Tag Archives: Math

Fascinating Data: One step closer to a thinking classroom


Are your students lacking motivation to begin a task?  Does their typical discussion involve headphones and iPhones?  Do they stop working at the first sign of trouble?

If so, read on…

As is often the case, I Googled one thing and found myself several links later reading  Peter Liljedahl’s research on thinking classrooms and was fascinated. His research included 300 teachers, the majority of whom taught 6th-12th grade, on the elements that supported or impeded a thinking classroom.

“A thinking classroom is a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together and constructing knowledge and understanding through activity and discussion. It is a space wherein the teacher not only fosters thinking but also expects it, both implicitly and explicitly.”   ~Peter Liljedahl, Associate Professor at Simon Fraser University, Canada

One of the elements Liljedahl found impactful was the student workspace.


And this is where it gets interesting.

Liljedahl  looked at five different workspaces.  He gave each group of 2-4 students only one pen to ensure group work, then gave students a task to solve.  The five workspaces included:

  • a wall-mounted whiteboard (vertical, non-permanent (i.e. easy to erase))
  • a whiteboard laying on top of their desks or table (horizontal, non-permanent)
  • a sheet of flip chart paper taped to the wall (vertical, permanent (i.e. can’t erase marks))
  • a sheet of flip chart paper laying on top of their desk or table (horizontal, permanent)
  • their own notebooks at their desks or table (horizontal, permanent).

Eight data points were collected to measure the effectiveness of each of the surfaces.

  1. Time to task
  2. Time to first mathematical notation
  3. Eagerness to start (A score of 0, 1, 2 or 3 was assigned with 0 assigned for no enthusiasm to begin and a 3 assigned if every member of the group were wanting to start.)
  4. Discussion (A score of 0, 1, 2 or 3 was assigned with 0 assigned for no discussion and a 3 assigned for discussion involving all members of the group.)
  5. Participation (A score of 0, 1, 2 or 3 was assigned with 0 assigned if no members of the group were active in working on the task and a 3 assigned if all members of the group were participating in the work.)
  6. Persistence (A score of 0, 1, 2 or 3 was assigned with 0 assigned if the group gave up immediately when a challenge was encountered and a 3 assigned if the group persisted through multiple challenges.)
  7. Non-linearity of work (A score of 0, 1, 2 or 3 was assigned with 0 assigned if the work was orderly and linear and a 3 assigned if the work was scattered.)
  8. Knowledge mobility (A score of 0, 1, 2 or 3 was assigned with 0 assigned if there was no interaction with another group and a 3 assigned if there were lots of interaction with another group or with many other groups.)

Here is the data:


Non-permanent surfaces outperformed permanent surfaces in almost every measure. Are students more willing to take risks when they are working on non-permanent surfaces?

Vertical surfaces outperformed horizontal surfaces in almost every measure.  The act of standing reduces the ability to hide.

Vertical whiteboards decrease the amount of time it takes students to get something on their surface;  from almost 2 1/2 minutes down to 20 seconds!  Eagerness increases when moving to a vertical whiteboard – a perfect 3!   And, Participation and Discussion jumped from less than 1 with a notebook to close to 3 with a vertical whiteboard.

How cool is that?!  This is impactful data!  

Get some white boards, people! Get them on the wall! Get them now!!  

And, let me know how I can help!


P.S. If you are in need of super cheap whiteboards, just laminate a piece of construction paper or tag board!

P.S.S. Find a concise summary of Liljedahl’s research of the 9 elements of a thinking classroom here.


Warming-up: Just do it

As I age, I’ve noticed that warming up before exercise is becoming more and more important. Walking a couple of blocks before I start jogging really does help me get my body and mind ready for the task ahead and avoid injury. When I think about applying the metaphor to school, it makes sense: I may not get “injured” in the academic aerobics in which I participate, but it’s certainly helpful to take a bit of time to reset and remember, before engaging in new learning. Thus, whether anticipating physical or mental exercise, I am a proponent of warm-ups.  Like preparing myself for a run, I find them to be a way of refocusing my students’ minds on math during the first five minutes of the period.  The key word in that sentence is “five.”  Unless I set a timer for myself, that five minutes can easily turn into 10 or 12. Like when I run, if I don’t push a little harder, my progress is going to be slow.

Forty-five minute class periods do not allow teachers to be as effective as we would like to be with our students.  When trying to figure out how to structure that precious time, we have to think hard about what which pieces have the greatest impact on student learning.  One piece that is often under fire is the daily warm-up; I’m here to remind you: it’s worthy!

...but warm-ups are!

…but warm-ups are!

Beyond using warm-ups to review the concepts of the previous day and/or to preview the day’s upcoming lesson, Jessica Bogie a high school level Geometry and 6th grade math teacher (and blogger – Algebrainiac),  proposes that warm-ups are good for conversation about math ideas – a worthy idea! Jessica hosted an episode on  Global Math called Warm-ups = What Are They Good For? .    She suggests a two-week rotating schedule of warm-ups:

Two-week warm-up rotation idea.

Two-week warm-up rotation idea.

I’ve blogged before about my love for Estimation 180 here and here  and Would you Rather here, so I was happy to see both in her 10-day rotating schedule.  I love the Visual Patterns site and Math Mistakes well and will likely blog about them in the future.

With much buzz about Carol Dweck’s Mindset theory applying in education, having a Mindset Moment Monday every month is  a great way to continue the conversation all year.  Check out the list of short videos that Marisa from the blog, La Vie Mathématique, posted on the topic.

Another warm-ups resource Jessica mentions, is Lisa Bejarano’s Filing Cabinet of Warm-ups on the Crazy Math Teacher Lady blog.  Lisa teaches Geometry and blogs about the lessons she teaches. This list repeats some sources I’ve listed but offers new ones as well.

So, praise be to warm-ups; just as physical ones get our bodies ready for the exertion of exercise, these mental ones get our minds ready for the hard work of learning!

Academic Vocabulary, Language, and Math

Ask yourself this seemingly simple question: What’s the difference between Academic Vocabulary and Academic Language? They sound similar, right?  We not only need to recognize the similarities and differences of these two concepts, we need to provide opportunities for all students to engage in both in our classes, including Math. Earlier this month, I had the privilege of collaborating with our district’s Sheltered Instructional Coach, Maranda Turner, and six of our talented high school math teachers doing just that.

Our work was centered on deepening our understanding of instruction that supports all students, particularly students struggling with language and literacy in math. We walked away with a solid understanding of the difference between academic vocabulary and academic language, along with engagement strategies to help us plan specific activities aligned to research based strategies to help all learners succeed.

Vocabulary Word Box used during the session.

Illustrated Vocabulary organizer used during the session.

The first part of our morning was spent making sure our next unit was aligned to the Common Core Learning Targets in our pacing guide. Research by John Hattie shows students make academic growth when we communicate and engage our students in learning targets. Besides students being able to really answer their parent’s age-old question: “What did you learn at school today?” they also have a defined purpose for doing the work being assigned to them.

Algebra 1 Semester 2 Unit 7

Algebra 1 Semester 2 Unit 7

The rest of the day we zeroed in on the particulars of academic vocabulary (words specific to the content area) and academic language (how to communicate the vocabulary in a “math way”). Each teacher pair chose 2-3 priority academic vocabulary words for the next unit, determined an appropriate activity, identified opportunities to practice the academic language, and establish where it would fit into the unit. Activities included the use of Illustrated Vocabulary Boxes, Frayer Models,  (low/no prep) Word Sorts, and Sentence Stems and Sentence Frames.  We discussed the importance of allowing students plenty of written and spoken rehearsals as they worked to use new academic language.  

Academic Language Frame for Math.

Academic Language Frame for Math.

After each instructional component (learning targets, academic vocabulary, academic language), teachers were provided 30-60 minutes to plan these tools into their instruction. It wasn’t nearly enough, considering the work needed in all units, but it’s the right work, and teachers appreciated the supportive start.  We know that when change is hard, we must narrow the path.  Consider this: take one strategy; use it once or twice a week in one course. How soon would the strategy become automatic and in all courses?   

Let me know if you are interested in spending some time working on how to support kids in their math language and literacy. I know a coach….


Salmon Tracking: Problem-Based Learning in Math

I had the opportunity to work with a handful of math teachers in our district this last week to create a problem-based unit.  The catalyst was the result of a paradox: an emphasis on problem solving vs. the time to teach the vast math standards at the depth of knowledge required.  We are experts at teaching procedural fluency.  Some of us try to incorporate conceptual understanding – if there is time.  Few, if any, of us incorporate an application piece – again because of time.

So a question surfaced: What if we created a problem-based unit that would allow us to bring balance to our instruction?  

After getting principal approval by way of funding, we met to try our hand at creating a problem-based learning unit.

As luck would have it, earlier this week Robert Kaplinski wrote a blog titled, “Problem-Based Learning FAQ.”  It referenced some of my concerns,  like how long to spend on an application problem (1-2 days at each end of the unit).  And what about the procedural fluency (“equal intensity“)?  He even suggested that our time should NOT be spent CREATING these problems.  Instead, he gives several sites with problems ready to use (here).

I’m hooked (pun intended).

The unit we drafted incorporates a fair balance of procedural fluency, conceptual understanding, and application.  It is likely not the way the problem-based gurus of our day (i.e. Kaplinsky, Emergentmath, etc) would have planned the unit, but we were happy with the outcome – especially since it was our first go-round.  We feel like it’s better than what we’ve done before.

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  • It’s more engaging.
  • It allows for the whole class to share a common background story.
  • It will not take as long to teach the targets as the pacing guide has suggested.
  • It is both content and practice standards-based.

If you are curious about trying your hand at creating or implementing some problem-based learning, shoot me an email.  Let’s give it a go together!


Geeking out on the (Math) Holidays

In an effort to keep our post light this week since winter break is right around the corner, I wanted to share clever but relevant (read “standards-based”) ways in which to infuse a holiday theme into your math lessons this week.  If you know me at all, you know that I am NOT one for taking a day off from math instruction regardless of the calendar.  After all, I don’t need a special day to be fun; I’m sure my students would say my math lessons are always fun!

Upon further reflection, I realized that I actually do infuse holiday related themes into my math class, just not the winter holiday ones.

For instance, what math teacher has not celebrated March 14th, Pi Day? In fact, this school year is particularly special because we actually get to celebrate more of Pi’s extrapolation: 3.14.15!  Last Spring, I visited a teacher who celebrated by purchasing several pies for her classes, while another teacher allowed students to decorate his room.  The students wrote as many digits of Pi as they could around the room, then found a Pi rap to share with class. Other teachers I know have a specific outfit or t-shirt they wear in honor of the day. Pi Day is a special day you can count on occurring every year.

But have you ever celebrated Fibonacci Day?   It occurs on November 23 (1,1,2,3,…).  I know this day was a couple weeks ago, but put it on your calendar to celebrate next year.  Watch Vi Hart’s You Tube or Arthur Benjamin’s Ted Talk on the Fibonacci Sequence.  And by all means check out #Fibonacciday on Twitter for more Fibonacci Fun and geekiness!

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There are some special dates we have time to plan a real party for as they don’t come around very often. One is on November 13, 2015,  National Odd Day (11/13/15).  In fact, I believe it is the last one for the rest of this century.  We better make this a big one!

Another elusive math holiday is Square Root Day on 4/4/16.  To celebrate you could purchase the Square Root Puzzle or play the Square Roots Game online.  Don’t forget to bring root vegetables to eat, too.

Thinking far ahead, it’s never too early to plan for Pythagorean Theorem Day on 8/15/17 (the next Pythagorean Triple). Check out The Best Pythagorean Theorem Rap and a demonstration of the Pythagorean Theorem by folding a circle.  Both would really add to the hoopla you are sure to be planning.

I’m sure there are other dates out there that should and could be celebrated (i.e. e Day).  So don’t feel too bad if you’re not taking a day this week to graph a Gingerbread Man or determine how much money was spent in the song “The Twelve Days of Christmas”. However, if you do have time (insert laugh track here), take a look at Pascal’s Triangle 12 Days of Christmas. You know we math geeks all love Pascal’s triangle.  This blog post  inserts the patterns the triangle creates into the 12 Days of Christmas.    My favorite patterns highlighted are the Intertwining Petals and the Powers of Eleven.  Way Cool!

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Last but not least, Beth and I want to wish you and yours a safe and relaxing break and warm holiday wishes for whatever holidays you may celebrate in the next couple of weeks, math related or not.

