Posted in Assessment, Curriculum Reform, Mathematics education

Assessment for learning – its genealogy

IN the beginning there was only diagnostic and summative assessment. Diagnostic assessment was supposed to share power with summative assessment in the classroom but never really attained equality with it not because teachers did not want to give diagnostic assessment but because stakeholders (parents and state) are more interested with statistics and well-defined label of students’ level of learning as measures of return of investments.

One glorious day, the education community had a dream. It dreamt that in the teaching and learning process, the students have much to contribute especially in the what and in the how they will learn! Thus formative assessment was conceived and born. Formative assessment shifted the focus of assessment from simply a process for collecting information about the learner to it being an integral part of the teaching- learning process. But the education community discovered a bug in the formative assessment.  At the end of the day, it couldn’t tell whether there was really learning that occurred or not because the teacher did not have the data of students’ initial understanding.  Actually they do only that the collection of the data is not “scientific” enough for educators. Thus, diagnostic assessment was resurrected and assessment for learning, was born.

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 Mathematics education

What is scaffolding in education?

Scaffolding is a metaphor for describing a type of facilitating a teacher does to support students learning. Some educational paper lists some of these scaffolding like “breaking the task into smaller, more manageable parts; using ‘think alouds’, or verbalizing thinking processes when completing a task; cooperative learning, which promotes teamwork and dialogue among peers; concrete prompts, questioning; coaching; cue cards or modeling”. Visual scaffolding is also popular in teaching mathematics.

Scaffolding is the latest buzzword in education community. In an international conference I attended recently for instance, I heard the word in almost all the parallel paper presentations.

There was a demonstration lesson for teaching English during the conference. I am not an English teacher so I asked the person seated beside me, who happens to be an English teacher, to tell me what the teacher was doing as she hopped from one group of students to the other. She said with authority that the teacher was doing a lot of scaffolding. I didn’t know what to make of her statement. Was it a positive or a negative comment? Is it a good idea to do a lot of scaffolding or is it something that should be given sparingly? Where do you draw the line?

Scaffolding can be traced back to Lev Vygotsky’s idea of ZONE of PROXIMAL DEVELOPMENT (ZPD). Vygotsky suggests that there are two parts of learner’s developmental level: 1) the Actual developmental level 2) the Potential developmental level

The ZPD is “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance”. This is where scaffolding is crucial.

scaffolding in education
Note that the activity students should be engaging in is problem solving. A problem is a problem only when you do not how to solve it right away. So when scaffolding deprives the students from thinking and working on their own way of solving the problem then scaffolding has not helped learn how to solve problem. It only helped them to solve problems using the teacher’s method.

You may want to read the different interpretations of zone of proximal development in research.

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