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?

Don't you want math to be better for your kids?

That's not the way I learned it! And if it was good enough for me, then it's good enough for my kid! (Along with either: "I was bad at math." or "I was good at math.")

That's how I interpret some of the posts trying to pass themselves off as examples of "bad Common Core math problems" (Google it and take a look at some of the images). Justin Aion has a great post that points out the problem with associating these examples with the Common Core State Standards in Mathematics (CCSSM). However, even if these examples are decoupled from the CCSSM, there's still the sentiment that these new math approaches are flawed.

Take this post, for example. The parent's letter says it all:

From Jeff Severt (some context)

"simplification is valued over complication" writes the Frustrated Parent. But is the parent's approach the simplest way to compute the difference between 4,000,002 and 3,999,999? As math educators, we encourage young mathematicians to build up a variety of computational tools so that they can attack any problem with confidence and phronesis.

Recently, my class explored the thinking inherent in the work of these third grade girls.

From The Big Dinner 

This was a Big Idea on the Multiplication and Division Landscape, Proportional Reasoning, that was new to nearly all of my preservice elementary teachers. Consequently, I followed up with a Think Aloud to reinforce the Big Idea and connected it to the CCSSM 3.OA.B.5.

Afterwards, one of the preservice teachers said, "I've never seen this before. Why?" Why, indeed. 

What did/does the 'A' say to me?

In the previous post, I used the song What Does the Fox Say? to frame a discussion about what grades communicate to various stakeholders in education - in particular students and teachers. It is my view that we do not share a common understanding of what a grade means and this impacts learning. I suggested some of the different ways we interpret an A by modifying the song's lyrics. Readers responded in the comments with what an A grade meant to them as a student and what they hope it says to their students as a teacher. Now is the time for me to share my perspectives about what did/does the A say to me.

When I got an A, I thought that I had pleased my teacher. I distinctly remember having a conversation with a friend who was struggling in school about how I achieved success. I tried to find out what the teacher wanted and then went about meeting that vision. The grade I got would tell me how close I came to giving the teacher what he/she had in mind. It did not matter the subject, the teacher was the all-knowing arbiter of my work. For me, school was not so much about learning as it was mind-reading.

As a new teacher, I simply flipped this perspective. An A meant that the student had done at least 90% of what I expected (i.e. what I would do). To my credit, I did not want my students to read my mind so I made my expectations very clear. I got a lot of student-work that looked just like my work. Instead of mind-readers I was fostering mimics.

Now, I see an A as representing what Joyce and Showers (2002) called Executive Use. The student has demonstrated complete content competency (I was uncomfortable with idea that there might be a 10% gap in a teacher's knowledge) and an ability to analyze under what circumstances the learning could be applied appropriately (phronesis) or how to adapt it to new situations. Granted, because I mostly teach teachers, this standard might be easier to implement now than when I taught middle school math. Still, I have applied a similar idea in a College Algebra course with some success - it was a tough sell.

Basically, I want an A to say to students that they have achieved sustainability in the topic being graded. They can apply what they have learned beyond what we talked about in class, and they can learn more on their own if needed. The teacher (me) has become obsolete.

What do teachers do?

The above Tweet was my take on a comment made at the Michigan Council of Teachers of Mathematics (MCTM) Conference. I agree with the President's point. An unintended consequence of compulsory education in the United States is that everybody thinks they know what it takes to teach but these people have only experienced education from the student-side of the desk. It is important to remember that those interested in becoming a teacher also fall into this category. Teacher preparation needs to make explicit what it means to be a teacher.

I am in the process of redesigning a probability and statistics course for preservice elementary teachers and I want to be sure that the activities in the course reflect the actual work of teachers - especially those aspects that go on behind the scenes. In order to set the intention that they will be doing the work of teaching, I am going to call my students teachers and group them according to grade levels and schools. Furthermore, the course will concentrate on the three areas of the Teaching-Learning Cycle that are often invisible to the casual educational observer: Assessment, Evaluation, and Planning.

Because I just finished reading Hattie's (2011) Visible Learning for Teachers, I want the overall theme of the course to be "Teaching is Learning." Too often, people interested in teaching think it is about telling or controlling or managing or ... because that is what they saw. And while teaching may involve these behaviors, if the teacher is not learning about the content and the learners along the way to inform instructional choices, then the teacher's actions will be haphazard and likely ineffective.

The course is separated into three projects. Each project is worth 30 points toward the teacher's final grade. Engagement Exemplars make up the last 10 points.

In the first project, schools will create 6-8 curricula using the Standards for Mathematical Practices (SMP) and the content standards for Probability and Statistics from the Common Core State Standards (CCSS). This is intended to make it clear that the CCSS do not represent a curriculum but require teachers to create units appropriate for their students. The 6-8 curricula will build on a K-5 curriculum using the SMP and Data and Measurement Standards that we will create together beforehand for practice.

The second project will require each teacher to demonstrate competency in the content associated with the standards in the curriculum. It is important that teachers are fluent with the mathematics they are teaching in order to assess understanding and select appropriate learning trajectories. Sometimes teachers encounter new content during their planning that they have to learn for themselves first. I remember having to teach myself about box-and-whisker plots when the topic showed up on the Eighth-grade Michigan Curriculum Framework. In order to demonstrate competency in the content, my teachers can choose between creating a problem portfolio or taking traditional exams.

For the final project, teaching pairs will conduct formative assessments on middle school students. They will gather data related to the students' understanding in Probability and Statistics and ability in the Standards for Mathematical Practice. In order to determine students' fluency, the teaching pairs will use the evaluation framework from the Teaching-Learning Cycle: What can they do; What are they trying to do; and What comes next?

The goal is that the teachers will leave this course with a better understanding of what teachers do and a set of tools that will empower them to teach effectively regardless of the circumstances they find themselves in during their career. Again, it comes down to phronesis.

So what do you think? Besides drinking large amounts of coffee, have I forgot anything that teachers do?

Why didn't you tell us about that?

I never heard about any of this in my school of ed

Statements like this occasionally pop up in my Twitter stream - usually during something like #edchat. Given my involvement in teacher preparation, it is easy to get defensive about these Tweets. So I try to ignore them. Last week, however, was Twitter Math Camp (TMC) and the following was Tweeted at the end of the sessions: "None of this stuff we've talked about these last three days came up in my education program." Sorry, I couldn't ignore this one.

I am not here to defend the current state of teacher preparation. There are certainly issues that need to be addressed. But introducing the latest popular instructional approaches is not the answer. In fact, chances are your education classes were talking about ideas that were popular at some point in time.

Hattie (2011) suggests that schools of education often put too much emphasis on particular methods of teaching, thereby ignoring what is most important - learning. Reading this reminded me of a time when I made sure all of my preservice teachers were proficient in writing Launch, Explore, Summarize (LES) lessons. What I considered to be one of the best middle school mathematics curriculum, Connected Mathematics Project, used this format and I wanted my students prepared to use it. Then I got an email from a former student who had just got a teaching position in a middle school that used Saxon Math which follows a much more traditional approach. She was at a loss about how to apply the LES format. I was preparing teachers to use a particular method but not necessarily to support learning.

For Hattie, teaching for learning requires educators to gather data on and analyze the effectiveness of the instructional methods being used. Reading this was affirming since after several episodes like the one described above I moved from pushing the LES approach to encouraging a certain stance. I wanted the preservice teachers to see themselves as educational researchers in order to determine what was and wasn't working for their students.

The push-back from some of these future teachers has been interesting. I usually get a few each semester who ask, "Why don't you just tell us the best way to teach math?"

I respond, " I don't know what grade level you're going to teach. I don't know where you're going to teach. I don't know what text you're going to be using. Deciding the best approach to use is dependent on these factors and many more. My goal in this class is to provide you with opportunities to practice using the tools that will support you in making those decisions. I want to help you to develop educational phronesis: practical wisdom that allows you to consider what's currently available to foster learning and what's worth doing under the circumstances."

