Tag Archives: 8th

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.


Two trains are heading in the opposite direction…: Teaching problem-solving

I’ve always hated the train problem; you know, the one in the title?  I’m not sure why; I think because I would typically read through it once and become defeated since I couldn’t immediately come up with an equation I could apply. Therefore, it must be hard.  My experience has been that most students feel this way about problem-solving.  The STEMtistic below says it all.


Our education standards for mathematics expect that students are engaged in problem-solving.  The National Council of Teachers of Mathematics (NCTM) Principals and Standards for School Mathematics has verified that problem-solving is an integral part of learning.  OK, then how do we rewire students’ brains so that they don’t fear math problems?  How do we help students access the math?

To start, we need to teach our students how to read like a mathematician. 

We need to teach our students how to bring the linguistic and math clues to the surface.

We need to engage students in understanding the problem before they try to solve it – or worse, freeze and dismiss it all together.

Close reading is eduspeak for reading deeply for a purpose. It is more than just skimming. In the literacy world, close reading includes three phases:

  1. Reading for Key ideas
  2. Rereading for craft and structure
  3. Rereading to integrate knowledge and ideas.

I’ve been to several trainings on how to close read science and social studies texts and articles, but I have never been trained on close reading in math; I didn’t think it even applied.

Then came those 5th grade classroom teachers I was privileged to visit last month. During one of those visits, I saw close reading . . .  in math!

The close reading strategy was called the 3 Reads Protocol. In this protocol, students read the problem three times, each with a particular focus. While the strategy was used in a 5th grade classroom, it can be used just as effectively at the middle school and high school levels.

Below was the problem given to the 5th graders:

On the first read, the problem was read chorally (as a class) then covered up and students were asked, “What is the problem about?” Students talked to their study partners about what they remembered of the situation – not the math, just the context. Students answered that the boy in the problem was doing some of his homework before dinner and some after.

Next, students read the problem aloud a second time with their partner and were asked to determine the key quantities and key words from the problem.  After 2 minutes of partner time, the teacher listed the quantities and words on the board.  The students answered with the obvious fractions, then included a smattering of words such as completedbefore, after, remaining, dessert, and the rest. Students were asked why these words were important and how the quantities were related.

The third read,  again with a partner, focused students on the question, What is the problem asking us to find out?  After determining this, students were asked to draw a diagram that included the quantities and their relationship. Some students started with a tape diagram, others with an area model.

Only after these three readings and active thinking did students begin to actually solve the problem in partners. There was a lot of math discourse happening around the room that continued for a good 15-20 minutes.  During this time, the teacher walked around the class, listened in, asked questions about student thinking, and noted which students she would call on during the whole class discussion.

Students were then brought together as a class.  The teacher asked specific partners to share the models they had drawn.  As each model was shown, the teacher asked questions of the class such as:

  • How does _________’s diagram show 3/7?
  • Where is ¼ in ________’s diagram?
  • How are ¼ and 3/7 related in this diagram?

She made sure to ask students who had made models that had taken a divergent path to explain their thinking and asked the same questions she had asked the students with the correct models. Once the teacher went back to the class list of key words, the class came to an agreement about which model (s) made the most sense and the answer that was correct.


After whole group discussion

We want kids to learn how to solve problems on their own. We need to give them strategies to do it!   Like any skill we want to master, teaching kids how to read closely in math will take time, practice, and coaching – especially if we want to change attitudes about word problems in general.

So, try the 3-reads protocol in your math class! Like most new strategies, it is not likely to go super smooth the first time. Don’t give up! Try it again and get your students out of the bathroom and cleaning up in math instead!

BTW:  I solved the train problem…eventually!

Train Problem - SOLVED!




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….


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!”


Padlet.com – Instant feedback

You are in front of your class, and it feels as though you’re pulling teeth to get your students to give you feedback about their thinking. Are they with you? Then you remember the very cool online tool, padlet.com, and you think, “You’re mine! Instant engagement!

Padlet is a free website that works like a bulletin board.  The teacher creates the wall and the question, gives the students the website, or has them scan a QR code that takes them right to your wall. They answer the question on their electronic devices: phone, tablet, or computer.  Students do not have to log in, or have any specific application, just access to the internet.

My wall from the April 25 Math Session

A padlet wall with a few responses from the April 25 Math Session.

Students have the ability to post text, a link, or an image. Teachers have the ability to approve student responses before being posted.

Students double-click anywhere on the wall, and insert text, a link, or upload an image or video

Students double-click anywhere on the wall, and insert text, a link, or upload an image.

