The mathematics that engineers, accountants, etc and teachers of mathematics know are different. They should be. There are some engineers, accountants, chemists, etc who become very good mathematics teachers but I’m sure it is not because they have ‘math knowledge for engineering’ for example but because they were able to convert that knowledge to ‘math knowledge for teaching’.
What is math knowledge for teaching?
It includes knowledge of mathematics but on top of that according to Salman Usiskin, it should also include knowledge of:
ways of explaining and representing ideas new to students;
alternate definition of math concepts as well as the consequences of each of these definitions;
wide range of application of mathematical ideas being taught;
alternate ways of approaching problems with and without calculator and computer technology;
extensions and generalizations of problems and proofs;
how ideas studied in school relate to ideas students may encounter in later mathematics study; and,
responses to questions that learners have about what they are learning.
I don’t know why some people especially politicians think teaching is easy. Surely college preparation is not enough to learn all these. You certainly need to be a practicing teacher to even start knowing #1 and #7. Teachers need more support in acquiring these knowledge when they are already in the field than when they are still in training.
I started this blog to contribute towards helping teachers to acquire the seven listed by Mr. Usiskin. After 250 posts, it looks like I have not even scratched the surface 🙂
Variation theory of learning was developed by Ference Marton of the University of Gothenburg. One of its basic tenets is that learning is always directed at something – the object of learning (phenomenon, object, skills, or certain aspects of reality) and that learning must result in a qualitative change in the way of seeing this “something” (Ling & Marton, 2011). Variation theory sees learning as the ability to discern different features or aspects of what is being learned. It postulates that the conception one forms about something or how something is understood is related to the aspects of the object one notices and focuses on.
Here’s an example: In linear equations you want your students to learn that a linear equation in one unknown can only have one root while an equation with two unknowns can have infinitely many roots. You also want them to learn that in an equation of one unknown, the root is represented by x only while in equation with two unknowns, the root is represented by an ordered pair of x and y. It is also important that students will see that while both roots can be represented by a point, the root of the equation in one unknown can be plotted in a number line or one-dimensional axis while the root of the equation in two unknowns are plotted in two-dimensional coordinate axes. Will the students discern these particular differences between the roots of the two types of equation in the natural course of teaching linear equations or should you so design the lesson so that students will focus on these differences? Variation theory tells you, yes, you should.
At the World Association of Lesson Studies (WALS) conference in HongKong in 2010 most of the lesson studies presented were informed by variation theory. The teachers reported that students achievement showed significant increases in the post test. Everybody seemed to be happy about it. I think it is not only because of its effect on achievement but it also gave the teachers a framework for structuring their lesson particularly on the design and sequencing of tasks. This sounds very simple but it is actually challenging. The challenge is in identifying the critical feature for a particular object of learning – what is it they need to vary and what needs to remain invariant in the students experiences. Variation theory asserts that change in conception can occur by highlighting critical elements of the object of learning and creating variation in these while all other elements are held constant.
Variation theory directs the teacher to focus on the critical aspect of the object of learning (a math concept, for example), identify differing level of conceptions, and from each of these conceptions identify the critical elements (core ideas) which needed to be varied and those that will remain invariant. In mathematics, these invariants are usually the properties of the concept. In the case of the angles for example, in order for students to have a ‘full’ understanding of this concept they needed to experience it in different forms – the two-line angles, the one-line angles, and the no-line angles.
‘Types’ of Angles
What they need to learn (abstract) from these is that they all consist of two linear parts (even if they are not visible) and they cross or meet at a point and that the relative inclination of the two parts has some significance – it defines the sharpness of the corner or the their openness. Given these, the teacher now has to design the lesson/ tasks that will provide the necessary variation of learning experiences. You can read my post Angles aren’t that Easy to See for further explanation about understanding angles. Check also my post on how to select and sequence examples to see how variation theory is useful for thinking about examples.
