Being Imaginative in Problem Solving

Sometimes my ignorance with respect to Twitter just floors me. Today alone I made two huge faux pas (is that plural?) with two people that I really respect and just made a fool out of myself – typos, misinterpretations, and misunderstandings abound in my tweets. But I have to say I press on – because I have found so much that informs my teaching and learning that I can handle the fool-making and embarrassment.

So here’s one thing that I did a few weeks ago – someone tweeted about this great article that I, of course, then went and read, took a picture of the great diagram in the article – but forgot to “like” or “favorite” or whatever it’s called now. So now I can’t give credit to whomever brought me to this wonderful enlightenment about which I will now write. So if you are reading, sir/madam, who tweeted this article, please forgive me.

I read this short blogpost entitled “Brennan’s Hierarchy of Imagination” and immediately made the connection to PBL.

The author, John Maeda, wrote about a conversation he had with Patti Brennan about Maslow’s famous Hierarchy of Needs of students in their learning. These two were talking about the fact that teaching creativity is really hard and Patti Brennan was thinking that it was a bit easier to teach someone to use their imagination. She was talking about this in the context of the health care field – trying to help people empower themselves to help themselves.

Of course, the first thing I thought of when I saw this pyramid was problem solving. I thought this was brilliant! The foundational, lower level of reflex or instinct is analogous to doing problems that you have seen before. Students love this instinct – the idea that if you can do a problem that someone has shown you how to do, that you are problem solving, – it gives them a reaction of completing something, some kind of satisfaction.

The next level which is appropriately called problem-solving is when you are actually solving a problem that has occurred but is constrained and you are executing skills. I love this. It still takes some talent and analysis, but you are still just reacting to a given situation. Perhaps putting together two different methods that someone showed you and seeing what happens?

The third level that she calls creativity is the first step to unique ideas and methods. The first attempt at doing something in a different way – saying what if we tried this? Has anyone every done this before? Why not? It’s still bound by the reality that we experience, but seeks to move past the knowledge that we have.

And finally imagination is what I think happens when my 4-year-old niece explains how the clouds got up in the sky because the stars moved so fast pieces came off and clumped together, or when a student can’t figure out if there are more real numbers between 0 and 1 as there are integers and they try to describe the size of those sets to me with things they imagine.

If I’m lucky, I’d say some of the kids in my classes get to creativity – in fact I think that’s been my goal as of late. To get them at least to experience it with some projects, assignments and good problems. To get them to realize that mathematics is more than just a reflex or even just reiterating a process.

Hopefully, this coming year the number of kids who will get to that third level will increase, but who knows? I do love this framework because at least I know I don’t want them stuck on that bottom level for level for long.

Journals: Paper vs Digital: The Pros and Cons

I was totally honored the other day when I saw some tweets from TMC16 from @0mod3 and @Borschtwithanna


And yes it’s true, I’ve been writing and practicing the use of metacognitive journaling for very long time – probably since 1996 ever since I read Joan Countryman’s book about mathematical journaling and heard about it in many workshops that summer.  I wrote a rubric (make sure you scroll to the 3rd page) while I was at the Klingenstein Summer Institute for New Teachers (that’s how long ago it was) and since then I’ve been refining that rubric based on feedback from students and teachers. A few years ago, I finally refined a document called How to Keep a Journal for Math Class to a degree that I really like it now.  However, please know that lots of math teachers do journaling differently and without the metacognitive twist. I do believe that metacognitive writing is essential to the PBL classroom (read more here)

So this morning, I was asked this question on twitter


Which is something that many people often ask so I thought I’d respond with a more in-depth answer.

Here are the pros, I’ve found over the years of having students journal digitally:

Speed/complexity: Students are used to typing, using spell-check, inserting pictures, graphics and naturally including documents, links and thinking in the complex way that digital media allows them to.  It allows their journal to be more rich in content and sometimes connect problems to each other if their journal is say on a google doc that can connect to other html docs.  If they create, for example, iBooks or Explain Everything videos, there is even a lot more richness that can be embedded in the file as well – their creativity is endless.

Grading/Feedback: I found grading in Notability or on Google docs or some other digital platform really nice that allowed you to add comments with a click or audio extremely easy and quick.  I did not receive feedback from the students very often about how the feedback helped them though.  If you use an LMS like Canvas that integrates a rubric or integrates connection to Google it’s even nicer because you can have those grades go right from your assignment book to your gradebook.

