I just finished a very language-heavy mini-unit on sequences in my Math 1 course, with my 9th graders. This is our first year with the CPM Integrated Math 1 curriculum, and for reasons of pacing, we pared down the unit a bit. We elected to not cover writing formal recursive rules, or writing explicit rules for geometric sequences. This left pretty much only the central question, "How can we identify and describe patterns in numbers?"

Since we were only looking at arithmetic, geometric, and quadratic sequences, we basically just needed to be able to precisely describe and extend any of those families of sequences. One of my focuses was on using vocabulary like

*common (nth) difference/ratio, arithmetic, geometric, quadratic*,

*term*. Having a unit with such a big language focus was a great learning experience for me, and in particular, it has led me to think hard about what it means to assess language and vocabulary. Here are some thoughts I have at this point:

- I have come to understand this difference as a manner of "scale." I think that vocabulary is word-ish-level, and language is pretty much everything bigger. An example of a language form is something like
*sequential sentence frames*, which a student might use to describe a process: "__First__, I did this.__Then__, I did that.__Finally__, I did this."

- The big question I was trying to work through during this unit was how to know when and how I was assessing vocabulary, language, both, or neither. Consider these different assessment questions:
- "Make an example of an
*arithmetic*sequence." - I'm expecting them to write down an arithmetic sequence.
**Vocabulary:**Here, the student has to know what the vocabulary word*arithmetic*means, and how it creates a sequence.- "Find the next three terms of this sequence, assuming that it is an arithmetic sequence: 2, 4,..."
- I'm expecting them to produce the next few terms of the sequence, no word writing necessary.
**Vocabulary:**Depending on your students, this is assessing if they know what the word "arithmetic" means.- "Describe this sequence: 2, 4, 8, 16, ..."
- I'm expecting them to accurately use vocabulary like "geometric," "common ratio," and "first term," all in a mostly-grammatical sentence or two.
**VOCABULARY & Language:**They need to know those words off the top of their head, and how to use them in a sentence. They have to identify the context, and recognize that I'm not looking for "exponential," "increasing," "rate of change," or any of the more function-oriented language. Like before, you need word-level understanding of precise mathematical vocabulary (Tier 3 words). You also need language level understanding to be able to write the description. But this assessment is skewed towards vocabulary (hence the all caps), because I am thinking of language as necessarily requiring facility with the vocabulary.- "Use the some of these words to describe this sequence: 2, 4, 8, 16,... {
*arithmetic*,*geometric, common, difference, ratio, term}."* - I'm expecting them to accurately use vocabulary like "geometric," "common ratio," and "first term," all in a mostly-grammatical sentence or two.
**Vocabulary & Language:**Here the students don't have to decide what vocabulary cluster of words to pick from (e.g., slope vs. common difference). Nor do they have to remember all the relevant words off the top of their head. But they DO have to know how to put them together into a sentence, which is VERY language-assess-y. But they also have to know what the words in the word bank even mean, which is a non-trivial degree of vocabulary assessment--albeit, less than if we asked them to come up with the words on their own.- "Describe this sequence: 2, 4, 8, 16,..."
- I'm expecting mostly-grammatical sentences that include accurate, if not exactly precise/formal language like I've expected before. For example, a student could write something like this:
*"You start with 2, and then to get from one number to the next you have to multiply by 2. Then you can go on and on.*" **vocabulary & LANGUAGE:**The vocabulary demand has gone down, because most of these words are pretty much what you would expect any English-speaking student to be able to use (Tier 1 and 2 words). But it also has the mathematical language rigor in that is accurately and pretty-completely describes what's going on in that particular list of numbers.- When is vocabulary assessment not just VOCABULARY assessment?
- In each of the quizzes above, for my students, these vocabulary quizzes are still like...15% computation--I am necessarily assessing numbers sense and arithmetic skills. This is because students have to do some subtraction and addition to extend the sequence, or identify the common difference. Depending on your students, the computations or language themselves might be especially demanding, at which point this moves across the scale between "vocab assessment" and "computation assessment."
- For example, if I have a bunch of students with dyscalculia, these computations could become VERY cognitively demanding, VERY quickly, making this much more of a computational assessment than a vocab/language assessment. I could move this back towards language by offering use of a calculator, numberline, or annotating the sequence for them.
- Alternatively, if I have a bunch of ELD1 students, who are in their first year (or months) of learning English, having to work with any words in English can be very demanding. Or, if I have a student with a reading or language disability, I might have to modify my expectations so that it is developmentally appropriate.
- Questions I still have (note my blog title)
- What do we have to do to make a vocabulary assessment JUST a vocabulary assessment?
- When/why could we do this, or not?
- When/why
*should*we do this, or not? - What language/vocabulary do I want which kids to actually
*memorize*? - If kids had a glossary in the back of their notebooks, is that something I'm okay with them using during assessments? Or if it was on an anchor chart or something?