One of my favorite CI tasks is the Ordering Numbers Task (See tasks for download). This task is easily adapted for different content and grade levels: I have recently used a multiplication version in third grade (see my blog about Spectacular Jumps and Modest Movement) and a large numbers version in fourth grade. As I have used the task, I have learned some things about how students engage in the task (and I would love it others also added what they have learned). I will use the large numbers task to illustrate what students have been teaching me about this task.
“Solving” Representations rather than Comparing Representations
My goal in this task is for students to push their mathematical thinking by engaging in comparisons of representations that have been specifically designed to evoke particular mathematical ideas. For example, one card has 309 thousands – 1 hundred thousand and another has 309 thousands – 99,999. I hoped that students might try to compare the subtrahend, realize that 99,999 is less than 1 hundred thousand and reason that 309 thousands – 99,999 must be greater than the other quantity.
However, students tend to first “solve” the representations on their own cards. (At the start of the task, each student receives one or more cards that they write their names on. They are the only person who can touch the card with their name on it – so these cards tend to become “their cards”.) For example, if a student has a card with 309 thousands – 99,999, they write out these quantities and subtract them using the standard algorithm, usually without even looking to other representations on other students’ cards.
After seeing this “solve-first” tendency many times with different variations of the ordering numbers task, I now realize that comparing representations requires a fairly sophisticated problem solving strategy that is somewhat counter to the usual activity in mathematics lessons: Students are used to solving the problem in front of them, not looking across an entire “problem set” to design a problem-solving approach for the larger task. Yet, looking across a sequence of representations (whether they are the representations in this task or a series of problems on a worksheet) is an important mathematical practice, one that I would encourage every student to try.
For now, I am content to let students, especially those encountering an ordering numbers task for the first time, solve the representations first as that gives students a way into the task. However, for teachers who will use variations of the ordering number task with students across a year, I suggest that you point out when students start to look across all representations. Ask students which representations might be particularly easy to compare and why. Point out, at the end of the task, that comparing representations before solving is, in some cases, a useful activity. Repeat many times until students start to take this up on their own.
I have also started to launch this activity in ways that focuses on comparing representations. I ask two students to write their names on the front of 8.5 x 11 papers that each have different representations. I then ask the class which representation is greater. Students inevitably solve the representations first and explain the sequence of the representations by telling me which solution is greater. I note this on a large piece of paper as one strategy. I then ask the students for another strategy. I continue asking for strategies, sometimes bringing up a third student to hold a third representation, until I have several different ways of comparing the representations.
This has helped students try different strategies as they work in small groups – but so far, these attempts at comparing representations have happened after students have solved their own representations.
Working with Visual Representations
This lesson is more specific to the large numbers task. I adopted the visual representations on these cards from Engage NY (that was the curriculum used by students I was working with). These representations consisted of ovals of quantities like 100,000 or 1,000. One card had 19 ovals with 100,000 in the ovals, arranged in columns of four ovals (see task card). I anticipated that students might count 10 ovals, realize the quantity was 1 million, count the remaining 9 ovals, and add 90,000 to 1,000,000. However, a number of students counted up to 12 ovals before stopping and then announced they had reached 1 million. These students frequently paused, realizing that they had counted past ten and stated that 10 ovals was one million. I realized that the students were counting to 12 ovals as that brought them to bottom of a column, so the students were using visual cues to stop their counting. They seemed to get into the rhythm of counting and, rather than attend to when they reached ten, they focused on the rhythm.
Another interesting student strategy was to add each column of quantities or to record all quantities and add them separately.
Students who turned to addition strategies seemed less comfortable working with large numbers – which I found ironic as I had hoped that the visual representations would provide support for these students by providing a less numerically-focused representation.
Wrapping Up focused on Strategies
While the goal of this task is to order the quantities on the cards, I do not focus on the sequence of the cards (i.e. the “right answer”) during my wrap-up. Instead, I select student strategies that point to central struggles with the quantities and their comparisons. For example, I asked a student who had counted the ovals on the visual representation to share her counting strategy as I wanted students to consider counting rather than adding as a strategy for reasoning with visual representations. By focusing on strategies, rather than on the final answer, I support students in better understanding strategies and in developing these as tools for future mathematical work. I also convey the message that the right answer is not the goal, the reasoning is the goal.
However, I recognize that there is something satisfying in knowing that you have sequenced the cards in the right order and something motivating in trying to figure out the right order. One fabulous teacher (Becky Cavazos), recommended that the cards become a math center, allowing students to return to this task over time. The center might include a place for students to post their strategies so that other students visiting the center could also explore those strategies. The final sequence of representations could also be posted at the center (or posted later) so that students can check their sequences.
