanding when introducing new words.

 Students are encouraged to be quick in their exams and tests which encourages students to skim the English to get on with the maths of a particular question. However, only by reading a question carefully will the student answers it correctly. 

Students of Non-English speaking backgrounds may need assistance in comprehending what a question is actually asking as the grammar and wording may be confusing for them.

SECOND READING – Booker

Students need to become highly numerate or they will be unable to communicate effectively in today’s emerging technological world. The foundations of computing and technology are mathematical ability. It is now more important than ever that students develop deep understanding of mathematical concepts.

As students are introduced to new math ideas it is important that they learn it thoroughly the first time around. Early misconceptions can cause confusion in later topics of learning. (Ma,1999) Students must not only recognise symbols but understand all that the symbol represents.

 Thinking mathematically is considered equal, if not more important, than just doing the algorithms to get a correct answer, thus the big push on “Working Mathematically” in state-wide exams. 

The naming of numbers mostly follow a pattern, but unfortunately there are exceptions to the general rules of naming which can cause some confusion to students. The naming of mathematical procedures and symbols, such as “put down” and “carry” do not allow students to pick up on cues for the required algorithm. Students have a “pseudo-conceptual” understanding which they try and use to solve problems, but it is only by understanding a concept thoroughly that they can make sense of it. Students should be introduced gradually to the different operations they will be required to do with meaningful language. Further, being automatic with basic facts, frees up the mind to focus on the steps within an algorithm.

Students need an understanding of mathematical processes and concepts to make their way in today’s society. Using language and meaningful symbols that allows fundamental maths to make sense and to be useful and useable is essential for all students.

Graphics Calculators.

My initial reaction to the graphics calculator was not really a positive one. It is a new piece of equipment that I am not familiar with. It seems that I needed a degree just on how to use the thing. To me it seemed to be an overly complex wigit.

Realistically my opinion of these calculators matters little. The question that needs to be asked is “Do they add value to a math student’s ability to learn?” If it does, then my nearly 40yo brain needs to get into gear and learn how to use this tool effectively.

It would be of benefit to visit a school that actually uses them to gain an informed opinion of their value. Unfortunately, I know of no school that uses them. Why would this be so? There would be many good reasons for schools to give as to why they do not have them; cost, students loosing them, waste of time, to complex, no BOS compulsory programming requirement. I would suggest that the old cycle of “I learnt it this way, so current students can too!” would be a factor here also. Until mathematic educators demand appropriate technology in their classrooms, teachers will be confined to using the old methods of direct instruction and boring textbook lessons with the occasional use of the good-old worn-out overhead projector.

The Federal Government is looking at putting a computer for each student in all schools. I see this as a real bonus as websites have easy-to-use programs which cover all the capabilities of the graphics calculator and more on them. Such websites include  http://www.expertmathtutoring.com/Plot-Graphs.php for plotting functions, http://www.quickmath.com/www02/pages/modules/equations/index.shtml for solving equations, http://www.internet4classrooms.com/skills_5th_original.htm for quick skills testing and  http://www.funtrivia.com/quizzes/sci__tech/math.html   for fun quizzes in specific topics. These tools can assist students to have a better intuition for math problems and how functions and other answers should look like. I would like to see greater technology used in classrooms, either graphic calculators or computers, as students need to be familiar with it in today’s world and can have their learning greatly enhanced by using them.

 The Two Articles by Zevenbergen.

The biggest challenge for any teacher is pitching their lessons at a level where their students are at, and trying to cater to the differences within each classroom. The teachers language use within a classroom can affect the lower end of students more adversely, as more commonly, professional teachers are from middle class backgrounds and so inadvertently use language in a form that they are familiar with but not as familiar to students with a lower SES. Teacher awareness of this should enable some language modification to assist all students understanding.

The second reading was just plain annoying. A picture is worth 1000 words. The diagrams reveal the authors opinion of math teachers; that they are morons, which I found to be offensive. I question the intellect of the authors as they bumble on about how students confuse words such as “odd numbers because they are strange”, rulers being about kings and Queens and volume being about the noise of a box. Only dyslexic students or authors of such dribble would have this problem. The reality is that teachers do explain new words. If some students don’t hear it the first time, they will pick it up very quickly by doing relevant exercises and using the appropriate terms. Mathematics does have its own language including symbols to represent ideas and they are learnt as students progress. Care needs to be taken by students to read questions carefully as they need to understand what a particular question is actually asking them to do, which can sometimes be misunderstood.

The two most important thoughts from the readings is firstly to be clear and simple when speaking and giving explanations so all students have the greatest opportunity to understand. Secondly, ensure that when new words are introduced, that they have been properly explained in the maths context, with even a written explanation left on the board for reference.

During my reading this week I found myself wondering what all the fuss was about, why all this quibbling about the meaning of a few words that people have slightly different meanings attached to them. It appeared that they were all pretty much saying the same thing anyway. As I read on it became clearer as to why it is important to distinguish between these meanings. It is desirable for research to be conducted to enhance the knowledge we have on how people learn, think about, understand and apply mathematical knowledge. Mathematical knowledge is built upon itself; further ideas and concepts built upon previously learnt knowledge. Similarly, research also is built upon previous learning’s. Research is conducted across the world and it is not until common word meanings are agreed upon can this research move forward with any significance.  From my readings I understand the following to be reasonably close to what is loosely the general understanding of these words:  

Literacy is having the skills to understand the ideas of others and have your ideas understood by way of written skills. Primarily “literacy” is English text based, but must also include the understanding of symbols in whatever sphere your communicating about (in the maths area this would be mathematical symbols). 

Mathematical Literacy is the ability of an individual to solve real life problems in his/her real life using their mathematical knowledge. This may include their knowledge and use of technology to solve mathematical problems. 

Numeracy is the same as Mathematical Literacy in that an individual can solve their own personal mathematical problems but it has an additional element in that, a person can communicate their mathematical ideas to others and understand others’ mathematical ideas by means of symbols and/or jargon. 

Quantitative Reasoning is the ability to be able know which mathematical method/s to use and in which combination to be able to solve real life problems where different situations arise in real life 

Quantitative Literacy includes Quantitative Reasoning in that it is the ability to understand mathematical concepts to apply correct methods to obtain the desired mathematical result. It has a further element in that a person is able to understand and interpret the results obtained, to scrutinize them for the purpose required, including flaws, robustness of useability, to predict further problems and further solutions. It is a deep understanding of the, who, what, how, when, where and why of all things pertaining the mathematical situation at hand.   

Another Question – Is learning to use technology to solve mathematical problems considered legitimate “learning maths”, or is obtaining the knowledge of how, why and the relationships between maths concepts to be considered learning maths? Are you good at maths if you can get a right answer from technology or are you good at maths if you know how and why the answer is what it is? Can we make up some new words to describe these? (Just to muddy the waters a bit more).