Vertical Learning

Vertical Learning
Vertical Learning

Last week I had the opportunity to visit a Downers Grove middle school for #samricamp. I’d like to give an appreciation shout-out to the DG58 staff and administration for putting on another excellent PD event. I’m always impressed with their ability to organize events and invite all interested educators and administrators to their school.

All of the sessions that I attended were insightful and I have many ideas to think about for the new school year.  I found the last session with Matt and James to be especially useful. See their presentation here. They facilitated a session on the idea of vertical learning. The session started off with questions related to why schools group students by age.  We then delved into what matters as educators engage students in learning. Both questions spurred conversations about differentiating instruction so all students can grow. The consensus was that all of our students come into our classrooms at different levels.  Regardless of their age, students have a “starting point” as they begin and progress through a school year. Sure, educators may group students at similar levels but in reality all students are at different levels of understanding. If all students are expected to show growth, how do educators show and document that growth? I organized my thoughts and came away with something to consider for the upcoming school year.

There’s a need to reorganize our resources

Typically I find most elementary resources labeled by a grade level, not a learning target. For example, a fourth grade class will use a fourth grade district-adopted math and reading text. The English and Science texts are matched to fourth grade based on a publisher’s recommendation. More often than not, teachers are trained to use these types of resources with a certain grade level. Most of the workbooks, worksheets, and journals are all associated with that grade level.

I feel like issues arise when students aren’t ready or have already learned concepts for a particular grade level. What do teachers do then? Instead of “covering” the curriculum teachers should be inclined to emphasize learning targets. What learning targets have students met? Some teachers use pre-assessments and pull groups or use a form of a workshop model. Teachers then scaffold the skills that match learning targets. Formative checkpoints along the way help align instruction, but without resources a workshop model has limits.

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Teachers need to have access to resources that match their students’ needs. This can look different depending on your school. Some elementary schools have a resource room with K-5 content spread throughout an array of cabinets. In those cases, teachers can visit the resource room to grab a needed text for a group of students. This can be a valuable resource for teachers although having a resource room isn’t always an option. Also, what happens when students are showing mastery of skills above fifth grade?  What then?  Other schools have staff development educators, resource teachers, or instructional coaches that can point classroom teachers towards resources that might be helpful. Again, this isn’t always the case. Sometimes teachers create or find resources outside of their school to meet the diverse needs of their students. Many educators on Twitter locate and share resources that help with student differentiation. Other educators visit different schools to observe instructional practices that can be incorporated in a different setting.

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Regardless of how resources are organized/obtained, I believe there needs to be a plan to communicate the organization to staff. Grade level texts are one resource that can be utilized to help students master concepts and grow. Supplementary resources exist and online/print material should match the curriculum being implemented. I believe the onus shouldn’t necessarily be placed directly on a school district’s shoulders, but they play a pivotal role in what resources are used in the classroom.

Students are expected to show grow regardless of their starting point. Having conversations about how to achieve that vertical learning and how to access resources is important.  I believe having these discussions with staff will benefit students long-term.

Student Surveys and the Reflection Process

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Yesterday was the last day of the school year for my students.  The end of the school year tends to be filled with excitement and pride as students transition from one grade to another. During this time of the year I usually give my students a feedback survey. I tell the classes that I’ll be using the information to change next year’s classes for the better.  I’ve been using this method for the past few years and find it valuable in preparing for the fall.  Most of the questions that I ask tend to stay the same while I add a few others depending on what I’m focusing in on for the year.  This year I asked a few questions related to feedback and student refections.  These particular questions stem from some of the district’s initiatives, as we’re emphasizing Hattie and Dweck’s research.  Next year we will be focusing on them even more and I believe they’ll be part of a formal walk through process.  So I gave the survey to 50 3 – 5th graders and collected the data.  The survey that I used can be accessed here.

I took the 50 students responses and had Excel calculate the averages for all of the questions. Below are few highlights from the feedback and reflection questions.  I used a 1 – 10 rating, with 1 being all the time and 10 being never.

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My takeaways:

I have to keep in mind that elementary students are taking this survey.  It’s valuable, but I feel like a third grader will perceive a question possibly different than a fifth grader.  Regardless, the data is valuable in my mind.  I looked over the numbers and shared this information with another class.  After showing the data we had a great conversation about reflecting on our learning.  Our conversation looked at the connection between allowing reflection opportunities and how they impact our learning.  We started connecting parts of the survey as a cause/effect scenario.  The conversation wasn’t too deep, but worthwhile as students made connections.  We decided that reflecting on our learning can be impactful, but not necessarily help a person understand a particular concept.  Feedback, reflection and opportunities to take action need to all be place. What seemed to be lacking this year were opportunities for students to reflect AND take action based on that reflection.  It’s important to reflect, but without any action or change in perception the act might not be reaching its full potential.  I decided to write an informal flow chart indicating the process that the classes tended to use.

