Posted in Curriculum Reform, Mathematics education

My issues with Understanding by Design (UbD)

Everybody is jumping into this new education bandwagon like it is something that is new indeed. Here are some issues I want to raise about UbD.  I am quoting Wikipedia in this post but this is also how UbD is explained  in other sites.

Understanding by Design, or UbD, is an increasingly popular tool for educational planning focused on “teaching for understanding”.

Is not teaching for understanding been the focus of all curricular reforms, then and now? No curriculum reformer wants to be caught in the company of rote learning, never mind that it’s how curricula are implemented, regardless of its form, kind and  substance in many classes. Teaching for understanding is not something new.

UbD expands on “six facets of understanding”, which include students being able to explain, interpret, apply, have perspective, empathize, and have self-knowledge.

I wonder which of these facets has not been a part of what it means to understand then. I’m not sure in other subject areas but these facets of understanding such as explain, interpret, and apply does not capture what it means to understand mathematics.

To facilitate student understanding, teachers must explain the “big ideas” and “essential questions” as well as the requirements and evaluative criteria at the start of the class.

Back in college we attribute it to Ausubel who promoted the idea of using advance organizers.  Of course, you don’t tell your students right away how they will be assessed. They don’t have those rights, then. Also, this method only works for some topics. In mathematics if the approach is Teaching through Problem Solving or Discovery method, this is a no-no as it might limit the students thinking in exploring their own ways of working with the task at hand.

The emphasis of UbD on “big ideas” is welcome development but shouldn’t this be contained in the curriculum framework? The “essential questions”, those elusive questions that teachers have difficulty formulating since probably the time the  education community was talking about “art of questioning” are also good reminders to all of us that ‘hello, processing questions before or after any activity are what make and unmake a lesson’. But isn’t it that one can only identify the enduring understanding required and formulate good questions if he/she has a very good content knowledge (CK) and pedagogical content knowledge (PCK)?. Shouldn’t the money and time for training teachers how to design a lesson using UbD be spent instead on deepening their understanding of CK and PCK? Shouldn’t we make sure first that we have a good curriculum framework that articulates what are important for students to know and understand in each subject area and in each content topic?

The emphasis of UbD is on “backward design”, the practice of looking at the outcomes in order to design curriculum units, performance assessments, and classroom instruction.

In my part of the globe, there is a national curriculum which is a collection of SMART objectives. These learning objectives have always been stated in terms of outcomes. Weren’t they called competencies? Aren’t these competencies tell what to assess? The trouble is, our list of competencies consist of factual and procedural knowledge and very little on problem solving and reasoning which never really get taught because they are all found at the end of each chapter!

According to Wiggins, “The potential of UbD for curricular improvement has struck a chord in American education. Over 250,000 educators own the book. Over 30,000 Handbooks are in use. More than 150 University education classes use the book as a text.”

That explains everything. Everybody is hooked on the book that no one found time to do research if it works or not. Of course, on this part of the world where I come from I could not possibly have full access to current studies in educational planning and curriculum conducted elsewhere. I’m pretty sure though that we don’t have a study here yet. This is actually my issue. We’re jumping on a bandwagon created elsewhere without checking first if it will run on our roads.

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Understanding by Design recommends a structure for curriculum planning, for designing instruction. It is not surprising that this is a welcome development because of lack of the same when it comes to this area. College education and in-service programs have failed to equip teachers the knowledge and skills to identify the important ideas in their major field of study.

Click here for the proposed stages of lesson development by UbD (thanks Jimmy Wysocki). Imagine it in the hands of our classroom teachers. Imagine how their faces will look like if you tell them “these elements should be in your written lesson plans”! And when they look for resources, all they have is an anemic curriculum framework and textbooks teaching facts that can be Googled. They will follow the directives, of course, as they have always done in the past in this part of the globe. They won’t just have time anymore to study and prepare  for the actual teaching of the lesson, especially in examining how their students learn specific topic. Surely, they will have a very neat plans complete with the elements. But lest we forget, learning is still more a function of the experiences students engages in, that is the lesson, and not in the lesson plans format.

Lastly, UbD is a one size fits all for all subject areas. That’s what make it highly suspect. Click here and here for sequels of this post.

Posted in Assessment, Curriculum Reform

Features of good problem solving tasks for learning mathematics

To develop higher-order thinking skills (HOTS) the mind needs to engage in higher-order learning task (HOLT). A good task for developing higher-order thinking skills is a problem solving task. But not all problems are created equal. Some problems are best suited for evaluating learning while others are best suited for assessing learning that would inform teaching. This post is about the second set of problems.The difference between these two sets of problems is not the content and skills needed to solve them but the way they are constructed.

What are the features of a good problem solving task for learning mathematics?
  1. It uses contexts familiar to the students
  2. What is problematic is the mathematics rather than the aspect of the situation
  3. It encourages students to use intuitive solutions as well as knowledge and skills they already possess
  4. The task can have several solutions
  5. It challenges students to use the strategy that would highlight the depth of their understanding of the concept involved
  6. It allows students to show the connections they have made between the concepts they have learned

It is this kind of problem solving task that is used in the strategy Teaching through Problem Solving (TtPS) which I described in the previous post. Here is a sample task:

Students solutions to the task can be used to teach area of polygons, kinds of polygons, preserving area, and meaning of algebraic expression. You can use the task to facilitate students construction of knowledge about adding, subtracting, multiplying and dividing algebraic expressions. Yes, you read it right. This is a good problem solving task for introducing operations with algebraic expression through problem solving! The problem above is also an example of a mathematical tasks that links algebra and geometry. Good mathematics teaching always links concepts.

