Category Archives: Math

Visual Thinking

I’m kind of a New York Times (NYT) junkie. A few years ago, we bought a subscription to the Sunday Times at home, which additionally provided us unlimited digital access. I signed up for news alerts and a few news digests, like “Education” and “Tests and Assessments,” so I’m sent short abstracts of articles that pertain to those categories.

It was in one of these digests, recently, that I encountered the headline: “40 Intriguing Photos to Make Students Think.” Not the catchiest of titles, but the photo accompanying it piqued my interest:screen-shot-2016-10-04-at-9-27-02-pmSo, I clicked. That’s where I was reminded of another great service The Times offers: The Learning Network. In essence, they deliver education resources like lesson plans, news quizzes and the like all based on NYT articles, photos, graphics and more. Better yet, you can access the Network’s many features without a digital subscription.

Back to what brought me to The Learning Network in the first place: the above picture. It’s part of a weekly feature called, “What’s Going On in This Picture?” Readers are asked three standard questions about a visual text:

  • What’s going on in this picture?
  • What do you see that makes you say that?
  • What more can you find?

Then, they are encouraged to acutely observe (a.k.a. closely read) the image, find evidence, and post and respond to comments of others online. Partners, Visual Thinking Strategies facilitate the online discussion Mondays between 9a – 2p Eastern Time (6a-11a Pacific). On Thursday afternoons, Learning Network writers reveal a caption and backstory below the photo on the site.  Check out this week’s photo below (week of October 17):

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As for classroom applications, I immediately thought of a warm-up. Whether you participate live through the Learning Network or want to spark an in-class dialog, there are images that relate to nearly any subject are. Five minutes at the beginning of class is plenty to take in the photo and respond, turn to a partner and compare perceptions. Cogs are turning, awareness is heightened and students are engaged in analysis–great practice for getting into the day’s content. And, what a great way to build a community of inquiry, teacher included!

In 21st Century literacy, when we hear or say text, we need to be mindful of the myriad forms that word may conjure. Our students are consuming these many texts in record numbers, daily. Teaching the tools to become informed about what they consume doesn’t get more real.

 

Fascinating Data: One step closer to a thinking classroom

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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.

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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:

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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!

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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.

THIS IS A PROBLEM!

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.

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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!

 

 

 

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!

Growing our mindset

Embracing challenges vs. avoiding them; expending effort vs. settling into complacency; learning from vs. avoiding feedback. These contrary actions describe the differences between Growth and Fixed Mindset.  Growth Mindset is becoming a popular initiative of teachers and teacher leaders around the United States and the United Kingdom. If you’ve not read  Mindset by Carol Dweck, I highly recommend it.  A quick read, it’s impactful on many levels–as an educator, partner, parent, coach, and the like.  The book, accompanied by an Educational Psychology class I took almost two years ago that discussed the Power of Yet and How We Learn, has my mind spinning about how we can help our students succeed. First, we need to believe our students can succeed.  And second, our students need to believe they can succeed.

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There are many resources I’ve looked at over the last year or two that have continued to give me insight into the importance of growth mindset. The following are three of the resources that I find exceptionally valuable:

  1.  Jo Boaler’s YouCubed  website is the first.  She has short video snippets that can be shown to students to help explain how their brains learn, why mistakes help fire the synapses in the brain, and the plasticity potential of the brain. Her website also include tasks, research, and a Mooc for students with 6 sessions on How to Learn Math.
Short Video Snipit from YouCubed about the Black Cab drivers in London.

Short video snippet from youcubed about the Black Cab drivers in London.

2.  Marissa of La Vie Mathematique shows a video (see example below) from her Mindset Moment List  once or twice a month as a warm-up to spark conversations about having a Growth Mindset. She sees value in connecting traits like perseverance, effort, and educational risk-taking to all students, regardless of content area.

Kid President with a Pep Talk

Kid President with a Pep Talk

3.  Mike Mann from Dexter McCarty shared an extensive list of resources on a google doc  last year.  In it, are  articles and videos lending themselves to close reading/viewing techniques, as well as graphics to post in your room.

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I would love to hear what are you doing in your classroom to help develop a growth mindset in your students.

Summer Opportunities Coming Your Way!

Whether you’re more of a fan of The Jamies (Summertime, Summertime) or Alice Cooper (School’s Out for Summer) or fall somewhere in the middle, one thing is clear, Summer is fast approaching! For our final post of the year, we’re offering a little something for you (check out our Events page for a host of classes/workshops), and if you read on, a little something for your students.

In my literacy world, one of the greats of summertime is the Summer Reading Program run by Multnomah County Library a cornerstone of our district’s Middle School Summer Reading Program, offering schools a variety of adventures for students!

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Why summer reading, you may ask? Well, a big reason is heading off the summer skills slide. A landmark study done by Barbara Haynes (1978) who followed 6th & 7th graders for two years revealed the following effects of summer reading participation:

1. The number of books read in summer is consistently related to academic gains.

2. Children in every income group who read six or more books in summer “maintained or improved their reading skills while kids who didn’t read any, saw their skills slip as much as an entire grade level.”

3. The use of the public library during the summer is more predictive of vocabulary gains than attending summer school.

4. The major factors determining whether a child reads over the summer were: whether the child used the public library; the child’s gender (girls read more than boys); socioeconomic status; and the distance from home to a library.

5. More than any other public institution, including the schools, the public library contributed to the intellectual growth of children during the summer. Moreover, unlike summer school programs, the library was used by over half the children and attracted children from diverse backgrounds.

There have been notable studies since this original publication, if you’re looking for additional research, such as the 1982 “Beginning School Study,” by researchers Karl Alexander and Doris Entwisle of the National Center for Summer Learning and a 2001 report, “The Role of Public Libraries in Children’s Literacy Development,” by the University of Michigan’s Susan Neuman and Temple University’s Donna Celano. The bottom line: summer reading, including free, independent reading of a student’s choice is a must! And, remember, reading doesn’t need to be limited to physical books. Some of our more reluctant readers can be turned on by an e- or audio book, a graphic novel, or Zine. There’s also a whole informational text side to reading that appeals to some, which could include anything from newsy articles to infographics to “reading” a museum exhibit. Truly, the possibilities are limitless! Check out the summer reading programs in your community and get your kids pumped to include reading in their down-time!

As we bid you adieu for the summer months, we hope you will spend 5 minutes browsing the Events page and find a class or workshop that resonates with you. If you know of anything we may have missed, send us a note and we’ll add it.

Until September….

 

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….