Category Archives: Whiteboarding

Whiteboard Experiments: Modified Mistake Game

I have used Mistake Game a lot in class. Students write up the solution to problems on whiteboards and purposely make a mistake in the solution. Then they present their solutions to each other, presenting their mistake like they meant to do it. Then, students ask thoughtful questions to try to find the mistake.

This works great with topics that are conceptually rich, but less so in topics that are more mechanical, where mistakes tend to be a bit harder to see and are less rich to talk about, like implicit differentiation for example. I did a modification of the Mistake Game that worked really well for this:

  1. In groups of 2-3, students write solutions to a problem on a large whiteboard. After checking their correct answer with me, they go back through and make a mistake in their solution.
  2. Students then flip over the sheet I gave them that had their answer and write what there mistake is, kind of like a mini answer key.
  3. Groups then rotate around the room and try to find the mistake in the solutions in front of them. Once they find the mistake and check their answer with what the group wrote, they move on to the next board.

I wanted to train them in the art of looking over a solution and checking its correctness, and I think that this did that well. Compared to the mistake game, I felt like more students were active at any moment, more students could carefully follow complicated work, and it took much less time (20 minutes as opposed to 40)… but we also didn’t have the great mathematical discussions that we normally have during mistake game. I guess it really depends on the topic at hand which version is more appropriate, so I’m definitely going to keep this one in my teaching toolbox.

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Whiteboard Experiments: Practice & Reflect

I tried something new in class this week that I think solves a few random problems:

  1. Sometimes, when working on whiteboards, one student hogs the marker and does a lot of the work (and thus the learning)
  2. With whiteboard work, students don’t have anything in their notebooks to study later
  3. When we practice things like derivatives in our notebooks, I feel like their notes become almost useless because it is a mess of 15-20 examples.

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Practice and Reflect

I put the whiteboards out on the desk and left them there the whole period. We were learning the derivatives of exponential and logarithmic functions of bases other than e so I wanted to go back and forth between the whiteboards and their notebooks a few times.

After teaching them a derivative rule as a whole class, I gave them a sheet of 15-20 problems that definitely got more difficult as they went along. I gave them 12 minutes to practice (I put a timer on), and they worked on the problems with partners on the whiteboards, which gave them a chance to discuss, erase mistakes and see problems in large format with different colors.

Then, I asked them to put the markers away and open up their notebooks and gave them 3 minutes to reflect. I told them they could do whatever they want with this – copy down a few problems that were tricky, write down some things that they want to remember, write down steps for the problem. With this, I feel like their notes were a bit more focused and useful. I also felt like the whole routine was efficient, in that it kept a vast, vast majority of the students moving and engaged. I’ll definitely try this again.

Whiteboarding Mode: Simultaneous Show and Tell

Side note: Simultaneous Show and Tell is a terrible name for this whiteboarding mode (because it kind of sounds like a lot of whiteboarding). Forgive me, I cannot think of anything better. So… propose a better name?

[update 11/25: Andrew in the comments suggested “Function Iron Chef” which is definitely the winner. That’s what this whiteboarding mode is called now]

Students are in groups of two at a whiteboard with a VERY LARGE set of 3 X 3 axes drawn up on the board. They are sitting in a U shape so that if everyone put up their boards, every student could theoretically see everyone else’s. I put up a prompt like this:

Draw a function such that…

  • \lim_{x \to -2}=3
  • f(-2) does not exist
  • \lim_{x \to 1} does not exist
  • and f(1)=-3.

I put the timer on. Students are given a few minutes to draw a function (any function, lots of correct answers!) that fit the prompts. Then, at the end of the time, everyone puts their markers down and puts their board up. We spend a minute silently looking around the boards to look at everyone else’s work. Then, after a minute is up I allow the students to ask questions of each other (i.e. not just say “THAT ONE IS WRONG”). If they don’t ask questions about some that are suspect (or some that are totally correct), I will ask questions at the end to talk about specific boards. We then do 5 or 6 other rounds like this.

POSITIVES: We have done this so far with limits, continuity vs. differentiability and will do it in a few weeks with graph sketching – I think that making them do things the other way around, making them create (instead of just identifying limits or whether a function is continuous) really forces them to think harder. I also like this because when students have to show their work to their classmates, they often put a little bit more focus into making sure they are proud of what they have (and just about every student is engaged in the process, especially if you make them switch markers). I also love times to showcase mistakes as part of the learning process – we try to be as open and supportive as possible in correcting the boards. Lastly, having a discussion in a math class is always a really nice change of pace.

ISSUES: Students can get a little crazy during the discussion process and some can phrase things negatively. Not all students are good at following along verbally when discussing, and will wait for others to point out mistakes in the board. A few times the whole thing has taken a long time with all the transitions, but it has gotten better every time. I’m not sure how the weak students feel about this activity (having their work showcased and critiqued). Also, I’m not sure that this type of activity would be great for anything but a topic where the students already have some fluency and mastery.

Starting Class with Whiteboards

As my students know, I am a bit of an efficiency freak (read: I’m kind of impatient and actively organize my classroom to avoid patience problems). I love having students work with students other than the ones that sit right around them ( I don’t have assigned seats), but I can’t stand the time it takes to reorganize into groups. So whenever we do this, I do it right from the beginning of class. Students walk into the classroom to find the whiteboards laid out on the tables like this:

I usually stand around the door just instructing them to sit where their names are (even though most figure it out anyway). What usually happens is about half of the kids show up a few minutes early and sit behind the boards twiddling their thumbs. But then the problem catches their eye… And you can see them looking back and forth, fiddling with the markers, trying to resist the urge to DO MATH! Inevitably someone asks: “Mr Bowman, can we get started?” I’m like the coyest math teacher in the world, so I always respond “Uh, yeah, um sure, I guess” while my mind is deviously tapping its fingers together like Mr. Burns. Then the other kids trickle in, and immediately engage because they see someone else working (and haven’t really missed anything). Every time I do this, every single student in class is engaged in math by 30 seconds into the period with little to no cajoling from me. Then after a few minutes, I might give a few more instructions (like, this is MISTAKE GAME!) I realize that this may seem trivial/common sense, but the thing that makes me happiest is using every minute of every class!

Variations: If you don’t want to assign groups (which I usually do to get specific students working together) you can do this randomly with a deck of cards, or something similar.

 

My 3 Favorite Math Whiteboarding Modes

GOAL: develop frameworks and modes appropriate for MATH specific Whiteboarding.

I did a ton of experiments this year with whiteboarding and a lot of brainstorming, but here are my three favorite modes of math whiteboarding that I tried (some writing copied from previous posts). A good whiteboarding mode for me can be applied to many different topics and takes advantage of everything whiteboarding has to offer: collaborative, interactive, promotes risk taking and visually stimulating.

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Guess and Check with a Partner

Students try to solve problems that take a certain amount of intuition or guesswork (like antiderivatives or factoring) by having one person write down a guess, and the other person check if it is correct.They would then keep doing this until they get a correct answer. After a certain number of problems solved, the two students switch roles. For example, above the students are looking for the antiderivative of x^2 – the guesser writes down 2x and the checker takes its derivative to see if that is correct. Since 2 does not equal x^2, the guesser tries again. They continue this process until they finally get that x^2 back again. This mode is great for showing students that a great way to do math (at first) is to just try things and adjust their answer; it’s great for getting students to converse together about how to get a solution; and it’s great to get them in the habit of always checking their answers. I had a really hard time getting some students to follow the procedure for this one, but the ones that stuck to their roles got a lot out of it.

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Color Coding Problems

Before solving a problem, students rewrite it using different colors to help them understand its important parts. For example, above is a whiteboarding exercise I did with the Chain Rule. Students were in groups of threes – for each problem, one person had to rewrite the problem in different colors to indicate which was the outside and which was the inside function, the next person had to differentiate it still using the colors to point out where each part of the new expression came from, and then the last person had to rewrite the expression in a simplified form. This was perfect because the hardest parts of the chain rule are recognizing when you need, seeing inside vs. outside and then seeing where the parts of the new expression come from.

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The Mistake Game

Groups present solutions to semi-complicated/involved problems on whiteboards, but while presenting their solution, they purposely make a mistake (and not an silly arithmetic mistake like 3+4=8 – a real hardcore-misconception-style mistake). Then, they present their work to the other students in the class, trying to sell their mistake as having been made for real. Other students ask thoughtful questions about the presenting group’s solution to try to help everyone find the mistake. This is always great with a quick class followup at the end collecting the most common mistakes. Check out the Guide to the Mistake Game from Kelly O’Shea, who introduced me to this game.

P.S. I’m realizing now that the example above actually isn’t a great example of a time to use this game… Some topics that it worked well for this year were graph sketching, solving for limits algebraically, using the quotient rule, implicit differentiation, related rates and using infinite limits in graphing exponential functions.

MATH Whiteboarding

I make it standard practice of mine to steal as much as I can from science teachers out there. One of the best things I have borrowed from Physics Teachers has been whiteboarding, popularized by the rise of Modeling Instruction. I feel like lots of math teachers have these smaller whiteboards in their classroom, either individual ones or a good size for groupwork, but they maybe aren’t all that sure how to use them. Physics teachers have figured out a way to use them effectively that makes sense for them. From what I gather, most of these Physics teachers (at a basic level) have students whiteboard problems that they have previously worked out on homework assignments or worksheets and then present those to each other in “board meetings” as part of the Modeling process (sorry if I totally butchered that). It makes sense with Physics and other teachers can join in on the fun because they can see immediately how to use them.

But as great as stealing is,we need a whiteboarding framework for math, so that math teachers can see how to use them immediately and also join in on the fun. We need non-content specific techniques that teachers can use day to day so that whiteboards don’t just get added to that pile of crap in the back of the classroom. There are math teachers in the blogotwittersphere – like Anna (@borschtwithanna) and Timon (@MrPicc112) – who are working on this too, and I hope we can get more collaborators. I am going to devote the next few posts to math whiteboarding, but please join in on the fun if you have done something super cool that might be helpful for others (i.e. me).

but first, some observations from this past year when I started using whiteboards…

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Why Whiteboard?

WHITEBOARDS PROMOTE COLLABORATION. My whiteboards are about 2 x 3 feet, which gives plenty of room for at least 3-4 students to work together on a problem. There is something about working on the same surface (as opposed to working on the same problem in individual notebooks) that gets students talking way more. Whatever written is owned by the whole group, so there is more of a natural desire for everyone to explain to each other and help each other understand, and to healthily debate various aspects of problems.

STUDENTS ARE MORE LIKELY TO TAKE RISKS. The magic of a whiteboard marker is that it erases easily. It’s really not a big deal to make a mistake on the board because it is so easy to change. Students will try things that they never would with a pen and paper.

WHITEBOARDS ARE A NICE CHANGE OF ROUTINE. My students loved using whiteboards just because it was something different. They would work with different groups of students, we would sit in a different spot in the classroom, and the onus of the learning would be on them instead of on me. I don’t think that the idea of changing up the routine is trivial.

ALL TYPES OF LEARNERS ARE TARGETED. With different color markers, verbal exchanges between students, lots of time where students are being active and opportunities for creativity, almost every type of learner is engaged by whiteboarding.

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Issues to Work Out

STUDENTS DON’T HAVE ANYTHING TO REVIEW LATER. Lots of great learning happens during the whiteboarding, but for those that need to review later, they don’t have anything written in their notebooks. I tried taking pictures of the whiteboards and posting online, but I doubt that any students ever really looked at these. A solution that I heard from a colleague this summer was to give them 5 minutes at the end of class to copy down a problem that they would like to look at later. I like this solution, and would like to try it out this year.

SOMETIMES ONE STUDENT WILL TAKE OVER. I found that occasionally whiteboarding would turn into one student writing while a few others sat back and watched. This is a fallback with almost every type of group work. Though they felt a little forced, the best way I found to avoid this is to have specific roles, or structured ways for all students to get themselves involved. This is something I am going to try to document and work on

WHITEBOARDING TAKES A LOT MORE TIME. I’m sure some pros have managed to work whiteboarding into their curriculum without sacrificing pace, but I definitely did not. In fact, I sort of used it to slow things down. If you’re looking to power through material (like I was at times in my AP class), I don’t think whiteboarding is the solution, because its strengths are in allowing students to communicate, construct their own concepts, and spend more time exploring a concepts deeply.

IT IS TOUGH TO GRADE WORK DONE ON WHITEBOARDS. Some teachers have expressed this concern to me – they aren’t handing anything in and the work is done by everyone, which makes it difficult to grade. My solution? Don’t grade it. Not everything that happens in the classroom needs to be graded.

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Getting the Materials

GETTING THE WHITEBOARDS: If you can, I think the easiest way to get whiteboards is to have your department order some, but you can also easily just head down to home depot, buy a huge piece of whiteboard and have them cut it for you. @borschtwithanna describes how she did this here, and @fnoschese describes a few different ways to get them in his classic post describing whiteboards here.

OTHER MATERIALS NEEDED: I found that for my whiteboards, the normal erasers just don’t work well. @mgolding suggests using black socks as erasers (white socks get gross) and @misscalcul8 suggests putting the markers right in the socks as an easy way to distribute the markers and a way to avoid students fighting over which color marker they get. With the whole class using markers instead of just you, I also found that you need tons of markers. Luckily, I don’t have a quota at my school, so I just always go nab some more from the supply closet, but @kellyoshea has a pretty good solution to this problem with refillable markers and marker buckets for each group.

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Student Feedback

Want to know what students think? I collect student feedback as often as possible, so I went into my most recent document and pulled out every comment having to do with whiteboards. Interestingly, most of the negative comments came from my AP class and very few negative comments came from my non-AP class. (emphasis below is mine)

I LOVE when we use the white boards because I get to see how my peers think and compare their thought process to mine. And then when we discuss it with the class its even more helpful. There is so much opportunity to lean with this exercise.

I really liked graph sketching on whiteboards, and then hiding a mistake in it. It was helpful because it helped me remember what common mistakes I should avoid.

You’re doing a great job with the white boards, it give us a chance to work together which is fun and helpful at the same time, we can talk a little but we generally do the work and its more of a competition to me instead of just work so that is an extra motivation

For me, graph sketching on whiteboards and working in groups on packets isn’t really helpful and it’s not your fault. Some kids in my class lack motivation and manners and a desire to improve; so I guess it’s really frustrating to work with them.

One learning activity I found particularly helpful is the activity in which we drew derivatives on whiteboards. I tend to find drawing graphs from functions to be difficult at times, especially when we are focusing on derivatives and so this activity helped me a lot. It allowed me to see common mistakes and different ways of drawing functions and the discussions we had while sketching graphs allowed me to realize my mistakes.

graph sketching on whiteboards [is helpful]. Since it was interactive I learnt much more

I really like graphing sketching on the whiteboards. I enjoy working in groups, then looking at what everyone did. It’s a good way of practice, and we learn from our mistakes.

when we use a white in groups [isn’t helpful] because i get confused when were than one wants to solve it on the board rather than if we work in groups and each one with his/her own paper.

Drawing derivatives on whiteboards [isn’t helpful], because I find that the diffusion of responsibility between the team members decreases their productivity in class.

I think the most helpful class activity that has been very helpful is sketching on the white boards leaving mistakes for others to pick, in this case we can learn where are the possible mistakes occurring and allow you to avoid them when dealing with your own.

I found that [it was helpful] when we graphed derivatives on white boards and slowly drilled each step aswell as common mistakes to avoid. It was perfect to clear any doubts both visually and algebraicly.

