Where's the math?

In this final post on the Five Practices, we look at the final Practice - Connecting. Previous posts from this series explore preservice elementary teachers work on Anticipating, Monitoring, Selecting, and Sequencing related to this lesson from Context for Learning Mathematics (CFLM - Fosnot et. al.). Because elementary children are in short supply at our university, the future educators use professional development materials gathered from a third-grade classroom working on the turkey-cost problem as a substitute.

Image used with permission of authors 
New Perspectives on Learning

In this part of the exploration, my teachers watch video of the third-grade teacher wrapping up the lesson. They look for evidence of the teacher Connecting (1) the various approaches used by the students, (2) key mathematical ideas, and (3) a new, related problem. Connections between the students' approaches are especially strong given choices she made in Selecting and Sequencing the work to share. 

The transition to the next problem also seems well-thought-out as students determine how long to cook the 24-pound turkey if the cookbook suggest 15 minutes per pound.

Image used with permission of authors 
New Perspectives on Learning

The connection to the turkey-cost problem is obvious to the teachers.

The part of the Connecting sequence that seems most lacking is Connecting the students' work to key mathematical ideas. I try to make it clear that it is unfair to criticize the teacher because we do not know what happened before or after the lesson or, for that matter, other contextual factors that might have influenced her instructional decision making. However, to ensure that the teachers recognize the big ideas associated with the different approaches, I offer the following additions to the lesson.

The first three students are applying the Distributive property to the cost per pound decomposed into it's whole number and decimal parts. In order to compute 24x0.25, the students are essentially factoring 24 into 6x4 (4 being the multiplicative inverse of a quarter, 0.25) and then using the Associative property to regroup the order of multiplication. Consequently, the original expression, 24x1.25, simplifies to 24+6 or $30 - the answer to the turkey-cost problem.

The last pair of students used a more efficient approach that bypassed the need for the Distributive property. They still factored 24 into 6x4, but then they used the Associative property to multiply 4x1.25 first, to get 5, and then 6x5. Again, the answer is $30.

While some of the teachers think the key mathematical ideas presented in these series of expressions might be beyond the third-graders' current understanding, the Associative and Distributive properties are found in the Grade 3 Common Core State Standards (3.OA.B.5).

What do you think? Is this too much to expect of third-graders? Do you have another way that these key ideas might be connected to the students' work?

How did the teacher organize the turkey-cost discussion?

My preservice teachers have reached stage four, Sequencing, of the Five Practices (see previous posts for Anticipating, Monitoring, and Selecting). Because these teachers have no direct experience facilitating a discussion involving students reflecting on their mathematical thinking, we return to the third-grade class and watch a video of how the teacher Sequences the work of the students on the turkey-cost problem. I ask the future educators to watch the consolidating conversation and hypothesize why the teacher decided to order the student-work the way she did.

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All images are from New Perspectives on Learning
used with the permission of the authors

Emma and Emma take the $1.25/pound price and start with a friendlier number - $1 per pound. If that were the price, then the 24 pound turkey would cost $24. But they recognize they need another twenty-four $0.25 to find the total costs. So they count by 25s, keeping track of how many 25s they have counted underneath. They find they need another $6 for a total cost of $30.

Harry and Ese use a similar strategy of breaking the price per pound into a dollar and a quarter. However, instead of counting by 25s, they gather the quarters in groups of four. Each group of four quarters is a dollar. There are six groups. Therefore, $6 must be added to $24 to get the total cost of $30.

The next pair, Nellie and Nate, also start by taking off the 25 cents to get an initial cost of $24. Then they group the quarters, but they do it differently than Harry and Ese. Instead of showing "pictures" of quarters, they use a table to represent the relationship between pounds and dollars at $0.25 per pound. The table shows that 24 pounds requires an extra $6. Again, the total cost is $30.

