Thursday, December 1, 2016

Promoting Productive Struggle & Implementing Formative Assessment Lessons

We had an amazing meeting yesterday with our district's Math Design Collaborative yesterday for training on implementing FALs. One of the resources we went through was this article of 8 Teaching Habits that Block Productive Struggle in Math Students. It's kind of a what-not-to-do guide to teaching math. I also like that they paired it with this infographic poster of what to do instead. 

Our district is involved in a 3-year initiative with SREB, and I really couldn't possibly be more excited about this. Our best teachers have been selected (2 per school) to participate. We have teachers from every grade level 6-12, and from all types of schools. (Even our alternative school is participating!!)

During year 1, we have 8 days together as a group, and yesterday was Day 5. PD focuses on how to implement Formative Assessment Lessons (FALs), which are housed at

Now, when I was in the classroom, I was aware of this website, but hadn't implemented any of the lessons in their entirety. (To be REALLY REALLY honest, I had just stolen a few tasks and card sorts, and not used anything else.)

I had NO IDEA that all of the resources on this website are research based and are most effective (by far!) as complete lessons. 

Here's the basic idea, but you'll really get a better picture by reading through one of the scripted lessons. 

  • FALs are separated into 2 categories:
    • Concept Development (named based on content)
    • Problem Solving (name based on context)
  • Concept Development FALs are meant to be used one-half to two-thirds of the way through a unit. The goal is to figure out what kids know, what they don't know, and then use that to guide instruction (both through the remaining part of the unit, and to change how you teach that content next year). Sometimes these FALs also work at the beginning of a unit to review prerequisite content and guide the transition into new content. 
  • Problem Solving FALs can be used any time during a unit, and are structured around really great problems with plenty of arguing potential. ;)

Here's the basic process to go through a FAL (I'll use a concept development FAL as an example, because so far it's the one I've worked with the most)

The day before the FAL: 
  • Give the pre-assessment as an exit slip.
  • That afternoon, sort the pre-assessments into 1,2, & 3 point piles (1 - little to no understanding, 2 - demonstrates some understanding, 3 - demonstrates understanding). Then use these piles to create homogeneous pairs of students (so the top 2 kids are paired together, then the next highest 2, etc). This isn't a formal grading process. 
  • Once pairs are developed, and while the results on the pre-assessment are fresh in your mind, choose (or create) some feedback questions from the script. They should be based on the major misconceptions, obstacles, or gaps in learning you observed on the pre-assessments. 
  • Make sure you have all materials & cards prepped and ready.
Day of the FAL:
  • Follow the lesson script through the whole class intro, collaborative activity, sharing & whole class discussion, and then administer the post-assessment. 
    • Whole class intro: usually involves white boards and some powerpoint slides. During this portion, you're just reminding kids of the work they did on the pre-assessment, and not "teaching." Just ask them some guiding questions to get them to notice differences in each other's responses. 
    • Collaborative activity: when students work on activity (usually a card sort) in the pairs you designed based on their pre-assessment. Just give the time allotted in the script, and let go of the idea of completion. Just let each pair get as far as they can in the time given. During this time, project the feedback questions developed the day before.
    • Sharing/Whole Class Discussion: usually involves some time to combine/change groups and compare answers, then a return to the whiteboards to discuss as a class. 
    • Post-assessment is "graded" (but not really) the same way that the pre-assessment was, so that you can measure growth for each class.
My personal favorite FAL? Right now, it's Generating Polynomials from Patterns
Students use dot patterns to develop polynomial expressions for the white, black, and total dot patterns and WOW do they have to do some serious work with this one! It seriously challenges the advanced kiddos without being inaccessible for lower-achieving students.

Second runner up is Applying Properties of Exponents. I had a huge "AHA" moment with this one. How many times, when we're teaching laws of exponents, do we pretend like addition and subtraction of terms just cease to exist for a week or two? I'm definitely guilty. Here are the first few cards from this FAL so you can see what I'm talking about:

But if we choose not to shy away from types of problems that aren't immediately simplified using one application of one exponent property, our students are all the better for it. And this FAL does a phenomenal job of facing those obstacles, misconceptions, and gaps in learning square in the face.

Each of the FALs is designed to promote productive struggle in students. Each one is also designed to promote valuable, serious mathematical discourse. And that's something we should all strive to include more of in our classrooms.

And don't forget about the What-Not-To-Do and What-To-Do-Instead for Promoting Productive Struggle that I mentioned in the beginning of the post from the MIND Research Institute blog

Tuesday, September 27, 2016 - Keeping up with the internet

There is just so much "out there" in the interwebs. 
Articles, tweets, ideas, pins, blog posts...

I use Bloglovin to keep up with the blog posts, but I only read through every couple of days. Which means some things fall through the cracks. Not to mention, it's only grabbing content from blogs I've already found. 

