This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.
You may want to check out Blogineering.com, a new blog by my colleague Guillermo Bautista (author of Mathematics and Multimedia) and Riley Ayes. Blogineering is a site teaching the basics of blogging, tips in growing and promoting a blog, search engine optimization, and step by step tutorials on using different platforms. The authors have already completed the WordPress.com Blogging Tutorial Series. The plan to develop other blogging platforms such as Blogger, Tumblr, Typepad, and others.
Assessing understanding of mathematics can also be done by asking students to write test items. Here’s my favorite assessment item. I gave this to a group of teachers. 
Possible answers/ questions.
Year level: Third year (Year 9)
Question 1 – What is the distance of P from the origin?
Question 2 – What is the area of circle P with radius equal to its distance from the origin?
Question 3 – With P as one of the vertex, draw square with area 2 square units.
Year level: Second year (Year 8 )
Question 1 – Write the equation of the line that passes through P and the origin.
Question 2 – Write 3 equations of lines passing through (2,1).
Question 3 – Write the equation of the family of lines passing through (2,1).
Year level: First year (Year 7)
Question 1 – What is the ordinate of point P?
Question 2 – Locate (-2, 1). How far is it from P?
Question 3 – Draw a square PQRS with area 9 square units. What are the coordinates of that square?
How about using this exercise to assess your students? Ask them to construct test items instead of asking them to answer questions.
Here are a few more assessment items which I constructed based on the TIMSS Framework:
One of the most difficult items for the Philippine sample in the Trends and Issues in Science and Mathematics Education Study (TIMSS) for Advanced Mathematics and Science students conducted in 2008, is about comparing the slopes of the tangent at a point on a curve. The question is constructed so that it assesses not only the students understanding of tangent lines to the graph of a trigonometric function but also students’ skill to use mathematics to explain their thinking. The question is one of the released items of TIMSS Advanced 2008 so I can share it here. The graph actually extends beyond point B in the original item.
Sophia is studying the graph of the function y=x+cos x. She says that the slope at point A is the same as the slope at point B. Explain why she is correct.
I don’t have information if the students’ difficulty has to do with their mathematical understanding or it is the way the question is asked. I have a feeling that had the question been ‘What is the derivative of the function y = x + cos x?’, the students would have been able to answer it. But of course, the item is also assessing students’ conceptual understanding of derivative as the slope of the tangent line at a point on a curve.
The TIMSS Advanced tests were given to Year 11/12 populations. Because the country does not have senior high schools, the Philippines sample were Year 10 students from Science High Schools where calculus is a required subject. The group of teachers we were discussing this question with said they are only able to cover up to the derivative of polynomial functions although the syllabus cover derivative of trigonometric functions. Indeed, the problem should not be difficult to those who have taken calculus or at least have reached the topic about the derivative of trigonometric functions. The solution is pretty straight forward. The derivative of the function y = x + cos x is 1+-sin x so the slope of the tangent at ? and 2? is 1.
Covering the syllabus is really a problem because of lack of time. Even if the students are well selected, I think it is still a tall order to cover topics what other countries would cover with an additional two years in high school. Quality of teaching suffers when teachers will teach math at lightning speed. One is forced to do chalk and talk.
The TIMSS item shown above can still be solved with basic knowledge of trigonometric function and slopes of tangent lines. The function y = x + cos x is a sum of the function y = x and y = cos x. The slope of y = x is 1. That slope is constant. The function y = cos x has turning points at ? and 2? hence the slope of the tangents at these points is 0. So, Sophia is correct in saying that the slope of the tangents at ? and 2? in y = x + cos x are the same. Students are more likely to analyze the problem this way if they have a conceptual understanding of the functions under consideration and if they are exposed to similar way of thinking, especially of expressing representations in equivalent and more familiar form. This way of thinking need to be developed early on. For example, learners need to be exposed to tasks such
1) Find as many ways of expressing the number 8.
2) What number goes to the blanks in 14 + ___ = 15 + ____.
3) Solve 3x = 2x – 1 graphically.
You may want to read my other posts to items based on TIMSS framework here and proposed framework for analysing understanding of function in equation form and sample problem on sketching the graph of the derivative function.
References for understanding the idea of derivatives
1.Students’ conceptual understanding of a function and its derivative in an experimental calculus course [An article from: Journal of Mathematical Behavior]
2. Calculus: An Intuitive and Physical Approach (Second Edition)
This post shows how we can help students make connections among counting principle, the Pascal’s triangle, and powers of 2. I have tried this lesson in an in-service training program but I’ve yet to test it with students in high school. The lesson uses the strategy Teaching thru Problem Solving.
A piece of knowledge is powerful to the extent to which it is connected to other piece of knowledge. The more connections there are, the more powerful it becomes. Mathematics teaching therefore should always aim to help students make connections among the different concepts of mathematics. You may want to read my article about understanding as making connections.
The Problem: Trace the paths that will spell “MATHEMATICS” starting from the letter M on top moving only downwards, either to the immediate letter to its right or to the immediate letter to its left. How many different paths are there in all?
After a few minutes and the class is seem getting nowhere you may suggest to students to try simpler case first like trying the word MATH. Trying simpler case is a good problem solving strategy and habit students need to learn.
Suppose we spell the word “MATH” only. From M we can move downwards and may either choose the A at the left or the A at the right. Having chosen an A we can either choose the T down left or the T down right. And having chosen one we can either choose the H down right or the H down left. Each time we only have two choices. Thus, the number of ways of tracing the word “MATH” in the above figure is
2·2·2 =23=8
Using the same line of thinking, the total number of paths which spells “MATHEMATICS” is
2·2·2·2·2·2·2·2·2·2=210=1024.
Notice the number of arrows that converges to a particular letter. It tells the number of paths that pass through it. Thus, to count the number of ways of tracing the word “MATH” we only have to add the total number of arrows that point to the H’s. There are
1 + 3 + 3 + 1 = 8.
Count the number of arrows converging to each letter in MATHEMATICS . You will generate the triangular array of numbers below.
The number of arrows converging to S is
1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 1024 or 210.
The solutions showed two important principles of counting.
The Multiplication Principle. If one task can be done in m ways and then another task can be done in n ways, the pair of tasks, first one and then the other, can be performed in
m • n ways.
The Addition Principle. If one task can be done in m ways and another task in n ways, then one task or the other can be done in
m + n ways.
Anyone who wants to understand permutations, combinations and anything that involves counting should first understand these principles.
The triangular array of numbers generated above is one of the most influential number patterns in the history of mathematics. It is called Pascal’s triangle after the renowned French mathematician Blaise Pascal (1623-1662) who discovered it. The triangle is also called Yang Hui’s triangle in China as the Chinese mathematician Yang Hui discovered it much earlier in 1261. The same triangle was also in the book “Precious Mirror of the Four Elements” by another Chinese mathematician Chu-Shih-Chieh in 1303.
The Pascal triangle yields interesting patterns and relationships. Some of the obvious ones are:
Recommended readings on combinatorics:
