## Posts tagged ‘assessment’

### How Would You Answer These Questions?

The following is one of my all-time favorite assessment items: Which of the following is the best approximation for the volume of an ordinary chicken egg? 0.7 cm3 7 cm3 70 cm3 700 cm3

The reason it’s one of my favorites is simple: it made me think. Upon first look, I didn’t immediately know the answer, nor did I even know what problem-solving strategy I should use to attack it.

I used estimation, first assuming that the egg was spherical — and, no, that is not the start of a math joke — and then by attempting to inscribe the egg in a rectangular prism. Both of those methods gave different answers, though, and not just numerically; each led to a different letter choice from above.

Not satisfied, I then borrowed a method from Thomas Edison — I filled a measuring cup with 200 mL of water, retrieved an ordinary egg from my refrigerator, and dropped it into the cup. The water level rose by 36 mL. This proved unsatisfying, however, because although choice B is numerically closer to this estimate than choice C — only 29 mL less, compared to 34 mL more — it was five times as much as B but only half as much as C. For determining which is closer, should I use the difference or the ratio?

It was at this point that I decided the answer doesn’t matter. I had been doing some really fun math and employing lots of grey matter. I was thinking outside the box, except when I attempted to inscribed the egg in a rectangular prism and was literally thinking inside the box. And, I was having fun. What more could a boy ask for?

On a different note, here’s one of the worst assessment questions I’ve ever seen:

What is the value of x?

3 : 27 :: 4 : x

I can’t remember if it was a selected-response (nee, multiple-choice) item, or if was a constructed-response question. Either way, it has issues, because there are multiple possible values of x that could be justified.

On the other hand, it’s a great question for the classroom, because students can select a variety of correct responses, as long as they can justify their answer.

The intended answer, I’m fairly certain, is x = 36. The analogy is meant as a proportion, and 3/27 = 4/36. (Wolfram Alpha agrees with this solution.)

But given the format, it could be read as “3 is to 27 as 4 is to x,” which leaves room for interpretation. Because 27 = 33, then perhaps the correct answer is 64 = 43.

Or perhaps the answer is x = 28, because 3 + 24 = 27, and 4 + 24 = 28.

Don’t like those alternate answers? Consider the following from Math Analogies, Level 1, a software package from The Critical Thinking Company that was reviewed at One Mama’s Journey. If this analogy represents a proportion, then the correct answer is $10.50, but that’s not one of the choices. Instead, the analogy represents the rule “add$1,” and the intended answer choice is $10.00. What amazing assessment items have you seen, of either the good or bad variety? ### WODB, Quora Style The following puzzle was recently posted on Quora: Which of the following numbers don’t belong: 64, 16, 36, 32, 8, 4? What I liked about this puzzle was the answer posted by Danny Mittal, a sophomore at the Thomas Jefferson High School for Science and Technology. Danny wrote: 64 doesn’t belong, as it’s the only one that can’t be represented by fewer than 7 binary bits. 36 doesn’t belong, as it’s the only one that isn’t a power of 2. 32 doesn’t belong, as it’s the only one whose number of factors has more than one prime factor. 16 doesn’t belong, as it’s the only one that can be written in the form xy, where x is an integer and y is a number in the list. 8 doesn’t belong, as it’s the only one that doesn’t share a digit with any other number in the list. 4 doesn’t belong, as it’s the only one that’s a factor of all other numbers in the list. I suspect that Danny has visited Which One Doesn’t Belong or has read Christopher Danielson’s Which One Doesn’t Belong. Or maybe he’s just a math teacher groupie and trolls MTBoS. But then Jim Simpson pointed out the use of “don’t” in the problem statement, which I had assumed was a grammatical error. Jim interpreted this to mean that there must be two or more numbers that don’t belong for the same reason, and with that interpretation, Jim suggested the answer was 32 and 8, since all of the others are square numbers. Don’t get me wrong — I don’t think this is a great question. But I love that it was interpreted in many different ways. It could lead to a good classroom conversation, and it makes me consider all sorts of things, not the least of which is standardized assessments. How many times have students gotten the wrong answer for the right reason, because they interpreted an item on a state exam or the SAT differently than the author intended? And how many times have we bored students with antiseptic questions, only because we knew they’d be free from such alternate interpretations? Both scenarios make me sad. ### All Systems Go I noticed the boys having an intense conversation in front of this sign at our local pizza shop: When I asked what they were doing, they said, “We’re trying to figure out how much one slice and a beer would cost.” As you read that, there were likely two thoughts that crossed your mind: • Why can’t these poor boys look at a pizza menu without perceiving it as a system of equations? • Why are eight-year-olds concerned with the price of beer? The answer to both, of course, is that I’m a terrible father, and both beer and math are prominent in our daily lives. But you have to admit that it’s pretty cool that my sons recognized, and then solved, the following system: $\begin{array}{rcl} 2p + s & = & 6.00 \\ 2p + b & = & 8.00 \\ p + s & = & 3.50 \end{array}$ They didn’t use substitution or elimination because they didn’t have to — and, perhaps, because they don’t know either of those methods yet. But mental math was sufficient. If two slices and a soda cost$6.00, and one slice and a soda cost $3.50, then one slice must be 6.00 – 3.50 =$2.50. Consequently, two slices cost $5.00, so a beer must be 8.00 – 5.00 =$3.00. A beer and a slice will set you back \$5.50.

I remember once visiting a classroom in Somerville, MA, and the teacher was reviewing the substitution method. My memory is a bit fuzzy, but the problem she solved on the chalkboard was something like this:

Mrs. Butterworth’s math test has 10 questions and is worth 100 points. The test has some true/false questions worth 8 points each and some multiple-choice questions worth 12 points each. How many multiple-choice questions are on the test?

The teacher then used elimination to solve the resulting system: $\begin{array}{rcl} t + m & = & 15 \\ 5t + 10m & = & 100 \end{array}$

The math chairperson was standing next to me as I watched. “Why is she doing that?” I asked. “You don’t need elimination. It’s clear there have to be 8 or fewer multiple-choice questions (8 × 12 = 96), so why not just guess-and-check?”

“Because on the MCAS [Massachusetts Comprehensive Assessment System], if they tell you to use elimination but you solve the problem a different way, it’ll be marked wrong.”

So much for CCSS.Math.Practice.MP1. Although most of us would like students to “plan a solution pathway rather than simply jumping into a solution attempt,” apparently students in Massachusetts need to blindly follow algorithms and not think for themselves.

The following is my favorite system of equations problem:

I counted 34 legs after dropping some insects into my spider tank. How many spiders and how many insects?

Why is this my favorite system of equations problem? Because there is a unique solution, even though it results in just one equation with two unknowns. Traditional methods won’t work, and students have to think to solve it. Blind algorithms lead nowhere.

Other things that lead nowhere are spending your leisure hours reading a math jokes blog. But since you’re here…

Why did the student put his homework in a fish bowl?
He was trying to dissolve an equation.

An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.