Nature and Pedagogy of Mathematics

This article is a briefing of the math workshop titled “Nature and Pedagogy of Mathematics” held at the ERC in Lawspet in the month of May, 2018.

The session began with a brief questionnaire to know the participants’ ideas about the nature and pedagogy of mathematics and their expectations from this workshop. Many of the challenges and queries raised by them were addressed in the workshop, a few of which are presented in this article.

Low Floor High Ceiling Pedagogy

The next part of the session was focused on understanding the Low Floor High Ceiling Pedagogy, Differentiated Teaching Strategy and Higher Order Thinking Skills.

Low Floor High Ceiling Tasks are those that all students can access but that can be extended to advanced levels. These tasks are important because all classes are heterogeneous.

LFHC tasks allow students to work at different paces and take work to different depths at different times. The low floor high ceiling tasks preferred are visual and thus can lead to rich mathematical discussion.

Benefits of using Low Floor High Ceiling tasks

• Enables the teacher in understanding the level of each child in their classroom.
• Provides differentiation to nearly all learners, high flyers can explore and challenge themselves and less confident students can consolidate their thinking
• Promotes positive classroom culture
• Offers many possibilities for learners to focus on more sophisticated process skills rather than more knowledge.

Here are some of the Low Floor High Ceiling tasks that were dealt in the workshop.

Visualisation of concepts

The challenges faced in visualisation of concepts and arithmetic operations can be addressed through the activity - Make Triangles

In this activity, several coloured buttons were provided to teachers. Probing questions like - “Will you be able to make a triangle from 6 buttons?” and “Will you be able to form a triangle using 7 buttons? If not, why?”, were asked.

Rationale: These activities will let a child visualize the formation of triangle and will also enable the child to think on reasons why a triangle cannot be formed with 7 or 22 buttons. This will let the child think and also help the teacher in understanding the thinking level of each child.

This activity can be further modified to introduce number patterns to the students. This will also enable the students to overcome visual difficulties while keeping the class easy for students and interesting at the same time.

Extending the activity, a question aroused on whether they could form a square using 10 or 6 coins. The teachers were also asked to come up with reasons and generalize their own formula to the formation of square.

How Many Rows? How Many in Each Row?

Number of Players: 2

Player A rolls a die twice. The first roll determines the number of rows and the second roll determines the number of squares in each row. Player A draws a rectangle that corresponds to the rolls in any location on the grid on the recording sheet, then writes the number sentence (for example, 3×4 = 12) in the rectangle. Player B follows the same procedure.

Each rectangle drawn cannot overlap a previous rectangle.  Each player continues by taking turns until he or she is unable to place a rectangle on the grid.  At this stage, the player records both the total number of squares covered by rectangles on the grid, as well as the number of uncovered squares.

Through this, the student will be able to understand multiple concepts like area and multiplication. Following the discussion, a teacher came up with an activity to create right triangles from a sheet of paper which was an example of differentiated strategy thinking.

The common challenge in math classroom is to make the child speak. This strategy will help students think differently and makes them comfortable with math. This in turn will make the classroom interactive making it easier for the teachers to assess the child.

Logic behind the algorithm of finding the Square root

A teacher raised a query that textbooks recommend following a certain algorithm in order to solve a particular type of problem. A square root algorithm was taken as a case study. We always follow the usual long division method to find the square root but do not understand why such steps are followed. We end up memorizing the method indirectly. This activity will address this issue related to the square root algorithm.

To find the square root of 64 009 we will start with the group of digits at the far left. We will therefore start with the group whose only member is 6. This represents the square of “a”. We know that the largest perfect square less than 6 is 4, and that the square root of 4 is 2. Since 2 must be placed in the hundreds position, we know that a = 200. The area of region 1 is 200x200 = 40 000. We then subtract this area from the total area of 64 009. By looking at the diagram we realize that we should next remove the areas of the two regions whose sides are “a” and “b”. To find the length of “b”, we must find the quotient of 24 009 divided by 400. The 400 is arrived at by recalling that two regions, each of which has a length of 200, would have an overall length of 2(200) = 400.

From this, one can clearly understand how rules and procedural knowledge are not sufficient in learning mathematics.

This activity helped us to explain the procedural and conceptual understanding of the process of long division of 1004/10 and similar problems which had a 0 in the quotient.

Open ended questions:

The last part of the session was on how an open ended question can be taken to students of various grades to improve their logical thinking and reasoning skills.

For instance, arrange the eight digits 0, 1, 2, 3, 4, 5, 6, 7 in different place values to create two four digit numbers, where each digit is used only once. The arrangement needs to be done so as to find the smallest answer possible while subtracting the two four-digit numbers. No negative numbers are allowed.

Teachers were very engaged with this session and by the end of which created their own open-ended questions. For example:

__ - __ = 26 i.e. Listing down the various combination in subtraction to obtain the result 26.

How many rectangles are there in each figure?