Finally, Suzanne and Rose share a unique strategy that does not break up the $1.25. They know that "4 pounds is 5.00" and use this to jump by 5s on the open number line. There are six jumps because there are six 4s in 24 pounds. Although they use a different approach, Suzanne and Rose also find the cost to be "30 $ in all!"

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After observing the Sequence of approaches, the preservice teachers offer their hypotheses about why the teacher put the student-work in this particular order. Some suggest it might have progressed from "most popular" to "most unique." Others think it was based on increasingly sophisticated structures. A few wonder if it might be related to the different representations being used.

Why do you think the teacher ordered the work in this way? And where do you think the teacher goes next to complete the last Five Practices stage, Connecting? As always, your participation in the comments is appreciated.

Who should share their turkey-cost solutions?

All images are from New Perspectives on Learning
used with the permission of the authors

In preparing future educators to facilitate productive math lessons, we provide opportunities for them to apply the Five Practices to problems like the one shown here. Teachers began (in the first post of this series) by identifying a Standards for Mathematical Practice (SMP) to focus on and Anticipating student solution strategies related to this SMP. Next (in the second post), they watched video of third-graders solving the problem in order to Monitor their efforts and compare them to the Anticipated responses. The teachers were also given copies of the students' work to examine. Teachers use this work to begin the process of Selecting who might share during a whole class discussion.

For example, the teachers focusing on SMP 5 might Select these students to share because they all used the open number line as a tool.

The discussion could revolve around how this tool was used appropriately and strategically.

Another group of teachers, focusing on precision could Select the work of the third-graders shown below since they seemed to arrive at different answers.

The class could work together to determine what question each pair of students was answering and how to move toward a correct solution.

Finally, teachers wanting to focus on structure might Select these four student-pairs.

This work included different ways students used the structure of money (decimals) to arrive at a solution.

By strategically Selecting student work to share, a teacher does not leave the  ensuing discussion to the vagaries of volunteers. Consequently, the resulting classroom conversations becomes more purposeful and productive. But first, the teacher must apply another of the Five Practices - number four, Sequencing. 

How might a teacher organize the Selected student-work for each SMP in order to maximum learning and why?

How did those third-graders determine the turkey's cost?

All images are from New Perspectives on Learning
used with the permission of the authors

Having anticipated third-graders' thinking for the scenario provided above (the first of the Five Practices which was attended to in the first post in this series), my preservice elementary teachers are ready to engage in the next Practice - Monitoring students' thinking. Unfortunately, the university has yet to meet my request for a lab school which means that elementary-aged children are in short supply in my classroom. Given my desire to create as authentic experience as possible for my teachers, this creates a problem.

Luckily, Dr. Catherine Fosnot and her colleagues have gathered classroom videos and student-work from elementary kids working on problems from their Context for Learning Mathematics (Fosnot et. al.) series (including from the turkey cost lesson). While it's not the same as monitoring actual students, it does represent the same experiences inservice teachers might have in Professional Development (PD) sessions using Dr. Fosnot's materials.

This PD involves helping teachers to develop phronesis. "Phronesis is situation-specific knowledge related to the context in which it is used—in this case, the process of teaching and learning." (from p. 147 of Young Mathematicians at Work: Constructing Multiplication and Division) By watching the video and examining the students' work, teachers are able to observe an authentic lesson and reflect on the teaching moves that support students who are immersed in doing mathematics.

The preservice teachers in my class watch the video and observe the students finding the turkey cost. Just as many of them predicted, the third-graders are splitting the $1.25 per pound into a dollar and a quarter. Students' papers (like the one on the right) show that they understand the 24-pound turkey will cost $24 plus 24 quarters. Pairs of students use a variety of approaches to determine the total cost of the turkey. As my teachers review the third-graders' efforts, the teachers move toward the next phase of the Five Practices - Selecting students to share their work based on the Standards for Mathematical Practice (SMP) selected at the beginning of this process. 

