# Consistency for Flexibility?

The last unit in my Pre-Algebra classes involved many related topics, but a large portion of the errors and misconceptions arose as we learned slope.  Students who rocked out rate of change early in the unit kept flip-flipping things when we dropped context and looked at ordered pairs or a line on the coordinate plane.  Students who were easily able to catch their mistake when they wrote hours/dollar now were oblivious when they had run/rise instead of rise/run.

In some respects, I think some of the errors were related to my consistency.  I kept switching my wording for slope in hopes students would get flexible thinking about slope in multiple ways.  Rise/Run.  Change in y-values/Change in x-values.  Change in Dependent Variable/Change in Independent Variable.  Steepness of a line.  All of these definitions are helpful, but switching the wording of my questioning time after time is not helpful for students working with slope extensively for the first time.

As what tends to be the case in reflections, I noticed that my questioning was probably the largest contributing factor in all of confusion I witnessed in my students.  After rethinking how I would approach slope in the next couple days, I decided one of the best ways to address student concerns would be to use a set of questions consistently.

The next day, students walked in the room and I shared my observations.  We watched Slope Dude to revisit the more basic elements of slope (+, -, 0, Undefined), completed a couple examples, then I set students loose to practice.  Throughout the lesson (and the remainder of the unit), I kept using the same battery of questions when I encountered a student exhibiting misconceptions:

• How are you reading this graph/table/set of points?
• For the graph/pair of points/table, how would you describe the slope?
• How are the y-values changing?
• How are the x-values changing?
• Did we write y-values over x-values?
• Are we forgetting anything?

I chose these questions based on the success I saw students have with rate of change (change in y-values/change in x-values is the working definition we developed), but I also wanted to stress the importance of reading before calculating and self-questioning.  Thinking about the majority of mistakes students made with slope, most of the hiccups were due to students skipping the basic step of reading a graph/table/set of ordered pairs and deciding if the slope was positive, negative, etc.  After making a decision about the slope, it’s really just a matter of asking how the y-values are changing as the x-values are changing.  Not surprisingly, students began to make errors less frequently when I began using these questions consistently and encouraging students to adopt these questions to guide their thinking.

Deciding upon a uniform set of questions also made the remainder of the unit easier to plan.  When I began planning to teach linear functions and all of the elements associated with this concept (rate of change/slope, initial value/y-intercept, tables/graphs/equations/descriptions, increasing/positive slope, decreasing/negative slope, constant function/slope of zero, linear relation/undefined slope), I was overwhelmed by the various ways mathematicians talk about singular ideas.  Considering the way I kept switching the wording of my questions with slope, I recognized that I was probably causing the same amount of stress in my students.  Not good.  By streamlining my questions and sticking with the same wording, I was giving students the consistency they needed to get fluent with the topics we were exploring.  Beyond this comfort, I was providing students with more incentive to adopt and use questions I was offering to prevent mistakes and guide thinking.  Once students recognized that I was thinking about a problem in a similar fashion regardless if we were looking at a graph, set of points, table, or description,  students were able to see the meaning of my exhortation that, “It’s the same idea, just presented a different way.” Basically, students were able to build the flexibility that was my initial goal.

My experience with teaching slope and my use of questions leaves me with some unresolved thoughts.  People crave consistency, whether or not they are vocal about this desire.  As teachers, we also know that a hallmark of problem-solving is being flexible in one’s thinking.  Based on the struggles and success of my students with slope, is consistency is needed to build flexibility?  What elements of my teaching need to be consistent to encourage and build flexibility?  Is it a matter of consistent questions?  Is it a matter of routine experiences like asking students to develop questions to ask themselves or providing questions to adopt for personal use?  Do my students and I need a problem-solving framework?  Since I have more experience, I often don’t consider how I am solving a problem.  Am I consistent in the way I approach problems?  Do I need to be consistent?  Is my use of flexibility consistent?  How flexible is my thinking?  How does the connect to the way my students are thinking about problems?  Are my students consistent in their approach to problems?  Do my students ask themselves questions about problems?  What questions are the most consistent for my students?  Do my students recognize when they need to flex their thinking when they hit a cognitive obstacle?  How consistently are my students changing their approach to a problem?