Graphing Math Stories – Video Style

I think it is fair to assume that analyzing the relationship between variables on a graph is an important math skill.  Making sense of the enormous amount of data coming at us and needing to make a decision based on the data necessitates a way to organize and understand it.   Graphing helps with that need.

The Common Core standards support this. Analyzing the relationship between variables on a graph is included in a priority standard in every math course (Geometry excluded) from 6th grade to high school.

  • Sixth graders are asked to Analyze the relationship between the dependent and independent variables using graphs and tables….
  • Seventh graders must Explain what a point on the graph…means in terms of the situation….
  • Eighth graders must Describe…the relationship between two quantities by analyzing a graph….
  • Algebra 1 students must Relate the domain of a function to its graph and to the quantitative relationship it describes….
  • Algebra 2 students continue their work with linear and quadratic functions, and begin their work with exponential and piecewise functions.  They are asked to Construct and compare linear, quadratic, and exponential models….

Engaging students in the process of analyzing the relationships between variables through the use of stories via video is a way to get even the least motivated student interested.  Many such videos already exist and are available as a free resource.

An early set of videos were created by Dan Meyer and are found in a 2007 blog post.  In the post, Meyer shares ten videos he created to introduce a linear unit to his Algebra 1 class.  The scaffolded videos are engaging, and allow entry into the skill. Below is an example taken from this set.

The second set of videos were created when Meyer teamed with the BuzzMath Website and created “Graphing Stories: Fifteen seconds at a time.” These videos have been created in the same format that Meyer used in his original videos, however they were crafted by an assortment of people.  Each video is tagged with the type of relationship being highlighted: linear, piecewise, parabolic, increasing, or decreasing. Graphing Stories: Fifteen seconds at a time is good for all levels as the purpose is to help students conceptualize the relationships between the variables.

Other teachers, inspired by Dan Meyers, created their own videos and shared them on You Tube.  They include: Figaro, Biking, Dunk Tank, The Slide, Hill, and Canoe Distance.  A couple of folks even made videos to graph systems of equations.  Take a look at Man and Girl and Running.

As we’ve observed by attending data team meetings this fall, all staffs are emphasizing common instructional strategies.  Peppering your lessons with sporadic video clips or providing them for homework support is a means to deliver non-linguistic representation two-fold, through the activity of graphing and video.  In the end, you’re targeting a priority standard with a high-yield instructional strategy.  In the immortal words of Matthew McConaughey in his Academy Award acceptance speech: “Alright, Alright, Alright!”


Number Sense: I don’t like this game anymore

I have a twitter account.  I don’t tweet much, but I do follow a fair number of those who do. One of those I follow is Andrew Stadel.  I have posted about him before; he’s the one who started Estimation 180, an engaging website that helps students improve their number sense (NS). Interestingly, NS is difficult to put your finger on, so after consulting several sources, I’ve landed on an intuitive understanding of numbers, their uses, interpretations, magnitude, and relationships. Definitely a needed area of growth for our students.

Estimation 180 is an activity to deliver Generating a Hypotheses,  one of the high-probability instructional strategies Marzano and Hattie reference. According to their research, the process of explaining students’ thinking helps to cement their understanding.  In Estimation 180, students are asked to explain how their observations of an image (height, weight, length, etc.) supports their estimation.  The discourse around their observations and reasoning has some rich potential.

Example of what students would see.Stadel presented an IGNITE talk called Number Sense: I Don’t Like this Game Anymore during the California Math Council North Conference in December 2013.  In an IGNITE Talk, each speaker gets five minutes and 20 slides that advance automatically every 15 seconds to teach, enlighten, or inspire the crowd.  In Stadel’s talk, he explains how he uses Estimation 180 in his classroom.  Rather than try to paraphrase him, when you can take 5 minutes, check out his video.

BTW:  Andrew Stadel will be at the NW Regional Math Conference  presenting a workshop titled, “Modeling Mathematics Using Problem-Solving Tasks”  on Friday, October 10 from 12:00 – 1:30. The Conference is also holding a tweet-up on Friday at 6:00pm in the lobby of the hotel and an IGNITE on Friday at 7:00pm featuring 10 speakers.  I’ll be there.  Look for my tweets!