Posted in Algebra

Math knowledge for teaching tangent to a curve

I am creating a new category of posts about mathematical tasks aimed at developing teachers’ math knowledge for teaching. Most of the tasks I will present here have been used in studies about teaching and teacher learning. Mathematical knowledge for teaching was coined by J. Boaler based on what Shulman (1986) call pedagogical content knowledge (PCK) or subject-matter knowledge for teaching. I know this is a blog and not a discussion forum but with the comment section at the bottom of the post, there’s nothing that should prevent the readers from answering the questions and giving their thoughts about the task. Your thoughts and sharing will help enrich knowledge for teaching the math concepts involve in the task.

The following task was originally given to teachers to explore teachers beliefs to sufficiency of a visual argument.

The task:

Year 12 students, specializing in mathematics, were given the following question:
Examine whether the line y = 2 is tangent to the graph of the function f, where f(x) = x^3 + 2.

Two students responded as follows:

Student A: I will find the common point between the line and the graph and solving the system

math

The common point is A(0,2). The line is tangent of the graph at point A because they have only one common point (which is A).’

Student B: The line is not tangent to the graph because, even though they have one common

tangentpoint, the line cuts across the graph, as we can see in the figure.

Questions:

a. In your view what is the aim of the above exercise? (Why would a teacher give the problem to students?)

b. How do you interpret the choices made by each of the students in their responses above?

c. What feedback would you give to each of the students above with regard to their response to the exercise?

Source: Teacher Beliefs and the Didactic Contract on Visualisation by Irene Biza, Elena Nardi, Theodossios Zachariades.

Posted in Algebra, Math blogs

Math Teachers at Play Blog Carnival

Math  Teachers At Play- Blog Carnival #49 of Let’s Play Math is now live in TeachBesideMe. Go check-out the fabulous submissions and of course the photos and images.

Mathematics for Teaching will be hosting the 50th edition of MTaP. You may use the  Math Teachers at Play Blog Carnival — Submission Form to submit your posts or email it to mathforteaching@gmail.com. MTAP 50 will go live on 2nd week of May.  Looking forward to your great articles on teaching and learning mathematics . Thank you.
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 Math blogs

Math Blog Carnivals

A  math carnival is a one-stop shop of math ideas from bloggers all over the world. Here are the latest edition of  three Math Blog Carnivals in English. Of course, you can always translate them to your language.

And if you are looking for more you can go to the mother of all math carnivals – Math Blogging.org.