Category: Assessment

Posted in Algebra, Assessment, High school mathematics

Algebra test items – Graphs of rational functions

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TIMSS (Trends in international Math and Science Study) classifies test items in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

In my earlier post about this topic on using the TIMSS Assessment Framework for constructing test items I presented a set of questions about zeros of cubic polynomial function. Here are three more test items about graphs of rational function based on the framework. Note that questions should be independent of each other, that is, an answer in one item should not serve as clue to the other items. I only used the same rational function here to highlight the differences among the cognitive domains – knowing, applying, reasoning.

Knowing

What may be the equation of the graph below?

 

Applying

The graph above the x-axis is function f and the graph below the x-axis is function g.  Which of the following equations describes the relationships between f and g?

a. g(x) = f(-x)              b. g(x) = f-1(x)                c. g(x) = f-1(-x)                d. g(x) = -f(x)              e. g(x) = /f(x)/

Reasoning

Carlo drew the figure below by graphing two functions on the same coordinate axes. The graph on the left is f(x) = 4/x2. Which of the following function is represented by the other graph on the right (the blue one)?

a. g(x)=\frac {4}{x^2}        b. g(x)=4+\frac {4}{x^2}        c. g(x)=\frac {4}{(x-2)^2}       d. g(x)=\frac {4}{(x-4)^2}                                   e. g(x)=\frac {4}{(x+4)^2}

All the graphs in these post were made using Geogebra graphing software. It’s a free graphing tool you can download here.

Posted in Algebra, Assessment

Algebra test items – Zeroes of function

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I find the Trends in International Math and Science Study (TIMSS) Assessment Framework useful for constructing test items. TIMSS classified the questions in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

Here are three items assessing students’ learning about zeroes of function using each category in the framework. I used the same polynomial function to illustrate the differences among the three. In the real exam use different polynomial for each category so it won’t stand as hint to other questions.

Knowing

Which of the following is a zero of f(x) =6x3 – 17x2 – 5x + 6?

a. -6                b. -3               c. 0           d. 3        e. 6

There is no way apart from luck that students will choose the correct answer in this question if they don’t know what a zero of a function is. There are many ways of getting the correct answer of course (graphical, applying factor theorem, definition of zero of function).

Applying

What is the value of k if 3 is a zero of f(x) = 6x3 – kx2 – 5x + 6?

Questions about applying usually include standard textbook problem like the one shown above. It involves knowledge of a a fact/concept or procedure to complete the task. It does not only involve straightforward application of concepts unlike those under Knowing questions.

Reasoning

If 3 is a zero of a third degree polynomial function f, which of the following statements can never be true about this function?

a. f(0) = 3.

b. f(-3) = 0.

c. (0,0) is a point on the graph of f

d. (-3,3) is a point on the graph of f

e. (3,-3) is a point on the graph of f

Unlike questions under Applying which are standard or routine tasks, tasks under Reasoning category are usually non-routine and involves decision-making.

Click link to view another set of test items about graphs of rational functions.

Posted in Assessment, Curriculum Reform, Elementary School Math, High school mathematics, Number Sense

Assessing conceptual understanding of integers

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Assessing students’ understanding of operations involving integers should not just include assessing their skill in adding, subtracting, multiplying and dividing integers. Equally important is their conceptual understanding of the process itself and thus need assessing as well. Even more important is to make the assessment process  a context where students are given opportunity to connect previously learned concepts (this is the essence of assessment for learning). Because the study of integers is a pre-algebra topic, the tasks should also give opportunity to engage students in reasoning, number sense  and algebraic thinking. The tasks below meet these criteria. These tasks can also be used to teach mathematics through problem solving.

The purpose of Task 1 is to encourage students to reason in more general way. That is why the cells are not visible. Of course students can solve this problem by making a table first but that is not the most ideal solution.

adding integers
Task 1 – gridless addition table of integers

A standard way of assessing operations involving integers is to ask the students to perform the operation. Task 2 is different. it is more interested in engaging students in reasoning and in developing their number and operation sense.

subtracting integers
Task 2 – algebraic thinking and reasoning in numbers

Task 3 is an example of a task with many possible solutions.  Asking students to find a relation between the values in Box A and Box B links operations with integers to the study of varying quantities or quantitative relationship which are fundamental concepts in algebra.

Task 3 – Integers and Variables

More readings about algebraic thinking:

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Posted in Assessment, Curriculum Reform, Mathematics education

Assessment for learning – its genealogy

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IN the beginning there was only diagnostic and summative assessment. Diagnostic assessment was supposed to share power with summative assessment in the classroom but never really attained equality with it not because teachers did not want to give diagnostic assessment but because stakeholders (parents and state) are more interested with statistics and well-defined label of students’ level of learning as measures of return of investments.

One glorious day, the education community had a dream. It dreamt that in the teaching and learning process, the students have much to contribute especially in the what and in the how they will learn! Thus formative assessment was conceived and born. Formative assessment shifted the focus of assessment from simply a process for collecting information about the learner to it being an integral part of the teaching- learning process. But the education community discovered a bug in the formative assessment.  At the end of the day, it couldn’t tell whether there was really learning that occurred or not because the teacher did not have the data of students’ initial understanding.  Actually they do only that the collection of the data is not “scientific” enough for educators. Thus, diagnostic assessment was resurrected and assessment for learning, was born.