Teachers must always remember however that “even if they aware of the need for the appropriate pattern of variation and invariance, quite a bit of ingenuity may be required to bring it about. Providing the necessary conditions for learning does not guarantee that learning will take place. It is the students’ experience of the conditions that matters. Some students will learn even though the necessary conditions are not provided in class. This may be because such conditions were available in the students’ past, and some students are able to recall these experiences to provide a contrast with what they experience in class. But, as teachers we should not leave learning to happen by chance, and we should strive to provide the necessary conditions to the extent that we are able” (Ling & Marton, 2011). I think we should also remember that the way the learners are engaged is a big factor in learning. You may have addressed the critical feature through examples with appropriate pattern of variation but if this was done by telling, learning may still be limited and superficial.
8. Look out for such features of the problem at hand as may be useful in solving the problems to come – try to disclose the general pattern that lies behind the present concrete situation.
9. Do not give away your whole secret at once—let the students guess before you tell it—let them find out by themselves as much as feasible.
10. Suggest it, do not force it down your throats.
I got this from the plenary talk of Bernard Hodgson titled Whither the mathematics/didactics interconnection? at ICME 12, Korea, where he highlighted the important contribution of George Polya in making stronger the interconnection between mathematics and didactics and between mathematicians and mathematics educators.
If it’s too hard to commit the 10 commandments to memory then just remember the two rules below which is also from Polya. Combine it with Four Freedoms in the Classroom and you are all set.
The teachers at Level 1 can only tell students the important basic ideas of mathematics such as facts, concepts, and procedures. These teachers are more likely to teach by telling. For example in teaching students about the set of integers they start by defining what integers are and then give students examples of these numbers. They give them the rules for performing operations on these numbers and then provide students exercises for mastery of skills. I’m not sure if they wonder later why students forget what they learn after a couple of days.
Levels of teaching
Level 2 – Teaching by explaining
Math teachers at Level 2 can explain the meanings and reasons of the important ideas of mathematics in order for students to understand them. For example, in explaining the existence of negative numbers, teachers at this level can think of the different situations where these numbers are useful. They can use models like the number line to show how negative numbers and the whole numbers are related. They can show also how the operations are performed either using the number patterns or through the jar model using the + and – counters or some other method. However these teachers are still more likely to do the demonstrating and the one to do the explaining why a particular procedure is such and why it works. The students are still passive recipients of the teachers expert knowledge.
Level 3 – Teaching based on students’ independent work
At the third and highest level are teachers who can provide students opportunities to understand the basic ideas, and support their learning so that the students become independent learners. Teachers at this level have high respect and expectation of their students ability. These teachers can design tasks that would engage students in making sense of mathematics and reasoning with mathematics. They know how to support problem solving activity without necessarily doing the solving of the problems for their students.
The big difference between the teacher at Level 2 and teachers at Level 3 is the the extent of use of students’ ideas and thinking in the development of the lesson. Teachers at level 3 can draw out students ideas and use it in the lesson. If you want to know more about teacher knowledge read Categories of teacher’s knowledge. You can also check out the math lessons in this blog for sample. They are not perfect but they are good enough sample. Warning: a good lesson plan is important but equally important is the way the teacher will facilitate the lesson.
Mathematical Proficiency
The goal of mathematics instruction is to help students become proficient in mathematics. The National Research Council defines ‘mathematical proficiency’ to be made up of the following intertwined strands:
Conceptual understanding – comprehension of mathematical concepts, operations, and relations
Procedure fluency – skill in carrying out procedure flexibly, accurately, efficiently, and appropriately
Strategic competence – ability to formulate, represent, and solve mathematical problems
Adaptive reasoning – capacity for logical thought , reflection, explanation, and justification
Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (NRC, 2001, p.5)
I think it will be very hard to achieve these proficiencies if teachers will not be supported to attain Level 3 teaching I described above. No one graduates from a teacher-training institution with a Level 3 expertise. One of the professional development teachers can engage to upgrade and update themselves is lesson study. The book by Catherine Lewis will be a good guide: Lesson Study: Step by step guide to improving instruction.
I don’t know why some people especially politicians think teaching is easy. Surely college preparation is not enough to learn all these. You certainly need to be a practicing teacher to even start knowing #1 and #7. Teachers need more support in acquiring these knowledge when they are already in the field than when they are still in training.
