You do not have to be a master psychologist to understand that if you tell a teenager that he may choose from tasks with different levels of difficulty many teenagers will choose the easiest ones. Already from this point we can draw the conclusion that many students will not reach their full potential since they were offered an easy way out.
Why is it then that many books in mathematics choose to have chapters with tasks with different levels of difficulty? One reason might be that the publishers of a book want the book to be made in such a way that it can easily be used by anyone, meaning that they have a potentially strong market value. But in order to really understand what these levels of difficulty do for students’ learning we have to dig deeper into the philosophies of education.
The behavioristic model
The behavioristic model has had the most influence on education throughout the last hundreds of years. In this model it is not a requirement for the curriculum to be coherent since mathematics is split into pieces. Each lesson consists of a teacher going through theories, showing examples and afterwards students perform repetative tasks. Since this model does not connect different areas in mathematics, it is possible to have tasks with different levels of difficulty. It does not matter if a student has learned how to solve equations or handle fractions properly before starting with, for example, system of equations or perhaps percentage since these areas are simplified to the point that they do not feel like genuine mathematics. What are the odds that system of equations will only have integer solutions? And when introducing percentual change you do not use equations to solve problems but instead tell students to use a formula “the difference divided by the original value”.
Teaching in this way can, however, be seen as an efficient way of teaching. You might think that one advantage in this model is that no matter how poorly students performed in previous chapters they will get a fresh start in the next. You might also think that another advantage is that students will be practising on a suitable level. Even if you think that the points made above are valid there is a fundamental flaw in this model. The big problem is that when students do not apply their knowledge in future chapters it will quickly be forgotten. Research show that students learning this way can have excellent test scores in small areas of mathematics immediately after the content was tought, but already after a two week break their test scores will drop significantly. This might also help explain why students underperform in tests like PISA. Another problem is that if you cannot combine the set of skills you possess you will never be able to research nor solve complex problems. Not to mention that this model does not encourage collaboration nor use of technology since the teacher is the one source of information.
The constructivistic model
The constructivistic model is the most credible and valid method when it comes to education. The basic idea is that when teaching, you start from what students already know, and present problems just beyond their scope of knowledge. In the process of trying to grasp new problems, students are to collaborate and learn how to use many different sources of information to get a broad understanding of the subject. Then, when students have cracked this new problem and learned something new, this new information is to be applied on their former knowledge and so the new information will become a part of the students knowledge.
In this model there cannot be tasks with different levels of difficulty. If students are to collaborate it really helps if they work on the same task. Also, students need to be able to do arithmetical calculations and have a good understanding in algebra in order to fully use it for learning new areas in a scientific accurate way. With this said it is not a problem if a student have not learned everything from before since he will get to practise it over and over again in new situations since it is applied in all that is to come. It is the teachers’ job to make sure that everyone knows enough to be able to move on and this is why formative assessment is of great importance in this style of teaching.
One thing that is possible in this model, and almost impossible in other, is weak students making enormous leaps forward when they suddenly start to understand how areas in mathematics are connected. Practising and realising the use of algebra while learning new content makes sure that students’ foundation gets stronger each day. This makes it possible for every student to reach a higher potential that it would ever be if mathematics was split inte pieces and simplified with easy tasks.
Simplifying mathematics with easy tasks is dangerous since students will not be able to apply their knowledge and therefore they see no point in learning it. You should instead work in a scientific mathematical way, meaning the learner should use his existing knowledge to grasp new content. In this way students will strengthen what they already know in the learning process. Working in this way, students are ensured to get a real understanding of what mathematics is all about, and in time students will develop a strong knowledge that can be used for real applications and at the same time ensure them that they are prepared for high-level studies.
Are you looking for material in maths designed according to the constuctivistic model? If you are, please visit Ma.fi (available in English, Swedish and Finnish) and learn about a digital math book for junior-high.