This is a collection of math lessons posted in this blog. Most if not all of the lessons use the strategy teaching through problem solving or through mathematical investigation. I believe that school mathematics is about teaching students how to think mathematically first and learning the mathematics second so math lessons should be designed so that students are engaged in thinking mathematically. This is something that should not be left to chance.
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”.
image from art.com
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.
The chapter is divided into four parts: First part- Explore; Second part- Firm Up; Third Part – Deepen; Fourth Part – Transfer. There are many unit topics in a chapter so it means for example that what is being ‘deepened’ is a different topic to what has been ‘firmed-up” or “explored’! I think this is a mortal sin in teaching.
EFDT is used in all subject areas. The nature of each subject, each discipline, is different. I don’t know why some people think they can be taught in the same way or to even think that within a discipline, its topics can be taught in the same way. Or that the same style of teaching is applicable to all year levels in all kinds of ability. UbD, the real one, not our version, does not even promote a particular way of teaching but a particular way of planning. Stges 1 and 2 dictates the teaching that you needed to do.
Activities for Explore part always have to be done in groups and with some physical movement. A math teacher was complaining to me that her students no longer have the energy for their mathematics class especially during the “explore’ part because all subject areas have activities and group work so by the time it’s math period which happens to be the fourth in the morning, students no longer want to move. The explore part alone can run for several days. All the while I thought the “explore part” of EFDT can be done with a mathematical investigation or an open-ended problem.
The prepared lesson plans given during the training consists of activities from explore part to transfer part and teachers implement them one after another without much processing and connection. Most activities aren’t connected anyway.
The teachers can modify the activity but they said they don’t have resources where to get activities.
The teachers cannot modify the first two parts of the UbD plan. The teachers said they were told not to modify them. I asked “how does it help you in the implementation of the lesson?” They said “we just read the third part, where the lessons are. We don’t really understand this UbD. Our trainers cannot explain it to us. They said it was not also explained well during the training.
The teacher have this cute little notebook which contains their lesson. So I asked “so what is your lesson at this time?” She said it’s 3.5. Indeed that’s the little number listed there. So what’s it about. I think we are now on Firm-up. I have to check the xerox copy of the lesson plan distributed to us. Well, I thought UbD is a framework for designing the lesson. It was proposed by its author with the assumption that if teachers will design their lesson that way, then perhaps they can facilitate their lesson well. How come that teachers are not encourage to design their own lesson? How come we give them prepared lesson plans which have not even been tried out?
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.
In this post I propose a way of teaching the concept of triangle congruence. Like most of the lessons I share in this blog, the teaching strategy for this lesson is Teaching through Problem Solving. In a TtPS lesson, the lesson starts with a situation that students will problematize. The problems either have many correct answers or have multiple solutions and can always be solved by previously learned concepts and skills. Problems like these help students to make connections among the concepts they already know and the new concept that they will be learning in the present lesson. The ensuing discourse among students and between teacher and students during the discussions of the different solutions and answers trains students to reason and communicate mathematically and thereby help them to appreciate the power of mathematics as a language and a way of thinking. In mathematics, language is precise and concise.
Here’s the sequence of my proposed lesson:
1. Setting the Problem:
Myra draw a triangle in a 1-cm grid paper. Without showing the triangle, she challenged her friends to draw exactly the same triangle with these properties: QR is 4 cm long. The perpendicular line from P to QR is 3 cm.
Pose this question: Can you draw Myra’s triangle?
Give students enough time to think. When each of them already have at least one triangle, encourage the class to discuss their solutions with their seat mates. Challenge the class to draw as many triangles satisfying the properties Myra gave.
2. Processing of solutions: Ask volunteers to show their solutions on the board. Questions for discussion: (1) Which of these satisfy the information that Myra gave? (2) What is the same among all the correct answers? [They all have the same area]. Possible solutions are shown below.
3. Introducing the idea of congruence: Question: If we are going to cut-out all the triangles, which of them can be made to coincide or would fit exactly? [When done, introduce the word congruence then give the definition.]
Tell the class that Myra only drew one triangle. Show the class Myra’s drawing. Question: In order to draw a triangle congruent to Myra’s triangle, what conditions or properties of the triangle Myra should have told us?
Myra’s triangle
Possible answers:
QR is 4 cm long. The perpendicular line PQ is 3 cm.
QR is 4 cm long. PQ is 3 cm and forms a right angle with QR.
PQR is a right triangle with right angle at Q. QR is 4 cm and PQ is 3 cm.
4. Extending the problem solving activity: Which of the following sets of conditions will always give triangles congruent to each other?
In triangle ABC, AB and BC are each 5 cm long.
ABC is a right triangle. Two of its shorter sides have lengths of 4 cm and 5 cm.
I would appreciate feedback so I can improve the lesson. You feedback will inform the sequel to this lesson.. Thank you.
