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.