EDUC4105

Sum of Interior Angles of a Polygon

The sum of the interior angles of polygons appears at various times in the 7-10 syllabus. At SGS 4.3 students look at the relationship between the interior angles of a triangle and then at SGS 5.2.1 students look at the angles of all polygons.

The positive aspect of this lesson are that it is very explicit with its instructions and has step by step instructions with pictures to avoid student confusion. This is very important especially with junior classes as they often get confused when working with interactive software. The negative aspects of this lesson are that students are simply told what to do for the majority of the lesson making this a very teacher centred  lesson.  The lesson also attempts to be enquiry based which is not the case.

Week 5 Tutorial Activity – Professional development Internet Resources Based on Video

Seeing Math (www.seeingmath.concord.org)
According to their website Seeing Math™’s online professional development programs use interactive software, illustrative video, guided discussion and standards-driven content to:
* Equip teachers with the knowledge and instructional strategies to engage, motivate and lead students to math success.
* Provide schools and districts a flexible and cost-effective solution to address rigorous standards, meet staff development needs and improve student achievement.
* Help new and veteran teachers gain insight into how students think about mathematics.
An analysis of the website found that this site is very effective in meeting it’s goals. The website is well set-out making navigation very easy and all the resources (those viewed), which include interactive Java programs, case studies and information videos were very helpful and most importantly relevant to students and teachers needs.

Lesson Study Group at Mills (http://www.lessonresearch.net)
Lesson study is a process in which teachers jointly plan, observe, analyze, and refine actual classroom lessons.
In Lesson Study teachers:
• Think about the long-term goals of education – such as love of learning and respect for others;
• Carefully consider the goals of a particular subject area, unit or lesson (for example, why science is taught, what is important about levers, how to introduce levers);
• Plan classroom “research lessons” that bring to life both specific subject matter goals and long term goals for students; and
• Carefully study how students respond to these lessons – including their learning, engagement, and treatment of each other.
This website focuses solely on elementary (primary) mathematics thus is not of a great deal of use to high school teachers, except possibly those teaching lower classes. It has a list of resources including articles (based on classroom studies), DVD excerpts, lesson worksheets and lesson plans, but as mentioned they are all based on primary school classrooms which offer little help.

Week 10 – Key Features of Teachers e-Portfolios

The key features of any teacher’s e-portfolio would be their teaching philosophy, pedagogical goals, strengths, qualifications and experience. An e-portfolio is in my opinion a miniature resume that includes a series of sample lesson plans, pedagogical tools, strategies and techniques. The purpose of an e-portfolio is to not only provide a sound overview of the teacher, their philosophy and qualifications, but also to provide resources that other teachers that may view or incorporated into their own lessons.
Social bookmarking, like an e-portfolio is a tool for the sharing of specialised professional resources. This can lead to professional development and pedagogical improvement. Social bookmarking allows industry professionals to share, analyse and add to professional resources such as lesson ideas, lesson plans, strategies and techniques in a pressure free environment.

Week 11 – Interactive Whiteboards

Interactive whiteboards have many potential pedagogical benefits in education. They are a great tool for improving student engagement within the classroom and also work as an effective tool in the teaching of students with special needs, such as visual or hearing impairments. Lessons incorporating the use of an interactive whiteboard are also (from experience) much easier to prepare as a logical lesson sequence can be easily prepared and adjusted at short notice.

Interactive Maths Game
As is the case with all technology in teaching, the interactive whiteboard is not a complete solution to all problems within the classroom. Lessons still require the direct incorporation of the NSW QTM’s principles and effective teaching strategies. Interactive whiteboards are a good tool for the classroom provided they are not relied on solely and they are used in conjunction with a variety of teaching strategies which require students to demonstrate different types of knowledge and understanding.
The main benefits of an interactive whiteboard is that they are able to be used not only to present various types of lessons, but students are also able to interact with the lessons through its various programs and software. The interactive white board may also be used as a screen to view movies, Java programs and presentations.

