October 21, 2014

NWMC Days 2-3 Recap

This is the second post reflecting on the Northwest Mathematics Conference I attended in Portland, OR last weekend.

I See It: The Power of Visualization
Marc Garneau, Chris Hunter

This session started with a great exploration of divisibility using base 10 blocks. It isn’t strictly applicable to my grade level (high school) but was a good puzzle that I will try to keep in mind for all the crazy shortened schedule days in case one falls between units. Kids of all ages find divisibility rules fascinating because they seem like magic, I enjoy finding the math that make tricks work.

I’m most excited about the introduction to radicals. The approach is to focus on the geometric interpretation of a square root – it’s the root (side length) of a square! Take your basic perfect square, say 16. What side length gives the square an area of 16? Four, so four is the square root of 16. Now, what side length gives the square an area of 18? Not a whole number, but that number is called the square root of 18. Maybe we could simplify that a bit? Dividing the 18 into four squares doesn’t really help but nine squares comes out nicely – each side of the small squares is the square root of two, thus the side of the square with area 18 is three square roots of 2. I’m still amazed and I’ve talked through this a few times since the presentation. I was not looking forward to the radicals unit this year, but I am now!

Finally, we looked at dynagraphs. I was just talking to my coworker today about comparing linear and exponential functions. I think this is a great way to see the relationship between input and output – especially since we spend so much time focused on comparing two outputs (many of us in attendance were momentarily confused when the linear graph with a slope not equal to one didn’t step along the way that x+3 does).

Now What? 
Danielle Maletta

The majority of this session was a refresher on things I’d heard at other PD, but those reminders are important because anything not immediately applicable gets lost in the shuffle. She spent some time talking about how to give low status kids a boost. One way was to explicitly mention all the skills needed for a task – we’ll need help organizing and recognizing patterns and … so kids can recognize at least one skill as something they’re good at. Related to that is a “How are you math smart?” poster that I’d love to have in my classroom.

One task I loved was something Danielle calls a treasure hunt. She gives groups of students a pile of cards (equations, graphs, descriptions etc.) and gives them the first clue. Students work with their group to find all the cards that fit the clue and send one person up to the teacher with the clue and the cards. If they found all the solutions they get clue two, if not, they get sent back to their desks. I like this because students have to look at characteristics of your choosing (ex: Is 4 in the domain?) rather than matching by one or two aspects.

Creative Integration: CCSS Math Practices, Quality Activities, 
Excellent Questioning Techniques and Technology
Tom Reardon

We started right off with a rich task:
In how many ways can you use only the digits of 8 and plus signs ( + ) to create an expression whose sum is 1000?
I didn’t think it was a particularly rich task to start (a great sign - it was definitely low entry) but as we went through all sorts of patterns emerged. This is another problem that doesn’t fit particularly well in a particular spot in my curriculum, but this one is great for the last ten minutes of class when my students need something a little different.

The painted cubes problem was also fascinating.
A cube with dimensions n x n x n, that is built from unit cubes, is dipped into a can of paint. How many unit cubes will have paint on zero faces? one face? two faces? three faces?
This one could fit more directly into my curriculum as there end up being four different patterns – a constant function, a linear, a quadratic and a cubic. It is approachable in Algebra 1 and if I use it during the quadratics unit it would reinforce the characteristics of each type of function.

Modeling Mathematics Using Problem Solving Tasks
Andrew Stadel

It’s always a debate for me when I attend a conference – do I want to attend sessions on things I’m familiar with or try to learn about something totally new? This session was a bit of both. I’ve read about most of what Andrew does and I use his Estimation 180 weekly, but I haven’t done many 3 act tasks before. I appreciated the opportunity to play the role of student in this class. Andrew did a nice job of modeling the role of the teacher, but then stepping outside of that role to highlight some choices he made and answer “What if students…?” questions. Even if I don’t jump into 3 act tasks with both feet, I’ll definitely take away this idea: For a sequel/extension question, flip the math. If you gave radius and asked for circumference in the original question, give circumference and ask for radius. Doing the same process with bigger, different or ‘harder’ (fractions, decimals) numbers isn’t much more challenging. Nor is doing the same process in a new context. Instead, change the givens and solve for a new unknown in the same context.

Better Than Engagement
Dan Meyer

I took away two main ideas from Dan’s session:
Dial up the math slowly.
Create the need for notation and vocabulary.
We so often jump directly to the abstract and the precise. We need to specifically ask kids to estimate, to sketch, to predict. Then once they have a baseline, they’ll want to know if they are right. This allows us to dial up a notch and try some examples. Dan asked “If math is the aspirin, what is the headache?” The headache should be something tedious or repetitive or cumbersome (ex: so many examples they wish for a generalization). It’s the teacher’s job to create a headache, then offer strategies and vocabulary as aspirin. This process of exploring first and formalizing later is something I’m good at in geometry. I am still working to find explorations for some aspects of Algebra but I’m getting there – this conference certainly helped build my pool of resources!

Bringing Problem-Solving Into Your Math Classroom
Fawn Nguyen

I had to join this session late as it overlapped with my own. I really would have loved to attend the whole thing. Two quotes struck me in particular. The first, “When you do a problem solving task, choose a good problem from a chapter other than the one you’re currently studying.” Fawn argues that problem solving tasks should not be on the exact same thing that students have been completing, otherwise it will be obvious how to solve them and students lose the experience of problem solving. I agree with this sentiment, but I am not sure how to sell my administration on it. They expect everything to match the objective, the objective to align with the standards and the standards to fit into the unit plan on the district website. As an intermediate step I can use problem solving tasks at the beginning of a unit – this way students don’t yet know what we are studying – and I can use problem solving tasks that bring together a variety of topics – students will know how to do the one aspect we are currently studying but they will need to use problem solving skills for the rest of the task. The second quote, “Not telling an answer is showing respect for students” reminds me that I need to return to my goal for students – “you are responsible for your own understanding.” When I first made this goal my intention was to get students to advocate for themselves when they were stuck. Now I realize this also means students need to be responsible for themselves and not take away someone else’s opportunity to understand. Collaborating is a good choice, but telling answers ruins the experience for the person who hasn’t had a chance to reason their way to an answer.

October 18, 2014

NWMC Day 1 Recap

I attended the Northwest Mathematics Conference in Portland, OR last weekend and it was amazing. I got to hang out with so many awesome people! It was a crazy mix of tweeps and PCMI reunion and people I've never spoken to who have heard of Nix the Tricks (something that never fails to astonish me). I had so much fun that it was totally worth the stress of prepping two days of sub plans and being exhausted and (super) behind for nearly a week.


Friday I finally felt a normal amount of behind, so I started going through my notes and writing them up. Portland State is very generous and they are offering two grad credits to anyone who attended all three days of the conference and writes about it. There were some specific prompts to write to and I learned that my writing voice for papers is different from my blogging voice. At least it feels different. NWMC was great but not so life changing that I sound different, apparently I just write differently when typing into Word than when I compose in Blogger.

On Thursday I attended two workshops:

What makes algebra hard to learn?
Steve Rhine

This session was full of resources and links to even more resources! I look forward to thoroughly exploring the AlgebraicThinking.org website, especially their database of problems used in research. Upon return to my classroom I was pleased to discover some of the apps loaded on our school iPads are from Steve Rhine. Hearing him discuss the intentionality behind these programs makes me much more likely to use them with my students. One in particular is Point Plotter. I never would have understood the goal of this app if I just tried it – a common misconception students hold is that there are a limited number of points between two points on a line (ex: only lattice points count) – this app pushes students to find as many points as they can between two points. This addresses a misconception while simultaneously encouraging students to use the equation of the line and/or definition of slope to calculate points.

I am teaching Algebra 1 for the first time in five years and it’s been an eye (re)opening experience. I had forgotten how challenging it is to explain the basics; I wish I’d attended this session (as well as many others) during the summer so I wouldn’t have made some of the mistakes I already have with my students. Luckily it is still early in the year so I have time to address their misconceptions. One of those is the difference between an expression and an equation. As a mathematically proficient adult, it is quite obvious to me that equations and expressions are very different objects, but when I asked my students about it this week they struggled to differentiate between the two. We have all seen the mistakes where students try to ‘solve’ an expression using inverse operations – Steve Rhine shared that this can be due to students feeling “lack of closure” when their answer is an expression rather than a number as it has been for the rest of their mathematical experience.

Another place where students struggle is in understanding variables. The idea that a variable changes with different contexts but is constant within a context isn’t one I’d wrestled with, let alone helped students wrestle with. The x in problem three has no relation to the x in problem four, however, if problem five is 4x + 4 = 5x students must realize that the two x’s in that equation are the same. As a group we decided that a variable is an unknown quantity because that definition encompasses x=5 as well as x=y+3. Steve emphasized the importance of a variable representing a quantity and not an object. H can represent the height of Harry, but H doesn’t equal Harry. Instead we should refer to variables as containers. It worked out nicely that I had started equations with this pennies lesson because now I can refer to variables as cups of pennies – a mystery quantity.

Our final topic of the morning was graphing. Students struggle to understand the difference between discrete and continuous graphs. One suggestion was to ask students who connect discrete data points what the midpoint represents. For example, if they connect Sally’s height to Ted’s height, is the midpoint meaningful on that graph? Lastly, lines having constant slope is a big idea, but one that students don’t often wrestle with. If we only give students tables where the patterns jump out at them (ex: x values always increase by one) then they don’t get the opportunity to engage with the concept. We need to make sure to give students that challenge them to think and that bring out misconceptions so we can address them!

Fostering Algebraic Thinking and the CC Math Practices
Irving Lubliner

I thought the last session was full of resources, and then I got to this one – we received a bound notebook of activities! I have yet to go through the entire thing and I can’t wait to have a chance to do so. Irving Lubliner mixed teaching practices with content throughout the workshop. He made his teaching moves explicit so we could reflect on those as much as the activities. When someone gives a wrong answer he finds the question that their response correctly answers, in other words, he finds something right about their solution. We practiced for a few responses and it was a fun challenge that reminded me of My Favorite No, an activity I’ve used weekly. It would be great to infuse that spirit throughout the course. He used tickets as rewards for participation and great ideas. I am generally hesitant to give students extrinsic rewards but my school is using the PBIS model so I need to consider it. I appreciate that he rewarded bravery rather than only correct answers.

Just like my morning session, we spent time talking about expressions vs. equations. My notes say *Spend time on this!* so I had better do just that. One example in this session was about language precision – you can’t double an equation, you double each expression. When evaluating expressions he gave a great tip for getting students to remember to use the order of operations – look at the expression in chunks. This process will help students think about terms which is helpful for expressions with variables as well. Underline the expression until you see a + or -, then write “later.” Repeat until you reach the end. 200*2 + (3+4)*53*6. Next evaluate each term (in whatever order makes you happy). Then rewrite each subtraction as adding the opposite. Finally, evaluate the expression (in whatever order makes you happy). This method pushes students to think about the structure of the expression, and also allows them to use their number sense – if the expression has 15 and -15 use those opposites instead of going from left to right.

We played with a great model for solving equations. I’ve never seen the utility of a function machine model until this one: 


Earlier this year I tried to use a representation where we evaluated at a specific value of x to show the ‘forward’ steps, then worked back up to determine the ‘backward’ steps. This diagram is much more intuitive and shows the ‘socks and shoes principle’ clearly. During the session he showed an example with 7 or 8 steps, I wish that example had made it into the book – it made a very complicated equation seem easy. I will have to search for or recreate it since my students still need more practice solving equations.

Update 10/20/2014: The internet is awesome. Someone from NWMC saw this post and sent it to Irv and he responded with the image I was wishing for!

October 2, 2014

Notebooks and Binders

Apparently I have a lot to say after the first month of school! The last four posts have been about things that are going well. Today I'm trying to figure some stuff out that hasn't gotten as smoothly as we'd hoped.

In the past we've used a binder system where students have sections to organize their papers and a specific system for notes in their reference section but the rest of the classwork was disorganized and frequently ended up in the recycling bin. This year we wanted to keep the binders for organizing assignments and reference materials but add a spiral notebook for classwork. We came up with a plan to merge several routines into the notebook. Then we started...

First, students struggled with the instructions "on the left hand page." So we drew a sample notebook on the board to model leaving it open and writing on the left hand side. Every time they are supposed to write a new heading in their notebook I add it to the model on the whiteboard. And sometimes I provide sentence frames (because IEP/ELL/9th grade).

Second, students don't want to write with their notebook flat, they always fold it back. This sadly defeats the purpose of starting on the left hand page - if they write their goals on the left we can still stamp while they're writing on the next page. But if they fold it back the spot to stamp rapidly disappears.

Finally, putting everything in the notebook means we have to look at the notebooks. We thought it would be convenient to have the stamps, quizzes and journals all in one place. However, we did not think about how annoying it is to get out the crates, open up their notebooks, find the correct page, and then finally be able to grade their work. I love the convenience for the kids of having everything in one place but I'm not finding it convenient for me.

The things that are currently in the notebook:
Stamps (ideally I'd like to tally them at the end of the week)
Do Now (not graded)
Classwork (not graded)
Quiz (graded)
Journal (read and responded to)

Last year:
Stamps were on small papers that got lost and not tallied
Do Now and Classwork were on lined paper that ended up in the recycling bin most days
Quizzes were on quarter sheets of scrap paper that got handed in, graded and then kids frequently lost them
Journals were on a full sheet that we collected on Fridays, read and responded to, handed back and then kids put them in the recycling bin (or in their binder, I should give some kids credit, but they never looked at them again either way)

Thought that occurred to me just now:
Stamp chart and space for the journal on a half sheet. Days with quizzes they'd take the quiz on the back. Then I'd actually look at all the things regularly. And a half sheet is big enough to hole punch and put in their binder but small enough to fit on the corner of their desk. Plus if the reflection is attached to the quiz they might even look back at it? Real possibility here! I'll offer this suggestion to my co-teacher and brainstorm further.

So I'm left with a question:
How do you use notebooks? Do you grade anything in them or have them do work in the notebook and hand in graded assignments separately?

October 1, 2014

Number Sense, Logic, Perseverance

Four of my five blocks this year are dedicated to teaching Fundamentals of Algebra 1. It's a double block course so I have two classes. They are filled with students with IEP's (one class is all IEP's and the other is mixed but the kids turned out to be about the same level despite a technical difference in the labels on the courses) and so I get to work with my awesome co-teacher from the special education department (this is our fourth year teaching together!). Since we have so much time we knew we wanted to dedicate time to the essentials of number sense, logic and perseverance in addition to the core concepts of Algebra 1. Students in the fundamentals courses tend to struggle with being students, those are skills we wanted to teach. These students have frequently lost their curiosity, we wanted to re-awaken it. Students with disabilities need to learn how their brains work, we wanted to help them discover techniques that help them learn.

Do Nows:
Mondays: Mental Math
Tuesdays: Visual Patterns
Wednesdays: Math Arguments or Would You Rather?
Thursdays: My Favorite No (this is the only day where the do now is related to the week's topic of study)
Fridays: Estimation 180

Some kids have already learned to tell the day of the week from the type of problem on the board. "I didn't know it's Friday! Woo!" And they know what's coming: "Are you going to pick your favorite wrong answer?" There's a balance of structure and variety. While each day focuses on a different skill, they all focus on the math practices which are an essential foundation for math class. When we do these problems they have to complete the task to the best of their ability individually. Then I take contributions from most if not all of the class (everyone's estimate, everyone's vote or everyone's prediction). Finally, students are asked to explain their answer. Even in the first month they've made great strides in respectful disagreement and sharing their reasoning.

