This announcement is for science and mathematics teachers and educators working in the Philippines.
The University of the Philippines National Institute for Science and Mathematics Education Development (UP NISMED) will hold a national seminar-workshop in science and mathematics education on October 26-28, 2011 at UP Diliman, Quezon City.
For details go to the seminar website.
When we were just being trained to be teachers of mathematics it was emphasized to us that in planning our lesson we should think of manipulative activities whose results will eventually lead to the concepts to be learned. The teacher will make use of the students results to introduce the new concept through another whole class activity to tie together the results or through question and answer discussion. This leads to the definition of the concept by the teacher or to a certain procedure or calculation with the help of the students, depending on the topic. The teacher then gives exercises so students can hone their skill or deepen their understanding of the concept. A homework, usually a more difficult version of the one just done in the class, is given at the end of the lesson. I don’t remember my supervising teacher requiring me to always give a test at the end of my lesson. I think I was on my third year of teaching in public school when this ‘bright idea’ of giving a test at the end every lesson was imposed. Failure to do so means you did not have a good lesson because you do not have an evaluation part! Anyway, let me stop here as this is not what I want to talk about in this post. I want to talk about the latest ruling about “Ubidized lesson pans”.
When I first heard about the DepEd’s “Explore-FirmUp-Deepen-Transfer” version of UbD I remember the framework I followed when I was doing practice teaching at Bicol University Laboratory High School. The lesson starts with activities, process results of activities to give birth to the new concept, firm-up and deepen the learning with additional exercise and activities and then use the homework to assess if students can transfer their learning to a little bit more complex situation. So I thought EFDT must not be a bad idea. I have observed as a teacher-trainer that over the years teachers have succumbed to the temptation of talk-and-talk method of teaching. Reason: there are too many students, activities are impossible; too many classes to handle, too many topics to cover. With this scenario I thought EFDT may turn out to be a much better guide in planning the lesson that the one currently being used: “Motivation-LessonProper-Practice-Evaluation” because EFDT actually describes what the teachers need to do at each part of the lesson. But it turned out that EFDT was very different what I think it is and is being implemented per chapter and not per topic or lesson in the chapter!
I don’t know if the teachers simply misinterpreted it or this is really how the DepEd wants it implemented. If this is how UbD is being done in the entire archipelago then we have a BIG problem.
Here’s my wish Explore, Firm-up, Deepen, and Transfer be interpreted in mathematics teaching.
Explore – students are given an open-ended problem solving task or short mathematical investigation and they are given opportunity to show different ways of solving it.
Firm-up – the teacher helps the students make connections by asking them to explain their solutions and reasoning, comment on other’s solutions, identify those solutions that uses the same concepts, same reasoning, same representation, etc.
Deepen – the teacher consolidates ideas and facilitates students construction of new concept or meaning, linking it to previously learned concepts; helps students to find new representations of ideas, etc.
Transfer – teacher challenges students to extend the problem given by changing aspects of the original problem or, construct similar problems and then begin to explore again.
The above descriptions corresponds to a way of teaching called teaching mathematics via problem solving which this blog promotes.
Credits: image from art.com
As a result of her research, Liping Ma developed the notion of profound understanding of fundamental mathematics (PUFM) as the kind of mathematical knowledge teachers should possess. She discusses this kind of knowledge in her book Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series)
Here’s what Liping Ma says in the introduction:
Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers seem far less mathematically educated than U.S. teachers. Most Chinese teachers have had 11 to 12 years of schooling – they complete ninth grade and attend normal school for two or three years. In contrast, most U.S. teachers have received between 16 and 18 years of formal schooling-a bachelor’s degree in college and often one or two years of further study.
In this book I suggest an explanation for the paradox, at least at the elementary school level. My data suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than that of most U.S. elementary teachers. Their understanding of the mathematics they teach and -equally important – of the ways elementary mathematics can be presented to students continues to grow throughout their professional lives. Indeed about 10% of those Chinese teachers, despite their lack of forma education, display a depth of understanding which is extraordinarily rare in the United States….
Why the word ‘profound’? Profound has three related meanings – deep, vast and thorough – and profound understanding reflects all three. From the paper delivered by Liping Ma and Cathy Kessel in the Proceedings of the Workshop on Knowing and Learning Mathematics for Teaching conference, Liping and Cathy offered the following explanation:
A teacher should see a ‘knowledge package’ when they are teaching a piece of knowledge. They should know the role of the current concept they are teaching in that package and how that concept is supported by which ideas or procedures.
To further explain the kind of mathematics knowledge a teachers should possess, Liping and Cathy used the analogy of a taxi driver who knows the road system well. The teachers should know many connections so that they are able to guide students from their current understandings to further learning.
I think this is how designers of curriculum, writers of curriculum materials, and teachers should interpret the standard “Making connections”. It is not simply linking.
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)
