Mathematical Practice #1 in Action (for teachers)

Mathematical Practice #1 is: Make sense of problems and persevere in solving them. This time of year, PUHSD teachers across the district and multiple content areas have the opportunity to model this practice...it is scanning time!  As you begin to scan those precious district assessments, here are some tips for "perseverance" while under the gun to finalize your grade books and ensure that the data is in D2SC. 

Mathematically proficient "teachers" start by explaining to themselves the meaning of a problem and looking for entry points to its solution.

• Are the bubble sheets printed with page scaling NONE? 
• Are the bubble sheets clean of stray marks and excess eraser gunk? 

 They analyze givens, constraints, relationships, and goals.

• Is the score I gave the student(s) reflective of the score that the assessment tool says?  Why or why not? 

They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.

• Did the students bubble in dark enough? 
• Did the student number get bubbled in correctly? 
• Did I use the correct FORM ID for the pre-slugged documents? 

They consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution.

• Did I scan just one of them to see the similarities or differences in the results?  Trevor Browne found that if a student did not bubble in dark enough, the scanner did not pick up the mark- Way to go!

They monitor and evaluate their progress and change course if necessary.  Mathematically proficient "teachers" check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Here are some other tips for scanning:

1. If you do not see the test in the assessment tool, most likely you need to release the test to your students.  Be sure to release the test, prior to scanning.
2. When you go to scan, alphabetize the scan sheets and also alphabetize the students on the assessment tool screen.  When scanning, you can easily see which students forms were missed, and you can easily grab them from the pile and rescan. 
3. If at first you do not succeed, try, try again!!! 

I know that this is a busy time of year so if you need any assistance, just let me know.  :)

Meeting of the Mathematical Minds

November 9^th the PUHSD  Math Team Leaders had their second gathering of "Mathematical Minds". We started the session with "accountability time" and the team leaders did not disappoint!!  Each Team Leader brought evidence of assessments from the August Team Leader training (See August blog for details on the assessment homework). 

The Algebra, Geometry and Algebra II team leaders shared samples of assessments and strategies for engaging students in self-reflection on their assessments.  They shared ideas on how to identify what the students know versus what they do not know on each assessment and, more importantly, how teachers engage students in the process.  They also shared strategies of how their PLCs are working to implement  immediate, specific, and corrective feedback for students as part of a Formative Assessment cycle for learning.

The example below models the quality, "Be specific so students are aware of the point value assigned to  each question." 



Maryvale Algebra Team


The example below models the quality, "Establish common grading rubric." 

Metro Tech Algebra 3-4

I posted a folder on SharePoint where you can find the artifacts that were shared from each team (Algebra, Geometry, Algebra 3-4). 

The second half of the training was dedicated to defining relevant versus meaningful mathematics and a crucial component to lesson design: Mathematical Task selection.  Specifically, what is the difference between a low and a high cognitive demand task? How do these tasks affect student achievement?   

Course level teachers participated in a  Task Development activity. A team of 3-4 teachers identified a standard and created a low cognitive demand task. 

Then, they composed a high cognitive demand task that would assess the same standard. After creating the high cognitive task, teams were asked to develop questions that either 1) assist students to get “unstuck” (assessing questions)  or 2)  challenge students to think more in depth about the task (advancing questions). 
Teams shared their tasks and we collected all of the examples into one document (Click HERE for the examples). 

The team leaders' Homework for the January meeting is:

• Bring evidence of a high cognitive demand task (common task) that your team has created. 
• We will focus on designing lessons for 2
  nd Semester and using a planning template that will help you to achieve the goals of the CCSS Mathematical Practices.

Dr. Kanold also summarized the meeting on his blog. Please view the PowerPoint for the Team leader meeting for details of the session. 

Happy December! 

PLC planning for CCSS Student Mathematical Practices

As your PLCs are working on lesson planning using the ACT lesson plan template, take a moment to un-pack each of the mathematical practices to describe what each practice will look like in your course. 

For each practice, answer the following questions: 

·        What does this (practice) look like?

·        What does this (practice) sound like?

·        What will students need in order to do this?

·        What will teachers need in order to do this?

·        Implications for assessment?

·        Implications for grading?

Below is a link to Visioning Grids that teams can use as a template. 

http://www.mictm.org/attachments/184_Standards_for_Mathematical_Practice_Visioning_Grids.pdf

Upcoming blogs will look at individual practices and lessons that PLCs are developing. 

ACT Training Day 1 and Day 2 Summary

I spent Friday working with Gwen Lee and 40+ PUHSD Math teachers exploring and constructing a deeper understanding of the ACT Framework. A big thank you to Gwen for her commitment and her willingness to share her experience with our teachers.  It was truly a day of learning!

Teachers focused on answering the following essential questions: 

·    What does an ACT instructional unit look like? 

·    How can we evaluate our current units and lesson to ensure that our learning plans meet the demand of CCSS? 

·    What high-leverage actions can our PLC use to improve student achievement? 

We reviewed the goals from the first day of training which included reviewing strategies from the ACT Educator’s Toolbox. Teachers shared ideas on how they are integrating strategies from the educator’s toolbox and discussed the strengths and challenges of each strategy. We spent time looking at the CCSS Mathematical Practices and identifying which specific strategy from the educator toolbox can support the development of each student mathematical practice.  Teachers reviewed the “CCSS Mathematical Practice Student Look-Fors” to assist with lesson design. 

We walked through the “Looking at student work” protocol for the Pancake Problem. Analyzing student work is one of the high-leverage actions that PLCs can use to improve student achievement.  As part of the protocol, Team members share a common level of expectation that will create an equitable learning plan: 

•    Teachers determine the degree to which student work meets the criteria of high-quality work (five finger rule).

•    Teams collaborate and reflect on the instructional process.

•        What were the students’ strengths or weaknesses? 

•        What scaffolding is needed to support learning? 

•        Do we need to reteach? 

•     Promote consistency in evaluating student work.

•        How are we grading the same assessments or tasks? 

The Pancake problem is a Mathematical Task that can be used to assess student understanding.  We evaluated  this task and other sample tasks against the 4 cognitive levels from Stein and Smith (1998) research on cognitive demand tasks, “The Task Analysis Guide.”   

The afternoon session was dedicated to tying all of the instructional components (Educator’s Tool Box, DOK, