Tag: conceptual understanding

Posted in Algebra, Mathematics education

When is a math problem a problem?

Posted on by

One of the main objectives of mathematics education is for students to acquire mathematical habits of mind. One of the ways of achieving this objective is to engage students in problem solving tasks. What is a problem solving task? And when is a math problem a problem and not an exercise?

What  is a problem solving task?
A problem solving task refers to a task requiring a solution or answer, the strategy for finding such is still unknown to the solver. The solver still has to think of a strategy. For example, if the task,

If x^2 - 7 = 18, what is x^2 - 9 equal to?

is given before the lesson on solving equation, then clearly it is a problem to the students. However, if this is given after the lesson on solving equation and students have been exposed to a problem similar in structure, then it cease to be a problem for the students because they have been taught a procedure for solving it. All the students need to do is to practice the algorithm to get the answer.

What is a good math problem?

The ideal math problem for teaching mathematics through problem solving is one that can be solved using the students’ previously learned concepts/skills but can still be solved more efficiently using a new algorithm or new concept that they will be learning later in the lesson. If the example above is given before the lesson about the properties of equality, the students can still solve this by their knowledge of the concept of subtraction and the meaning of the equal sign even if they have not been taught the properties of equality or solving quadratic equation (Most teachers I give this question to will plunge right away to solving for x. They always have a good laugh when they realize as they solve the problem that they don’t even have to do it. They say, “ah, … habit”.)

Given enough time, a Year 7 student can solve this problem with this reasoning: If I take away 7 from x^2 and gives me 18 then if I take away a bigger number from x^2 it should give me something less than 18. Because 9 is 2 more than 7 then x^2 - 9 should be 2 less than 18. This is 16.

Why use problem solving as context to teach mathematics?

You may ask why let the students go through all these when we there is a shorter way. Why not teach them first the properties of equality so it would be easier for them to solve this problem? All they need to do is to subtract 2 from both sides of the equal sign and this will yield x^2 - 9 = 16. True. But teaching mathematics is not only about teaching students how to get an answer or find the shortest way of getting an answer. Teaching mathematics is about building a powerful form of mathematical knowledge. Mathematical knowledge is powerful when it is deeply understood, when concepts are connected with other concepts. In the example above, the problem has given the students the opportunity to use their understanding of the concept of subtraction and equality in a problem that one will later solve without even being conscious of the operation that is involved. Yet, it is precisely equations like these that they need to learn to construct in order to represent problems usually presented in words. These expressions should therefore be meaningful. Translating phrases to sentences will not be enough develop this skill. Every opportunity need to be taken to make algebraic expressions meaningful to students especially in beginning algebra course. More importantly, teaching mathematics is not also only about acquiring mathematical knowledge but more about acquiring the thinking skills and disposition for solving problems and problem posing. This can only happen when they are engage in these kind of activities. For sample lesson, read how to teach the properties of equality through problem solving.

Finally, and I know teachers already know this but I’m going to say it just the same. Not all ‘word problems’ are problems. If a teacher solves a problem in the class and then gives a similar ‘problem’ changing only the situation or the given ‘numbers’ but not the structure of the problem or some of the condition then the latter is no longer a problem but an exercise for practicing a particular solution to a ‘problem’. It may still be a problem of course to those students who did not understand the teacher’s solution. I’m not saying that this is not a good practice, I am just saying that this is not problem solving but an exercise.

You may also want to read How to Solve It: Modern Heuristicsto further develop your problem solving skills.

Posted in Trigonometry

Slopes of tangent lines

Posted on by

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)