Posted in Mathematics education, Teaching mathematics

Three Levels of Math Teachers Expertise

Level 1 – Teaching by telling

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:

  1. Conceptual understanding – comprehension of mathematical concepts, operations, and relations
  2. Procedure fluency – skill in carrying out procedure flexibly, accurately, efficiently, and appropriately
  3. Strategic competence – ability to formulate, represent, and solve mathematical problems
  4. Adaptive reasoning – capacity for logical thought , reflection, explanation, and justification
  5. 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.

Posted in Mathematics education

What is abstraction in mathematics?

Abstraction is inherent to mathematics. It is a must for mathematics teachers  to know and understand what this process is and what its products are. Knowledge of it can enrich our reflection of our own practice as well us guide us and make us conscious of the type of learning activities we provide our students.

All the definitions below give emphasis about abstraction as a process. Note also that the direction of the abstraction is always from a set of contexts to an abstract concept. Abstraction is related to generalization which I discussed in another post.

Abstraction –
  • the omission of qualities from concrete experience – Aristotle
  • the process of separating a quality common to a number of objects/situations from other qualities – Davidov  (1972/1990, p. 13)
  • the act of detaching certain features from an object – Sierpinska (1991, p 61)
  • Abstracting is an activity by which we become aware of similarities … among our experiences. An abstraction is some kind of lasting change, the result of abstracting, which enables us to recognize new expereinces as having the similarities of an already formed class. … To distinguish between abstracting as an activity and abstraction as its end-product, we shall … call the latter a concept. – Skemp
Empirical vs reflective abstraction (Piaget et al)
  • Empirical abstraction is based on superficial similarities and is the type of abstraction involved in everyday concept formation.
  • Reflective abstraction is, according to Piaget, based on reflection one one’s actions. For example when one object and two objects are put together you always get three objects. This leads to recognition of invariance (later expressed as 1+2=3). These objects of invariance become concepts (the numbers 1, 2, and 3) and the invariant action becomes a relation between these concepts (addition). In reflective abstraction, concepts and relations are abstracted together.
Abstract-apart vs abstract-general (Mitchelmore et al)
  • abstract-apart:  concepts formed that exist apart from any contexts from which they might have been abstracted
  • abstract-general: concepts that have been abstracted through the recognition of similarities between contexts. These concepts derive their general meaning from the set of contexts from which it has been abstracted
Stages of abstraction
  • a cycle of interiorization-condensation-reification – by Sfard 1991
  • generalization-synthesis-abstraction cycle – Dreyfus (1991)

Reference: The Role of Abstraction and Generalization in the Development of Mathematical Knowledge by Michael Mitchelmore – paper presented during 2nd EARCOME.

You may want to read my post about assessing understanding of function in equation form for an example of abstracting based on Sfard’s interiorization-condensation-reification cycle.

Posted in Algebra, Mathematics education

When is a math problem a problem?

One of the main objectives of mathematics education is for students to acquire mathematical habits of mind. One of the ways of achieving this objective is to engage students in problem solving tasks. What is a problem solving task? And when is a math problem a problem and not an exercise?

What  is a problem solving task?
A problem solving task refers to a task requiring a solution or answer, the strategy for finding such is still unknown to the solver. The solver still has to think of a strategy. For example, if the task,

If x^2 - 7 = 18, what is x^2 - 9 equal to?

is given before the lesson on solving equation, then clearly it is a problem to the students. However, if this is given after the lesson on solving equation and students have been exposed to a problem similar in structure, then it cease to be a problem for the students because they have been taught a procedure for solving it. All the students need to do is to practice the algorithm to get the answer.

What is a good math problem?