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I told the students that one of my homework assignments over the summer is to provide ways to make student reflection opportunities more efficient.  This is something I’ll be revisiting in the fall with my new classes.

Surface Area and Conceptual Understanding

Surface Area

My fourth grade class has been exploring a measurement unit for the past few weeks. We’ve been discussing the difference between area and volume. This has been a bit challenging as many students can apply area and volume formulas but struggle when finding surface area. Students were confusing area and volume and weren’t sure when to use a specific formula.  The idea of area being squared and volume cubed has been emphasized but still not cemented.  It seemed that students knew much more about the formulas and not as much about the conceptual understanding. To strengthen students’ understanding my class started a surface area activity late last week.  Click the image below for the template.

Sheet

Students were asked to pick one box in front of the classroom. I had many different boxes to choose from. Many of them were board games or boxes I borrowed from other teachers. It’s near the end of the school year and some teachers are moving classrooms so there were plenty of boxes. All of the boxes were rectangular prisms. Once students picked a box they took a picture and then found the dimensions. Students then took one piece of butcher paper and created a net based on the dimensions found earlier.

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Students created the net and then wrapped up the box. Students were able to immediately identify whether their measurements were off or on target. It took some groups multiple attempts to find a correct solution. After students wrapped up their box they took a picture. Before and after pictures were sent to me via Showbie. I printed them out and the students placed them on their sheets.

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It would have been great to print these out in color, but at this time of the year our school’s color printer is out of ink.  After the activity students reflected on how their perception of area has changed over the past week.  After listening to a few student reflections I’m deciding to keep this activity for next school year.

Higher-Level Math Tasks

A few days ago I started reading Principles to Actions Ensuring Mathematical Success For All as part of a book study. As I was reading in preparation for our first session I came across a few ideas worth highlighting. Pages 18 and 19 discuss the four levels of cognitive demand in math classes.   Along with expectations, these demands are often revealed in tasks or assignments that students are asked to complete.

The book describes lower-level demands as tasks related to memorization and procedures without connections. Memorizing rules/formulas and following procedures is often related to lower-level demands. Students often understand what’s expected when lower-level demands are required. Generally one answer or procedure is evident with this type of task. Worksheets that have students practice rote computation skills without words could fall into the lower-level demand category. Higher-level demands are procedures with connections and often require considerable cognitive effort to achieve. Anxiety is often a part of higher-level demands, although this may be because students don’t see these types of tasks as often.

After reading this page and looking at the different examples I started to reflect on how elementary math classrooms are organized. Math practice is needed, but students should also be given time to explore, discuss and make connections in a low-risk environment. I find more lower-level demands in math classrooms than higher-level, but an ideal ratio is challenging to ascertain.

So after reading pages 1-35 I decided to use an example of a higher-level demand activity with a fifth grade classroom. This particular class is learning about fraction multiplication and division. Students have learned in the past to multiply the numerators and denominators to arrive at a solution. To delve a bit deeper in their understanding I decided to use and adapt one of the tasks in the book. I first grouped the students into teams and gave each team 12 triangular blocks and a whiteboard marker.

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Students were asked to show a visual model of 1/6 of 1/2. Some students knew the answer already but seemed unsure of how to show the answer visually. Many of the groups weren’t quite sure on how to approach the construction of the fractions. They understood the abstract and procedural but had a challenging time visualizing the fractions.

After seeing the students struggle a bit I’m glad that I decided to have them work in pairs. Students started to build models of 1/2 using the 12 triangles. Some of the groups came to a conclusion that two different sets of six triangles shows half. Then from there students started to think of what’s 1/6 of the 1/2. Students took out 1/6 but then debated on the value. Some groups said that the answer was 1/6 while others were confident that it was 1/12. Eventually the students decided that 1/12 was the correct solution.

 

I went around the classroom and took some pictures of the different creations. Not everyone created the same type of model. This was a great opportunity to highlight some of the different models that arrived at the same solution.

 

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Afterwards, I thought that offering exactly 12 triangles helped but limited the choices for a visual model. The student models were somewhat similar as a result of the level of scaffolding. As students reflected on their actions in this activity I heard some interesting conversations. Students were aware of the procedure to multiply fractions less than one, but started to visualize the model through this activity. I thought this might be one way to introduce fraction multiplication at the fourth grade level.  I also thought that this activity was well worth the time and I’m looking at incorporating additional high-level cognitive demand activities in the future.

Teaching Algebra Through a Different Lens

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I recently taught a lesson on pan-balance equations.  In my curriculum pan-balances are taught as a precursor to more in-depth pre-algebra.   My students seemed to understand simple pan-balances and found that the balances (like an equation) needs to be balanced to work.  The majority of students had no problem with questions (like these) involving oranges, apples, paperclips, etc.

