Posted in Teaching mathematics

What is Universal Design in Learning?

This is a perfectly good knob to use. Grab it, turn it, pull (or push), and the door swings open. So it meets your needs. Or does it? Does it meet ALL your needs, your universal set of needs, needs that arise in different situations, different contexts?

Well,  suppose you are rushing down the corridor in your office building, with a cup of coffee in one hand, the other clutching a folder or file. You are late. You run up to the door and see the same round knob as above. Can you open the door with it when both your hands are “occupied”?

Or would this design do a better job:

Because with this, you can lean down a bit, and push the handle with your elbow. If neither hand is occupied, you still grab, turn, push (or pull) as always. The handle meets your diverse needs, needs that rise in different situations. It has, what we call, a “universal design”.

It took a long time for people to become conscious of needs that go beyond those that are mainstream and taken for granted. Like people using wheelchairs. When they first built houses and office buildings, people used stairs or steps to climb up the building to get to upper floors. This alienated a population of people who were unable to use their legs. They used wheelchairs in hospitals. Maybe a few in their own homes. But by and large, they were kept out of office buildings and even prevented to do social visits. They couldn’t pay their bills, draw their money from banks, or perform any transactions. Or see visit relatives and friends.

It has been only 4 decades that their needs were acknowledged. In the beginning, it was expensive to redesign and rebuild buildings and homes for people with such special needs. So ramps were “attached” on the side of the buildings for such “special” people. Like this:

“We” soon became associated with the “normal” population that could use steps, and  “They” with “those” people who couldn’t.  Handicapped people. Those “poor people” who couldn’t walk. This led to the exclusion of a part of mankind to a lower, somewhat lesser conceptual level where the handicapped felt like outcasts in their own midst. They were made to enter from the side rather than from the front, “like the rest”.  It compromised their self-dignity.

The Inclusion Movement rallied against the tradition of “Exclusion”  for decades. Until the day came when leading architects and designers began envisioning buildings in which various options were made available to negotiate higher floors. Technology, too, came to the rescue. So elevators and escalators took over. No side-entrances for special needs was necessary. ALL needs were human needs. And ALL needs had to be equally respected, equally addressed. Everyone deserved to enter from the front. And if ramps were needed to enter a building, then the ramps would be integrated into the design of the building from its very conception. They would run alongside the steps. All who enter a building should enter as equals. And all buildings and homes should be designed for such “universal access”.

This same idea applies as much to education as to buildings and door-knobs. Learners have different needs. And these may vary among individuals of different ages and genders as much as within anyone’s given lifetime.

Can educational courseware be designed in such a way that they address the needs of (a) gifted learners (b) disadvantaged learners (c) and all learners that fall between these two extreme poles?

I have tried to meet the demands of Universal Design in Learning in the educational courseware I share in Karismath Insights Videos.

I suggest you read my two other posts on What is visual mediation? and Teaching mathematics by visual scaffolding to fully appreciate the theory behind the videos.

Shad Moarif
Founder-Developer
Karismath

About Shad:

Shad, a Harvard graduate, has a background in Science, Psychology, Reading and Mathematics. He has also developed a comprehensive theoretical perspective of his Five Stages of Math Achievement that awaits publication. 

His work has been influenced by his 35 years of teaching Mathematics and Language to children (and adults) with Mathematics and Language-learning difficulties in Asia, Canada, US and the UK. He has conducted numerous teacher-training seminars and workshops at conferences  in the US, Canada, UK, Singapore, Bangladesh, Pakistan and Kenya.

Posted in Teaching mathematics

Use of exercises and problem solving in math teaching

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:

  1. Teaching triangle congruence through problem solving
  2. Teaching the properties of equality through problem solving
Click the links for more readings about Problem Solving:
Posted in Elementary School Math, Teaching mathematics

What are the uses of examples in teaching mathematics?

Giving examples, sometimes tons of them, is not an uncommon practice in teaching mathematics. How do we use examples? When do we use them?  In his paper, The  purpose, design, and use of examples in the teaching of elementary mathematics, Tim Rowland considers the different purposes for which teachers use example in mathematics teaching and examine how well these examples were achieving the objective of the lesson. He classified the use of examples in two types – deductive and inductive.

Types of examples

Examples are used deductively when they are given as ‘exercises’. These examples are usually given after teaching a particular procedure. The initial purpose is to assist retention by repetition of procedure and then eventually for students to develop fluency with it. It is hoped that through working with these examples, new awareness and new understanding of the preocedure and the concepts involved will be created (I’m not sure if many teachers do something to make this explicit). In using examples for this purpose, the teachers should not just haphazardly give examples. For instance, practice examples on subtraction by decomposition ought to include some possibilities for zeros in the minuend. For practice in subtracting integers, the range of examples should include all the possible cases such as minuend and subtrahend both positive; minuend and subtrahend both negative with minuend greater than subtrahend and vice versa, etc.

The second type of examples is done more inductively. Here, examples are used to teach a particular concept. Their role in concept development is to provoke or facilitate abstraction. The teacher’s  choice of examples for the purpose of abstraction reflects his/her awareness of the nature of the concept and the category of things included in it, which of these categories may be considered exceptional and the dimensions of possible variation within a particular category. In other words, teachers must not only give examples but give nonexamples of the concept as well.

Sequencing examples

It is not only the example but also the sequence that they are given that affect the kind of mathematics that is learned. Rowland reports in his paper a Grade 1 lesson about numbers that add up to 10. The teacher asked “If we have nine, how may more to make 10?”. The subsequent examples after 9 are as follows: 8, 5, 7, 4, 10, 8, 2, 1, 7, 3. This looks like random examples but in the analysis of Rowland it was not. The teacher had a purpose in each example. It was not random.

  • 8: the teacher knows that the pupils usually uses the strategy of counting up so they will have success here
  • 5: this will bring up the strategy of a well-known double – doubling being a key strategy for mental calculation
  • 7: same as in 8 but this time, pupils have to count up a little bit further
  • 4: for the more able students
  • 10: to point to the fact that zero is also a number which can be added to another number
  • 8: strange to repeat an example but the teacher used this to ask the pupil who answered 2 “If I’ve got 2, how many more do I need to make 10?” which was the next example.
  • 2: here the teacher said based on previous interaction “2 add to 8, 8 add to 2, it’s the same thing (commutative property and counting up from larger number)
  • 1: the teacher did not ask how many more to make 10 as this will trigger counting up but instead related it to 2 and 8 to make obvious the efficiency of the strategy of counting up from a bigger number and perhaps to make the children be aware of commutativity.
  • 7 and 3: to reinforce the strategies made explicit in using 8 and 2 as examples.
Let us be us more conscious of the kind of examples we give to our students in teaching mathematics.