Week 12 – Graphics Calculators
I must firstly admit that I am not very comfortable working with graphic calculators. With a little research however I was able to see that they do offer more opportunities to assist learning and understanding within the mathematics classroom. Graphics calculators can be programmed, so that users can customise the calculator to perform particular tasks of interest to them. They are also capable of both graphing and calculating difficult functions which would take much longer to work with without a graphic calculator. As is the case with all technology within the classroom, a graphic calculator is not a perfect solution it is just another tool that a teacher can have in their arsenal to assist students with the grasping and understanding of mathematics. I have included in this blog a link to an article that dispels many of the myths associated with graphic calculators.

Week 13 – Webquest
Webquest lessons use an inquiry-based method to help increase connectedness between students and the knowledge they are attempting to divulge. This means that students are genuinely solving real-world problems when they are completing a webquest lesson. The only problem with this is that many of the webquest lessons are solving problems that are not really relevant to student needs. Many of the ideas are really clutching at straws to maintain their connectedness to real-world maths making students less likely to want to complete the work as first hoped. The main point of this rant is to hope that when teachers choose to incorporate webquest into their lessons that they choose or design webquests that are relevant or at the very least interesting such that connectedness is established but interest is maintained.

Week 14 – Dynamic Geometry & Excel Spreadsheets
Dynamic geometry programs allow teachers and students to see and interact with many various shapes and angles in a way that is much less time consuming than the old methods of construction. It also eliminates the need for compasses and protractors for teaching some topics (which saves a lot of hassle within the classroom). They are however, like al programs only another tool to help build knowledge and understanding amongst students.
A fundamental question when using excel is whether or not we are teaching students for understanding of mathematical concepts or are we teaching students how to use computer programs to solve problems for them? Excel allows us to store and retrieve many statistics which can be very useful. It also allows us to graph this data to both show and interpret it. Excel spreadsheets can be very useful but the main focus should still be on fundamental concepts such that students can build their understanding.

Schleppergrell Reading
This article points out many of the linguistic difficulties encountered by many students when studying maths in schools. Linguistic difficulties such as coming to terms with the structure of mathematics and its tendency to take words with everyday meanings and give them totally different meaning in maths. These difficulties require students to use multiple semiotic (meaning creating) systems to express meanings, which go beyond that of everyday language. This provides an immense challenge for classroom teachers who need to help students move from everyday informal ways of constructing meaning to the technical and academic methods necessary for learning in all subjects. An example of this is that of a trigonometry problem where language plays a large part of explaining the problems but has no real ties to the mathematics involved when answering it. A very good point to come from this article is to teach algebra not as just substituting letters for numbers in solving equations but as a means of dealing with properties of numbers and quantities.
Mathematish
This article hopes to consider the language of mathematical symbols and their use as a fully developed language. It shows two sides to mathematics, its general and abstract concepts and its symbolic language. Thus mathematical texts are bilingual, having both written words and symbols. As such sentences or arguments in mathematish either make sense or they do not based on the structure of the language. Learning mathematish is not the same as learning another country’s language such as Mandarin, or German, as mathematish has a similar structure to English. It does require an understanding of both what each symbol means and as such how it can be entered into mathematical argument.
This article prompts me to reflect on a question posed in previous courses asking, is mathematics a language created by man? Or does it simply exist and man has simply created ways of expressing it? I’m really not sure.

John Gough Readings
Conceptual complexity and apparent contradictions in mathematics language
Gough points out the many differences between everyday language and our “artificially constructed” mathematical language and the cognitive conflict this causes many students. Many of which have gone to great lengths to discover and determine the meaning of many words only to discover that in mathematics they have totally different meaning. For example the borrowing of words that already exist with everyday meanings and redefining them. His point on the topic of fractions where something is “selling for a fraction of its normal cost” when the value of the fraction is actually greater than 1 is quite intriguing. Also incorporating his ideas on the inclusion in our counting and curriculum could be quite beneficial for many students.
Teaching square roots
I am yet to teach square roots, nor do I remember how I learnt them, but this article presents some very good ideas to help develop student understanding. For example starting with a number and imagining that it is the area of a square, the square root gives the side length of that square, which is something that gives students a real example of how to view the square root. This would also tie in with Pythagoras as we look at a right-angled triangle as three separate triangles when beginning it.

Starting to talk in the mathematics classroom

This reading seems to focus on interactions within the classroom that would only be suitable to a small percentage of students. Either that or a teacher would have to have a lot of control over students. This may just be my opinion, but it is very difficult to convince students to answer questions and participate in class discussions, even when what they are saying is relevant. The article does provide some effective tools for how to achieve a talking and learning classroom such as classroom organisation and an inclusive ethos, but the author seems to assume students would genuinely be interested in discussing mathematics and improving their general understanding, which is highly unlikely.   

The Language of Mathematics

The explicit nature of this article is very appealing. The fact that it is to the point on what problems are present in classrooms which are taken for granted such as the confusion and ambiguity about 2-d shapes having sides and 3-d shapes having edges, and pairs of words whose meaning are difficult to distinguish between, or that have multiple meanings in different contexts. This has led me to believe that all students, especially those in younger years should allocate a number of pages in their books for a mathematical dictionary and vocabulary.

Webquest Evaluation

Credit Cards

Affiliator- Good for a real-life situation.  Incorporates group interaction

However could utilise better group to group interaction.

Technophile- Very good on all aspects. Incorporates movies, websites, online tutorials and online templates for students work.

Efficiency- Practical, good glossary of definitions, but research is quite tedious-might as well present it could be altered for Australian syllabus.

Altitudinist- Good connectedness, required students to use synthesised info from many sources but needed too much information to gain deep knowledge and was not a higher order thinking task.

Language Pitfalls and Pathways to Mathematics

This article, like many we have already seen analyses the mathematical symbols and vocabulary used in modern maths teaching. It raises a number of issues mainly concerned with student confusion with mathematical symbols and an incorrect development of meaning for individual mathematical symbols. It also raises issues similar to Zevenbergen of homophones and synonyms, which need to be analysed by the classroom teacher within lessons.
Two issues raised in this article of much interest were that of discourse and internal chatter. Discourse is essentially students understanding of key ideas. The article points out that the development of student discourse is multifaceted and consists of basically all learning and interaction within the classroom. Thus a teacher must structure their classroom effectively for the learner, such that both students and their teacher are involved in both expressive and receptive communication. Internal chatter, I feel is an essential tool in the mathematics classroom. I know that to this day when trying to understand a problem I talk it over to grasp what is important and what is not.
I also like the point on not trying to oversimplify key words and definitions. It is often tempting to gloss over things to save time when realistically this is not acceptable.

Learning the Language of Mathematics

“This paper is about the use of language as a tool for teaching mathematical concepts”. It makes clear reference to the rhetoric often seen in Mathematics, which often causes to confusion for many students. It also places great emphasis on the rigorous nature of mathematical definition; that it must be clear, concise and effective. This is achieved through the discussion of the three features that distinguish mathematical language from ordinary language; nontemporal- there is no past, present or future in mathematics, it is devoid of emotional content and it is precise.
I also like the discussion of genus and species. For those studying higher mathematics these concepts are of much importance. Genus refers to the fact that every concept is defined as a subclass of a more general concept, where each subclass of the genus is referred to as the species. The clear nature of Jamison’s explanations would help all students to be clear in their minds as to how to read, write and compose a clear definition.

Included are two links, the first of which shows Euclid’s axioms, each of which are clear and concise (except for his fifth postulate, as discussed in the second). The second is a discussion of Euclid’s axioms. These are examples of the rigor and precision needed when defining things mathematically.

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html

http://www.geocities.com/CapeCanaveral/7997/noneuclid.html

Mathematical English (ME)

I have read and been informed in the past that English is quite a difficult language to learn due to the fact it is a combination of many (200) different languages. However it had not occurred to me what difficulties this would pose to students in the mathematics classroom whether they have an English or Non-English speaking background.

Mathematical English (ME) as introduced in this article is an interesting concept but essentially all the article is asking us to do as maths teachers is to carefully analyse all language we use in the classroom to avoid ambiguity and misunderstanding by students. This requirement seems to fall solely on the classroom teacher essentially making them responsible for all language within the classroom thus responsible for any misunderstanding within this setting. Thus teachers must be aware of students from NESB and their language needs prior to investigating problems involving language. Also we must be explicit and concise when designing problems involving language.

Valuing Language In Mathematics

Booker’s article moves away from mere definitions of literacy and numeracy to put into perspective the importance of a more numerate society as we progress into more technological times. Having also seen Ma’s work on the need for a profound understanding of fundamental mathematics i agree that understanding elementary ideas underpins the development of all mathematics.

His extended discussion on the symbols used in mathematics, their heritage, the value of zero, and also the reasons why each number is named the way it is within the Hindu Arabic numbers system. These seemingly logical yet almost never discussed reasons would prove very effective tools in the development of a PUFM for all students. They will also help many students to come to terms with the idea of place value which many students struggle with.  Ultimately integrating students into society who will thrive in the technology dominated world of today.

ZevenbergenLanguage implpications for numeracy

Many articles have been published exploring the issue socio-economic status and success at school. Generally the lower the SES or class the less successful the student performs at school. This has generally been explained, as Zevenbergen suggests by a lack of cognitive development. But as we move into a less theoretical more ‘real world’ style of mathematics classroom, I find her research into the implications that literacy and a lack of literary skills are having on these lower SES students. Is it possible that a lack of literacy is more to blame for their lack of understanding than a lack of mathematical ability? As she states more work needs to be done to explore this situation.

Language and mathematics

Although long-winded for the most part I cannot disagree with Zevenbergen’s focus on ‘making pedagogy explicit.’ Our need as mathematics teachers to be clear, concise and consistent is more important now than ever. As the focus on literacy in the classroom is near omnipresent we essentially cannot afford to confuse students and as such must use specialised explicit mathematical language at all times. Thus they have little or no chance to be confused about word meanings.

To help with the issue of ambiguity the list of homonyms, homophones and discussion of polysemy are great to focus on and try and make our mathematics classrooms more specific and concise

Language and Mathematics

Literacy, numeracy and quantitative literacy. What does it mean to be literate, numerate or to have a developed quantitative literacy skills? And from our perspective as beginning maths teachers what implications does this have on us?

All three terms as defined by various academics and academic groups seem have the same key focus area of providing students at all levels with the reading, writing and interpretation skills to become a productive member of society. The two most important and relevant ideas in the reading for us as teachers were “the development of practices and dispositions that allow individuals to function effectively both as learners and members of society…for Australia to reach it’s social and economic goals.” With the key word being effectively. Quality literacy, numeracy and quantitative literacy skills give all students the opportunity to prosper in our society. As teachers we are ultimately preparing students at all levels to integrate into Australian society for the benefit of the entire country (Why else would school be compulsory?).

Summing Up

This article still spends much of its time defining the concept of numeracy and ultimately comes up with a very similar definition to that of the first reading. Although it is nice to know how Australian students achieve in comparison to many others, simply stating statistics and results is quite boring.

The article did however get me thinking about how and why we teach maths in Australian schools. The idea that students be numerate is very important and achievable. The point on “whether maths teaching should be taught in context” as unresolved gave me the opportunity to reflect on the views of people I have spoken to about the subject. For me understanding fundamentals/concepts has lead to an ability to apply knowledge, whereas other individuals would rather focus on applications to help understand fundamentals. Surely pure mathematicians insist that fundamentals are more important for understanding and application and it we cannot practice every aspect of life in which maths is used within a classroom. Most students however are not mathematicians and the achievement of quality numeracy skills is more important no matter how we do it.

Technology and Teaching

My experiences learning maths with technology have been quite limited, apart from a few experiences with geogebra, maple and graphics calculators. However I do believe that in our ever-expanding technological society the use of and familiarisation with technology in education is going to become more important.

If we take a short look at the history of mathematics we can see for example from the ancient Greeks that a deep understanding of mathematics does not require technology. However, with the assistance of technology students grasp and understanding of fundamentals can be greatly improved. Computers are not a guaranteed solution they are as I see it another means to help and improve student learning.