Tag: PCK

Posted in Number Sense

Math Knowledge for Teaching Addition

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This post is the second in the series of post about the Math Knowledge for Teaching (MKT) where I present task/lesson that teachers and interested readers of this blog can discuss. The first is about Tangents to Curves, a Year 12 lesson. This second post is for young learners.

The task

How many small cubes make up this shape?

cubes

This is a pretty simple task.  Any Grade 1 pupil will have no difficulty giving the correct answer. All they need to do is to count the cubes. Yesterday, in one my workshop with teachers about lesson study, we viewed a Japanese lesson using the same task but was used in such away that children will learn not just counting.

The lesson

Before this lesson the class already learned that putting together concept and the symbol + and =.

The pupils were given small cubes to play with on their tables. After a minute, the 2x2x2 cube was shown on the TV screen and the teacher asked the class to predict how many small cubes make-up the shape. Some used their cubes to make a similar shape without the teacher encouragement to do so. The cubes were only there to help those who might have trouble imagining the bigger cube were some parts are not shown. The pupils counted the visible cubes one-by-one and then those not seen in the drawing (a drawing of the cube is posted on the board). But, the teacher was not just after the answer 8, he was after the learners’ counting strategy. So he asked: Can you use the + sign to show us your counting strategy? Some of the students answers were: 4+4 = 8, 2+2+2+2 = 8, 6+1+1=8. But, the teacher was not only after this, he wanted the class to realize that this number expressions may have come from a different way of looking at the cube. He started with those who wrote 4+4 to show the class how this counting was done. There were two different strategies: halving the cube vertically and the other horizontally which the students demonstrated using the cubes. All throughout the teacher was asking the class, “Can you follow the thinking? “Do you have a different idea?” “Who has another idea?”

After the summarizing the different ideas of the pupils in the first task, the teacher gave the second task:

What is your idea for counting the small cubes in this shape? Show your idea in numbers and symbols.

cubes

The shape was projected on the TV screen as the teacher rotated the shapes. The pupils came-up with different combinations of visible and not visible cubes like 7+3 = 10, 4+6 = 10, etc. They were invited to explain these expressions and their thinking using the drawing on the board. The teacher did not have any difficulty getting the answer he wanted from the pupils: “We already know that this shape (the big cube) is 8 so we just add 2  (8+2 = 10).

Questions for Teachers Discussion/Reflection:
  1. What about numbers will the pupils learn in the lesson?
  2. What is the role of technology and visuals in this lesson?
  3. What about mathematics is given emphasis in the lesson?
  4. What mathematics teaching and learning principles underpin the design of the lesson?

Remember this quote from George Polya: What the teacher says in the classroom is not unimportant, but what the students think is a thousand times more important.

math knowledge

For further reading:

Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (Studies in Mathematical Thinking and Learning Series)

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.

Posted in High school mathematics, Lesson Study

Pedagogical Content Knowledge Map for Integers

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I’m working with a group of Year 7 mathematics teachers doing Lesson Study for the first time. The teachers chose to do a lesson study for what they believe to be the most difficult topic in this year level – integers. Part of my preparation as facilitator is to draw a map of what I know about teaching the topic. The map is more than a concept map because it includes not just related big ideas or concepts but also how  these are taught and learned. Hence, I call this pedagogical content knowledge map (PCK map).

The PCK map I present here is a product of my own readings and my own experiences of teaching the topic. This means that it may not be the same as other teachers especially the ‘teaching part’ of the map, the ones in orange colors. For example, experience and research results back my claim that the number line is a very good way of representing the set of integers but not in teaching operations. Click here for my post about this. Notice that I gave emphasis on students knowing when a negative, a positive or a zero result rather than the rules for operation. I believe that without this, a conceptual understanding of the operation involving integers will be weak. Also, experience has taught me that although integers are numbers, the teaching of it must be algebraic. The instructions should be so designed so that students are learning algebraic thinking as well. I have noted this in the PCK map.

The map is not yet complete. I intend to include descriptions of effective activities and students’ learning trajectory of the concept after my research with the teachers. Please feel free to give your comments and share experiences for teaching integers that I could look into in my study.

pedagogical content knowledge
PCK Map for Integers

Please click the link to see my PCK map for Algebraic Expressions.