We will continue this work in the next post. But first, which SMPs would you say the work of Emma and Emma highlights?

How much is that turkey in the window?

Disclaimer: I receive no financial benefit for my endorsement of the Context for Learning Mathematics (CFLM - Fosnot et. al.) curricula or the associated professional development resources shared in this series.

In my efforts to make my classes for preservice elementary teachers more accurately reflect the work of teachers, I try to craft my lessons as professional development sessions. We spend a significant amount of our time applying the Five Practices for Orchestrating Discussions to activities from established K-6 mathematics curricula. Recently, we used a lesson from CFLM's The Big Dinner unit in anticipation of Thanksgiving.

From Professional Development Resources
used with permission of author

This introductory lesson asks students to find the cost of buying a 24 pound turkey. My teachers start by selecting a Standard for Mathematical Practice (SMP) as a goal for the lesson. (We could also look at content goals, but these teachers need more practice with the SMPs.) Having chosen a goal, the teachers begin applying the First Practice: anticipating possible student responses associated with the goal. The students in this scenario are third graders who are unlikely to use the standard algorithm for multiplying decimals.

Many of the teachers anticipate that the students will break the $1.25 into dollars and cents. For the teachers who identify "Look for and make use of structure" as their SMP goal, this prediction seems reasonable. Some of the teachers consider the models and tools (SMP 4 and 5) students will use to solve the problem. Other teachers, who are attending to precision (SMP 6), wonder where students might make mistakes in their computation. Nearly all the teachers are interested in the different strategies the students will use to solve the problem.

Before we move on to the Second Practice, monitoring students' work, I want to give you an opportunity to add the SMP you would choose to focus on for this problem and possible student responses you might anticipate. As always, please add your contributions to the comments.

In the next post, I will explain how preservice teachers can carry out the remain Practices of Orchestrating Discussions even though they are not actually in an elementary school classroom. 

Who would you want to work with?

We are in the process of making teacher-groups for Family Math Night. The teachers (MTH 221 students) will work together to develop an activity related to specific standards, try out the activity with K-6 students, and reflect on the activity's effectiveness. Throughout the project, teachers use frameworks from the 5 Practices and the Principles to Actions to inform their efforts. This is one of the ways I try to embed the work of teaching into the course.

Because I also want to prepare pre-service teachers to be your future colleagues, I am soliciting your help in identifying norms for collaboration. What are some things you look for in colleagues with whom you choose to work? I am trying to come up with five criteria that the teachers could consider as they evaluate their interactions with their peers.

I have a compulsion to use acronyms, so I made the checklist on the right using some suggestions shared on Twitter. Does this list work for you? If not, how would you adjust it? Please do not be limited by this format as you offer suggestions in the comments.

Thank you in advance for your contributions to the development of these future educators.

When should we intervene?

More on the session

During our session at NCTMNOLA, participants explored several games that offer opportunities to encounter mathematical content and processes associated with the Common Core State Standards for grades K-2.

As the teachers played the games, or observed as others played, we asked them to keep an eye out for meaningful mathematical moments that might be shared with the entire group.

One of the games introduced many of the teachers to a new manipulative - a rekenrek. 

A teacher in this group anticipated that students might have a hard time following the directions for this game and treat each row as a separate roll. She wondered when to intervene if a student did this. 

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I am sure I gave her a very unsatisfying answer, "It depends."

It depends on my goal for the lesson. If the lesson is about using the structure of the rekenrek to help students visualize groups of tens and fives in regards to place value understanding, then I might intervene. However, if I want the lesson to focus on decomposing numbers in order to make groups of ten, then I might wait until the whole class discussion (reflecting on the learning); this choice allows us to talk about it as a group.

It also depends on whether or not everyone is exhibiting the same issue. I hate putting out a lot of little fires. If I saw everyone doing this, then I might intervene with the entire group since there would be a lack of diversity in what students could share during the reflection. However, if it was a single student, then I could decide whether or not to select this approach for the reflection and where in the sequence (see Orchestrating Discussions).

So let's assume that my goal was about making tens and only Patsy played the game in this way. After having a few students who followed the directions as written share, I would move our attention to her "game board."

I want to share Patsy's work because she played a slightly different game. She answered a different question. If I wanted to know what Patsy rolled during each turn, I could find out from her rekenrek: 10 the first roll; 7 on the second; on the third a 3 (coincidence there, eh?); 9 on the fourth; 1 on the fifth; and 4 on the sixth roll. But the game wants us to say how many beads we have total and how many we need to get to 100. So with an elbow partner, I want you to devise a plan for finding these two numbers, the total and what's left to get to 100, but don't find them - yet. Ready? Go.

Although it is not what I expected (probably because it is not what I expected), I really like what Patsy's new game does for the lesson. In fact, I might tuck this example away for another time when we play the game. Then, if no one else plays it this way, I can still use it in our discussion because the game provides a shared context. This context, at least once, created an interesting problem for students to solve. And that was the main point of the session:

How Might I Orchestrate the Discussion?

In the last post, I shared how I used ideas from Orchestrating Discussions to:

• Set goals using established standards for a Family Math Night (FMN) activity;
• Anticipate learners' responses to certain tasks and questions aligned with those goals; and
• Create a monitoring sheet to collect information on learners' actions and answers.

When the time came to test out the activity during a mock FMN, I asked preservice teachers from each group to observe their peers carrying out the tasks and ask question that would make their peers' thinking visible. The observers recorded this information using the monitoring sheet, which I gathered onto a single form (provided below).

Selecting

Because my principle goal was associated with the target, Represent and Interpret Data, I concentrated on selecting work associated with the second task on the monitoring sheet. (I took pictures of the participants' efforts in order to share them here.) While some participants laid out the Pattern Blocks they grabbed into something resembling a real graph and others translated their handful to Unifix Cubes, everyone went on to create more permanent graphs. Therefore, besides acknowledging this step, I would not want to spend class time asking participants to share these real graphs.

Instead, I would focus on the permanent representations participants created and how these representations might be interpreted. In particular, the work selected would relate to the Common Core indicators 2.MD.10

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories

and 3.MD.3

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories.

With this in mind, I selected L1, T1, A1, and J2 to share their work.

Sequencing

L1's work reflects the representation closest to the actual activity, so I would ask her to share her work first. Then we could discuss ways it might be interpreted. If it is not raised by the class, I would point how the graph might be misinterpreted: "It seems like the graph shows there is more yellow hexagons than red trapezoids. Is that truly the case?"

Next, I would have T1 share his graph. He drew the shapes inside squares on a grid; this seems to address the possible misinterpretation that the more direct representation might foster. A person reading this graph can see immediately which shape occurred the most in T1's handful. But as the number of red trapezoids is so close to the top, it raises another question: "What would we do if there were more of a shape than squares in a column on the grid?" Two participants encountered this problem and I would ask them to follow T1.

As you can see by her graph, A1 dealt with the problem of having more blocks than squares by amending the bar graph. She added ellipses to the top of the graph to indicate that some items from the data set were omitted. Also, the number of total items in each of the "over-abundant" categories were written directly on the graph. While this graph conveys all the necessary information, it can impede quick interpretation of the data because some of the visual characteristics are lacking.

J2 addressed the issue of having too many blue rhombi for the column by splitting the squares in two. Each half represents one block. She standardized the unit for the other shapes, thereby making it easy to compare the quantities. This approach begins to get at the idea in 3.MD.3 of a "scaled bar graph." The sharing would end with this example and lead to a whole class discussion meant to bring the ideas associated with these models together.

Connections
Looking Back
I would want the participants to notice the multiple approaches that can be used to represent a data set and how the representations are related. For example, most of the graphs reflect a one-to-one correspondence. Discussing the limitations of this approach seems a natural next step as we could revisit the possible misinterpretations (L1's work) and the need for creative modifications (A1's work).

