Tag: math teaching

Posted in Teaching mathematics

Math Knowledge for Teaching

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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:

  1. ways of explaining and representing ideas new to students;
  2. alternate definition of math concepts as well as the consequences of each of these definitions;
  3. wide range of application of mathematical ideas being taught;
  4. alternate ways of approaching problems with and without calculator and computer technology;
  5. extensions and generalizations of problems and proofs;
  6. how ideas studied in school relate to ideas students may encounter in later mathematics study; and,
  7. responses to questions that learners have about what they are learning.

appreciating teachersI 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 🙂

More posts: teaching mathematics and levels of teaching mathematics

Posted in Algebra, Math videos

Teaching Math with Mr Khan’s Videos – Variation

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I’ve yet to read a math educator’s blog that endorses Khan Academy materials. Well, this blog does. Yes, you read it right. This blog endorses Mr. Khan’s materials for teaching mathematics. No, not by simply viewing the video but using the Mr Khan’s lecture as the object of investigation. Let’s take the video on direct variation. In the video, Mr Khan started with “varies directly” like it’s the simplest thing in the world to understand. Mr Khan then gave the sample problem and solved it as shown in the image below. Mr. Khan’s method is deductive and he uses lecture method. Click here to  view the video in YouTube then read on below to see how the same video can be used to develop the concept of direct variation with conceptual understanding by linking it to students previously learned knowledge about proportion and then as context to introduce or review the concept of function.

How to use Mr Khan’s videos in teaching math
  1. Show the video. It’s a short one so it will be over before your class will realise it’s math.
  2. Ask the class if they can solve the same problem without using Mr Khan’s solution. The problem is elementary school level so students can solve it using arithmetic. Since a gallon of gas costs 2.25 so all they need to do is to find how many 2.25 in 18. They can continue to add 2.25 until they get to 18; continue taking away 2.25 from 18; or just divide 18 by 2.25.
  3. Ask for another solution. Didn’t they do ratio and proportion in 5th/6th grade? So, with a little scaffolding, students can set up 1:2.25 = n:18. I’m not a fan of product of the means is equal to the product of the extremes since it has nothing to do with proportional reasoning but I’ll allow it this time.
  4. Ask for another solution. Again with a little scaffolding questions like “If 1 gallon costs 2.25, how much would 2 gallons cost? 3 gallons? Can you organise those data in tables? It’s important that at 4 gallons you asked the students to solve the problem. There’s no need to continue all the way to 18$. Asking students to predict will make them consider the relationship between pairs of values. This is an important habit of thinking and it is crucial to appreciating and understanding algebra. 
  5. Ask for another solution. With a little scaffolding again like “What do you notice about the values in the table? Can you imagine the arrangement of the points if you plot the values on the Cartesian plane? How will you use the graph to solve the problem?” Again there’s no need to plot the points all the way to 18. Students should think of extending the line to make the prediction. 
  6. Now, go back to Mr Khan. “Study Mr Khan’s solution. What are those x and y that he’s talking about? What does y = kx mean in relation to your graph? Where is it in your table? Anyone can explain what Mr Khan mean by varies directly?”
  7. Assessment/ Assignment/ Further discussion: “The following are questions other students posted in Mr. Khan’s direct variation video in YouTube. How would you answer them?”
    • Sorry if this question seems basic, but I don’t understand how this example relates to functions…could someone please explain? Thanks!
    • What is K in general?
    • Why do we always have to set x?
    • The practice for this video includes inverse variations, which are not yet covered. It would be great if there was practice specifically for direct variation only. Thanks!

George Polya on thinking

This style of teaching is called teaching math through problem solving. If you enjoyed  Teaching Math with Mr Khan, don’t forget to subscribe to this site. I will try to develop more lessons where I will be co-teaching math with Mr Khan’s videos.

Posted in Algebra

Math knowledge for teaching tangent to a curve

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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.