Does this mean that my students might not be familiar with 3 Acts or foldables or "My favorite no" or some other great ideas the mathtwitterblogosphere comes up with in the coming years? Probably. But if these preservice teachers become educators who attend TMC and critically consider how to adapt what they hear for their students, then I will be happy.

Wanna race?

It depends. I do not agree to challenges without more information. To do so would be, in my opinion, irresponsible. Sort of like using blanket statement to make some point.

Aquacar

For me, it seems to come down to phronesis: what's available, and what's worth doing?

Take, for example, the car to the right. It was built to get you from point A to point B over land or water. It seems to be an effective means of travel, and it could probably beat most regular modes of transportation in a race that involved a combination of land and water. But there are times when it would not be the best choice in a race. What if the race was all on land (or all water)? The best method of transportation depends on the conditions.

The ability to choose from a variety of methods came to mind today as I read The Faulty Logic of the 'Math Wars' (here). In the fourth paragraph, Crary and Wilson write:

The most efficient algorithm for addition, for instance, involves stacking numbers to be added with their place values aligned, successively adding single digits beginning with the ones place column, and “carrying” any extra place values leftward.

If I understand their assertion correctly, then I would like to challenge them to a race. First one to compute the following sum wins. They can use their efficient algorithm and I'll use some other method.

Ready?

Set?

Go!

999,999 + 41,562

Why did you do that?

We just completed another semester at Grand Valley State University, and once again I found myself asking our teachers-in-training some version of "Why did you...?" after many of the observations. I try to make it clear that this is an authentic question not some sort of accusation of wrong doing that demands an accounting. Seriously, I want to know the rationale behind some of their teacher moves. Also, I want them to be mindful of why they make certain decision during their lessons. Consequently, this is what I would call a win-win question.

For example, many student teachers struggle with time management during class, and on their Action Plan they will ask for suggestions about how to be more efficient while conducting a lesson. During the subsequent observation, I often see a typical lesson component. (Stigler and Hiebert were right about there being an often unconscious script associated with teaching math.) The teacher provides some time for individual or small group practice followed by a whole class discussion of the practice items. 

Afterwards, I ask, "Why did you go over all the items? What did you see as you walked around that made you decide that this was necessary?" Again, this is intended to be a serious question, and I sometimes get answers that teach me something about them as teachers and their students as learners. Some of the teachers say something like:

• "I noticed that most of them were struggling on [some aspect of the practice] and decided we needed to look at it as a class." or
• "There were a few different approaches and I wanted the class to see that problems can be solved in multiple ways." or
• "While they got all the practice items correct, I wanted to provide them an opportunity to communicate mathematically. They still struggle with vocabulary and precision and I thought this would be a good time to practice these given that they understood the concepts."

But occasionally, the response I get is, "What do you mean? We always go over the items after their practice. You mean I can skip this.?" The discussion that follows is nearly always goose-bump-inducing as the novice teacher begins to develop phronesis (what could I do here and what's worth doing?).

My goal as a teacher educator is not to develop a bunch of Dave Coffey clones. One is enough. Besides, my approach would not work for all the different teachers I work with over a semester. But if I become the voice in their head that asks after a lesson, "Why did you do that?" - I can live with that.

Is it worth it?

The second episode of Mystery Teacher Theater 2000 came out last week.

My partner in MTT2K, John Golden, did an excellent summary of his perspective on Khan Academy here. In this post, I want to give a bit of background on how we came to use this video and expand on John's point that "at some level this is two guys goofing around to make a point about good use of resources."

I first saw Khan Academy's video, Introduction to matrices, while doing an observation of a student teacher. As a part of the lesson, the student teacher decided to show this nearly twelve minute presentation to a class of precalculus students. Throughout the video, the students paid very little attention to the screen and the student teacher ended up going over the basics again.

The idea of having over 3,000 videos on a variety of subjects available to anyone with an internet connection is appealing to me. If the quality of the videos is suspect, it is less attractive but it really is none of my business. When those videos are assigned to K-12 students to watch either in class or after class, then it is part of my responsibility as a teacher educator to question their use. In this case my question would be, "Is this video worth showing in class or could the time be better used?"

