EDUC4105

The following link gives a mathematics lesson that can be used in a congruent triangles lesson. Of all the properties used in congruent triangles SAS needs the most emphasis as the angle must be an included one. This lesson helps the students see what happens when this angle is not an included angle.

I cannot see a better way to teach this concept. I have taught this by drawing two triangles that follow these properties on the board but the lesson was nowhere near as visually stimulating as the geometers sketchpad version. Not only did the students see an example where two triangles followed the SSA property and were not congruent but they were able to change the shape and size of the triangles and see that even though the two triangles were still not congruent the SSA property still stood.

Congruent triangles are a big part of the NSW syllabus and students are required to gain a deep understanding on this topic if they wish to reach or exceed the required outcomes. This lesson will enforce the idea that any two triangles with 2 corresponding sides and an angle can only be proven congruent if the angle is an included angle. This will encourage students to think critically when working with congruent triangles in exams.

This lesson would still need much support from the teacher. For instance two of the sides are equal since they are both the radius of a circle. It can be challenging for students to see this. Some scaffolding is required in the lesson.

There were a few interesting points given in the readings. The semiotic systems, dense sentences and ‘mathematish’ used in the mathematics classroom have areas of concern for me.

The semiotic systems used in mathematics such as graphing, equations and diagrams are not very well connected in the classroom. Students often do not understand the significance of graphing say y = x +2. I feel that the lack of connectedness makes it difficult for students to see the connection between these two semiotic systems. A graph illustrating the exponential dissipation of heat for example could help students make this connection. A graph illustrating the dissipation of heat of coffee in a mug could be converted into an equation. Students then can see how long it takes for their coffee to be half its original temperature or have a discussion about how the function never touches the time axes or could work out it’s temperature after ‘x’ seconds.

Dense sentences make written proofs confusing for students. In teaching students I am hesitant to put a written proof up because it seems that it only makes everything more confusing. I feel that it is better to discuss an important concept and then put the written proof up. After this discussion students have a chance to understand the meaning of the written proof.

The ‘mathematish’ used by teachers in the classroom can cause problems. For example the equation ax + by + c = 0 is always written in this form but not in the form xa + yb + c = 0. I feel that students need to understand why teachers write this equation in this order. These reasons are often only to present information in a visually pleasing way. When students explore other ways of writing out such equations they will understand why mathematical writing has such a set style.

Semiotic systems, dense sentences and ‘mathematish’ can cause problems for students only because it is so difficult for teachers to pass on their tacit knowledge to students.

The webquest I chose was one called “Break the Code.” In our group of 5 we made comments about different aspects of the webquest. The roles were the affiliator, the efficiency expert, the altitudinist, the technophile and the NSW syllabus adviser.

The affiliator has the job of making sure that the task requires or forces students to work together. The task outlined in the “Break the Code” webquest can easily be done by a student on their own. Thus the affiliator did not do their job in fulfilling their role in this task.

The efficiency expert values time. They like tasks that are small and still teach the concept well. It was commented that this project integrates matrix multiplication and inverses in an interesting context. Clear steps were given for students to follow. This task provided a deep understanding of matrix multiplication and inverses in a reasonably short amount of time.

The technophile values use of technology in the webquest. They value a well presented site with links to other useful sites and resources. This webquest lacked a good use of technology. It provided few links and only referred to general sites such as google. The page presentation was weak and could have been given as a handout with little change in affect.

The altitudinist values higher order thinking in the task. This webquest was very procedural a usual sign that higher order thinking is low in the task. However the task was challenging enough to require some higher order thinking from the students.

The NSW syllabus adviser makes sure that the webquest concentrates on knowledge that is only in the NSW syllabus. Matrices are not taught at all in NSW so naturally this webquest does not suit NSW students. However students from Victoria might benefit form this as they learn about matrices.

The webquest provided an interesting and fun activity. It would have been so much better if the above mentioned improvements were implemented.

If we read pages 10 to 19 of a book are we reading 9 pages? Well the answer is no! We are reading 10 pages. People commonly do the sum 19-10 and conclude that 9 pages have been read. What about if you are asked to read pages 1 to 9 of a book? I think most would realise straight away that they were about to read 9 pages but 9 - 1 = 8 not 9. So why does subtraction give us the wrong answer?

The answer lies in whether the first digit is included in the sum. For instance if we have the set of numbers 1,2,3,4,5,6 and we take away two of the numbers lets say 5 and 6. We end up with the set of numbers 1,2,3,4 thus we have 4 numbers left over, 6 - 2 = 4. So the subtraction method works. However if we start counting from the number 2 and end at the number 6 we get 2,3,4,5,6. Notice that we have 5 numbers not 6 - 2 = 4 numbers. This is because we included the number 2 in our count. We needed to add 1 to our final solution.

This sort of thing causes problems all the time with students. The following are situations relating to the subtraction problem:

• Reading pages of a book.

• When trying to draw fractions like 3/8 using a bar they commonly draw 8 lines in the bar rather than 7. Thus they end up with the fraction 3/9.

• If you need to build a fence 1000 metres long and you need to have a fence post every 10 metres how many fence posts do you need?

Students need to see the potential for mistakes in using subtraction. They need to think about whether there is an extra digit that needs to be included in the solution.

There are many words in the English language that have a different meaning in mathematics. Similarly there are words specific to mathematics that have different interpretations outside the classroom. This not only causes problems in the classroom but makes it difficult for students to relate mathematics to the real world.

In the first reading it discussed the problem of using word such as “up” or “down” in the math classroom. For example if I travel from Newcastle to Brisbane I might say, “I am going down to Brisbane this weekend.” Brisbane is North of Newcastle and so one might assume that “up” is North. In the reading it said that this kind of comment “teeters or registers a kind of mental jolt” into people. I personally seem to never struggle with this kind of comment. I instantly interpret the word “up” or “down” in these situations as “going to” or “travelling to.” However in the classroom it is easy for “up” or “down” to cause confusion. Especially when students need to express questions as diagrams.

When selling an item “at a fraction of the cost” everyone instantly expects a heavy discount but what if the fraction is 4/3? Students have been trained in the real world in many instances to think of fractions as something less than a whole. This can be very confusing when students learn about improper or “top-heavy” fractions.

In the second reading it was asked, “what does a square and a root have to do with square-roots?” Students could have a great understanding of what words mean if they were broken up and explained in pieces. For example “poly” in polygon means many and “gon” means sides. In this way students will remember what a polygon is. Similarly with square roots students could explore the similarity between squares and the square-root.

Teachers need to discuss mathematical applications and use of mathematical language. Students need to see where problems can occur in the use and talk of mathematics and find ways to conquer these problems.

www.mathforum.org/teachers

The following website has been established by Drexel University and is a very valuable resource. It has a large variety of teaching methods, proofs and other important information provided for math teachers by math teachers. With over 900 threads this website provides answers and ideas to accommodate much of the curriculum.

With threads dating to over 10 years old this website gives a feeling of being well established. Because of this many of the threads are well discussed among teachers providing further insight and answers for each thread. Those who use this site feel that any queries they put forth will be viewed by many fellow teachers and therefore receive input very quickly.

This site lacks the flashy look most commercial sites need in order to attract this customers. It makes up for this by its usefulness and long standing reputation.

http://illuminations.nctm.org/index.aspx

This site provides lesson plans and activities for students in the math classroom. The site is very well organised and has a professional feel to it unlike the previous site. Everything is very uniform. Every lesson plan is structured in the same way.

The mathematics activities were predominately java programs. One program explored the quadratic (a+b)^2 and how it relates to the area of a square. This program and many others would work very well on the smartboard. With 102 of these programs this site provides a great variety of interactive lessons.

With 515 lesson plans this site provided a great variety of lessons that did not rely so heavily on direct instruction. Some lesson plans also referred to the java programs provided on the site and worksheets.

The previous site is a collection of threads from different teachers. This strength of this site is its diversity. The illuminations site is very well structured and easy to understand and follow. It is recommended that teachers first check the illuminations site to get an idea for the outline of a lesson. The mathforum site is useful for teachers who wish to explore further ideas in order to improve the quality of the teaching.

English is a very complicated language and it is not surprising that this causes problems in mathematics. Mathematical English (ME) used in the classroom often becomes confusing since its meaning can be taken in totally different ways depending on whether students interpret it as ME or Ordinary English (OE).

By looking at Mandarin we can see how the language can influence student outcomes in the classroom. One example from the first reading is giving an equation for the statement, ‘There are 6 times as many men as women.’ It is quite easy to see how English speaking students give the answer as 6m = w as this follows the sentence structure. If this were read in mandarin it would be more like, ‘Male members are female members 6 times.’ Thus the correct equation can easily be derived from this statement.

It is interesting in the second reading about referring to 3/4 as 3 fourths or 3 over 4. It would be helpful to teach students to understand this quite complex language. They could be taught that the first number i.e. 3 is the numerator and the number that ends in ‘ths’ i.e. fourths is the denominator. They could then have a quiz that requires students to translate words into fractions and vice versa. They could also discuss what you would call the fraction 3/2. Is it three tooths? This kind of discussion can help students make sense of ME.

