There are two types of assessment based on its goals or use. One is what I call assessment in the service of teaching. The second is what I call assessment in the service of learning. Assessment in the service of teaching refers to the use of assessment information to improve teaching while assessment in the service of learning refers to the use of assessment information in the form of feedback to keep the learners to the task of learning. This post is about assessment in the service of learning. Continue reading “Six ways to give feedback to students to keep them in the task of learning”
Category: Assessment
Student Achievement in Mathematics – TIMSS Ranking
East Asian countries continue to lead the world in student achievement in mathematics. Singapore, Korea, and Hong Kong SAR, followed by Chinese Taipei and Japan, were the top-performing countries in TIMSS 2011 at the fourth grade. Similarly, at the eighth grade, Korea, Singapore, and Chinese Taipei outperformed all countries, followed by Hong Kong SAR and Japan. Here’s the result for 4th Grade and 8th grade achievement for 2011 released last December 2012. The number enclosed in the parenthesis is the average scale score of the country. The average scale centrepoint is 500 for both grade levels. TIMSS stands for Trends in Mathematics and Science Study.
You can access the full report in International Student Achievement in Mathematics.
| 8th Grade TIMSS 2011 | 4th Grade TIMSS 2011 |
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What are the big ideas in function ?
Function is defined in many textbooks as a correspondence relationship from set X to Y such that for every x (element of X), there is one and only one y value in Y. Definitions are important to know but in the case of function, the only time students will ever use the definition of function as correspondence is when the question is “Which of the following represents a function?”. I think it would be more useful for students to understand function as a dependence/covariational relationship first than for them to understand function as a correspondence relationship. The latter can come much later. In dependence/co-variational relationship “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (Sfard, 1991, p. 15)
The concept of change and describing change is a fundamental idea students should learn about functions. Change, properties, and representations. These are the big ‘ideas’ or components we should emphasize when we teach functions of any kind – polynomial, exponential, logarithmic, etc. Answer the following questions to get a sense of what I mean.
1. Which equation shows the fastest change in y when x takes values from 1 to 5?
A. y = 4x2 B. y = -2x2 C. y = x2 + 10 D. y = 6x2 – 5
2. Point P moves along the graph of y = 5x2, at which point will it cross the line y = 5?
A. (5, 0) and (0,5) B. (-5, 0) and (0,-5) C. (1, 5) and (-1, 5) D. (5, 1) and (5,-1)
3. Which of the following can be the equation corresponding to the graph of h(x)? 
A. h(x) = x3 + 1 B. h(x) = x3 – 1
C. h(x) = 2x3 + 1 D. h(x) = 2x3 + 4
4. The zeros of the cubic function P are 0, 1, 2. Which of the following may be the equation of the function P(x)?
A. P(x) = x(x+1)(x+2) B. P(x) = x(x-1)(x-2) C. P(x) = x3 – x2 D. P(x) = 2x3 – x2 – 1
5. Cubes are made from unit cubes. The outer faces of the bigger cube are then painted. The cube grows to up to side 10 units.
The length of the side of the cube vs the number of unit cubes painted on one face only can be described by which polynomial function?
A. Constant B. Linear C. Quadratic D. Cubic function
Item #1 requires understanding of change and item #5 requires understanding of the varying quantities and of course the family of polynomial functions.
Of course we cannot learn a math idea unless we can represent them. Functions can be represented by a graph, an equation, a table of values or ordered pair, mapping diagram, etc. An understanding of function requires an understanding of this concept in these different representations and how a change in one representation is reflected in other representations. Items #2 and #3 are examples of questions assessing understanding of the link between graphs and equations.
Another fundamental idea about function or any mathematical concept for that matter are the properties of the concept. In teaching the zeroes of a function for example, students are taught to find the zeroes given the equation or graph. One way to assess that they really understand it is to do it the other way around. Given the zeroes, find the equation. An example of an assessment item is item #4.
You may also want to read How to assess understanding of function in equation form and Teaching the concept of function.
Assessing understanding via constructing test items
Assessing understanding of mathematics can also be done by asking students to write test items. Here’s my favorite assessment item. I gave this to a group of teachers. 
Possible answers/ questions.
Year level: Third year (Year 9)
Question 1 – What is the distance of P from the origin?
Question 2 – What is the area of circle P with radius equal to its distance from the origin?
Question 3 – With P as one of the vertex, draw square with area 2 square units.
Year level: Second year (Year 8 )
Question 1 – Write the equation of the line that passes through P and the origin.
Question 2 – Write 3 equations of lines passing through (2,1).
Question 3 – Write the equation of the family of lines passing through (2,1).
Year level: First year (Year 7)
Question 1 – What is the ordinate of point P?
Question 2 – Locate (-2, 1). How far is it from P?
Question 3 – Draw a square PQRS with area 9 square units. What are the coordinates of that square?
How about using this exercise to assess your students? Ask them to construct test items instead of asking them to answer questions.
Here are a few more assessment items which I constructed based on the TIMSS Framework: