Here is another installment in my series reviewing the NY State Regents exams in mathematics.

Consider the following problem from the January 2013 Algebra 2 / Trigonometry exam:

What’s interesting about this problem is what it *doesn’t* tell you.

The student here is expected to assume that the terms in the sequence keep going up by 9. That is, the student is expected to assume that the sequence is *arithmetic.* Once the student makes that assumption, they can use the appropriate formula to sum the first 20 terms.

Encouraging students to assume that arbitrary sequences are arithmetic is bad practice. It can develop in students a sense that sequences are usually arithmetic, which will make it harder for them to understand non-arithmetic sequences later on.

More generally, we don’t want students getting into the habit of making unconscious assumptions about problems. By forcing them to make assumptions that fit the problem to the test, questions like this train students not to ask questions like “Are we sure the next term is 50? What else might it be?”. This was one of the main points of my talk “g = 4, and Other Lies the Test Told Me“.

It’s not outrageous that think that this sequence might be arithmetic. But the mathematical world is a rich and complex place. Are we really sure that this sequence is arithmetic? After all, maybe what we’re really looking at is the sequence of numbers whose digits add to five!

Am I thinking too deeply about this sequence? Maybe. But as a teacher, thinking deeply about mathematics is precisely what I want my students to do.

I’m curious what you’d recommend. Should questions like this one be scrapped altogether, labeled as arithmetic, or something else?

Sue-

I think there are a few reasonable things you could do here.

First, as you suggest, you could identify the sequence as arithmetic in the statement of the problem:

The first five terms of an arithmetic sequence are 5, 14, 23, 32, and 41. Find the sum of the first 20 terms..To deepen the question, you could add something like “

Explain how you know that the first 5 terms are in arithmetic progression“.To leave it more open, you could change the directive to something like this:

The first five terms of a sequence are 5, 14, 23, 32, and 41. Find the sum of the first 20 terms of the sequence. List any assumptions about the sequence you are making.Thanks for pushing me to be constructive.

“Are we really sure that this sequence is arithmetic? After all, maybe what we’re really looking at is the sequence of numbers whose digits add to five!”

A quick search on oeis returns 8 different sequences that fit the pattern. Truly intelligent students with some math experience outside of the basic standard would be at a disadvantage here because they would read the sentence and assume that the writers were INTENTIONALLY dropping the word “arithmetic”. They would then go off on wild goose chases that waste time and effort that less capable students wouldn’t bother with.

I doubt many strong students would get this problem wrong, but the tragedy here is how these tests train those students

notto go on those wild goose chases.Whenever an integer sequence arises in the course of investigating any mathematical question, I check to see if it’s polynomial, by looking at successive differences, and if it is, I calculate its formula by dividing the constant nth difference by n! and iterating. If it’s not polynomial, then I check to see if it’s [nearly] exponential by looking at successive ratios, and if it is, I look for a Fibonacci-like recurrence relation.

Failing those simple tests, I turn to OEIS. Even if those tests had succeeded, as they did in this case, I might still turn to OEIS to look for alternative sequences.

A search in OEIS for 5, 14, 23, 32, 41 gave these four results:

https://oeis.org/A017221 9*n+5

https://oeis.org/A052219 Sum of digits is 5

https://oeis.org/A043473 Numbers n such that number of 5’s in base 9 is 1

https://oeis.org/A044057 Numbers n such that string 1,2 occurs in the base 3 representation of n but not of n-1

In my opinion, this approach (or something similar to it) is a reasonable thing to begin teaching to kids as they learn and become familiar with algebra. I understand that some teachers may object to including the part about checking OEIS. I see the rationale for making kids prove they can answer questions without checking ubiquitous online resources. Certainly a kid with no interest in actually learning math will use the online resources to find answers, copy them down whether they’re right or wrong, and then learn nothing. But I maintain a kid with no interest in learning math will also learn nothing by memorizing formulas and performing arithmetic manipulation by rote. On the other hand, showing kids the online resources and giving them the theoretical tools to understand why the answers are what they are will instill (in some, not all, I admit) a true interest in really learning math. Such a kid will spend hours exploring OEIS, WolframAlpha, and other resources to check and solidify their understanding of math. I know, of course, because I was (and am) such a kid (now with kids and a grandkid).

Finite differences and recursion are wonderful topics, and should definitely be a part of any exploration of sequences and series.

As far as technology goes, it has to be addressed, regardless of how one feels about. The tools are out there and students know about them.

I agree completely. We should teach our students that it is important to state our assumptions. A good student would probably infer that an arithmetic sequence was intended, but we are setting a bad example for our students if we omit assumptions.

I love this post – it is so important that we teach our students to state assumptions or challenge assumptions that are not stated.

This post urged me to share a problem/example where the pattern is not what it seems:

http://mrwardteaches.wordpress.com/2013/02/21/patterns-and-the-need-for-proof/

This pattern appears to be a simple exponential: 2^n. However, the fifth number in the sequence is 31, not 32.

Thanks for sharing this exam question!

Yes, that’s a classic. A great problem to sneak into a “Pattern Finding” problem set!

I’ve seen questions like this where more than one answer on the test is a correct possible answer. Really, simply giving a sequence is never enough.

I mean, it could even be:

5, 14, 23, 32, and 41

(n)(n-1)(n-2)(n-3)(n-4)+9n+5

So technically giving a sequence without saying the type of sequence is not mathematically correct, because it doesn’t *ever* name a unique sequence.

My 13 year old daughter just found a mistake of unstated assumption on the 2013 Jan Geometry Regents. Question 17: how many points are 5 units from a line and also equidistant from two points on the line? My daughter circled the answer obviously expected (2) and wrote next to it, No! there are infinite points!

Still, for all the problems with the Regents, you should be grateful that NY students are even asked to solve multistep problems, actually prove things, draw constructions, graph functions, etc. Here in Virginia, proofs are optional for classroom teachers to cover and since students don’t have to prove anything on our entirely multiple choice SOL exams, they aren’t even covered in class, even in geometry. I feel like in NY at least they are trying and there is some motivation for students and teachers to learn/teach to something over minimum requirements, which is lacking in Virginia due to the simplistic SOL tests.

I guess the absence of the phrase

in the planedoesn’t even register with me anymore when I read this exam. It’s fairly clear that the authors of this exam do not think deeply about the geometry of space.It’s sad that the benchmark exams in Virginia are entirely multiple choice, and the fact that proofs become “optional” in Geometry class is a direct consequence of the absurd focus on high-stakes tests.