From the second reading it is apparent that students struggle to find meaning in the calculations they perform. Long division is a great example of this as few seem to understand why it works. It is important for students to understand how the base 10 number system works as this lays the groundwork for students to learn addition, multiplication and division algorithms.

Multi-based arithmetic blocks (MABs) have been used to help students develop a concrete understanding of whole numbers. Probably more effective is teaching students different number base systems. It is easier to explain the concept of the base-ten number system after they first learn that our number system is a concept first.

I feel that substantive communication, problematic knowledge and higher-order thinking are some of the most prominent elements of the quality teaching model for tackling the problems of literacy in mathematics.

I have already made comments about graphics calculators but have made a few extra points about the advantages and disadvantages of them.

 Advantages of graphic calculators:

• Graphing, probability, calculations and other difficult, long and tedious tasks can be done more quickly and efficiently.
• It will prepare students for certain careers that require their use.
• It can be used to provide students with charts, graphs and other visual media compared to the usual display of numbers, equations and other less stimulating forms of information.

Disadvantages of graphic calculators:

• Students can become lazy and stop drawing graphs and charts without the calculator.
• Students may become dependant on the calculator eg: in an exam when the graphics calculator is not allowed.
• Simple mathematical skills may be forgotten and replaced by the use of the calculator. In many cases this increases the time taken to find solutions.

The readings both pointed out some very real problems students face in the classroom in regards to literacy in mathematics. These problems were summarised very well in the second reading where 5 key areas were mentioned. I am not as interested in talking about the problems as I am about mentioning solutions to these problems.

After reading the two readings I felt that there were two ways a teacher can battle the problems faced with literacy in mathematics. Firstly the teacher can make sure that they are careful with the language they use in the classroom to provide greater clarity for the students. Second, the teacher could concentrate more on getting the students to a higher literacy level in regard to mathematics.

A good example comes from the second reading. Let’s take the two phrases, ‘25% of a price’ and ‘25% off a price.’ Both phrases have completely different answers but look and sound almost exactly the same. There are two things that we can do here. Firstly the teacher could ask the question and write it down underlining the word ‘of’ or ‘off.’ This is adopting the first method I mentioned. Alternatively if the teacher had been taking the second approach over the period of time the students had been with them a better result could occur. The students might approach the teacher and ask whether they meant ‘of’ or ‘off.’

I feel that teachers could give students work that requires them to think and discuss the language used in the classroom. They could hand out question sheets that have problems like the following:

Match the following question to the correct equation:

Susan has 2 apples and Aaron has 3 pears, how much fruit do they have altogether?

(a) 2 x 3 = 6

(b) 2 + 3 = 5

(c) 3 - 2 = 1

(d) 2 apples and 3 pears

Such questions like this used in math classrooms will have students not just practicing the language but talking about the language as well. This is what I feel is the problem. Students are only practicing using the language when they should be going a step further and talking about it as well.

As important as it is for teachers to be careful with the way they speak and write in the math classroom I feel greater emphasis should be put on training students to understand discussions and questions in the context it is given. This can be achieved by having students not just practice using the language but talking about it as well.

In the study of language and mathematics there are 5 concepts or terms that are used to describe the way we use, think and communicate mathematics. These terms are; literacy, numeracy, mathematical literacy, quantitative reasoning and quantitative literacy. Some of these concepts are difficult to understand especially since many people have different views on their meanings.

The concept of numeracy seems to be of particular interest to people and has been debated internationally. I would define it as the ability to use and communicate mathematics to meet the demands of home life, the work place and the community. Some people like to delve much deeper into the meaning of numeracy but I think things are better kept simple.

The definition for mathematical literacy seems almost identical to the definition of numeracy. I feel the only difference would be that numeracy refers to meeting the demands that life puts on our mathematical abilities while mathematical literacy refers more to our general ability as a mathematician.

Literacy refers more specifically to the use of mathematics in communication. This means recognizing and understanding characters, words and mathematical language. Alternatively you should be able to use characters, words and mathematical language to communicate as well.

The information on the definitions of quantitative literacy seemed to talk about the reasoning and thinking side of mathematics. I found it a difficult concept to grasp. Quantitative reasoning was an even harder concept to grasp. All I know is that it has something to do with interpretive meaning.

These definitions are still changing and under debate. This would explain the difficulty for me to develop my own meaning of these concepts.