Posted in Curriculum Reform, Mathematics education

Teaching through Problem Solving

Problem solving is not only the reason for teaching and learning mathematics. It is also the means for learning it. In the words of Hiebert et al:

Students should be allowed to make the subject problematic. … Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students. (Hiebert, et al, 1996, p. 12)

For years now, UP NISMED in-service training programs for teachers have organized mathematics lessons for teachers using the strategy we call Teaching through Problem Solving (TtPS). This teaching strategy had also been tried by teachers in their classes and the results far outweighed the disadvantages anticipated by the teachers.

Teaching through problem solving provides context for reviewing previously learned concepts and linking it to the new concepts to be learned. It provides context for students to experience working with the new concepts before they are formally defined and manipulated procedurally, thus making definitions and procedures meaningful to them.

What are the characteristics of a TtPS?

  1. main learning activity is problem solving
  2. concepts are learned in the context of solving a problem
  3. students think about math ideas without having the ideas pre-explained
  4. students solve problems without the teacher showing a solution to a similar problem first

What is the typical lesson sequence organized around TtPS?

  1. An which can be solved in many ways is posed to the class.
  2. Students initially work on the problem on their own then join a group to share their solutions and find other ways of solving the problem. (Role of teacher is to encourage pupils to try many possible solutions with minimum hints)
  3. Students studies/evaluates solutions. (Teacher ask learners questions like “Which solutions do you like most? Why?”)
  4. Teacher asks questions to help students make connections among concepts
  5. Teacher/students extend the problem.

What are the theoretical underpinnings of TtPS strategy?

  1. Constructivism
  2. Vygotsky’s Zone of Proximal Development (ZPD)

Click here for sample lesson using Teaching through Problem Solving to teach the tangent ratio/function.

The best resource for improving one’s problem solving skills is still these books by George Polya.

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Volume I

Posted in Curriculum Reform

Algebra vs Arithmetic Thinking

Algebra had always been associated with high school mathematics while arithmetic, the study of numbers, is associated with elementary school mathematics. One of the solutions to help students understand algebra in high school is to start the study of algebra earlier hence the elementary school curriculum incorporated some content topics traditionally studied in high school. However, I believe that more than knowledge of additional content, pupils can best be prepared for further mathematics work by engaging them in activities in deeper and more challenging ways using the traditional content of elementary school mathematics. I believe that children who become familiar with algebraic thinking from an early age and in meaningful contexts will do better in mathematics.

There is this study which I read in this paper titled A cognitive gap between arithmetic and algebra. This study distinguished algebra and arithmetic in terms of the type of equation tasks. According to them if the equation only involves one unknown then that is an arithmetic task. If the equation involves two unknowns then it is an algebra task. For example,

(1) 15 + ____ = 40 is an arithmetic task while

(2) ____ = 4 + _____ is an algebra task.

This distinction, in a way, makes sense. To answer Equation (1), a child only need to ask: What number should I put in the blank so that when I add it to 15, it gives 40? Equation (2) involves the concept of a variable. There are infinite values that you can  put in the two blanks. It also involves  the concept of function, the relationships between two numbers. The two numbers in the blanks cannot be just any number. The two numbers must differ by 4 and the number in the first blank should always be the greater number. This relationship is a “very algebraic” concept. But, even then, I’m still not very excited about this distinction between algebra and arithmetic!

I believe that one is engaged in algebra when one thinks relationally. Equation (1) for example is not necessarily an arithmetic task. If a student solves it by reasoning “because they are equal, even if I subtract 15 from both side of the equality sign then I still maintain the equality then he is doing algebra.  Another solution to Equation (1) is to express 40 as 15 plus another number, i.e., 15 + ___ = 15 + 25. This may be a simple solution but it involves another very important principle: Since the quantities on both sides of the equal sign are equal and 15 is equal to 15, then the blank must be equal to 25! This is algebraic reasoning! As for Equation (2) even if students can generate hundreds of correct pairs of values, if they cannot see the relationship between the two numbers that goes to the blanks then they are not yet engaging in algebra. So it is not so much the task or the problem but the solutions we use to solve it that could tell whether we are doing algebra or not.

What is algebraic thinking?

Algebraic thinking in working with numbers as described by Kieran is characterized by a focus on relation between numbers and not merely on the calculation; a focus on operations and their inverses and on the related idea of doing/undoing; a focus on both representing and solving problems rather than on merely solving it; a focus on the meaning of equal sign not as a signal to perform operation but as denoting equivalence.  Algebraic thinking involves deliberate generalization, active exploration and conjecture (Kaput, NCTM, 1993) and reasoning in terms of relationships and structure.

I suggest you also read Prof Keith Devlin What is Algebra?

More activities about algebraic thinking: