Whiteboarding activities in the MS/Algebra 1 classroom: Part 1

In my most recent post, I discussed how to get cheap whiteboards for student use in the classroom. I made large whiteboards (24" by 32") so that I could use them for group activities and investigations. The logical next question is what would be some activities and investigations that would make good use of this resource? Since my most recent experience is with MS Math classes, specifically Algebra 1, I'm digging through my files and seeing what would work well as a whiteboarding activity.

There are some awesome ideas at http://bowmandickson.com/2011/11/05/experiments-with-math-whiteboarding/. If you know of others, please link to them in the comments or ping me on twitter (@Borschtwithanna).

So, from my files from last year, here are some activities that I think can be adapted to whiteboarding in groups.

1. "1, 2, 3, 4 competition": Groups compete to see how many different ways they can generate the integers -24 through 24 using the numbers 1, 2, 3, and 4 exactly once each and any of the standard operations (addition, subtraction, multiplication, division), as well as parentheses. Sometimes, I let groups use square roots and exponents, as an extra challenge. So for example, doing 1 + 3 • (2 – 4) generates the integer -5. Each different expression is worth one point, even if it generates the same integer. Bonus points if they can generate all of the integers in the set. After time is up, all of the whiteboards will be revealed and groups have a chance to challenge an expression they think is incorrect, getting a point for each error they find. (Adapted from an IMP, Year 1 task)
2. "Number tricks with expressions": students can work on the following problem in their groups:
   
   A magician said to a volunteer from the audience, "Pick a number, but don't tell me what it is. Add 15 to it. Multiply your answer by 3. Subtract 9. Divide by 3. Subtract 8. Now tell me your answer."
   
   "Thirty-two," replied the volunteer.
   
   Then the magician immediately guessed the number that the volunteer had originally chosen.
   
   a. What was the volunteer's number?
   b. How did the magician know so quickly? (The magician couldn't possibly have worked backwards that fast.)
   c. Create your own impressive number trick. Write down the directions that you would give an audience member and explain what you would do to figure out the number that was picked. (Adapted from Discovering Algebra)
3. "Mystery Bags" investigation, which builds towards formal equation solving using the balance scale model. (Adapted from IMP, Year 2)
Mystery Bags
4. "Pauline's Run": fractions, rates, and equation solving. Students could use whiteboards to compare and contrast different methods. (Adapted from IMP, Year 1)
   
   One day, Pauline was walking through a train tunnel on her way to town. Suddenly, she heard the whistle of a train approaching from behind her! Pauline knew that the train always traveled at an even 60 miles per hour. She also knew that she was exactly three-eighths of the way through the tunnel, and she could tell from the train whistle how far the train was from the tunnel. Pauline wasn't sure if she should run forward as fast as she could, or run back to the near end of the tunnel.
   
   Well, she did some lightning-fast calculations, based on how fast she could run and the length of the tunnel. She figured out that whichever way she ran, she would just barely make it out of the tunnel before the train reached her. Whew!!
   
   How fast could Pauline run? Carefully explain how you found your answer.
5. Groups could use whiteboards to solve different versions of one problem so that the class can then discuss how they are related to one another. For example, one group could solve and graph the solution set for each of the following:
   a. 2x – 3 = 13
   b. 2x – 3 > 13
   c. 2x – 3 < 13
   d. |2x – 3| = 13
   e. |2x – 3| > 13
   f. |2x – 3| < 13
   
6. Sketching graphs to model situations and vice versa. For example, having students sketch a variety of graphs representing motion (walking away at a constant speed, standing still, walking away at an accelerating speed, walking toward at various speeds, etc) or having students create stories from a series of graphs representing different people's walks (there are lots of these types of activities in the book A Visual Approach to Functions). Here is a similar set of questions from the Exeter Math 1 packet:
   Graphing Functions
   
7. Introducing slope-intercept form of a line through changing patterns (adapted from The Pattern and Function Connection).
Intro to Slope-Intercept Form
8. Group investigations that can be presented and summarized. One idea is that groups can use graphing calculators and sketch a variety of functions and what happens when you take their absolute value in order to reach a conclusion about the nature of the absolute value function and how it operates on other functions. For example, one group would graph and , another group would graph and , a third group would graph and , and so on. Many types of graphic transformations can be examined by having each group present an example to the class and then synthesizing the investigation into a cohesive conclusion.
   