So I stumbled upon Paper.Li
I think someone must have tweeted it out with a #MTBoS hashtag, because I can't think of another way that I would have actually paid attention to it. But ever since, I keep finding new public paper.lis to follow!

Basically, lets you build a newsletter out of tweets, articles, or basically anything out on the internet. It's really just a content collector. But I really like it because other people have built paper.lis that align with my interests. (I could build my own, but I feel like the ones I've found online are already so great that I don't need to.)

I keep the links on my bookmark bar, and when I get a free moment, I browse through a couple. I've found some new blogs to follow, and plenty of great ideas. Here's the list from my bookmarks bar:

If you know of a great one I didn't list, please let me know! I really love finding new ideas, and this format just works for me. It kind of reminds me of Flipboard, but I never really found it useful because I didn't take the time to fill in all of my interests/fav blogs/etc. I like that with, I can just flip through everyone else's pages!

Friday, September 23, 2016

Content Literacy in Secondary Math

Wow - this blog's gotten a bit dusty. But here I am. Let's just pretend it's been 5 minutes instead of ... too long. 

I've been asked to work with a small group of teachers regarding content literacy strategies. We'll be meeting in groups of around 7 teachers (all math and science) during their planning periods. The school's improvement goal this year is to improve student performance on the ACT and English 2 state exam. The idea is to support that goal through implementation of some new strategies & some serious changes in expectations for reading and writing across the various content areas. 

I'll admit, I haven't done much work on this subject in math, but I'm excited to learn some new techniques and get to researching! Obviously, if we want students to have meaningful discourse and justify their thinking, we need to promote academic language and writing in our math classrooms.

Of course, my first thought was to check the TMC and GMD archives, but I didn't find very much. If you know of something I missed, please comment below!!

One of the articles I found listed 10 ways Literacy Can Promote a Deeper Understanding of Math.  I found it interesting that most of the items listed are things we do naturally in math class, but that it's important to formalize verbal explanations on paper. I thought a great "baby step" was to take discussions and put them online, which encourages students to write, edit, and read other responses. 

I also reached out to our county's SIOP & Differentiation coach. She does a training every semester on specific strategies for math to help our EL students. She has some incredible resources - check out her blog and resource site!

A couple of my favorite resources from her site are the Glossary of Strategies & Activities and the Bloom's Question Stems for ELs.  These resources are meant to support EL students, but the idea is that they're necessary for ELs, and beneficial for all students. I think so many times, it's easy to forget about this population within our schools. However, I'm trying to remember that ELs represent a microcosm of our larger student population when it comes to struggles with literacy and academic vocabulary. 

Link up any math content literacy resources/articles/etc you've got below! 

Saturday, January 30, 2016

Using Mobiles to Solve Equations

This is a great idea for students to really show what they know with numbers and solving equations. 

I found the link on one of Resourceaholic's Gem lists. The post led me to Don Steward's Median blog. This is his activity. There are 3 posts: 'mobiles,' then 'mobile inequalities,' and finally 'mobile moments'.

I really like these activities! They are so adaptable to just about anything. Elementary students through high school students could work with adapted versions of these. Students do so much halving, doubling, sequencing, comparing, etc.

I can also see adapting this for fractions or decimals.

Maybe radians? Definitely integers...

This could even become an activity framework. It could be used for something like basic function notation, students evaluate to a number, then use that number to balance. (Almost as a self-check method)

Have a great weekend everyone!

UPDATE: There's also a website called SolveMe, which has these same types of problems, without numbers or expressions. It uses the same logic and reasoning skills, but with a bit friendlier interface. Here's the link

Friday, January 29, 2016

Do you know where 2/11 is on a blank circle?

I've been going through the gem list from resourceaholic. I found it about a month ago, bookmarked it, created a checklist in Trello to keep track of my progress, and have been slowly working my way through all 40-something lists. There are so many great ideas, links, and other people to follow embedded in these short, 5-item posts. When I finish up a big project or need a brain break or just need to be inspired, I open up the next one and explore a little. 

Today, I got to Gem #15

The first item is this cute little fraction applet, called Slice the Pie
To play the game, you hover over a circle, and click where you want the shaded part of the circle to stop. (So you could shade a tiny slice, or a giant slice)

The object of the game is to shade the amount of the pie indicated by the fraction shown. However, there aren't any partitions shown (just a blank circle). You have to use your own number sense to get as close as you can. Do you know where 2/11 is on a blank circle? 1/12? 23/30?

I love the estimation skills, the number sense, & how relative number sizes are all a part of this easy, quick, little game. I can even see using something like this when starting the unit circle. (Can you shade for pi radians? pi/4 radians?)

My first try, I scored a 49. Can you beat that?

Tuesday, January 26, 2016

My Favorite: Trello #MTBoS Blogging Initiative

Warning: This post is LONG. If you hate reading & just want my favorite thing, go here and explore. Or scroll ALL the way down. Have fun. 

Time to circle back to prompt #2 from the MTBoS Blogging Initiative

Here's the prompt: Our week two blogging challenge is to simply blog about one of your favorite things.  Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well.  It does not have to be a lesson, but can be anything in teaching that you love!  It can also be something that you have blogged or tweeted about before.  Some ideas of favorites that have been shared are:
  • A lesson (or part of one) that went great
  • A game your students love to play
  • A fun and/or effective way to practice facts
  • A website or app you love to use in class
  • An organizational trick or tip that has been life changing
  • A product that you use in your classroom that you can’t live without!
My favorite thing lately is Trello, and it fits #2, 3, 4, & 5 on the list above!
Trello will change your life if you let it, I promise. ;)

I'll walk you through what it is, & the amazing education potential, and then I'll share the ways I like to use it. 

What is it?
From the Trello site:
Trello is the free, flexible, and visual way to organize anything with anyone. Drop the lengthy email threads, out-of-date spreadsheets, no-longer-so-sticky notes, and clunky software for managing your projects. Trello lets you see everything about your project in a single glance.

Trello is really the ultimate organization tool. You can very visually organize projects, lesson plans, ideas, or even use it like Pinterest. It's easy to collect and organize and reorder ideas, attachments, and links. It's incredibly easy to use on any device. Sharing is easy. Updating your progress on items is easy. And it connects with IFTTT. (Can you tell I love it yet?!? haha)

Trello has 3 parts: 
  1. Boards (think like a Pinterest board - these are general topics, like Unit 1) 
  2. Lists (columns, or sub-boards, such as topics within a unit)
  3. Cards (individual pieces of information, like activities for a specific topic)
I should mention that cards can include labels, color coding, due dates, attachments, links, comments, pictures, and can be assigned to certain people. 

Trello is free as long as you're ok with solid colors for backgrounds instead of custom pictures. Here's the link to the welcome board shown so the hands on peeps can understand what I'm saying. 

Ok, so what? Why should I care?
Trello has infinite potential! 
In my personal life, I have boards for grocery lists, organizing quilt projects in different stages, keeping track of what's in the freezer (we do OAMC in my house), keeping track of chores & other good habits, gift ideas for family, and so much more. My husband is shared on several of these, so he can add/edit as needed. I'm a list girl, and Trello helps me keep all my lists electronically.

Professionally, I find new ways almost every week to use Trello. 
Here's the rundown. 
  • Professional Development Approval & Progress
    • My dept shares a board to discuss PD opportunities we will offer. We attach handouts, make to-do checklists, and communicate approval of funding all in one place. As each PD session moves through various levels of approval, its card progresses into new lists on the board, until it's finally sent to True North Logic, our county PD software.
  • Brainstorming, & To-Do Lists
    • categorized, with attachments or links as needed 
    • I use this board as my "thoughtbox." 
    • Future blog post ideas, activities or foldable ideas, etc. stay here until I get to them. 
  • I have "Resource" Boards for various math topics
    • the lists are things like "quadratics" and each card is a link or idea
    • I also have a list of places I like to look for questions, activities, etc.
  • Sorting, ordering, etc activities. 
    • Example: Exponential Equations Warmup
      • Students drag & drop matching cards together into a new list.
    • I use Trello the same way I like to use the Post It Plus app (blog post for that is here). Trello has the advantage of being possible on ANY device, including Chromebooks. Post It Plus is iPad/iPhone only.
    • I want to make some ordering activities too - FDP, integer ops, etc. 
    • This is totally graph/picture/equation compatible - you just need a screenshot for the card image. 

I also use Trello to collect my favorite things - very much like Pinterest. Here's the link to my Review Games board. Most ideas are from other MTBoS blogs. 

Trello even has inspiration boards for you to look at, to get an idea of what it's capable of, in specifically education-related ways.  There are some really cool ones - I especially like the project board & classroom newsletter. 

Have fun exploring!

Monday, January 25, 2016

Better Questions #MTBoS Blogging Initiative

I may have skipped prompt #2 - I'll get back to it. :)

In the meantime, I have this great resource to share that fits perfectly with the prompt for this week

This resource was shared with me by one of our county's elem math curriculum facilitators. I've had a shortened version of this taped to my classroom document camera for several years, but never knew where it came from (I got it at a conference as a handout once upon a time, and didn't realize its value until I got home and looked at it). I am SO HAPPY to have the entire list, and to be able to give credit to Dr. Gladis Kersaint. This is a GREAT list!

100 Questions That Promote Mathematical Discourse

Dr. Gladis Kersaint

Help students work together to make sense of mathematics

  1. What strategy did you use?
  2. Do you agree?
  3. Do you disagree?
  4. Would you ask the rest of the class that question?
  5. Could you share your method with the class?
  6. What part of what he said do you understand?
  7. Would someone like to share ___?
  8. Can you convince the rest of us that that makes sense?
  9. What do others think about what [student] said?
  10. Can someone retell or restate [student]’s explanation?
  11. Did you work together? In what way?
  12. Would anyone like to add to this?
  13. Have you discussed this with your group? With others?
  14. Did anyone get a different answer?
  15. Where would you go for help?
  16. Did everybody get a fair chance to talk, to use the manipulatives, or to be recorded?
  17. How could you help another student without telling the answer?
  18. How would you explain ___ to someone who missed class today?
Refer questions raised by students back to the class.

Help students rely more on themselves to determine whether something is mathematically correct

  1. Is this a reasonable answer?
  2. Does that make sense?
  3. Why do you think that? Why is that true?
  4. Can you draw a picture or make a model to show that?
  5. How did you reach that conclusion?
  6. Does anyone want to revise his or her answer?
  7. How were you sure your answer was right?

Help students learn to reason mathematically

  1. How did you begin to think about this problem?
  2. What is another way you could solve this problem?
  3. How could you prove that?
  4. Can you explain how your answer is different from or the same as [student]’s?
  5. Let’s see if we can break it down. What would the parts be?
  6. Can you explain this part more specifically?
  7. Does that always work?
  8. Is that true for all cases?
  9. How did you organize your information? Your thinking?

Help students evaluate their own processes and engage in productive peer interaction

  1. What do you need to do next?
  2. What have you accomplished?
  3. What are your strengths and weaknesses?
  4. Was your group participation appropriate and helpful?

Help students with problem comprehension

  1. What is this problem about? What can you tell me about it?
  2. Do you need to define or set limits for the problem?
  3. How would you interpret that?
  4. Would you please reword that in simpler terms?
  5. Is there something that can be eliminated or that is missing?
  6. Would you please explain that in your own words?
  7. What assumptions do you have to make?
  8. What do you know about this part?
  9. Which words were most important? Why?

Help students learn to conjecture, invent and solve problems

  1. What would happen if ___? What if not?
  2. Do you see a pattern?
  3. What are some possibilities here?
  4. Where could you find the information you need?
  5. How would you check your steps or your answer?
  6. What did not work?
  7. How is your solution method the same as or different from [student]’s?
  8. Other than retracing your steps, how can you determine if your answers are appropriate?
  9. What decision do you think he or she should make?
  10. How did you organize the information? Do you have a record?
  11. How could you solve this using (tables, trees, lists, diagrams, etc.)?
  12. What have you tried? What steps did you take?
  13. How would it look if you used these materials?
  14. How would you draw a diagram or make a sketch to solve the problem?
  15. Is there another possible answer? If so, explain.
  16. How would you research that?
  17. Is there anything you’ve overlooked?
  18. How did you think about the problem?
  19. What was your estimate or prediction?
  20. How confident are you in your answer?
  21. What else would you like to know?
  22. What do you think comes next?
  23. Is the solution reasonable, considering the context?
  24. Did you have a system? Explain it.
  25. Did you have a strategy? Explain it.
  26. Did you have a design? Explain it.

Help students learn to connect mathematics, its ideas and its application

  1. What is the relationship of this to that?
  2. Have we ever solved a problem like this before?
  3. What uses of mathematics did you find in the newspaper last night?
  4. What is the same?
  5. What is different?
  6. Did you use skills or build on concepts that were not necessarily mathematical?
  7. Which skills or concepts did you use?
  8. What ideas have we explored before that were useful in solving this problem?
  9. Is there a pattern?
  10. Where else would this strategy be useful?
  11. How does this relate to ___?
  12. Is there a general rule?
  13. Is there a real-life situation where this could be used?
  14. How would your method work with other problems?
  15. What other problem does this seem to lead to?

Help students persevere

  1. Have you tried making a guess?
  2. What else have you tried?
  3. Would another recording method work as well or better?
  4. Is there another way to (draw, explain, say) that?
  5. Give me another related problem. Is there an easier problem?
  6. How would you explain what you know right now?

Help students focus on the mathematics from activities

  1. What was one thing you learned (or two, or more)?
  2. Where would this problem fit on our mathematics chart?
  3. How many kinds of mathematics were used in this investigation?
  4. What were the mathematical ideas in this problem?
  5. What is the mathematically different about these two situations?
  6. What are the variables in this problem? What stays constant?

Google Doc version here.
Stay warm, friends! (And enjoy your snow/ice day GCS friends!)