teaching through problem solving Archives

(Mis) Understanding by Design

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The country’s schools are now implementing ‘Understanding by Design (UbD) curriculum.’ Some private schools are implementing it at all levels while all the public schools are on its first year of implementation starting with first year high school subjects. I’m not a fan of UbD, especially in the way it is being implemented here but that is irrelevant. (If I have my way, I rather spend the money for Lesson Study.) But of course, I want UbD to work because DepEd is spending taxpayers money for it. But from conversations and interviews with teachers and looking at what they call call ‘Ubidized learning plans’, I am starting to doubt whether or not what we are implementing is really UbD. Here’s how UbD is understood and being carried out in some schools:

1. With UbD teachers will no longer make lesson plans. They will be provided with one. Here’s a comment on my post Curriculum Change and Understanding by Design: What are they solving? from a Canadian educator:

UbD may not be your priority–I gather that you see PCK and CK as the core issue. But at least UbD positions teachers as the decision-makers rather than imposing lessons on them…. I am not a UbD proponent, but I think it’s a structure I could work with, a structure I could infuse with my beliefs and goals, because it puts teachers at the center of the decision making, with student understanding as the target.

Indeed, nowhere in the UbD book of McTighe and Wiggins that they propose that teachers should no longer make lesson plans or that it is a good idea that somebody else should make lesson plans for the teachers. What they propose is a different way of designing or planning the lesson – the backward design. Continue reading “(Mis) Understanding by Design” →

Algebra

Teaching the properties of equality through problem solving

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I like to teach mathematical concepts via problem solving. It right away engages students mind. It creates a need for learning a more systematic way of doing things and hence a reason for learning the concept. It provides a context for making connections. Most important of all, it gives students opportunity to learn before they are taught.

One of the ways of creating a problem solving task for this kind of lesson is to start by giving the solution/answer. For example, a standard textbook task is to solve for x in an equation. Why not do it the other way and ask students to find the equation given the solution?

Here is my favorite problem for introducing the properties of equality:

Write three equations in x with solution x = 5.

Fresh from their elementary school math experience of solving equation of the type 15 – ___ = 20 or x + 12.3 = 20, students will generate equations by trial and adjust. That is, they think of an equation then check if it gives x = 5 by substituting the value of x to the equation. It will not take long for them to realize that this is something very tedious.

Other students will verify their equations by expressing the terms into equivalent structure. I explained this method in my previous post. This method is not also very efficient for some equations. But students have to experience these so that they will be able to appreciate the efficiency of solving equations using the properties of equality.

Usually when they already have at least two equations I will extend the task to:

Two of the equations should have x on both sides of the equal sign .

Challenge them further to think of a more systematic way for making the equations. You will not hear the bell ring before students will realize that all they need to do to have an x on both sides is to add or subtract x on both sides of the equal sign! (Multiplying or dividing by x is a different matter as there is a possibility that x can be zero). Now they know that those ubiquitous equations in their algebra textbooks did not just drop from they sky. They can even create one themselves. The students are applying the properties of equality before they even know what they are! Speaking of constructivist teaching here.

Teachers I shared this with were also very happy because they no longer have to do trial and adjust when they make exercises for solving equations.

Teaching the concept of function

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Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study. Function is the central underlying concept in calculus. It is also one of the key concepts of mathematics that can model many quantitative relationships.

Textbooks and teachers usually introduce function via a situation with the related quantities already identified. What is required of the students is to learn how to set up and represent the relationships in tables, graphs, and equation and analyze the properties. In the real world, when function is used as a model, the first thing that needs to be done is to identify the varying quantities. So, it is important to let students identify the quantities and let them determine which of these quantities may be related. This way they get a sense of what function really is and what it is for. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties, not finding y or f(x) given x and vice versa, not reading graphs, and not translations among the representations. These are important knowledge and skills, yes, but only in the context for investigating or learning more about the relationships between the quantities, that is, the function. Thus, for an introductory lesson for function, I find it useful to use a situation where students themselves will:

identify the changing and unchanging quantities;
determine the effect of the change of one quantity over the others;
describe the properties of the relationship; and,
think of ways for describing and representing these relationships.

These are the ‘big ideas’ students should learn about function. Of course, there are others like looking or dealing with function as a mathematical object and not only as a process or procedure for generating or predicting values. However, for an introductory lesson on function, teachers need not focus on this yet.

Sample introductory activity:

What are the quantifiable attributes or quantities can you see in the figure below? Which of these quantities will change and remain unchanged if GC is increased or decreased? Click the figure and move point C. Are there ways of predicting the values of these changing quantities?

teaching function — Identifying related quantities

Click here or the image above to go to dynamic window for the worksheet.

I like this particular activity because it gives students the opportunity to link geometry/measurement concepts to algebra and learn mathematics through solving problems.

Click link for a synthesis of the evolution of the definition of function and What are the big ideas in function?

Lesson Study, Mathematics education

Lesson Study in Mathematics

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From 2006 – 2009, the University of the Philippines National Institute for Science and Mathematics Education Development (UPNISMED) piloted a school-based professional development program for teachers called Lesson Study to two high schools and two elementary schools in Metro Manila. The project involved all the mathematics teachers of the schools. Lesson study engages teachers in creative and collaborative problem solving activity in designing a lesson that teaches mathematics through problem solving. The project is based on the following principles: (1) learners construct their own knowledge whether that learner is a teacher or a student; (2) learners learn most when they are engage in tasks that they view as significant to them and that presents a real problem for them; and (3) learning is a social activity whatever the object of the learning is.

The results of the lesson study project has been very encouraging. In terms of outputs, video lessons and lesson plans have been produced showcasing teaching mathematics via problem solving. These lessons were developed and implemented by the teachers collaboratively together with one UP NISMED mathematics education specialist per year level. The lessons produced show:

how to facilitate a problem solving lesson where students solve problems without being shown a solution first (the essence of the problem solving activity is preserved);
that a problem, traditionally given at the end of the chapter can be given at the start of the chapter;
that review of concepts, traditionally a separate part of the lesson and in drill type, can be integrated to the main lesson itself;
that lesson can be structured that would engage students to represent ideas mathematically, solve problems in different ways, and reason out;
that a problem solving task can be a rich context for learning new mathematical concepts and link these with previously learned concepts.

As a result of these, and this is perhaps the most important achievement of the project, is the change in the teachers’ perception about the role of problem solving in mathematics. During the planning meetings, the mathematics teachers I was working with expressed apprehension about the problem solving lesson they were developing. They said that “Work” problems are one of the most difficult types of algebra problems so they thought there is no way students can solve it by themselves without being shown sample problem and solution first and the even if these are shown, students still need to know how to solve rational equations. This is the reason why the problem is found at the end of the chapter! These were their impressions until they produced and implemented a lesson that challenged their own assumptions. They realized that problem solving can also be a means for learning mathematics rather than simply a reason for learning it; and, that students are more capable in solving problem on their own than they previously thought.

The teachers admitted that initially, they saw lesson study as another “burden” to them but as the project progresses they eventually appreciated it. They said that they learned a lot from each other and the post conference and discussion part became a venue for them to deepen their understanding of mathematics and how students understand mathematics. We also documented changes in the quality of teachers discussion during the post conference. Initially they were focusing on general pedagogy but towards the third cycle of the lesson implementation they were now more focused on the content and how their questions for discussion is affecting the quality of the students’ thinking.

This year we are working with another school with an improved design of the project. We just finished a three-day orientation seminar about lesson study and teaching mathematics via problem solving for the mathematics teachers of the said school. Goal-setting, the first step in the lesson study process was done during the seminar. The teachers agreed that their goal is to make students value mathematics by developing their thinking skills. Their sub goals for this year is to develop lessons that engage students in mathematical representations and solving problems in different ways. I will talk more about these in my next post.

Download full paper: Scaffolding Teacher Learning through Lesson Study.

Email me if you are interested to do lesson study in your schools (schools in Philippines only.) To give you an idea how lessons are planned and analyzed in a lesson study context view this presentation: Planning and Analyzing Mathematics Lessons in Lesson Study

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