In other words, how is this video better than:

• the teacher providing this information through a more interactive lecture;
• the students reading the textbook section on matrices; or
• the class working on an "archeology project" where they try to discover the  basics of matrices using artifacts strategically "found" by the teacher?

Grant Wiggins talks about the juice needing to be worth the squeeze when it comes to assessments. I would say the same could be applied to other instructional decisions like the use of Khan Academy videos as a part of a school's curriculum. In the case of the student teacher, I would say it was definitely not worth the time it took to watch the video and get the students focused back on the lesson.

Perhaps the problem is that the videos were not intended for whole group instruction. A recent post from a Teach for America staff member explains how he used Khan Academy to help differentiate his instruction.

I could point my most advanced students towards videos instructing them on multiple application of a particular theory while I simultaneously walked students struggling with the same notion through a lesson explaining its fundamental premises. It helped me to be a better teacher who reached more of my students more effectively.

Several promotional videos from Khan Academy offer the same testimonial. Teachers assign Mr. Khan's videos to students to watch while the teachers work with smaller groups. Here are two teachers from Eastside College Prep in East Palo Alto, California talking about using Khan Academy in their classrooms.

If Khan Academy helped these middle school teachers to break out of the mindset of simply giving notes to students, then I really do owe a debt of gratitude to Mr. Khan. However, because valuable school time is being used to watch these videos I must push these teachers to think about what comes next. Repackaging the lecture as a video is not a re-imagining of education nor is it true differentiation of learning.

The way they are using the videos sounds more like addressing a classroom management issue than truly facilitating learning. If these teachers are looking for something productive for other students to do while they work with small groups, then I would encourage them to talk to their colleagues who teach reading and writing and manage small groups regularly without assigning videos. I can think of at least a half-a-dozen things that are more worthwhile to do in class than watching a video like the one I sat through during that precalculus lesson. 

I will write another post about alternative activities (if there's any interest) but this post is getting long and I want to give those teachers who assign Khan Academy videos to their classes a chance to respond to the question, "Is it worth it?" Please post your comments below with the understanding that any off-topic comments will be deleted - I would not want you to waste your time.

How will it work?

I need your help. Due to circumstances beyond our control, the GVSU Department of Mathematics is looking at canceling two courses this fall semester. Teaching Middle Grades Mathematics [MTH 329] is required for undergraduates interested in teaching secondary mathematics (though some inservice teachers take it as part of adding an endorsement in mathematics to their certificate). Secondary Student Issues [MTH 629] is part of a College of Education's Masters of Education program. Because of the importance of these two courses to their respective planned programs, we are considering alternatives to canceling them.

One option is to combine the two courses in some way. This is of particular interest to me because of past successes with preservice and inservice teachers collaborating in MTH 329. As I said, we sometimes get inservice teachers taking this course for an endorsement, and I always try to include their perspective when discussing the realities of teaching and learning. I also connected 329 students with inservice teachers two years ago when a scheduling conflict meant that I needed to be teaching and conducting professional development at the same time. Participants report that the combined effect of preservice teachers' enthusiasm and inservice teachers' experience has been beneficial. 

All I need to do is come up with a proposal for how the combined course might work. My colleague, John Golden, and I sat down this morning to develop a draft, but I recognize that our plan would benefit from your feedback. Here's the idea:

Essentially, the content addressed will remain unchanged for MTH 329. The undergraduates in this course learn what it means to do, learn, and teach mathematics in the middle grades. In their course portfolio, they demonstrate their fluency of middle school-level mathematical content, their competencies in teaching and learning middle grade mathematics, and their ability to engage in their own learning.

Graduate students enrolled in MTH 629 will continue to focus on issues in teaching and learning secondary mathematics, but this will extend to mentoring the preservice teachers in mathematical pedagogy. The mentoring will involve supporting the undergraduates in the assessment and analysis of middle grade learners’ mathematical thinking and the design and implementation of mathematics micro-lessons. The graduate students’ portfolio will document their mentoring efforts, their results from an action research project, and their ability to engage in their own learning.

MTH 329 will meet from 6 to 7:50 on Tuesdays and Thursdays. MTH 629 will meet from 6 to 8:50 on Tuesdays. During the 6 to 7:50 overlap on Tuesdays, the class time will focus on developing a taken-as-shared understanding of pedagogical concepts through demonstrations, classroom dialogues, and collaborations. From 8 to 8:50 on Tuesdays, MTH 629 students will concentrate on aspects of effective mentoring and conducting action research that focuses on the artifacts of teaching (lessons and assessments). On Thursdays, the MTH 329 students will focus on the middle grades mathematical content typically addressed in this course.

It is my hope that this structure will help both groups to recognize that they can contribute to the positive development of the teaching profession. I want them to understand that they can improve teaching without having to wait for some outside force to tell them what that improvement would entail. In other words, I want to provide an experience that provides them with phronesis.

So what do you think? I really value your input in designing this combined course. Thank you in advance for your support.

When is it okay to use a calculator?

All too often, I run into teachers (both preservice and inservice) lamenting that kids are using calculators to compute something simple, like 6 x 7. These teachers express their frustration by threatening to not let kids use any calculators until the kids prove that they know their facts. And there will be no calculators for any simple computations. I understand this thinking but I am not sure it will achieve the desired result - wise calculator use (i.e. phronesis).

Problem is, if teachers are the ones deciding when their students can or cannot use calculators, then students are not able to practice this critical-thinking skill for themselves. Consequently, whenever a calculator is available in the future it can be used because that was what the students learned in school. As an alternative to controlling calculator use, I suggest a couple of activities that can help students to decide for themselves when it is appropriate to reach for the calculator.

The first activity comes from the article, John Henry - The Steel Driving Man. This article suggests several different experiments involving routines that can be completed by human or mechanical effort (eg: sharpening pencils). Students are asked to predict which method will take longer, gather data, and compare the results using box-plots. 

The experiment I am most interested in involves computing multiplication facts with and without a calculator. I give pairs of students four worksheets like the one shown below.

Each of the four worksheets is different. As one student completes the worksheet, the other one uses a stop watch to time the effort (errors add an extra five seconds to total time). This is repeated until each student completes two sheets - using the calculator for one but not the other. 

Classroom data are gathered and typically show that the students complete the worksheets quicker without a calculator. Discussing the results can provide students with an opportunity to reflect on whether or not it is efficient to use a calculator for basic facts. In the few cases when it is faster to use a calculator, the issue is usually that the student has a lot of wrong answers when computing without a calculator. Teachers must decide an appropriate course of action in these instances.

A second activity that I use to develop students' wise use of calculators involves a worksheet of multi-digit multiplication items. Instead of assigning the entire worksheet, I ask students to pick four items to compute without a calculator, four items to estimate, and four items to use a calculator on. The students are also expected to explain why they selected the approach to use with each item.

Calculator phronesis, the wise use of calculators, requires opportunities for students to experience activities that involve metacognitive aspects. We teachers will not always be there to guide students' choices, but this is not to suggest that we do not have a responsibility to help students to develop this ability. It is my hope that through these classroom experiences, students will be able to ask and answer for themselves the question, "When is it okay to use a calculator?"

Which tool makes sense?

Confession of a control-freak: I want lessons to run smoothly (exactly the way I envision them). I have written about this issue before and my efforts to give learners more control in the classroom. If the goal of my teaching is learners who possess phronesis, then I need to provide them with ample opportunities to practice making and evaluating choices. In this post, I give another example of my efforts to turn over more responsibility to my learners.

During a recent lesson, my preservice teachers were relearning what it means to add fractions and the role manipulatives can play in supporting understanding. In the past, I would have: 1) put out one manipulative (perhaps the pattern blocks); 2) explained the rules of using the manipulative (2 yellow hexagons represent a whole); and 3) provided them plenty of practice in using the manipulative to represent fraction addition. Then, I would have them put away the pattern blocks and grab some fraction circles and go through the process again. As I said, controlling. I came to realize that my management of the tools and the rules was disempowering my learners. This time I simply put out a variety of manipulatives and asked them to explore.