Drawing derivatives on white boards, because my classmates and I can discuss different methods of solving a question.

I like using the white boards because it’s nice when we all share our work and see everyone else’s work and compare it to ours and then we look at the mistake and fix it together.

The best activities which I felt that helped me a lot is the group work like the games and the white boards

I think that when we do goup work and work on answering a question on the white-board I feel that one student work and the rest just watch him working which is not as beneficial to everyone.

Drawing in Math Class

One of my favorite ways to start class is by putting out whiteboards with a problem paper-clipped at the top, and names of random groups.  I love it most  because every single person is engaged in mathematics within 30 seconds of class starting. In fact, students always ask me a minute or two before class starts “can we begin?” They can’t seem to resist the markers and the problem in front of them. Also, I found when I wanted to use whiteboards in the middle of class and put students in random groups that it just ate up a few minutes in each class, so this just feels more efficient (I’m kind of neurotic in terms of efficient use of class time).

Continuing my experiments with different modes of math whiteboarding, a great whiteboard warm up I tried was having them illustrate related rates type situations for objects that are changing in different ways. For example:

A pumpkin grows in a garden…
1. With a constant increase in the radius of the pumpkin
2. With a constant increase in the volume of the pumpkin

Then I had them describe what is happening to the rate of change of the important variables (so if dV/dt is constant, what is happening to dr/dt?). We then had a really good full class discussion where students explained their situation. I think this helped clarify for a lot of students the difference between “V” increasing and “dV/dt” increasing, or how just because “dV/dt” is decreasing it doesn’t mean the volume is decreasing.

This was part of a larger goal of mine to focus on big ideas and deep understanding this year – I’ve always asked students interpretation questions on tests (my final this past term had a crap-ton of writing) but I never felt like I actually directly taught them these sorts of things. For Related Rates, we solve all these problems and come up with all these numbers, but never actually talk about why they are interesting problems – the fact that as one aspect of a situation changes, another may change at a totally different rate, and that there is a relationship between all these rates that explain how things change the way they do. And honestly, I think this little activity made a huge difference – on the interpretation question on the Related Rates quiz, tons of students drew pictures to aid their explanations. 15 minutes well spent!

Experiments with Math Whiteboarding

Getting five mini-whiteboards was a real game changer for my classroom. It has completely changed the way that I go about skills-based instruction (which is what a large chunk of the first term of Calculus ends up being) and has added so many new tools to my instruction toolbox. I think that too often in the past I relied on variations of the learn-practice-apply model, and the “practice” part not only always seemed to make class drag, but never really felt effective. Well, doing practice on the whiteboards with some sort of extra little component to make it interesting has proven to be not only interesting (I’ve never seen students more engaged while practicing skill-based math), but far more effective at teaching skills. The only downside to the whiteboards is that they don’t have anything in their notebooks to study at home, but I’m brainstorming ways to fix that (mostly posting pictures of the whiteboards on our course website or not caring because they can always find examples of solved problems in our textbook – it’s the practicing part that matters).

Here have been my favorite things to do so far with the whiteboards and a few ideas for experiments that I want to try in the future. I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire.

1. UTILIZING DIFFERENT COLOR MARKERS

One of the great parts about whiteboards is that you can get get students to use different colors like you do on the board up front to get them to focus on different things. For example, above is a whiteboarding exercise I did with the Chain Rule. Students were in groups of threes – for each problem, one person had to rewrite the problem in different colors to indicate which was the outside and which was the inside function, the next person had to differentiate it still using the colors to point out where each part of the new expression came from, and then the last person had to rewrite the expression in a simplified form. This was perfect because the hardest parts of the chain rule are recognizing when you need, seeing inside vs. outside and then seeing where the parts of the new expression come from.

EXPERIMENT I WANT TO TRY: Mistake Marker. I want to have the students solve problems and then whenever they make a mistake, instead of just erasing part and fixing it, they write over their mistake with the mistake marker color (or make a note with the mistake marker if it’s not possible to write something in that color) and then continue the problem in a new line below with their original color. Then, at the end we can collect as a class the most common mistakes that are made when doing a complicated problem like, say, the Quotient Rule (not complicated you say? Then you have never taught students Calculus who have a terribly weak Algebra background).

