Entrance Slip: Mathematical understanding and multiple representations

The graphs, charts, visuals represent the external representations which stimulate to understand the underlying idea behind the math problems. I agree with the argument of the author that one can use representation to extend the knowledge to the real-world, for instance, bar charts can be used to see the yearly change in the population trend over time which would otherwise be a challenge to observe the trend. In addition to this, I am really impressed by the author's idea of considering representation as a “social activity”. It means that representation is not a “static end result” which the students have to follow but it is a process that requires a student’s active involvement to reach the representation.

This can be exemplified as the addition of the whole numbers using base ten blocks (each block represents 1, each I by 10 block make 10, each 10 by 10 block makes 100), requires social involvement of teachers and students to think, imagine, internalize, communicate and reproduce to the final answer. The pilot experiment conducted by Tchoshanov reassures me to investigate the student’s prior knowledge on any topic and then engage them to related hands-on or cognitive thinking activities (mapping, plotting, coding) so that they have chance to communicate, explore the underlying mathematical concept.

Examples of mathematical representations that are not included.

Adding ¾+ ¾ can be taught with the help of pies where each pie is divided into 4 parts. The number of parts is the denomination. Therefore, we get 4 quarters (I whole) + 2 quarters (1 half) = 6 quarters (6/4), teaching addition of integers with help of number line ( -12+11= -1 implies starting from -12 on the number line move 11 steps towards positive direction so we reach at -1). Factoring of the quadratic equations can be taught with the algebra tiles, finding the slope of the line by the study of the graph, finding the signs of the trigonometric identities by drawing 4 quadrants in a unit circle. For instance, all the trigonometric identities (sin, cos, tan) are positive in the first quadrant, (sine, cosec) positive in 2^nd qd. rest are negative, (tan, cot) positive in 3^rd qd. Rest are negative (cos, sec) positive in 4^th qd. rest are negative.