Day Two

During the next lesson I introduced the idea of variables with equations, like 2x + 4 = 18. Students seemed to understand, but less than the first lesson.  I brought everyone up to the classroom whiteboard and practiced many problems with the students.  The students who understood always seemed to raise their hands, while students who didn’t completely grasp the concepts tried to blend in with the carpet.  Students became less interested in what I was teaching when I started writing equations on the whiteboard.  Unfortunately, I felt like I was losing a battle here.  The more the students seemed to not understand, the more I felt the need for direct instruction.  Near the end of the lesson around half of the students seemed confident to proceed to the next algebra lesson – solving for x on both sides of the equation.

I had to change something.

Day Three

During the next day I decided to change up my instructional approach.  I remembered back to when I first learned algebra and the confusion that I used to experience.  My school memories of algebra started and ended by watching a chalkboard and overhead projector, as my teacher wrote and erased equations on the board.  This was the only way to learn algebra, or so I thought back then.  Using my experience  I decided to change the instructional medium.

I started my next algebra lesson with a quick review of pan-balances.  Students seemed to gain confidence as we had a conversation about the importance of using algebra in careers outside of the classroom.  We watched a quick BrainPop video on algebra and it’s uses.  Instead of using the whiteboard again, I decided to take out the iPads.  I already downloaded an app called Hands-On Equations a few weeks ago.  The students were quickly motivated and I modeled how to use the app under the document camera.

Hands-on-Equations

The class and I went through a few problems together until I thought they were ready to proceed.  I allowed the students 20 minutes to explore the app and lessons. The students were expected to complete at least three lessons and reflect on their experiences in their math journal.   What was interesting was that the students immediately took control of their own learning and utilized the app at their own pace.

Payoff

After approximately 20 minutes, I asked the students to write in their journal how they felt about their journey with pre-algebra.  The majority of responses were positive …

“I now understand why we use algebra”

“I never thought algebra could be so much fun”

“Having a picture of the balances helps me understand the concepts better.”

The Takeaway …

After hearing their responses and reflecting on the outcomes, I’m becoming more motivated to vary instruction to better meet the needs of my students.  Varying the instructional approach can give students multiple opportunities to grasp concepts that can be particularly challenging.  Your students may benefit from a bit of instructional change from time to time.


photo credit: ajaxofsalamis via photopin cc

Measurement and Mini Golf

Measurement Project
Measurement Project

Approximately a week ago I was paging through my math curriculum. Through a pre-assessment I found that students were in need of a review on angle classification and measuring skills.  The curriculum lessons offered a number of worksheets and angle measuring drills.  Although these lessons seemed beneficial, I felt the need to create a more memorable learning experience for my math students.   At this point, I decided to search for measurement projects. While following #mathchat, I came across this Edgalaxy site.  The project seemed to match many of the objectives that needed strengthening in my class.  I changed up the directions and modified some specifics in order to best meet the needs of my students.

So … a week has passed and almost all of the projects are complete.  I listed the project steps below.  Feel free to use any of the ideas below in your own classroom.

1.  Had out the direction sheet.  Here is a Word template (via Google Docs) for your use.

Directions in WRD

2.  Review many of the different vocabulary words associated with the project: acute, obtuse, right, parallel, perpendicular, trapezoid, etc.

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3.  Show possible examples.  I tend to show just a few examples as I don’t want to give them a mini golf course to copy.

4.  Group the students into pairs.  If you prefer, this project could be implemented as a collaborative group activity.

5.  Students choose their construction paper color (11″ x 20″)

6.  Students draft their course in pencil (on grid paper).  The draft gets approved by the teacher and then is transfered to scale on construction paper.

Sample

7.  Students present their final projects to the class.

iPad Apps for Math Intervention

IPad Apps for Math Intervention

Over the past few months I’ve been experimenting with guided math strategies in my classroom. One station in my classroom has been dubbed as the technology table. This table has been primarily used to differentiate  instruction to improve students’ understanding of mathematical concepts.  I’ve been using the tech table for the past few months with great success. There are five iPad apps that are used at this table.  Unlike many math apps that offer only demo versions, I’ve found the below apps to be useful in the classroom.


5 Dice

This app is the newest addition to my iPads for intervention list.  This app emphasizes order of operations for upper elementary and middle school students.  The game encourages students to use multiple dice to find the “target” number.  A whiteboard is built into the game for students to work out problem.  Progress reports can be emailed to the teacher for formative assessment data.

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Splash Math – Grade 3

This app is used to differentiate math instruction and assigned practice.  What I like so much about this app is the variety of concepts that I’m able to individualize.  For example, if a student needs additional work on the concept of time, then I can setup the app to only give questions related to time. Questions first appear simple, but then become more challenging as questions are answered correctly.  If you prefer, Splash Math will send you a weekly update indicating the progress of each student.