Looking Forward
In order to move toward 3.MD.3, I could ask them to consider which approach they might use if they were going to graph the "handful" results from their group or the entire class. Hopefully, increasing the size of the data set will push them to considering a scaled approach similar to J2's work. This would lead to the next task associated with this activity

Next Time
Glows
If I were to do this activity again, I would continue to provide participants with a variety of ways to represent their data sets. This results in a rich source of approaches to choose from when engaging in the Selecting phase. 

Furthermore, because I am not asking for the "best way," participant are free to select methods like the picture graph because they want to be creative rather than feeling the need to be efficient. Hopefully, this lessens the likelihood that participants will somehow link approaches shared earlier in the Sequencing phase with the creator's ability - it was simply a choice.

Grows
Next time, I want to make sure that the Pattern Blocks are equal in thickness. While having a combination of Pattern Block sets mixed together made for some interesting approaches, it could potentially be a distraction, as this stacking shows. It can create extra categories that are not always consistent across the buckets from which the participants grab a handful. The task is rich enough as it is without adding this extra variable.

Also, after graphing one handful, I would ask students to predict how many would be in two handfuls. This would hopefully, create interest in conducting another experiment. If they do grab two handfuls, this increases the likelihood that students creating a bar graph would encounter the issue A1 had with not enough grid space.

How Many in a Handful?

One of the projects in #MTH221 involves hosting activity stations during a Family Math Night at local schools. Our preservice elementary teachers are assigned a topic (patterning, measurement, data, or probability) and asked to find an activity related to that topic that might interest K-5 students. As a group, they decide on two that they want to run and begin gathering/developing the resource they need to make the activity work.

For example, I found I Have a Handful in the November 1999 issue of Teaching Children Mathematics in the Math by the Month department. I made a poster and began identifying Common Core State Standards in Mathematics (CCSSM) that this activity might address. We encourage our teachers to connect at least two Standards for Mathematical Practice and two content standards. For I Have a Handful, I decided to focus on:

Standard for Mathematical Practice:

• Model with Mathematics [SMP 4]
• Use Appropriate Tools Strategically [SMP 5]

Content Standards:

• Kindergarten: Classify objects and count the number of objects in each category [K.MD.3]
• Grades 1, 2, and 3: Represent and interpret data [1.MD.4, 2.MD.10, 3.MD.3]
• Grade 6: Develop understanding of statistical variability [6.SP.1]
• Grade 6: Summarize and describe distribution [6.SP.4, 6.SP.5a]

Next, I identified some questions I might ask to help make students' thinking visible during the activity and aligned them with the CCSSM.

1. Which shape shows up most often and how do you know? [K.MD.3]
2. How could you record your result on a graph? [SMP 4, 1.MD.4, 2.MD.10, 3.MD.3, 6.SP.4]
3. Why did you use this type of graph to represent your results? [SMP 5]
4. Which shape occurred the most? The least? How many more? [SMP 4, 1.MD4, 2.MD.10, 3.MD.3, 6.SP.5a]
5. What would happen if you grabbed another handful? Why? [6.SP.1]

I decided to concentrate on the first three questions, and using the framework from Orchestrating Discussion (5 Practices), I began to anticipate possible student responses (Practice 1). This lead to the monitoring sheet (Practice 2) that is provided below.

On the second page, I tried to arrange the responses in such a way that they represent movement from a concrete approach to an abstract one. In essence, a rubric reflecting various levels of comprehension related to the idea of creating a display of the results from the activity. It was not as evident, to me, how to break up the questions on pages 1 and 3 using this approach. Perhaps seeing people engage with the activity will make these responses easier to arrange.

In the next post, I will share the results of carrying out the activity and address the remaining Practices (Selecting, Sequencing, and Connecting).