Workshop

Schema Activation: All learners need time to explore the tools at their disposal. Please take five minutes to play with any of the manipulatives at your table.

I find that it is important to provide learners, regardless of their age an opportunity to play with manipulatives (they are going to anyways, I might as well embrace it). The learners spend the time building, organizing, and comparing the shapes. At the end of five minutes, I ask them to put the manipulatives back into their separate containers. This provides a break that I find helps learners to shift their vision of the manipulatives from toys to tools. I make sure to make this point explicitly.

Focus: Each of you is now going to pick a manipulative and consider what happens when you add two unlike shapes. For example, If I combine the blue rhombus and the red trapezoid from the pattern blocks, then what would I get? How did I get it? Why does it work? When does it work? And what if I combine other shapes, will my thinking still hold or do I need to adjust it?

This is usually where the manipulatives' rules are shared. Instead, I want to provide a framework of questions for them to keep in mind as they consider combining shapes. Not having done this workshop before, I am unsure this will provide them with enough structure. If there is any uncertainty, I am prepared to model how my wife had thought of combining the two pattern blocks by focusing on their side lengths: 4 units (rhombus) plus 5 units (trapezoid). There is no need, however, as the learners get right to work.

Activity: Learners work in groups on the task.

This is an opportunity for me to conduct some formative assessment. As I walk around, I try to focus on asking questions that check for understanding - any teaching/leading questions can wait until later. Based on the data I gather, I organize the remaining part of the lesson.

One group is working together with a single manipulative. They are discussing different ways to view the result. Another group has split the manipulatives between them and are seemingly working separately. But every so often, one learner shares with the others her conjectures and the others provide their feedback. From these observations, I ask two learners if they are willing to share their thinking with the rest of the class. They agree.

Reflection: Mathematician's Chair - learners share their thinking and any struggles with the rest of the class in order to expose their work to the larger learning community.

The first person I ask to share was working with her group to combine two shapes from the fraction circles. They had chosen to combine the two pieces shown at at the right. These seem pretty basic but the group had gotten into an interesting discussion about what represented the whole. The gist of the discussion, which she recounts for the class, was, "If we let the white circle be the whole, then the result is one-and-a-half white circles. If the orange half-circle is the whole, then the result is three orange half-circles. It all depends on what you denote as the whole."

Next, a learner shares what she found using the pattern blocks. It goes something like this. "I found it easiest to break down each shape to the smallest shape. So the blue rhombus is two green triangles and the red trapezoid is three green triangles. That means together they are five green triangles."

At this point, another member of this learner's group interjects. He had taken the idea and tried to apply it to the Cuisenaire Rods. He essentially points out, "It really comes down to names. I can break down any of the sticks to the smallest one, the white block, and then combine them because they have the same name."

Teacher's Reflection on the Workshop

I am pleased with the workshop. The learners were able to recognize the importance of identifying a whole and the need to find a common name (denominator) in order to combine fractional pieces. These are big ideas that are often reduced to rules without reason for students being instructed about fraction addition.

There are a few things I will do differently next time. First, I need to make even more explicit my decision to give them choice about which manipulative they would work with. Much of the work of teaching is invisible and I worry this was the case in this workshop. Next, I want to have a larger discussion about the wise use of manipulatives in math class. This might be a conversation around Deborah Ball's article, Magical Hope. Finally, I want to further expand on the idea raised during the reflection regarding naming. Using the Cuisenaire Rods, we could explore different names such as those shown below.

While this lesson did not go perfectly (they rarely do), I see real progress in my learners and myself. This experience will provide a pivotal experience that we can return to in subsequent lessons as we talk about appropriate use of educational resources. It also provides data that I can use in future classes. It is impossible to say how learners will respond, but now I have examples of how learners have responded. If necessary, I can always share these examples with as models of how others thought about the task. Often, one of my most effective teaching moves is to ask, "Would you like to see how other learners thought of this problem?"