Math Blaster Hyper Blast 

This app is used to improve computation fluency.  This interactive app has a quick tutorial to teach students how to move the main character through a variety of mazes.  Students control a space vehicle that inevitably encounters an octopus type of creature.  Students must answer computation (addition, subtraction, multiplication, or division) questions to defeat the boss.


Factor Samurai

Factor Samurai is an app geared towards emphasizing the concepts of prime and composite numbers.  Basically, numbers fly into the air and the student is expected to slice the composite numbers with their fingers.  If it’s a prime number, then the student leaves the number alone.  Some composite numbers can be sliced multiple times.


ScootPad

ScootPad can be used to individualize practice in your classroom.  I’m able to assign specific students certain Common Core objectives to practice. After a student completes an assigned section, they are allowed to see all of the correct answers.  Scootpad will also send the teacher a statistical report of the progress made by individual students.  I’d also like to note that Scootpad can also be used on a PC or MAC.


Honorable Mentions: 

 Math 7

 Sail Through Math

 Divisibility Dash

Equivalent Fractions

Rocket Math

update:  02/03/13

I’ve been asked by a number of people what apps I would recommend to an elementary teacher.  I decided to create a quick chart to help.

Elementary Apps



So, what math iPad apps do you use in your classroom?

Still Exploring Guided Math

Still Putting the Pieces Together

I recently participated in an afternoon professional development session led by Laney Sammons.  The session focused on how to implement guided math.  I’m still understanding the guided math process, as you can tell by the picture above.  I wouldn’t consider myself an expert in guided math, but I’m starting to use a few strategies that Laney discussed today.

A few takeaways from today …

  • Guided math can be similar to guided reading
  • Math games can be used in stations
  • Groups should consist of no more than six students
  • Groups can be used for informal assessments
  • There isn’t a “one size fits all” model for guided math

After the session I decided to explore guided math a bit further.  The links below have been vetted and may help shed additional light on guided math in an elementary setting.

Feel free to share any links or blog posts that you find relevant in the comments section.  Thanks!

* Picture credit to Janoon28

The Value of Self-Correction and Student Ownership


This year I’m continuing to find that student ownership plays a critical role in the learning process.  Students often become more responsible for their own learning when they are given additional opportunities to show their learning.  I’m finding that part of the key to increasing student responsibility depends on how it’s communicated by the teacher.  Students can’t be expected to own their learning without any guidance.  The gradual release of student responsibility can benefit the overal climate and achievement of a classroom.  In the past, I’ve used student journaling, plus/delta, surveys, choice boards, self-selected research projects, and other strategies to promote student ownership.  This past week I introduced another strategy that involves self-correction.  Here are the steps:

1.)  Students complete an assignment in collaborative groups or independently.

2.)  Students finish the assignment and self-correct using the Teacher’s Manual.  This can also be applied to digital progress monitoring tools.

3.)  Students independently use markers to indicate wrong/right answers.  If needed, students will write in correct answers.

4.)  Students utilize their math journals to reflect on the assignment and their feelings about the topic and achievement.

5.)  Student turn in their paper and journal to the teacher

6.)  Optional:  Students use multiple journal entries for individual goal setting

It might seem simple, but I’ve had terrific results from using this strategy.  Overall, I feel as though the students benefit from practices like this.  The self-correcting / journal process took modeling and practice at first, but the benefits are starting to become apparent.

Number Line Concepts

Image by:   D. Rizzuti


Lately, I’ve been having conversations with colleagues regarding how to communicate number line concepts in the classroom.  Specifically, I’ve been giving examples of how understanding number lines may lead to a more stable mathematical foundation. In the past, my class has created various products related to the number line.  My original inspiration came from this number line below.

The project in this post emphasizes the idea that percents, fractions, mixed numbers, and decimals are all related  This basic understanding helps develop number sense skills.  Here are the generic steps for this project:

  • Students cut out percents, decimals, percents, and fractions out of the template
  • Students draw a number line on a piece of construction paper
  • Students glue/tape each number on the number line

Here are a few sample photos (click to enlarge):

The project seems simple, right?  Well … it took about 20 minutes for the cutting, coloring, and gluing.  I then facilitated a classroom discussion after the number lines were presented.  The math curiosity (I really like that term) and discussion that followed the project seemed beneficial.  It’s truly amazing to see what type of concepts can be discussed when observing the number line through a variety of lenses.  Our conversations touched on the concepts of absolute value, positive/negative numbers, fractions and mixed number conversions, addition of negative numbers, and place value.  In fact, the math conversation lasted 30+ minutes.  Having these types of “math chats’ with third graders was a phenomenal learning experience.  All of the concepts discussed will be introduced later in their academic career, and hopefully I gave my students a quick preview to what is to come.

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