Set up is minimal: less than 3 minutes to create a wall.  And, you get instant feedback.  While you may not know exactly who posted what, you get a general feel of what the class is thinking, and if there are any misconceptions.

Your challenge? Try it, and not just once or twice.  It will take a few times to make it yours, and for the kids to be efficient with its use.

Some ideas…

ELA:  What is one change the protagonist has made?

Science:  List as many cycles as you can think of.

SS:  Name a cause of the Revolutionary War.

Math:  What is one thing you need to remember when graphing a line?

We are excited to hear how you use it in your classroom.  Let us know!

Estimation 180: Improving Number Sense

In October of 2012, Andrew Stadel, a middle school math teacher, created a website called Estimation 180, with the goal of helping his students improve their number sense. Andrew considers estimating key to developing number sense and reason (we are inclined to agree). He teaches Pre-algebra, Algebra and Geometry, so his estimation challenges are not limited to a specific grade level.

Every day, Andrew gives his students an estimation challenge around a theme.  He reveals a picture of an object, and students make an estimate about the object’s height, weight, or other measurement.  Each chooses a number that is too high, one that is too low, and finally their estimate, allowing ALL students entry into the challenge. In addition, he asks his students for their reasoning, a great way to explicitly teach – and for students to engage in – Math Practice #3: Construct viable arguments and critique the reasoning of others.   There is potential for some great math talk here.

Estimation Challenge - Day 1

Estimation Challenge – Day 1

He’s also created a worksheet for students to keep track of their estimates, their reasoning, and their error percentage.  It is a great way for students to track their own progress (a research-based strategy with a high effect size).

As an engaging extension, students and classes can submit their estimates right on the webpage, compare them to participants outside of their classroom, and see the reasoning behind others’ estimates.

Estimation Challenge – Day 53

Andrew has recently added a tab with lessons ranging from 4th – 8th grade that are focused on a particular Common Core Standard. His infamous File Cabinet task (picture below) is among those lessons he is sharing.   A fan of 3-act tasks, Andrew cleaned up many of those found in his original 3-act catalogue, tagged them with Common Core Standards, and now includes handouts to go with the lessons.

File Cabinet Task

Act 1 of the File Cabinet Task

When we think of someone who walks the walk with regard to what it means to teach with the math practices in mind, Andrew Stadel is one who comes to mind. He incorporates student collaboration and problem-solving into his lessons while balancing the need for individual think-time and accountability.

You’ll find Estimation 180, as well as a link to Andrew’s blog, Divisible by 3 on our Math Symbaloo. The math half of us would LOVE to model this in your classroom.

 We encourage your comments.

Online Math Resource Treasures

In staying with the same vein as last week, we wanted to share a few new math resources appropriate for middle and high school level.

1.  Symbaloo – This is a bookmarking website that has real potential.  We really like how visual it is! You can create a symbaloo for your students or parents of students of tutorial sites, virtual manipulative sites, and textbook sites.  Or, you can create a symbaloo for your own organization of sites you frequent regularly.  They can be organized on the board by color, icon, and location.  It takes very little time to create.   The math half of us started one called CCSS Math – Parvankin.  It is full of math sites that are often referenced.  And, not to be outdone, the Literacy half of us started one called CCSS Literacy – Elliot including Literacy Resources featured last week here, as well as some upcoming Literacy Events, and SBA Tools to peruse. We’ll add any worthy links as we encounter them.

2.  Dane Ehlert Videos – Dane Ehlert is a Pre-Algebra and Algebra teacher who has created videos he uses as both an introduction and review of a concept.  They are typically between 1-2 minutes long.  The best part about the videos is that they are done for you!  He also has created several 3-act tasks that you can also look at on his vimeo site.  And, he has a blog where he posts the lesson he taught that day and all of his assessments as well.  Take a look at what he has to share.

3.  Mathalicious – This is a website full of real-world problems for middle and high school students.  There are 8 lessons that are free to use, otherwise you pay a subscription of $185 per year.  Convincing your principal, Site Council, the PTC, or the Ed Foundation to give you funds shouldn’t be terribly hard, since it would help you apply the math practices in your classroom, as well as the many content standards that expect application and modeling.   You can search the site by grade level, topic, or standard.  Some lessons are meant to be introductory, while others are meant as practice.

We hope you find these resources helpful. We will be sure to include more in the future.  Feel free to share math resources you may have stumbled across in the Comments section below.