Since this blog entry is getting a bit long, I'll put the activities from the second half of Algebra 1 into a new post.

Whiteboards for groups & student roles - Made 4 Math Monday #2

Recently, @fnoschese linked to a blog post he wrote just about two years ago, called "The $2 Interactive Whiteboard," where he described the advantages of a large dry-erase board that students can use to work collaboratively and then easily share their findings with the class. You can buy these large whiteboards from various places on the interwebs, but I thought I would try the cheaper option explained in the blog and went to Lowe's, where a 48" by 96" piece of white tile board cost me $11.50.

They cut it for me into 6 pieces that are 24" by 32" each (for free!!), which are just about perfect for two or three students to use for a partner or group activity. The edges weren't sharp, but I put duct tape around them so that the cardboard backing wouldn't get fuzzy and disintegrate.

I'm thinking that students can just stand them up on the marker tray at the blackboard at the front of the room, but if that doesn't work, I can get all fancy and attach two metal hoops to the back of each whiteboard so that it can hang on the hooks at the top of my blackboard.

There are more whiteboarding resources at Whiteboards USA and Whiteboarding in the Classroom, as well as at Frank Noschese's blog, linked above.

The second thing that I worked on today was cleaning up classroom roles and expectations. I've had an assigned class leader for the past few years, who rotates every week, and it's been helpful both in making the class run more smoothly and in fostering students' independence (it's really nice to leave sub directions like, "1. Take attendance 2. Give all handouts to class leader and let her direct class for the rest of the period."). I've gradually moved away from group roles, which has made the class roles more meaningful as well. This year, I've decided to split up some of the jobs into two roles: a Class Leader, who sort of runs many of the starting and ending class routines, and a Materials Manager, who takes care of supplies and handouts/class organization.

Class Leader

Materials Manager

Three Templates - Made 4 Math Monday #1

A few fellow math teachers on Twitter (yay @druinok and @pamjwilson!) came up with the idea of making something for your math class, at least once per week, and sharing it on Monday and attaching the tag #made4math. I decided to put together two templates that have been bouncing around in my head for some time now. The first one is inspired by @mgolding, who has written a lot about interactive notebooks (for example, here and here). I will still have students use a binder system next year rather than interactive notebooks in order to save time and avoid cutting and gluing, but I wanted to move to a more interactive system for note-taking so I created a note-taking template that uses the "input" and "output" notions of interactive notebooks to help kids move from passive note-taking to processing and making sense of what they did in class.

Note Taking Template

The idea is that students would write down what's happening in class in the "input" (right) side, then write down reflections, comments, and questions in the "output" (left) side. I will also have them write an annotated example that either shows what they understand or what is still confusing to them from that day's lesson and to write a summary at the bottom of the last page in their own words of what they learned that day. My plan is to have them do this during the last few minutes of the class period since my experience has been that kids tend to blow these types of assignments off if they are asked to do them for homework.

The second template that I created for today was inspired by @samjshah's post on participation quizzes, where he talked about giving feedback to groups on the quality of their communication and ability to work together on a given mathematical task. The idea is that while students are working in their groups, the teacher is circulating and making notes on an overhead or SmartBoard of the things that are going on in each group, both positive and negative, as well as positive phrases or quotes that are promoting good groupwork. This gives groups instantaneous feedback and an opportunity to learn about group norms and what types of behaviors and ways of talking promote or hinder their ability to work together. I'm hoping to be able to use the template below to organize my feedback to each group and to be able to do this on my iPad while circulating around the room. I've done it before on an overhead and it can get a little hectic to both circulate and keep running to the front of the room to write down comments and observations.

Participation Quiz Template

The final template I made is inspired by stolen from @approx_normal's Homework SWHHW (See, what had happened was...) sheet, which she collects from students who don't have the homework due that day. I thought that this was a great way to keep a clear record for students who habitually neglect their homework beyond what I record in my grade book. This just seems so much more objective and will be in the students' own handwriting too, so no squirreling out of that one during a conference with parents. My optimistic plan, since I am terrible at keeping track of papers, is to take a photo of the form the student fills out in class and then upload it somewhere - possibly Evernote - where I can tag it with the student's name. Then, I can call up all of the missing homework forms for a particular student by searching. In a future world where every student has some sort of digital device in my classroom, I can just have them fill it out electronically, but I'm getting ahead of myself. So, here's my version of the form, just slightly altered from @approx_normal's document.

Missing HW Form

I would love any comments or ideas on how to improve these templates since this is just my first attempt at putting them together.

Summer Plans - Curriculum Remix

While this new blogging/twittering (tweeting?) thing has been wonderful, it has also made me overwhelmed with thoughts about what I want to change in my curriculum for next year. I've been working in a bunch of different directions for a few years (some quite contradictory), but I'm hoping to pull a lot of things together this summer and not feel like I'm tinkering at the edges. One of my sad realizations this year when we were moving offices and I cleaned out a giant box of crap I had lugged from my first school where I spent my first year teaching was all of the insane, cool, cutting edge stuff that I was doing then because I had no fear and didn't know any better and was doing the teaching thing completely solo and rudderless. The part that made me sad lay in realizing how much of that stuff has gone away since I've been at my current school and have become normal, established, and, gasp, respectable. In reality, a lot of that stuff did need to go away. I was giving kids really hard problems and open investigations with no scaffolding and support. I jumped from project to project with no follow through or continuity, which just created confusion. Everything was also handwritten and xeroxed (clearly, I hate trees). Not to mention the fact that my classroom management was just so ridiculous that first year that none of my grand curricular plans even had a chance. Since taking a curricular leap backward these past few years, I've been able to get so much better at organizing the day-to-day running of a class and my ability to interact with and lead students, which I know is really important for my development as a teacher. But I'm really ready to bring some of that crazy self back and inject some pow! back into my teaching.

To that end, I've been gathering resources and thinking how I want to restructure the various Algebra courses that I teach. Some things that I'm finding helpful are:

• Connected Mathematics
• Mathematics in Context
• IMP
• Mathscape
• CPM
• CME Project
• Exeter packets
• Park Math
• Harkness resources, such as this
• Stalking everyone else's wonderful math blogs
• Relooking at grad school materials, now with actual teaching experience to use as a lens

Mainly, I want to change from feeling like I'm teaching a traditional curriculum with interesting problems thrown in every once in a while to a legit constructivist approach. But I still want it to feel cohesive and be rigorous. And I want the students to be on board, which means that I have to meet them where they are when they arrive in my classroom and get them to follow me somewhere different. So this will have to be much, much slower than that erratic first year and more intentional too. My plan right now is to start with the big ideas/concepts from each chapter in the book (not ready to ditch the textbook quite yet), and decide on the best sequence (I'm fine with "going out of order") and problem-based entry point for each concept. I need to decide on how many days for exploring and problem-solving and how much procedural practice I still want to include. Another change that I want to make is to tie journaling more directly to our classwork and problem solving, rather than an add-on "reflection" that students do before a quiz or to wrap up the chapter. I am also thinking of diversifying assessments to include more writing and problem solving, which is mainly just a brain switch for me to stop thinking of tests and quizzes as the "real" assessments of what students know and of projects as the intermediate step where they're still learning and making connections.

The nice thing about doing things "normally" at my current school is that I have some cred with students and the department so that they will give me leeway to try things without assuming I have no idea what I'm doing. It also means that I don't have to think as hard about procedural/classroom management/dealing with teenagers stuff because it sort of starts to make sense after all these years (which is its own brand of crazy). I can just focus on my curriculum. So it's on!

 I would love to hear from others about how you've made your class more progressive or get recommendations for more curricular resources. As I start the actual reworking, I'll be posting ideas for problems and journal prompts for feedback too.

Bootstrap Programming Project

I wanted to reflect on a project that I did with my 8th grade Algebra students (the non-accelerated group) this year. Usually, the end of the year is filled with review and studying for the final exam, but this year, I decided to give the final before the 8th grade trip, which left 4 weeks at the end of the year for......? Exactly, I had no idea what I was going to do with that time, but even though I've never taught this particular class before, I had taught 8th graders before and remembered their "senioritis" and difficulty with doing any work after a week-long fun trip away from school with all of their friends. Sooo, my big question was, what activities/projects are a suitable and engaging way to celebrate and culminate the end of a 2-year Algebra course? By chance, I heard about a workshop being sponsored by Google about a programming curriculum that was very connected to Algebra, and I decided to attend. The curriculum (called Bootstrap) is completely open - all student and teacher materials are available online, and workshops are free for teachers. It basically walks students through the creation of a functioning video game using a language (WeScheme) that is much more mathematical than other languages and is very user friendly.

The creator of the curriculum is Emmanuel Schanzer, a Math teacher (who was a programmer before that) who is now working on his M.Ed. at Harvard. I did the programming project for 15 consecutive 45-minute class periods (with no HW assignments), which was quite a few less than Emmanuel recommended, but seemed to work fine for my students (a few complained that it was a bit too rushed, but others complained that it went too slowly, which probably means that it was okay). I didn't have a chance to do any of the extension activities, although a class that got ahead of the others did some design explorations, which was great fun. In reflecting on this project, I'm going to organize my thoughts into pros and cons.

Pros:

• Students really enjoyed it. It was one of the most mentioned activities in the end-of-year feedback form when asked about their favorite thing we did in class this year. And having our last day of class dedicated to student presentations was such a nice closing note.
• Concrete, fun end result (a working video game) that students could share with each other
• Student choice in the images and "storyline" for their game; they did feel like the game was theirs
• Seeing algebraic concepts (definition of function, domain, range, composition of functions, writing function rules, distance formula) in action, as well as previewing some geometric concepts that students will encounter next year
• All materials already exist and are broken down into lesson chunks. All I had to do was work through the directions to make sure that I knew how to code each part and decide on my tweaks (which parts to work through as a class, which parts students could do in pairs/groups, how many examples, etc).
• Emmanuel is super helpful - he answered emails within a few minutes and even called me up so that he could walk me through a part that I couldn't figure out on my own.
• Allows students to see a real hands-on side of mathematics, not something "real-world" that's been created in order to justify using math
• Fun for me! I haven't done any programming since college, so it was nice for me to stretch a bit and be frustrated/stumped in a different way, as well as lots of opportunity to be amazed by students' thinking because it was basically new to me.
• Allows different kids to shine than those who traditionally excel in formal math. I loved seeing kids who barely spoke (or passed a test) all year take dominant roles in the class and be respected as experts.
• It's kind of awesome (and really, really bizarre) not to assign homework for such a long stretch. Students were really appreciative given the high loads they seemed to be getting in all of their other classes at the time. The deal I made with them is that in return, they had to be productive in class, and this worked fairly well.

Cons:

• A lot of the code is already preset (which direction different components can move, structure of the game) so the kids in actuality are just tinkering at the edges. I felt like my students would have benefited from more challenge and opportunities to make the game more complex. It also means the end products are really, really similar.
• All materials already exist and are broken down into lesson chunks. I generally itch at constraints like this... I like to own a curriculum because I made most of it up or twisted it so much from the original version that it hardly resembles it. This felt a little prefabricated and "nice" for my taste.
• The way that the lessons are written is very heavy on teacher-direct instruction, which isn't really my style. I amended some parts to include more pair/group work, but it was hard for me to deviate too much since I don't know enough about programming to really set them/myself loose. It also means everyone is expected to move at the same pace, which is basically unreasonable. It meant that there were days where kids with a more intuitive understanding of coding sat around bored for half the class and other days where kids who were struggling felt lost because we just needed to move on.
• I needed to reserve laptop carts every day for 4 weeks straight. The IS department at school was not pleased, and it put a strain on other teachers' ability to use laptops at that time.

