Thursday, December 4, 2014

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?

Friday, November 28, 2014

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.
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!"

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.

Wednesday, November 26, 2014

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?

Tuesday, November 25, 2014

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?

Thursday, November 13, 2014

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. 

Sunday, November 9, 2014

What is this craziness?

From 1075 KZL Facebook Page
If we are to believe memes like this, the math currently being taught in schools has never been so meaningless. What exactly is this craziness? According to a recent NBC News story, it involves using number lines.

I understand that this approach can be confusing to adults who were taught an algorithm without really understanding why it works. Unfortunately, many of these adults have forgotten how memorizing rules without reasons made them feel as a kid.

As students, we wanted our mathematical efforts to make sense. But instead, we got little ditties like, "Mine is not to question why, just invert and multiply." So we made up that doing math simply meant following certain rules given by some authority.

It is reasonable that adults who grew up with this view of mathematics might be confused by instructional approaches meant to foster the development of mathematical understanding rather than rote rules. However, my primary concern is the confusion of the kids not the adults. To do otherwise is not only crazy, it is insane.


Friday, October 31, 2014

Will it fit?

I used this lesson in MTH 221 (Mathematics for Elementary Teachers) to address Common Core State Standard 7.G.B.4. It seems to have a lot of potential, but there are still some elements that I think need to be tightened up. These are written in red - along with some other thoughts. I would appreciate any feedback on how to improve this lesson.

[Schema Activation]
How many of you know your fitted hat size? For example, I wear a seven-and-seven-eighths. Today, you are going to find your hat size and what that number means. 
I like the DC cap for a couple of reasons. First, obviously, DC are my initials. Second, it foreshadows the circumference and diameter relationship we will explore in the lesson.

In the past, I have had students measure a bunch of circles to find the ratio between circumference and diameter. It has been a struggle to make it an interesting lesson. This connection to something personal (hat size) seemed like it might be an improvement.

According to, there are a couple of ways to determine your hat size.

We will use both a flexible tape measure and their printable ruler in order to...

... consider the following questions:
  • How are hat sizes and head-measures related? (In other words, if we didn't have access to their table, can we determine a hat size given a head-measure?)
  • We are supposed to be working on CCSSM 7.G.B.4. Is this connected in some way to circles? Too obvious?
  • Why don't hat makers just use the head-measurement as the size?
Measure your head using both the flexible tape measure and the printable ruler.

Feel free to wear the printable ruler as a stylish headband as you work. Optional

Place both your head-measure and hat size on a sticky note and place it at the proper coordinates on our graph.

What does our graph show? Is there a relationship? If so, what do you predict the relationship to be? (If we input 22 inches for head-measurement, what hat size is the output? What if we input C inches?)

Here are a couple of tables for hat sizes from Let's use them to see if we can determine the input-output rule they are using to find hat size from head-measurement. 

There are other hat size tables (for example), but I like that this one makes 22 inches a hat size of 7 because 22/7 is often used as an approximation of pi.


What did you find? What is the rule?

Des found the following: y = 0.333x - 0.333. Does it work for our table? If it's correct then what hat size does a person need for a head-measurement of 24?

Okay, I wanted to play with the new Desmos linear regression feature - sue me. Is it a problem that this line doesn't go through the origin? That the slope actually represent an approximation of 1/pi? During the lesson, it seemed like this portion required a lot of scaffolding.

So what does the hat size mean? Let's take our headband and place it on the table. Notice that it is nearly the same shape as a circle. Now measure the distance (diameter) across your headband (circle).

In my case, if I measure what is approximately the diameter of my headband, I find the length is close to my hat size (seven-and-seven-eighths). How about you? What does that suggest our hat size means?

The students were most impressed by this portion of the reflection. They liked that the hat size number was not some arbitrary value - that it was actually connected to something mathematical. I looked for some history of hat sizes to explain why this value is used instead of circumference, but Google failed me.

Now what fitted hat size should I buy from if my head measure is 24 inches? Seven-and-two-thirds doesn't look like it's an option.

In an earlier unit, students struggled with the idea of independent and dependent variables and creating graphs that accurately represent a real-life situation. Because a hat maker does not make all possible diameters, we decided it didn't make sense to connect the dots. Instead, we came up with the graph shown above.

One of the reasons I like this activity is because it does connect with so many other standards, like 6.EE.C.9 and 6.SP.B.4. What do you think? Does this lesson have merit - is it worth saving? If so, how? Please add your thoughts in the comments.

Updated: As much as I loathe Pi Day, this piece on Stormy Kromers (hats made in the Upper Peninsula of Michigan) might make a nice connection.