I love having kids use digital platforms for writing/creating in mathematics when it is for a project or big problem that I want them to include many pieces of evidence, graphs, geogebra files and put it together nicely in a presentation or portfolio.  Not necessarily for their biweekly journals. Some guys who make use of digital journals in interesting ways are @GibsonEdu and @FrasiermathPBL at the Khabele School in Austin TX.

Here are the cons, in my mind of using digital journals: (which might be the “pros” of paper journals) – which is the side I have come down on.

the “real” writing factor: there is some research about the actual physical process of writing and the time it takes for kids to process their thoughts.  I do believe that when i want kids to be metacognitive about their learning and also want them to be thoughtful and take the time think about their initial error, think about what happened in class discussion to clear up their misunderstanding and also then what new understanding they came to.  That’s a lot of thinking. So I want them to take the time to write all that down.  Sometimes typing (like what I’m doing right now!) is a fast process and I’m not sure I do my best writing this way.

practice in hand-writing problem solving: this is re-enacting doing homework and sitting for assessments (in my class at least) and I want them to do this more regularly.  If in your class kids take assessments digitally or do homework nightly digitally then maybe they should do their journal digitally as well. This also give me practice in reading their handwriting, getting to hear their voice through their handwriting and seeing what it looks like on a regular basis.  In a time crunch on an assessment it honestly helps me know what they are thinking.

Conversational Feedback: I feel that when I hand write my feedback to them I can draw a smilely face or arrows or circle something that I want to emphasize more easily than when it is on something digitally (this is also true in a digital ink program – so that is something to consider, like Notability for example). I give feedback (see some journal examples on my blog) that is very specific about their writing and want the to improve not only in the math aspect of their writing but in how they are looking at their learning.  I want them to respond and I want to respond in the hope that we are starting a mathematical conversation about the problem.  I have received more questions about the feedback in the paper journals (like “what did you mean by this?”) than on the electronic feedback – not sure why.

Portability: I find that small composition graph paper notebook is extremely portable and easy for me to carry home to grade.  The students bring them to their assessments and there is nothing else in the notebook (no homework at all and no access to the internet) so I am not worried about academic honestly.

There are probably more but this is it in a nutshell – please add your comments below or tweet me to let me know your thoughts!


Documents for CwiC Sessions at Anja Greer MST Conference 2016

Instead of passing out photocopies, I tried to think of a way that participants could access the “hand-outs” virtually while attending a session.  What I’ve done in the past a conferences is have them just access them on their tablet devices.  You can also go and access copies on the Conference Server if you do not have a device with you (you should be able to use your phone too).

These link to This is an Adobe Acrobat Documentpdf documents that I will refer to in the presentation about “Assessment in PBL”

Information on Spring Term Project and Spring Term Project Varignon 2015 (this document includes rubric)
Keeping a Journal for Math Class
Revised Problem Set Grading Rubric new
Rubric for Sliceform project and Sliceforms Information Packet

Page at my website with Rubrics and other guides for Assessment

Adventures in Feedback Assessment

On an assessment students did for me today I gave this question:

An aging father left a triangular plot of land to his two children. When the children saw how the land was to be divided in two parts (Triangle ADC and Triangle BDC), one child felt that the division of the land was not fair, while the other was fine with it. What do you think and why? Support your justification with mathematical evidence.

 So this student had a hard time with this question. Since there was no height given and the bases were different, she was unable to think about how to compare the areas. She was however able to say that it would be a fair split if the areas were the same. So since I am doing this work this year with giving feedback first and then grades (see past blogpost “Why teachers don’t give feedback before grades and why they should”) I wrote this feedback on the problem set: 
 I am trying to get her to remember a problem we did in class where there was a similar problem we did with an acute triangle and obtuse triangle that shared the same height:

The area of the shaded triangle is 15. Find the area of the unshaded triangle.

This idea of where the height of obtuse triangles are is a really tough one for some geometry students. But more than that the idea of sharing a height and what effect that has on the area is also difficult.

We will see tomorrow if this student is able to take my feedback and see what whether the division of the land is fair.

By the way, here’s a response that another student had:

Just in case you can’t read it:

“Because the height is the same, it’s the ratio of the bases that would determine which child would get the most land. I think the division of land was not fair, because the heights are the same so therefore the bases are determining the area of the plot. If x=5 then child one would get A=20, child 2 would get 12.5 and that makes the original plot of land 37.5. This means child 2 has a third of the land (12.5:25) (part:part) and half of child 1’s) Even without x=5, the child 2 would only get a third of the land.”

We’ll see what happens!

Can you undo an adolescent’s fixed mindset?

Yes, it is this time of year where I have to stop and wonder – what the heck am I doing wrong? Is it me?  Is it the kids? Is it the combination of us? In the spring, many of the kids are breezing through and finding ways to problem solve and have gotten really comfortable with being uncomfortable in doing their nightly struggle – they’ve learned to trust that when we get together the next day, their questions will get answered and all will come together, if not that day, then the next.

This year is somewhat more frustrating for me and I can’t figure out why.  I feel as if the students are still attempting to get everything right every night.  It’s as if they created habits that I did not see somewhere along the way.  Reading the beginning of Andrew Gael’s blogpost on Productive Struggle  made me realize this was true and I’m more frustrated than ever now.  I’ve noticed that the conversations that I am “facilitating” are actually either one student talking about their ideas (basically the kid who thought they got it right) and everyone listening intently checking if they agree with him/her or everyone remaining silent until the one or two kids who are willing to take the risk speak up and take the risk to see if they are right.  I’m not quite sure what this is about.

In prior years, there have been kids that really felt much more comfortable with attempting something and being wrong.  I am really wondering what I did differently this year.  There is much more of a feeling of holding back – many more caveats of “I don’t think this is right…” before someone puts their ideas on the board (even though I repeatedly stress that that is not important.)

I have in the past few years become very disillusioned with the idea that high school students are capable of undoing 12-14 years of fixed mindset.  I think I tweeted about this last year sometime when, after a conversation and exercise about Fixed vs. Growth Mindset a student said to me “Is this supposed to make us feel bad?”  I was in shock.  I couldn’t figure out what I had done to make him feel bad at all as I had done just what Carol Dweck suggests and presented the two mindsets as a continuum – a journey of learning about yourself and how you learn best.  Some of the kids saw it as a good tool to know about yourself, but many of them saw it as just one more thing they had to “overcome” in order to get in to a good college or to be the “best they can be” – because you know, if you have a fixed mindset, that’s not the “best you can be” – you have to change that too now.  Oh god, what have I done?

So, maybe there’s a little part of me that feels bad for them and truly understands the fear of being wrong. My goals are to prepare them for the thinking, for problem solving in life and their immediate goals are getting good grades, doing the best they can right now to get into a good college, etc.  Sometimes these goals are definitely at odds and it’s really tough to compete with the immediacy of what they perceive as success for them and those people they want so much to make proud. And as always when there are two parties who have goals that are at odds – there is ultimately conflict.  And the battle continues.

Getting Kids to Drive the Learning

It doesn’t always work this way, but it would be awesome if it did.  When PBL is perfect or ideal, the students are the ones who make the natural connections or at least see the need or motivation for the problems that we are doing.  Yeah, some of them are just really interesting problems and the get pulled in by their own curiosity, but as all math teachers know, we have a responsibility to make sure that students learn a certain amount of topics, it is quite that simple.  If students from my geometry class are going into an algebra II class with trigonometry the next year where their teacher will expect them to know certain topics, I better do my job and make sure they have learned it.

So how do I, as a PBL teacher, foster the values for the problem-based learning that I have while at the same being true to the curriculum that I know I have a responsibility to?  This is probably one of the biggest dilemmas I face on a daily basis.  Where’s the balance between the time that I can spend allowing the students to struggle, explore, enjoy, move through difficulty, etc. – all that stuff that I know is good for them – while at the same being sure that that darn “coverage” is also happening?

So here’s a little story – I have a colleague sitting in on my classes just to see how I teach – because he is interested in creating an atmosphere like I have in my classes in his.  We have just introduced and worked on problems relating to the tangent function in right triangle trigonometry in the past week and now it was time to introduce inverse tangent.  I do this with a problem from our curriculum that hopefully allows students to realize that the tangent function only is useful when you know the angle.FullSizeRender (3)

So as students realize they can’t get the angle from their calculator nor can they get it exactly from the measurement on their protractor (students had values ranging from 35 to 38 degrees when we compared), one of the students in my class says, “Ms. Schettino, wouldn’t it be great if there was a way to undo the tangent?” and the other kids are kind of interested in what she said. She continues, “Yeah, like if the calculator could just give us the angle if we put in the slope.  That’s what we want.”  I stood there in amazement because that was exactly what I wanted someone to crave or see the need for.  It was one of those “holy crap, this is working” moments where you can see that the kids are taking over the learning.  I turned to the kids and just said, “yeah, that would be awesome, wouldn’t it?  Why don’t you keep working on the next problem?” and that had them try to figure out what the inverse tangent button did on their calculator.  They ended up pressing this magical button and taking inverse tangent of 0.75 (without telling them why they were using 0.75 from the previous problem) to see if they could recognize the connection between what they had just done and what they were doing.

At the end of the class, the colleague who was observing came up to me and said, “How did you do that?” and I said, “What do you mean?” and he said, “How did you get the kids to want to learn about inverse tangent? I mean they just fell right into the thing you wanted them to learn about.  That was crazy.”  I really had to think about that.  I didn’t feel like I did anything honestly, the kids did it all.  I mean what made them all of a sudden care about getting the angle?  Why were they invested?   It doesn’t always happen in my classroom that’s for sure.  This is not a perfect science – there’s no recipe for it to work – take a great curriculum, interested kids, an open, respectful learning environment and mix well?

I do think however that a huge part of it is the culture that has been created throughout the year and the investment that they have made in their ownership and authorship in their own learning. We have valued their ideas so much that they have come to realize that it is their ideas and not mine that can end up driving the learning – and yes, I do end up feeling a little guilty because I do have a plan.  I do have something that I want them to learn, but somehow have created enough interest, excitement and curiosity that they feel like they did it.  It is pretty crazy.

Considering Inclusion in PBL

It’s always refreshing when someone can put into words so eloquently what you have been thinking inside your head and believing for so long.  That’s what Darryl Yong did in his recent blogpost entitled Explanatory Power of the Hierarchy of Student Needs.  I feel like while I was reading that blogpost I was reading everything that I had been thinking for so long but had been unable to articulate (probably because of being a full time secondary teacher, living in a dorm with 16 teenage boys, being a mother of two teenagers of my own and all the other things I’m doing, I guess I just didn’t have the time, but no excuses).  Darryl had already been my “inclusive math idol” from a previous post he wrote about radical inclusivity in the math classroom, but this one really spoke to a specific framework for inclusion in the classroom and how in math it is necessary.


In my dissertation research, I took this idea from the perspective of adolescent girls (which, as I think towards further research could perhaps be generalized to many marginalized groups in mathematics education) and how they may feel excluded in the math classroom.  These girls were in a PBL classroom that was being taught with a relational pedagogy which focuses on the many types of relationships in the classroom (relationship between ideas, people, concepts, etc.)  – I did not look at it from the perspective of Maslow’s Hierarchy of Student Needs and this is really a great tool.

Interestingly,  I came up with many of the same results. My RPBL framework includes the following (full article in press):

  1. Connected Curriculum– a curriculum with scaffolded problems that are decompartmentalized such that students can appreciate the connected nature of mathematics
  2. Ownership of Knowledge – encouragement of individual and group ownership by use of journals, student presentation, teacher wait time, revoicing and other discourse moves
  3. Justification not Prescription– focus on the “why” in solutions, foster inquiry with interesting questions, value curiosity, assess creativity
  4. Shared Authority – dissolution of authoritarian hierarchy with deliberate discourse moves to improve equity, send message of valuing risk-taking and all students’ ideas

These four main tenets were what came out of the girls’ stories.  Sure many classrooms have one or two of these ideas.  Many teachers try to do these in student-centered or inquiry-based classrooms.  But it was the combination of all four that made them feel safe enough and valued enough to actually enjoy learning mathematics and that their voice was heard. These four are just a mere outline and there is so much more to go into detail about like the types of assessment (like Darryl was talking about in his post and have lots of blogposts about) the ways in which you have students work and speak to each other – how do you get them to share that authority when they want to work on a problem together or when one kid thinks they are always right?

The most important thing to remember in PBL is that if we do not consider inclusion in PBL then honestly, there is little benefit in it over a traditional classroom, in my view. The roles of inequity in our society can easily be perpetuated in the PBL classroom and without deliberate thought given to discussion and encouragement given to student voice and agency, students without the practice will not know what to do.  If we do consider inclusion in the PBL classroom, it opens up a wondrous world of mathematical learning with the freedom of creativity that many students have not experienced before and could truly change the way they view themselves and math in general.

Revisiting Journals: Getting Kids to Look Back

I have been using metacognitive journaling in my PBL classroom since 1995.  I first learned about it the Summer Klingenstein Institute when I was a third year teacher and just fell in love with it.  At that time, the colleagues at my school thought I was crazy trying to make kids write in my classes – it was just “something else for them to do” and didn’t really help them learn but I did more reading on it and there was clearly more and  more research as time went on that showed that writing-to-learn programs especially those that prompt for metacognitive skills really do help in learning mathematics (see my metacognitive journaling link under the Research tab for more info and sample journal entries).

Every once in a while a student will write a journal entry that I think is so thoughtful that I will write about it like this one a few years ago that just impressed me with his insight into his learning process of a particular problem. But other times kids write about their understanding of their learning overall like one I’ll write about today and I am also blown away.

Here’s a student I’ll call Meaghan reflecting on a problem that she found challenging for her.  Really, it doesn’t matter which problem it was or what topic it was, just the fact that she had a hard time with it at first, right?  The most important part was that after she wrote about how to do it correctly, she then took the time to write this: (in case you can’t read her handwriting, I will rewrite it below).


Part of Meaghan’s Journal Entry

“This problem was a challenge for me.  When I saw the question, it didn’t look that difficult but once I was trying to solve by [sic my] brain wasn’t thinking on the right track, and it was trying to use prior knowledge that was irrelevant in this case.  I wasn’t making connections to the properties of triangles that I had recently learned.”

Why is this realization so important for Meaghan?  Polya’s Fourth Principle of Problem Solving is “Look Back” – why is this fourth principle so important?  In my mind, this is where all the learning happens.  The three other principles are very clear

  1. Understand the problem
  2. Device a plan
  3. Carry out the plan

These three are all very basic – if they work, right?  But most of the time they don’t work for kids.  It’s the fourth step that we know is the most important – it’s where the critical thinking and analysis takes place.  If this part isn’t taken seriously and the right steps within it are not taken nothing happens, no moving forward, no growth.

So what did Meaghan do?  She realized that she had not made a connection between the triangle properties that we had just learned and how it applied to this problem.  She had not use the correct prior knowledge.  She  just created more openings to other knowledge that she knows- and I know what you’re thinking.  Does this mean that next time she will use the correct prior knowledge in another problem?  From my experience with kids, no, it does not.  But honestly, what I have seen is that the more they realize that there are more possibilities and also that the option of just saying “I don’t get it” or “I can’t do this” is unlikely, the more they will keep trying.

So what did Meaghan do? By just being asked to write a reflection about one problem (every two weeks) she has reinforced her own potential in problem solving on HER OWN.  That she may, in the future, weed out the irrelevant prior knowledge and possibly see the connections to the relevant prior knowledge, with more practice.  I think it’s made her feel just a little bit more confident – and they said it was just “something else for them to do.”

Everything Old is New Again…(or why teaching with PBL is so great)

So I heard that what everyone is saying about the new Star Wars Movie, The Force Awakens, is that “Everything Old is New Again” – go ahead google it, there are at least 5 or 6 blog posts or articles about how “BB-8 is the new R2D2” or “Jakku is the new Tattoine” or whatever.  I actually don’t have a problem with J.J. Abrams reusing old themes, character tropes or storylines because I think that really great stories are timeless and have meaning and lessons that surpass the movie that you are watching.  I still thought it was awesome.

This concept of everything old is new again really hit home to me today in my first period class when I was having the students do a classic problem that I probably first did in 1996 while I was under the tutelage of my own Yoda, Rick Parris (who I think wrote the problem, but if someone reading this knows differently, please let me know).  The problem goes like this:

Pat and Chris were out in their rowboat one day and Chris spied a water lily.  Knowing that Pat liked a mathematical challenge, Chris announced that, with the help of the plant, it was possible to calculate the depth of the water under the boat.  When pulled taut, directly over its root, the top of the plant was originally 10 inches above the water surface.  While Pat held the top of the plat, which remained rooted to the lake bottom, Chris gently rowed the boat five feet.  This forced Pat’s hand to the water surface.  Use this information to calculate the depth of the water.

What I usually do is have students get into groups and put them at the board and just let them go at it.  Today was no exception – the first day back from winter break and they were tired and not really into it.  At first they didn’t really know what to draw, how to go about making a diagram but slowly and surely they came up with some good pictures. Some of the common initial errors is not adjusting the units or mislabeling the lengths.  However, one of the toughest things for students to see eventually is that the length of the root is the depth of the water (let’s call it x) plus the ten inches outside of the water’s surface.  Most students end up solving this problem with the Pythagorean Theorem – I’ve been seeing it for almost 20 years done this way.  Although I never tire of the excitement they get in their eyes when they realize that the hypotenuse is x+10 and the leg is x.

However, since everything old is new again, today I had a student who actually is usually a rather quiet kid in class, not confused, just quiet, but in a group of three students he had put his diagram on a coordinate plane instead of just drawing a diagram like everyone else did.  This intrigued me.  He initially wrote an equation on the board like so:

y= 1/6 (x – 0)+10

and I came over and asked him about it.  He was telling me that he was trying to write the equation of one of the sides of the triangle and then I asked him how that was going to help to find the depth of the water.  He thought about that for a while and looked at his partners. They didn’t seem to have any ideas for him or were actually following why he was writing equations at all.  He immediately said something like, “Wait, I have another idea.” and proceeded to talk to his group about this:

Jacksons solution to Pat and Chris

Jackson’s Solution to the Pat & Chris Problem

He had realized from his diagram that the two sides of the triangle would be equal and that if we wrote the equation of the perpendicular bisector of the base of the isosceles triangle and found its y-intercept he would find the depth of the water.  He proceeded to find the midpoint of the base, then the slope of the base, took the opposite reciprocal and then evaluated the line at x=0 to find the y-intercept.  I was pretty impressed – I had never seen a student take this perspective on this problem before.

This made my whole day – I was really dreading going back to work after vacation and honestly, first period was the best class of the day when this wonderful, new method was shown to me and this great experience of this student’s persistence refreshed my hope and interest in this problem.  Perpendicular bisectors are the new Pythagorean Theorem!

Someday I’ll get this assessment thing right… (Part 2 of giving feedback before grades)

So, all assessments are back to the students, tears have been dried and we are now onto our next problem set (what we are calling these assessments).  What we’ve learned is that the rubric allowed us to easily see when a student had good conceptual understanding but perhaps lower skill levels (what we are used to calling “careless mistakes” or worse). We could also quickly see which problems many students had issue with once we compared the rubrics because, for example, problem number 6 was showing up quite often in the 1 row of the conceptual column.  This information was really valuable to us.  However, one thing we didn’t do was take pictures of all of this information to see if we could have a record of the student growth over the whole year. Perhaps an electronic method of grading – a shared google sheet for each student or something to that effect  might be helpful in the future – but not this day (as Aragorn says) – way too much going on right now.

We also changed the rubric a bit for a few reasons.  First, we found that when students completed the problem to our expectations on the initial attempt we felt that they should just receive 3’s for the other two categories automatically.  We considered not scoring them in this category but numerically felt that it was actually putting students who correctly completed a problem at a disadvantage (giving them fewer overall points in the end). Second, we also changed the idea that if you did not write anything on the revisions you earned 0 points for the revisions columns.  Many students told me afterwards that they felt like they just ran out of time on the revisions and actually had read the feedback.  This was unfortunate to me since we had spent so long writing up the feedback in the hope that the learning experience would continue while doing revisions.

Here is the new version of the rubric: Revised Problem Set Grading Rubric new

What we decided to do was to try the revisions this time without the “explanation” part of writing.  I think it will keep the students focused on reading the comments and attempting a new solution.  I was frankly surprised at how many students stuck to the honor pledge and really did not talk to each other (as they still got the problem wrong the second time around – with feedback).  Truly impressive self-control from the students in my classes and how they were sincerely trying to use the experience as a learning opportunity.

I do think the second assessment will go more smoothly as I am better at doing the feedback and the rubric grading.  The students are now familiar with what we are looking for and how we will count the revisions and their work during that time.  Overall, I am excited about the response we’ve received from the kids and hope that this second time is a little less time-consuming.  If not, maybe I’ll just pull my hair out but I’ll probably keep doing this!