CSET Math Guru - THE Site for CSET Single Subject Math
How should Free Response Questions be written?
Each Single Subject Math CSET Subtest contains 4 Free Response Questions, each of which is graded on the basis of the answer demonstrating the following 4 performance characteristics:
* Purpose: the extent to which the response addresses the questions aim in relation to the relevant CSET Subject Matter requirements
* Subject Matter Knowledge: The application of accurate Subject Matter knowledge as described in the relevant CSET Subject Matter requirements
* Support: The appropriateness and quality of the supporting evidence in relation to the relevant CSET Subject Matter requirements
* Depth and Breath of Understanding: The degree to which the response shows understanding of the relevant CSET Subject Matter requirements
What does all this infernal gobbledygook (obtained from the CSET site!) actually mean?!
Simply that as a prospective teacher, you must know
a) the gamut of Math concepts being assessed on the CSET and the underlying skills to a considerable degree of proficiency
b) the relevance of each problem, as well as problems of a higher calibre that the current question suggests
c) real-world applications each question might indicate
d) how to lucidly and elegantly demonstrate a)-c).
Succinctly, then, while responding to Free Response questions, write in a manner you would employ to elucidate the problem to a student of appropriate ability.
For instance, for the resolution of a Subtest I question, write out a detailed and exhaustive algorithm as if you were attempting to illuminate a fairly competent Precalculus student! Likewise, your Subtest II Free Response solutions should be geared to make it readily comprehensible to a reasonably smart student of Geometry / AP Statistics (as the case may be).
Here are a few Tips and Pointers about writing your Free Response answers:
1. Make an OUTLINE of your answer: much like a man knowing that he is to be hanged in a fortnight - qv. Boswell's Life of Johnson - making an outline of concentrates the mind wonderfully! It promotes a logical thought process and leads to a clear systematic layout to problem-solving.
2. Use English extensively! Yes, yes, confound you, it is indeed a Maths exam and the fellow that grades your paper might verily grasp the nature of your work, but remember your implicit audience: a school student of only average ability! Clear explication of the solution requires one to use English to clarify the function of EACH step!
3. Define any variables (x. y, z, a, b, c, etc.) used at the outset EVEN if these are 'standard' variables like t for time, I for interest, C for circumference, and so on.
4. Do NOT take ANYTHING for granted ('the figure is a right triangle', 'the line intersects the parabola at 2 points', etc.) UNLESS you can and you actually do justify it on the basis of a Theorem, an Axiom / Postulate, or some known Principle!
5. Explicitly state ANY Formulae, Theorems, Axiom / Postulates employed to validate a step in your work. Put important or long formulas on a line of their own, and then center them; this enhances readability!
6. It helps to clearly restate the problem to be solved as a statement that begins with "Objective:" or "Aim:" or "Required: " or "To Show That:" or "To Prove That:".
All these phrases demonstrate to the grader that at the very least you know your destination (even if you are momentarily oblivious about the route!) Oftentimes, just rewriting the problem might serendipitously produce the necessary inspiration!
7. State the answer in a complete sentence which stands on its own at the termination of your solution ('The polynomial with the given roots is f(x) = ' or 'Thus the given proposition is proven' or 'The equation of the hyperbola is: ').
Try to avoid variables in your answer; otherwise, provide a reminder about what they stand for. Also, restate any significant assumptions made in the contrivance of the solution.
8. Clearly label diagrams, tables, graphs, or other visual representations. Label all axes, with words, for graphs.
Give diagrams a title describing what they represent. It should be clear from the picture what any variables in the diagram represents. The whole idea is to make everything as clear and self-explanatory as possible.
9. Use technical language and appropriate Math terminology extensively to exhibit your competence and knowledge! Curb use of words like thingummy, whatchamacallit, doodad, thingamabob and thingamajig!
10. Observe rules of grammar at all times! While you are NOT being assessed on the English, remember: effective communication is your goal here, so the writing should be clear and professional!
11. Show steps on SUCCESSIVE lines, preferably preceded by the '=>' symbol to indicate 'which leads to...' . This makes it easiler to follow!
12. Don’t use “=” sign outside of a formula as in 'Let V = Volume'! While one can devine the meaning of that, it strictly does NOT make sense!
13. Use the Phrases below to convey a logical flow in Free Response Q solutions: * Therefore (also: so, hence, accordingly, thus, it follows that, we see that, from this we get, then, it follows that) * I am assuming that (also: assuming..., where M stands for...; let..., given..., where M represents...) * show (also: demonstrate, prove, explain why, find ) * We use the formula, Theorem, ...since the conditions are satisfied. * While I am very glad to help you this time, you should be advised that my usual consultation fee is $85. * See the formula (1) or formula (A) above. Also: see * on the previous page. This means that . . . * If (also: whenever, provided that, when ) * Notice that (also: note that, notice, observe that) * Since (also: because )
In general, the test taker whose background is in Math / Science and/or whose inclination is simply to solve the question and leave it at that, should make a point of using sentences and grammatical English as well and, overall, explaining the problem while solving it.
On the other hand, the test taker whose background is not in Math / Science / Engineering and/or whose inclination is simply to explain the problem should make a point of using the formalisms of mathematics, and, overall, solve the problem as a mathematician would as well as explaining it.
Both camps should make a point of using the descriptive technical terminology of mathematics. Use of the appropriate technical terms is one of the classic means in other CSET exams to achieve high marks on the "knowledge" rubric.
I didn't mean it to be consumed in the strictest sense, just that even Math teachers prefer proper grammar and spelling, and that it facilitates comprehension.
[I imagine the more punctilious Math and Science teachers to be horrified when mangled English leads to conclusions contrary to what was intended to be conveyed!]
The issue is communication, and I acknowledge that quite a few non-native English speakers take the test. And surely one shan't be penalized for awkward construction of phrases.
Regarding writing English sentences as explanation / using phrases for logical flow, this is the sort of the thing I had in mind:
Since 2 + i is a root of the equation, the other root must be 2 - i by the Complex Conjugates Theorem.
Therefore, f(x) = (x - (2 - i))*(x - (2 + i)).
Now, this kind of detail is what a regular textbook would carry, too! I've used apposite terminology ["root"] and referenced a Theorem ["Complex Conjugates Theorem"] in support of my work.
And it's always a good idea to terminate the resolution of a problem by the statement of a "result":
Thus, the proposition has been proven, or
65% of voters support candidate A, or We have insufficient evidence at the 5% significance level to reject the Null Hypothesis that people like Coke and Pepsi equally.
The roots of the function, f(x) are -1, 4 +6i and 4 - 6i.
14. Break long explanations into several short paragraphs, each one with its own idea or step. Leave a blank line between paragraphs.
15. After solving the problem, reread the explanation you've provided to ask 'Does It Make Sense?!'
Furthermore, * Did you answer the right question? * Did you answer ALL parts of the question? * Does your explanation communicate what you were thinking? * Does it explain the math in a way that will help a novice comprehend how to solve the problem?