The following problem (or proving activity, if you like to call it that) is a typical textbook geometry problem. It is tough and guaranteed to scare the wits out of any Year 9 student.
When I used the given condition to construct the figure using GeoGebra, the only thing I can move is A or B and what it does is simply to reduce or enlarge the circle. Pretty boring. So I thought of making C dynamic. The way to do this is to construct point C along the circle and then construct a perpendicular line to AB. With C moving along the circle, the once static and close task is now a dynamic, exploratory and task.
Your students will observe that for triangle ABC to be an equilateral triangle, CD must be the perpendicular bisector of AB. You can now ask them the problem: Given that CD is the perpendicular bisector of radius AB, prove that ABC is an equilateral triangle, which is what the textbook is asking them to do.
In presenting the problem the way I’ve shown above, you did not only make the problem more interesting (hopefully) and accessible to the majority of the learners (I’m sure most of them can answer the questions), you have also given learners the chance to explore the problem first and be familiar with the situation.
Note that you will be doing a disservice to your students’ geometry life if you will stop at #5 and not give them the opportunity to prove. Proving is what makes mathematics different from other disciplines. It would be a shame if they will go through life only complaining about x and not of proving as well. I’m joking but you know what I mean. You may want to check some of my favorite post about teaching geometry through problem solving: Unpacking mathematics – a geometry example and Problem Solving Involving Quadrilaterals.
Mathematical tasks can be classified broadly in two general types: exercises and problem solving tasks. Exercises are tasks used for practice and mastery of skills. Here, students already know how to complete the tasks. Problem solving on the other hand are tasks in which the solution or answer are not readily apparent. Students need to strategize – to understand the situation, to plan and think of mathematical model, and to carry-out and evaluate their method and answer.
Exercises and problem solving in teaching
Problem solving is at the heart of mathematics yet in many mathematics classes ( and textbooks) problem solving activities are relegated at the end of the unit and therefore are usually not taught and given emphasis because the teacher needs to finish the syllabus. The graph below represents the distribution of the two types of tasks in many of our mathematics classes in my part of the globe. It is not based on any formal empirical surveys but almost all the teachers attending our teacher-training seminars describe their use of problem solving and exercises like the one shown in the graph. We have also observed this distribution in many of the math classes we visit.
The graph shows that most of the time students are doing practice exercises. So, one should not be surprised that students think of mathematics as a a bunch of rules and procedures. Very little time is devoted to problem solving activities in school mathematics and they are usually at the end of the lesson. The little time devoted to problem solving communicates to students that problem solving is not an important part of mathematical activity.
Exercises are important. One need to acquire a certain degree of fluency in basic mathematical procedures. But far more important to learn in mathematics is for students to learn to think mathematically and to have conceptual understanding of mathematical concepts. Conceptual understanding involves knowing what, knowing how, knowing why, and knowing when (to apply). What could be a better context for learning this than in the context of solving problems. In the words of S. L. Rubinshtein (1989, 369) “thinking usually starts from a problem or question, from surprise or bewilderment, from a contradiction”.
My ideal distribution of exercises and problem solving activities in mathematics classes is shown in the the following graph.
What is teaching for and teaching through problem solving?
Problems in mathematics need not always have to be an application problem. These types of problems are the ones we usually give at the end of the unit. When we do this we are teaching for problem solving. But there are problem solving tasks that are best given at the start of the unit. These are the ones that can be solved by previously learned concepts and would involve solutions that teachers can use to introduce a new mathematical concept. This strategy of structuring a lesson is called Teaching through Problem Solving. In this kind of lesson, the structure of the task is king. I described the characteristics of this task in Features of Good Problem Solving Tasks. Most, if not all of the lessons contained in this blog are of this type. Some examples:
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 , what is 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 and gives me 18 then if I take away a bigger number from it should give me something less than 18. Because 9 is 2 more than 7 then 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 . 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.
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.
, what is 
equal to?
and gives me 18 then if I take away a bigger number from 
should be 2 less than 18. This is 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 
to further develop your problem solving skills.
