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.

]]>I was in a third grade classroom yesterday, teaching a lesson on ordering representations of multiplication expressions. (See the Tasks for Download section for an example). The students needed to compare representations like *eight groups of six*, *7 x 8*, and *2 x 3 x 6*. The first inclination of many students was to solve these expressions and then put the products in order from least to greatest. The teacher and I pushed students to then find an alternative strategy, one that did not involve finding a product. The students really struggled to do this. They could represent *eight groups of six* as *8 x 6* and they could change *2 x 3 x 6* into *6 x 6*, but then they weren’t sure how to compare *7 x 8* and *8 x 6*, although they had many interesting and productive discussions about this comparison.

The teacher (who is an amazing, student-centered mathematics teacher) described a future task she might do with the students. She would make one of the factors a really large number (e.g., *8 x 6* compared to *8 x 1,000,000*). Her thinking was that students 1) would not be able to calculate the product of the large number and so would need to first turn to other strategies and 2) the really large number would help the students in using factors (rather than products) to compare the expressions.

I also found myself drawn to this revised task. Perhaps if the students worked with these kinds of extreme comparisons (i.e., those with one really large factor), they would then use the same reasoning in situations that involved comparing numbers that were more similar.

But then I realized that while extreme expressions might result in quick and correct comparisons, I was focused on the wrong thing. I was trying to get students to efficiently **do something**, rather than really valuing the less obvious and less flashy, but really productive mathematical ideas students were debating. In other words, in the task we presented, students were not making huge leaps to new solution strategies, but they were wrestling with important ideas like what the factors in a multiplication expression mean, how to use the commutative property of multiplication, and how great the difference needed to be between factors before you could say anything about differences in products (e.g., can you use comparing factors to determine whether 3 x 15 is greater than 8 x 6?). These were really productive mathematical struggles and the students were quite engaged in these ideas. Yet, it was harder to notice, value, and celebrate these struggles because they did not result in the bright flashbulb of the “Ahha!”, right-answer moment.

This isn’t to say that I might not use an example of an extreme comparison in order to get a student started thinking about comparing factors. However, I am realizing that I need to be less seduced by tasks that result in spectacular jumps to right answers. Instead, I need to continue to celebrate tasks that engage students in modest movements toward essential and exciting, but perhaps less conspicuous mathematical understandings.

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These ideas are not new to most of the teachers with whom I work. Many took jobs in urban public schools, because they live in these neighborhoods and they want to support the young people for whom Guinier is advocating. However, Guinier goes beyond the common claims about inequity. She claims, that yes, the overuse of standardized tests and only the use of standardized tests to identify talent in our country is inequitable, often bars access to universities and thereby impacts a person’s earning potential, job security, and access to affordable health care among other things. We know this argument well, and it’s often equal access to these very things that we are working so hard to ensure for our youth. But, then Guinier takes the argument further and explains that what our country needs in its efforts to solve the major problems of our time, is **not** a homogenous group of people who all had high scores on the SAT. “If we want to be successful, we don’t hire Steven and John who got the highest scores on the test. We hire Steven and Jane *if *Jane (who scored lower on the test than both Steven and John) got all of the problems right that Steven and John got wrong.” (Guiner cites Scott Page’s book, The Difference, in which Page argues that diversity trumps ability in problem solving.). In summary, if we want to reward Steven and John for their high test scores, then we hire Steven and John, but if we want a successful organization, then we hire Steven and Jane.

I have listened to Guinier many times, because her argument about the Black Agenda, problem solving and diversity resonates so strongly for me. As a result of an education system that identifies and rewards a very limited set of skills, talents, and strengths, too many our young people never come to know how they are smart and never get a chance to develop and use their talents for their individual and our collective benefit. We are all suffering because of this. Our education system is literally failing itself as it continues to purport very narrow definitions of success. If, for example, this country wants a strong economy, department of defense, health care and educational system, an educated citizenry and savvy consumers, then we have a responsibility to redefine merit and create schools and classrooms that identify and then support the development of a broad and diverse set of strengths. We owe this to our students and we owe this to ourselves. As Paul Wellstone, former U.S. Senator from Minnesota, stated, “Everyone does better when everyone does better.”

And so, as this new school year begins, I challenge each of us to consider how we define what it means to be smart in our schools and classrooms. What assumptions do we make about the “smart kids?” What messages are we sending our young people about who can be smart? What opportunities are we providing all of our students to see smartness in action and to get better at the many things that they are not good at YET? In our math classrooms, we might turn to the CCSM mathematical practices to develop our smartness repertoire, spend time brainstorming our “smart talk” in department meetings, or observe in classrooms outside of our content area and listen to our colleagues talk about what it means to be smart. Challenge yourself to go deep and beyond the talents of “Steven and John” with whom we might already be quite familiar and consider the many ways in which the Lucianos and K’Dishas make remarkable contributions to our classroom communities. We’ll all be better for it.

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I am a former elementary school teacher and a current mathematics teacher educator, so I have opportunities to teach students at all levels. In my own teaching, I have found that CI teaching moves are at times magic bullets, quickly engaging formerly uninvolved students. For example, I spent one summer teaching eighth grade math. One of my students, I’ll call him Alonzo, was frequently off-task and disruptive – breaking pencils and throwing them across the room among other unhelpful behaviors. Along came a day in which Alonzo was working on the math problem and had the best solution of anyone in the class. It took some coaxing to get him to present and he suffered some taunting from his peers, but soon everyone, including Alonzo, realized the important mathematical ideas in Alonzo’s answer. After this public recognition of Alonzo’s mathematical skills, Alonzo was a different math student. He was much less disruptive and even remained in his seat during breaks, voluntarily working on math problems. While I don’t know what the long term outcomes were for Alonzo, I was pleased that he had become a productive, contributing member of our summer learning community. And I attribute this shift in his academic focus to my use of CI strategies. In particular, I had an assumption that each of my students had mathematical strengths that may not have been recognized. This assumption meant that I was constantly seeking instances of mathematically productive activity and then publicly recognizing and labeling that activity. By helping individual students AND the whole class recognize each others mathematical skills, students more readily worked on math tasks. This CI activity, called *assigning competence,* was somewhat straightforward and had enormous impact.

While CI seemed fairly easy to implement in my summer class, I have frequently found that I’m saying and doing things in my teaching that are not at all consistent with the philosophy of CI. I’ll find myself frustrated with a student and I’ll focus on how quickly he is picking up (or not picking up) ideas. I’ll talk about elementary students as high or low performing. In each of these moments, I’m drawing from typical ways of talking about students that focus on how quickly they can do mathematics – rather than focusing on student’s mathematical strengths and areas of growth and considering how a task might draw upon strengths and encourage growth in other areas. Thinking about students through the lens of CI means reframing what I notice and how I respond. And this is not easy work, especially when so many aspects of school ask teachers to consider how students quickly respond to somewhat narrow mathematical tasks.

What we would like to do with this webpage is provide a forum for ideas and support for folks who are exploring ideas related to CI. We welcome your input, ideas, tasks, and questions! Just drop us a line at cimath@cimath.org.

Thanks,

Marcy

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