In past years my co-teacher and I have kept a pile of logic puzzles, connect the dots, find the hidden object and other similar sheets. When kids finish an assignment early or need a break (say, after taking a test) we offered them the choice of any of those pages. They promote attention to detail and perseverance among other things. This year we knew we'd have freshman who would need to develop these skills and also might need a break more frequently than our students have in the past. We managed to nab a table and a couple extra chairs to set up in one corner. We took all of our games and puzzles and pages of challenge problems and put them together on this table. Students are motivated by the idea of having time to do puzzles and get their work done efficiently. We also direct kids there for a break if they need one. They don't think of it as developing their logic skills, they think of it as a game to play!

Last year we gave kids stamp charts to give them regular feedback on whether or not they were meeting the goals we set. This year we are having them write the goals in the margin of their notebooks and stamping there. I'm undecided if that's better or worse than the small charts which would get lost between papers. We started with just "ready, on task and off task" and now we're adding "on topic and off topic" to get them to work together. But it's hard to stamp their notebook unless they're still on the first page of the day. Things we're still figuring out...

September 30, 2014

Long Blocks, Low Tolerance for Stillness

I have some seriously antsy students. They are pencil tappers, foot jigglers, pen clickers, finger strummers... You name it and they will find a way to wiggle it and make noise. On Friday afternoon they were bursting with energy. They were supposed to be taking a test and everyone was squirming in their seats. Some successfully settled by taking a walk to the water fountain, others needed a stress ball to move without making noise and the final student needed to sit out in the hall to stay focused.

I teach kids with ADHD. I teach kids with learning disabilities that make focusing for longer periods of time tiring. I also teach kids. I get bored sitting and listening without doing anything. Kids shouldn't have to sit still and be quiet. It's not natural or healthy. Here's how I help my kids balance learning with their inclination to wiggle.

A 90 minute block is a long time. My co-teacher and I happily allow students to use the bathroom or get some water during work time. The school has a rule of no passes during the first 10 minutes or the last 10 minutes of class which means they won't miss the do now or exit. Otherwise, we let students go whenever they ask. Luckily we haven't had any issue with students wondering the halls - the bathroom is close to our classroom but even that brief chance to stretch their legs is often enough to get blood flowing and refocus. Some kids are in the habit of asking to "go for a walk" which we do not allow. They are welcome to walk but they need a destination.

Fiddle Toys:
I tried telling students to strum their fingers rather than tap their pencils, but they were just as loud with that so we needed to find some alternatives. Stress balls are working great so far (I have a variety - plastic-y, cloth, cloth and fuzz combined - they are all squishable and quiet). If students struggle with these (some are tempted to throw them rather than keep them in their left hand while they write with their right hand) we will try sticking some velcro to the desks next. If you put the two pieces side by side the texture difference between the loops and hooks can be enough stimulation to quiet the busy mind.

Balance of structure and flexibility:
At the beginning of every class students come in, get their binders from the crate, put their homework on their desk (assigned seats) and set up their notebook according to the goals on the board (ready, on task, off task go down the margin). Then they start the do now. A minute after the bell rings I walk around the room, check homework and stamp everyone's 'ready' if they have the goals written and have started the do now. This section is very structured so students know what to do when they arrive and can transition into math class.

In the middle of class students can choose to work with other students or independently. They can stay at their assigned desk, join another person at their desk (this part of the room has desks positioned in rows to face the front) or move to one of the groups of desks. We frequently set up stations so they have to move from one group of desks to another. This part of class is self paced. When students finish the assigned task(s) they are allowed to take something from the puzzle table. We have decks of cards for playing integer war, jigsaw puzzles, math bingo and more. They help kids develop logic and perseverance but the kids think they're getting away with not doing math!

At the end of class students return to their assigned desks to reflect. They have to respond to two questions - one about goals or more general things and one about a specific math concept from the day. Then they need to clean up their areas and return to their assigned desks at which point they receive their homework. 

Some things are still works in progress but overall this flow is working for us, and more importantly, it's working for the kids. They are learning and the pen tapping isn't driving me insane!