The ideal math problem for teaching mathematics through problem solving is one that can be solved using the students’ previously learned concepts/skills but can still be solved more efficiently using a new algorithm or new concept that they will be learning later in the lesson. If the example above is given before the lesson about the properties of equality, the students can still solve this by their knowledge of the concept of subtraction and the meaning of the equal sign even if they have not been taught the properties of equality or solving quadratic equation (Most teachers I give this question to will plunge right away to solving for x. They always have a good laugh when they realize as they solve the problem that they don’t even have to do it. They say, “ah, … habit”.)

Given enough time, a Year 7 student can solve this problem with this reasoning: If I take away 7 from x^2 and gives me 18 then if I take away a bigger number from x^2 it should give me something less than 18. Because 9 is 2 more than 7 then x^2 - 9 should be 2 less than 18. This is 16.

Why use problem solving as context to teach mathematics?

You may ask why let the students go through all these when we there is a shorter way. Why not teach them first the properties of equality so it would be easier for them to solve this problem? All they need to do is to subtract 2 from both sides of the equal sign and this will yield x^2 - 9 = 16. True. But teaching mathematics is not only about teaching students how to get an answer or find the shortest way of getting an answer. Teaching mathematics is about building a powerful form of mathematical knowledge. Mathematical knowledge is powerful when it is deeply understood, when concepts are connected with other concepts. In the example above, the problem has given the students the opportunity to use their understanding of the concept of subtraction and equality in a problem that one will later solve without even being conscious of the operation that is involved. Yet, it is precisely equations like these that they need to learn to construct in order to represent problems usually presented in words. These expressions should therefore be meaningful. Translating phrases to sentences will not be enough develop this skill. Every opportunity need to be taken to make algebraic expressions meaningful to students especially in beginning algebra course. More importantly, teaching mathematics is not also only about acquiring mathematical knowledge but more about acquiring the thinking skills and disposition for solving problems and problem posing. This can only happen when they are engage in these kind of activities. For sample lesson, read how to teach the properties of equality through problem solving.

Finally, and I know teachers already know this but I’m going to say it just the same. Not all ‘word problems’ are problems. If a teacher solves a problem in the class and then gives a similar ‘problem’ changing only the situation or the given ‘numbers’ but not the structure of the problem or some of the condition then the latter is no longer a problem but an exercise for practicing a particular solution to a ‘problem’. It may still be a problem of course to those students who did not understand the teacher’s solution. I’m not saying that this is not a good practice, I am just saying that this is not problem solving but an exercise.

You may also want to read How to Solve It: Modern Heuristicsto further develop your problem solving skills.

Posted in Mathematics education

Learning research study module on analyzing understanding of function

Just recently the National Centre for Excellence in the Teaching of Mathematics (NCETM) of the UK released a set of learning research study modules to support continuing professional development of mathematics teachers online. One of the modules is based on my paper Growth Points in Students Understanding of Function in Equation Form. This paper was published in 2009 in Mathematics Education Research Journal (MERJ). MERJ is an international refereed journal that provides a forum for the publication of research on the teaching and learning of mathematics at all levels. It is the official journal of the Mathematics Education Research Group of Australasia, Inc. (MERGA). The papers in the journal used to be downloadable for free. Since last year I think they are now published through Springer link, no longer free. I have tried to share ideas from the paper in my posts How to assess understanding of function and What is a function in my attempt to provide teachers another thinking tool by which they can analyze students understanding of function and of mathematics concepts in general.

The learning research module about analyzing understanding of function was developed by Anne Watson, professor of Mathematics Education at Oxford University. NCETM presented it in power point presentation platform. I love the way Professor Watson turned my otherwise boring research paper into a thought provoking learning module for teachers. I have downloaded the presentation and sharing it here so other teachers can easily access it. Of course you can also go to the NCETM site. Teacher educators can learn from the style of presentation of the research study modules. This is a great way of making research results accessible to teachers. Kudos to NCETM for this project.
Professor Anne Watson is also one of the editors of the book New Directions for Situated Cognition in Mathematics Education (Mathematics Education Library). The book is a great reference for those doing research about mathematics teaching and learning.