Out of all of the questions I just wrote, the question that interests me most is whether or not consistency is needed to build flexibility.  I’m inclined to say yes, but I’m still uncertain about how that process looks in the classroom in relation to my teaching and student activity.  I’m guessing that there’s information out there about this idea and for all I know I might have just missed the session(s) at Twitter Math Camp that were focused around this idea.  If you have any suggestions, links, articles, or knowledge about using consistency to build flexibility, please leave this information in the comments!

# Conversations with Individuals

I think everybody is prone to some looking back and looking forward this time of year, so I might as well add to the deluge of posts that are reflective in nature.

In the midst of a school year that can only be described as unprecedented, I have managed to try out some new strategies and methods of reflection.  One of the most helpful tools I’ve used in reflection is the Teacher Report Card from Matt Vaudrey.  At the end of first quarter and second quarter, I gave students in my math classes time to complete a slightly modified version of the form Matt created.  The results are interesting, so read through Matt’s posts to get a more detailed idea of what you can expect to gain from allowing adolescents to anonymously analyze a teacher.  Rather than describe all of the results of the Teacher Report Card I gave just a couple weeks ago (and proclaim myself president of the Matt Vaudrey fan club), I’m going to focus on one question I included in my version of the report card.

As I was setting up the Google form for the second quarter Teacher Report Card, I decided to add a couple questions based on the differences I observed between first and second quarter.  Among the largest differences I noted was how I chose to conduct conversations.  I was wondering if my gut feeling would match student thoughts on the survey, so I added the item, “I think that Mr. Hall talks to me individually (one-on-one).”  Here are the results (1 is not at all and 5 is definitely):

The percentages were somewhat surprising, but the percentage of 3 or above was not that shocking.  During second quarter, I built more time into lessons for students to work with math.  Most of the time, these activities were in groups and I circulated to question students and check progress; however, I shifted to talking with individuals just as often as I talked with groups.  This difference might seem small, but I think it’s important for teachers to recognize the power of the individual conversation.

It’s easy to assume an entire group is comprehending a concept or skill, when in reality one or two people of a group are just following along and hoping to learn by repeatedly mimicking everyone else.  By questioning individuals as much as groups, I was able to better see which students actually knew what they were doing (and the depth of their knowledge).  I was able to offer help to students who just going through the motions with their group, then let them work with their group when they were feeling more confident.  I also liked the way I could praise students and encourage them to share when I called the entire class to attention to talk about problems.

Beyond talking with students during group work, I employed more individual conversations in the hallway or after class during second quarter.  The expectations students bring to these conversations is striking.  In most cases, a student had a look of dread of his/her face upon hearing my request for them to go in the hall or stay after class for a moment.  This expression slowly relaxed or a small smile appeared (a rare win when working with 8th graders) as the student heard what I had to say.  Most of the time, I was either celebratory (praising or encouraging hard work, thoughtfulness, leadership, etc.) or inquisitive (questions about group dynamics, level of understanding, comfort with the topics, etc.).  Even when I was celebratory, I included a question or two for the student to answer to make it a conversation.  I wasn’t talking to a student.  I was talking with a student.

I think most students associate hallway (or after class) conversations with negative behavior or poor grades.  When I flip the script students expect for these conversations, it creates moments where I get to know my students in detail.  I see which student responds better to praise for work, which student wants to be a leader, which student recognizes he/she needs to be more confident, and which student secretly cares more about his/her grade than he/she would ever admit in front of peers.  I get feedback about what’s working or not working for a student.  I chip away at the mental armor that so many students put on every time they walk into a classroom full of peers.  Basically, I like the way individual conversations allow students and myself to see the human element to the learning process.

I hope to continue my use of individual conversations during the second semester.  Hopefully, the number of 1s and 2s in the third quarter Teacher Report Card will be lower than the results for second quarter.  More importantly, I hope my students will recognize that I teach them as individuals just as much as I teach entire classes of students.  I also hope that I can change the stigma of the hallway conversation.  The fact that almost every student I talked with in the hallway this quarter was full of dread is an unfortunate product of how most teachers use these conversations.  Why does the deserted hallway during 3rd period (or any period) need to be a place for the uncomfortable conversations?  Can we make 2017 the year when teachers changed how they use the hallway for conversations with students?

Another quick post about a tool that’s making my life a little easier this year.

At the start of this year, another math teacher in my grade decided to start making some of his quizzes multiple choice.  While neither me nor said teacher is the biggest fan of the level of rigor that’s possible with multiple choice, these quizzes have been a nice change of pace and a way to build some self-correction into the quiz taking process.  I’ve also been using some multiple choice activities are partner practice and using the tool I’ll describe to check student progress.  In addition, the quizzes have been a way to save time thanks to the app both of us have been using to grade.

ZipGrade is a free app for Apple and Android, but a annual subscription is needed if you end up scanning more than 100 papers.  On the full website, it’s easy to type or upload class rosters and print bubble sheets.  I’ve been using the ID numbers the website generates, but I’m guessing it wouldn’t be too much trouble to change the ID numbers to match school-issued numbers. Blank bubble sheets can be printed from the site, but I’ve been printing the bubble sheets from my classes so student names and generated IDs are prefilled.

On the app, the menu offers the ability to create quizzes and manage classes.  In the quiz menu, all you have to do is create a multiple choice key for a new quiz and choose the classes for the quiz.  When you’re ready to scan, hit scan papers.

After lining up a bubble sheet with the phone camera, the app takes a photo of the page and scores the paper using the key.  I tend to review the papers, which gives a snap shot with the correct and incorrect marked (see below).

What I’ve found helpful with this feature is the ability to immediately pinpoint which problems students missed.  When I’ve been using multiple choice activities for partner practice, I’ve used this feature to let students know what to revise or provide scaffolding with specific topics if a certain mistake is being repeated.

Another feature of the app I appreciate is the stats and item analysis that are generated from the scanned papers.

While I don’t like using multiple choice for everything, I’ve liked being able to use ZipGrade to provide students with quick feedback and encourage revision.  For instance, I scanned student bubble sheets during a recent partner activity and told students which problems to revise.  I made students turn in work with their bubble sheet, so I was pleased to see students talking about missed problems and revising work before asking me to scan their papers again.  I think the students also appreciated getting to see how their score was improving with each revision.

Check it out for yourself and let me know what you think in the comments.

# Quick Post: Crowdsourcing Notes

It’s been a long time.  I’ve been really busy with my schedule and I haven’t been able to write as much because of fatigue or (I’ll admit it) feeling like I got nothing worthwhile to post.  This blog is more about my reflection than anything, but I feel like I haven’t done anything special that was worthy of a lengthy exposé.  Then again, I think I’ve been noticing more of the small things that make or break lessons as this year has progressed.  In that spirit, I’m going to begin posting about the little things in my classroom that have been saving my sanity and (hopefully) making my students think more and take more ownership of learning.

Today’s quick post: Crowdsourcing Notes.

I’ve used this strategy in the past, but I’ve barely mentioned on this blog.  It’s not the most creative practice in the world and it’s definitely not meant to be used with every concept.  When I use this activity, I like how it promotes clear communication from students and gets everyone involved in creating notes.  Beyond the benefit to students, I like how I can focus on asking questions as students work through problems as a class.

For my set up, I use SMART Notebook and an IPEVO interactive whiteboard pen system.  One could use a document camera and paper, but the technology makes it easy to print copies of notes for the class (which is the product of the activity).

Basically, I think up some examples of a topic that progress from what students currently know to what’s going to be new or require some reasoning.  For example, my 8th grade math classes were working on solving multistep equations last week.  After a few days reviewing  two step equations, I wanted to work ease students into multistep equations with variables on one side.  Students settled into class and I projected the following sequence of equations on the board:

Only the first equation was visible, since I did not want students to blow through the problems without describing their thinking (or avoid contributing to the class).

I asked, “What can I do to simplify the left hand side of the equation?” for the first problem.  A student mentioned combining like terms, so I gave her the interactive whiteboard pen and had her write down what she was thinking.  Other students in the room gave nods of approval or looks of confusion, so I asked the student to explain why she combined 8x with 3x and 6 with -2.  After her explanation and general agreement from the class, I had her hand off the pen to another student to complete one additional step in solving the equation.

The activity continued in this fashion for about 25 minutes.  I questioned the class throughout and students even jumped into the questioning when they noticed errors.  When mistakes happened, we discussed the errors and identified how we could prevent them in the future.  For instance, we changed -x to + (-1x) in the third equation when multiple students added an x.  Overall, it was a good flow of students posting work and me questioning student reasoning.

By the end of the activity, we made the following notes:

It’s definitely not not the most spectacular series of notes, but it’s awesome to point out to students how much of the math and the thinking they did as they crowdsourced the work for the problems.  Beyond this success, my 8th graders were pretty stoked about not having to write anything down and gluing a copy of the notes into spirals the following day.  Most importantly, I liked how the activity stresses verbal communication for everyone in the class.  I need to be specific in how I word questions so that students are able to make progress when they get stuck, but students also have to be specific when I question their reasoning or ask them to describe what they are doing for the rest of the class.

It’s not the greatest thing in the world, but crowdsourcing notes is one the small things that make some of my lessons better than average this year.

# Clothesline: Irrational Numbers

In my Pre-Algebra classes last week, we worked on developing an understanding of irrational numbers.  Specifically, we focused on irrational numbers like √2 and -√73.  Among the various elements to developing this concept, I had students work with finding square roots of perfect squares and estimating the closest integer for an irrational square root.  Plotting approximate locations for irrational numbers was also a part of the scope of this concept, so I knew right away what I wanted to do with this topic when I was thinking about it at the start of August: Clothesline math.

I used the Clothesline activity last year for my 6th grade students when we learned about integers.  The activity was the perfect way to have students see negatives as opposites of positives and introduce absolute value.  It was awesome to see a student place -2 far away from 0, only to have another student protest saying, “-2 needs to be just as far from 0 as +2.”  I knew using a Clothesline activity with irrational numbers would be just as valuable for my 8th grade students.

The set up:

• Cheap clothesline (\$1.98) from a hardware store hanging from the front and back staple strips above the whiteboards in my room
• Clothespins from a dollar store
• Cards with integer values -7 to +7
• Cards with perfect square roots (-√25, +√16, etc.) and other rational numbers (1.5, -2.5, etc.)
• Cards with irrational square roots (-√22, √2, √3, √43, √48, etc.)
• Purposely vague directions

Implementation:

I started the lesson by asking students what they thought the clothesline was going to be used for that day.  Not surprisingly, a student guessed a number line because we drew a couple in our notes the previous day.  I told students the activity would take a while and require everyone to critique each other’s work, then I directed the people with -1, -2, 0, +1,  and +2 to hang their cards with a clothespin.

The 4 students made quick work of this task and I noticed two of them using their hands as measuring tools to place the cards about a half foot apart from each other.  I asked them to explain to the class what they were doing and a student responded, “The numbers need to be the same distance apart.”  I moved the cards slightly so there would be plenty of room in between the numbers (and so the final number line would use the whole length of the rope), then I moved onto the interesting part.

I directed anyone with a card with an integer value to put it up on the clothesline.  Immediately, the students with integer cards (-5, -4, +3, etc.) got up and started coordinating to place their cards at the appropriate locations.  A moment later, a student said to me, “My card has √16.  That’s just 4.  Am I supposed to put my card up there, too?”

In keeping with my style, I answered with a question of my own, “Does your card have an integer value?”

Realizing the vagueness of my direction, the student got up and clipped his card over +4.  A couple other students asked me similar questions and I gave similar responses.  Within a minute or two, all of the students in the room with perfect square roots realized that they needed to put their cards up on the clothesline.  It was pretty cool to see the realization dawn on student after student.

After a little discussion about what students did and asking about some cards (a couple students with perfect square roots clipped their cards next to an integer, but were able to tell me it should be in the exact same place as an integer), I directed any student with any remaining card to put hang it up on the clothesline.

I only took a picture of the final product after class, so here’s a drawing of that the activity looked like as it was happening.

I didn’t say much as students moved around and I watched intently as I saw students stopping, putting a card for an irrational square root between the two closest integers, then moving it closer to a particular integer.  One student put √48 really close to +7, while another put √43 close to halfway between +6 and +7. A student with -√11 changed his mind about whether it should go between -2 and -3 or -3 and -4.  One student still working on the square root thing put √5 on 5.  It was really interesting to see two students repeatedly swapping the order of √2, 1.5, and √3 over and over again.  They finally settled on placing √2 closer to +1, 1.5 halfway between +1 and +2, and placing √3 near where 2/3 would have been.

I tweeted a picture of part of the final product:

When it was all settled, I brought everyone to attention and began asking about specific cards.  Everyone was in agreement about some cards.  For example, everyone agreed √48 should be slightly before +7 because +7 is √49.  Similar reasons were given for other cards, but some cards led to disagreement.  For instance, I asked about the placement of √2, 1.5, and √3.  One student said √3 should be on 1.5 because 1.5 was half of 3 (we’re still working through that misconception).  There was some buzz in the room about the cards until one person raised a hand and calmly said, “2 is closer to 1 than 4, so √2 needs to be closer to √1 than √4. It’s the same idea for √3.”

There was at least one audible, “Oh, I get this now,” after that statement.

Discussing a couple more cards, I concluded by emphasizing to students the thinking process they did for plotting irrational numbers.  They created a number line with equal intervals.  They matched perfect square roots with integers, then they used those values to estimate where an irrational square root needed to be placed.  It wasn’t random.  Even though we’ll never be able to place the irrational square roots in their exact locations on the number line, we got close because of the thinking we did with the number line we made.

Future Improvements:

Some students were slipping into social conversations as the activity passed the 20 minute mark.  Next time, I think I’ll either use more cards (some students only placed one card the whole activity) or set up multiple clotheslines in the room.  Some of the discussion I had with the whole class might have happened naturally if I challenged groups of 4 to create their own clothesline.  Also, I want to slide into approximating irrational square roots with mixed numbers on the same day.  I needed to pick up with approximating the following day, which was fine; however, I think the process would have been easier if I was able to ground it in the concept of plotting on a number line.

Overall, the Clothesline activity was a good way to get kids thinking about irrational square roots using a number line.  There were elements of concept building, practice, and revision built into the activity.  I also liked the way I was able to see the product of kids thinking as they moved cards they originally placed elsewhere.

# Square Roots: Do you want to fix your work?

In some respects, today was the first day since the start of school (August 17th) where I was totally confident in what I was going to do with my 8th grade math classes.  I did not feel rushed.  I was fine with things taking longer than expected.  I knew how I was going to arrange the lesson to require more explanation from students. Adjusting to a new grade (8th), new preps (prealgebra, science, and PE), and new schedule (60 minutes periods) has made the first couple weeks a blur, so I knew I had to commit the moments of today to writing to help me keep everything in perspective.

The goal for this prealgebra lesson was to get students reacquainted with exponents and introduce them to square roots.  After taking care of housekeeping for the day (bellwork, homework questions, changes to the schedule for the week), students picked up a copy of this page.  I gave everyone about 10 minutes to read the blurb and complete the exponent problems without a calculator.  Once we compared work (everyone was pretty solid up to 12 squared), I defined perfect square and square root.  I kept my definitions short and emphasized that square roots are essentially the same as asking the question, “What number squared is [fill in the blank]?”

With the stage set, I turned students loose on the Figure It Out problems for square roots.  I encouraged students to talk with their neighbor if they were unsure about a problem and to keep asking themselves questions.  I circulated and questioned students as needed.  For instance, some students asked about finding the square root of 4/9.  I responded by questioning, “Can you find root 4? Can you find root 9?”  Once I noticed most students wrapping up the problems, I gave random students dry erase markers and told them to post an answer on the board.

I don’t have students post work as often as I would like and I don’t want anyone to be embarrassed by an incorrect answer, but I knew what I wanted to do with the answers students posted.  For each problem, I read the offered solution and asked, “Do you agree or disagree everyone?”  I say, “everyone,” when I want all students to call out a response.  When there was a chorus of agrees, I followed up by asking, “How could we check this answer?”  When the results were mixed, I asked people who disagreed to explain their thoughts.  This prompting was especially enlightening when the square root of -16.  Many students agreed with the proposed solution (-4), but a couple students protested that it didn’t work.  When I asked for an explanation, one student offered, “-4 squared is positive and so is 4 squared.”

I loved the flow, questioning, and responses that were happening during this lesson; however, what happened next is what had me thinking the rest of the day about how I could improve my future teaching.

When I reached the second to last problem (the square root of 16), the student who posted her answer (8) said, “Oh, I know that’s wrong now.”  Rather than move along and ask the class about the answer, I asked her, “Do you want to fix your work?”  She came up to the board and altered her answer to 4, then returned to her seat.

I could’ve let things go and moved on, but I didn’t.  I asked, “Why did you change your work?”

The student responded by saying, “Because it was wrong.”

I wasn’t going to let this student off the hook that easy, so I said, “You already said that, but that doesn’t tell me how you knew it was wrong.  What changed your mind?”

The student explained, “I realized 8 couldn’t work because 8 x 8 is 64.  4 x 4 is 16, so it had to be 4.”  It was a compact explanation, but it was accurate.

This moment was great for two reasons.  First, I loved how the student was able to state aloud how they realized a mistake and fixed it.  Hopefully, everyone else in the room got a another lesson about the value of learning from mistakes.  Second, I liked that I was insistent about getting the student to explain her thoughts.  Was the momentum of the lesson?  Was it that I don’t accept, “I don’t know,” or, “Because that’s the way it is,” as responses anymore?  Perhaps, but I think the biggest difference was the offer I made the student when she realized her mistake.  When I asked the student if she wanted to fix her work, I opened the door to discussing why she fixed it in the first place.

Now I’m wondering how I can open this door in future lessons…

# Squad Goals: The First Three Days

School starts on a Wednesday this year (a change from the last two years).  I’m adjusting to a new grade (8th), new content (science and PE join my repertoire this year), and a team of coworkers (who I’m sure will be great).  It’s a lot of grasp.  I would be lying if I said I never felt like there was an elephant sitting on my chest as I was setting up my room this week, thinking about the year to come.  In order to alleviate some of that stress, I thought it would be helpful to talk about what I have planned for the first three days.

First, I got to talk about my squad goals.  I know at some point I’ll have to drop using the latest and lamest slang terms, but this term actually felt appropriate as I was thinking about what I want this year to be for myself and my students.  According to Google, squad goals are what a person wants their group of friends (squad) to accomplish.  I’m modifying this definition to be what I want my classes (squads) to accomplish. They’re not friends, but I can still have aspirations for my students, right?  Anyway, the squad goals I have for students this year are more refined than in year’s past and are best presented in bullet form:

•  Do as much for themselves as possible
• Learn/Reinforce the idea that all work is valuable and expect the best from themselves
• Help others

I included the last bullet point because students often struggle with self-questioning and often are unclear about what it means to ask themselves questions about problems.  The other goals are a reflection of my philosophy of education and expectations for students.

Now that my squad goals have been defined, here’s a rundown of activities I plan to use during the first three days of school to establish these goals and get students to see what these actually look like in the classroom.  Commence more bullet points!

• Random Questions Introductions – I plan to ask students random questions during attendance on the first day, then have students stand up and ask each other random questions at various times during the first week.  By random questions, I mean creative questions like, “If you could rename Illinois, what would you call it and why?”   I want to students to get to know each other, get used to asking questions, and get used to thinking of questions.
• Introduce and practice music cues – I know I keep bringing up this topic, but the first week is a prime opportunity to tell students why I use music cues (to promote independence) and get students used to carrying out tasks when certain music clips are played.
• Noah’s Ark and Marshmallow Tower – Noah’s Ark is a good introduction to problem-solving strategies (asking questions and helping each other) that my math classes will complete.  The Marshmallow Tower activity will serve a similar purpose for my science classes with the added element of me introducing the 8 Science and Engineering Practice from the NGSS.
• Fill-in-the-blank Syllabus – This activity is mainly meant to make sure students get all the basic information about my class and familiarize themselves with my expectations and routines.  The added benefit of this activity is that it’s the first of the millions of times students will hear the phrase, “Everyone does everything.”
• My Job, Your Job, Parent’s Job, Our Job – I like this activity because it reveals how much students already know about what is expected for their learning (expecting the best) and it brings the element of parent involvement into the picture.

Overall, I want the first three days of this school year to be ones that set a productive tone for the year.  While we’re not diving into content straight away, I know these activities (and what I say about these activities) will get students familiar with my classroom and my squad goals.