Pilnik - Week 8 - Mathropology and Place Value

My first full day of teaching solo came earlier than anticipated: Last Friday, my teacher was out and the sub the school had scheduled was reappointed to another, intern-less classroom (a decision of somewhat dubious legality). My teacher had written out her plan for the day, which included 45 minutes of Math for students to complete two worksheets on place value.

During my prep hour, I deliberated over how to present the worksheet to the class. I thought back to the Kabiri & Smith article ("Turning Traditional Textbook Problems into Open-Ended Problems"):

Many teachers have found success by "basing instruction on problems and activities that invite different solution approaches and many levels of solution so that less talented students can participate in the task with more talented individuals and all can experience individual success . . ." (Bley and Thornton 1994, p. 158).

The worksheet I decided to focus on follows:

 I focused first on the objective of the lesson. The students had been practicing mastery of identifying ones, tens and hundreds place values for the past week. This worksheet was meant to reinforce that understanding by using different symbols -- indeed, a different system -- to represent place values.
The connection between the Ancient Egyptian and the Arabic numeral place-value systems  would be lost on students who struggled already with place value, if the "code" were simply given to them and they were asked merely to reproduce it.

Rather than giving them the worksheet, modeling one problem and asking them to complete the rest of the given problems, I decided to try introducing it as a puzzle. That way, I reasoned, no one student would automatically be an "expert", and all students (given that none had had exposure to this system before) would have a chance of solving it. I would briefly introduce the idea that other civilisations have been using place value for thousands of years, that our way of representing numbers is not the only way, and that the students would soon have a shot at being mathematical anthropologists themselves. I wrote one number on the board and gave students 30 seconds to think about what that number might be, and to share out.*

Students were engaged and nearly all -- from various skill levels -- wanted to share their ideas. There were a few instances when students shouted "ooh!" and threw their hands in the air again to change their answer after hearing another student's hypothesis, which indicated to me that they were listening actively and using each other's responses to inform their own.

After many hypotheses were shared, I confirmed the correct one and wrote the Arabic numerals beside the hieroglyphs. I gave the class another example, and chose specific students to share their reasoning based on their reasoning from the previous attempt. Students listened keenly and volunteered their own guesses again, based on my feedback of the previous guess.
After three problems, the class had "cracked the code" by themselves, and were able to articulate it. One student came up and wrote out the code using the HTO notation they were already familiar with with Arabic numerals. Three students of varying ability were able to come up to the board to write a three-digit spoken number (provided by me) in hieroglyphs.**

After the whole-class exercise, I handed out the worksheet to students and allowed them to work with a partner to complete it.*** Students who completed it could flip it over and work on the abacus-based problems sheet my MT had included.

The difference between the students' engagement (and the variety of students engaged) in this activity and most of their math worksheets was palpable. This activity gave me a concrete understanding of the importance of different entry points for students, and dissipated any lingering thoughts I had of some of my lower-performing students as being just not math people. All the students, including my ELL student, were engaged, because all were able to muster the confidence to complete the task.

It was a fly-by-the-seat-of-my pants kind of day, and I was beyond relieved to have been able to tap into the readings and ideas we've talked about so much in class to improvise a higher-level lesson.

There are definitely approaches to this lesson I would have changed, however:

* I wrote one number on the board and gave students 30 seconds to think about what that number might be, and to share out.

It would, I think, have been better had I asked students to write their guesses down so I could wander about the room and choose my volunteers more strategically.

**Three students of varying ability were able to come up to the board to write a spoken number in hieroglyphs.

I wish I had asked students to come up and give us their own hieroglyph so that the class could guess it.

***After the whole-class exercise, I handed out the worksheet to students and allowed them to work with a partner to complete it.

If I could have modified the worksheet to allow them space to write their own, too, the level might have been amped up a bit higher.

I noticed, too that some students were noticing the conspicuous absence of Zero in the hieroglyph system. I wish I'd had time to ask students to share out their Zero observations, and perhaps to make observations between the Ancient Egyptian and Arabic numeral systems, too (e.g., that both were base 10, both used place value, etc.).