So overall, the project was definitely a success, and I hope that with more experience, I can address some of my cons next year. For example, I will communicate with Emmanuel to find out if there are ways to change the preset backbone code in order to allow for more complex variations of the game. I will also work this summer to revise the lesson outlines and make them more student-directed and find more opportunities for student exploration (there are extension activities on the Bootstrap website that I haven't looked at in great detail yet, but which may be promising). If you are interested in using Bootstrap in your class, they seem to be running workshops all the time, and it's a nice chunk of a curriculum that can fit with almost any level of math experience of middle and high school students. They also have a Facebook page and Google group, which has an updated list of upcoming workshops. And write me a note if you have any questions or comments about my experiences with it this year.

Assessing projects

Now that the last day of school is here, it's finally time for me to grade students' final projects, and I'm realizing that this is really hard for me. For Algebra 1, students had quite a few options to choose from, of varying difficulty and mathematical content.

Some students worked on the same project for a week, while others worked on two or three during that time. Some created a product (video, book, poster), while others explained their process verbally to the class and/or annotated their work on paper.

I feel that this year more than in the past, students engaged fully with their projects, worked productively, and accomplished something meaningful in the last week of school, but I have no idea how to fairly grade this work. Should a weaker student who picked a project that was challenging for them and put a lot of effort into it be graded lower than a stronger student who completed a more challenging project with perhaps less effort? What about a kid who tends to be easily frustrated, but worked through his sticking points and got himself through areas of frustration? I'm not even sure that I can quantify their effort... some students chose to do extra work outside of class (this was discouraged, but not banned), while others didn't. Some volunteered their time to help their peers while others preferred to work alone. How can I tell how hard a student is actually working? And should it matter in assessing a project or do I grade on mathematical content alone? I do have a rubric (attached to an earlier post), so that will help somewhat, but I'm still mulling over if each category should be referenced to some norm (what I think an Algebra 1 student should be doing) or compared to that particular student's previous progress and how much they learned doing this project. I would love to hear from others who have thoughts on how to assess projects.

P.S. All of the photos are so boy-heavy because this class literally has 3 girls in it. Did I mention that it's also the last class of the day? Sigh.

 

Student feedback

Going through 8th graders' feedback from an anonymous Google form that I had them fill out last Friday is kind of amazing. It's funny to me how few of the comments have anything to do with Math, and everything to do with our personal connection or the community they felt in our classroom. I was also reminded that what they most remember from the course are the hands-on activities, projects, and lab experiments. Far and away, their favorite activity this year was the programming project (using the Bootstrap curriculum, I blogged about it here). But they really did take in a lot more stuff than they appeared to at the time. This is my first year in a long while teaching the "regularly paced" group (we have "accelerated" and "regularly paced" for all Math classes starting in 7th grade), and I am actually quite impressed with the maturity and attitude toward learning. We get bogged down in the day-to-day and forget how much growth these kids are making in a given school year and that they give as much back to us as we give to them.

Some of my favorite quotes:

"You were very open to questions and were wise with your responses not to give away the answer." Seriously impressed by this kid's maturity.
"i learned alot this year but I don't remember that much of it."  This is more like what I'd expect from an 8th grader :)
"Algebra is something that I can actually see using in real life situations." I love that someone is taking this away from my class
"Even when mistakes were made, you still respected me and never held it over my head." Okay, this one just made me aww. We should pat our own backs every once in a while.
"I agreed with all the grades I got. Though I still honestly don't understand why i need to label my axes." I'll let next year's Math teacher continue fighting that good fight.
"I enjoyed taking the final test." I really want to know who this kid is! This is what he/she listed as their favorite part of my class!! <mind blown>

Some recommendations for next year:

"let the students give more feedback" Yes, I do need to incorporate this more into daily classroom routines. I'm curious to hear how others incorporate student feedback into their classes.

"Do more projects." Totally agree... will be working on developing more this summer.

"On ocassion play a game that has nothing to do with math to change things up." Probably more games in general would be a good thing. I would love input on this one too.

My Google form survey for students

I also thought about soliciting feedback from parents, but didn't organize myself enough to do it this year. I thought that I would ask just a few questions (Did you feel that there was enough communication from me? How could I have better supported your child? Any other suggestions for improvement?) and email it out via a class mailing list. Any feedback about parent questionnaires would be appreciated!!