2. THE MISTAKE GAME

I know I have mentioned this like twelve times already, but I absolutely love it. This is stolen from Kelly (read her description here), but the basic idea is that groups present solutions to semi-complicated/involved problems on whiteboards, but while presenting their solution, purposely make a mistake (and not an silly arithmetic mistake like 3+4=8, real hardcore-misconception-style mistake). Then, they present their work to the other students in the class, trying to sell their mistake as having been made for real. Then other students ask thoughtful questions about the presenting group’s solution to try to help everyone find the mistake. This is always great with a quick class followup at the end collecting the most common mistakes.

3. ROTATE

When I have wanted to show how a topic from my AP class could be applied to many different situations, I have done some sort of rotation so that students could be exposed to a wide variety of problems (without taking the time in class to do ALL of them). The first exercise with this is a simple gallery walk – each group of 2-3 students solves a problem individually. Then, when everyone is done, the groups rotate around to each station, taking a few minutes at each one. This is far more successful if you give them specific tasks, like “First, check if their answer makes sense, then see how they set up the limit definition of the derivative” or something concrete like that. The second exercise is when each group starts doing a problem, and then the groups rotate after about 5 minutes and they pick up where the last group left off (someone on Twitter gave me this idea, sorry, I forget who!). Then, they rotate every 4 to 5 minutes until all of the problems are completed. I tried to emphasize that while doing this, you must show your work neatly and clearly (an important skill for all math, but especially an AP test) so that the next group can quickly see what has been done to solve the problem and what still needs to be done. The thing that I liked about this was the meta cognitive mapping out of problem solving, though I don’t think I left enough time for students to really think about each one.

EXPERIMENT I WANT TO TRY: Rotate Marker. I didn’t love the rotating problems mid-problem solving, mostly because it stopped students in the midst of great problem solving, but I think one thing I am going to try is having the students rotate within their group who is writing. So a group would be solving a problem and every 2 minutes, the next person in the group would become the writer. This would ensure that all the students in a group are engaged in the problem solving process and that they are all talking math with each other.

4. SORTING

One of my many goals this year is to step back and focus more getting students to figure out how to go about problem solving. One thing I did that I really liked was I photocopied a few pages from the book and cut out like 50 functions for each group that all required a variety of differentiation rules. Then the students made categories on their whiteboards and sorted the functions based on which differentiation rule they needed. It was a really interesting process, especially because many of the functions needed more than one rule. I really enjoyed seeing how students solved this – most just made a bunch of different categories (like Quotient & Chain, Product & Chain, Product & Quotient & Chain), but one group made a crazy complicated Venn Diagram and another made a table kind of like one of those that shows the distances between cities (so like Quotient, Product and Chain both across the top and down the side) then placing the functions at the intersections of the rules they needed. It only took 15 minutes, but after learning so many differentiation rules, I think it was great to give them a chance to step back and figure out what types of rules they needed to use and where. The next day on the quiz, I saw tons of students circle parts of functions and write “Product” and “Chain,” which is something I have never seen them do before. To me, this is a wonderful problem solving strategy that was explicitly identified and strengthened by a quick activity.

 

Overall, I just think the added presence of the whiteboards has given my classroom a much more dynamic feel. When I asked for feedback from my students about how class was going these were my two favorites:

I like how we change up the routine. We do not sit and do the same thing over and over again, its changes up and keeps me interested.

 

One positive thing is the different types of work you give us because it is not all the same thing it is diverse so it keeps things interesting.

I guess it was a good indicator to me that switching up the routine for switching-up-the-routine’s sake is not a bad thing. Knowing some basics about how the human brain works, keeping the kids from sinking into a comfortably numb routine will certainly make everything a little bit stickier.

And in case you forgot in all of my blathering, I’ll repeat my plea from above…  I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire!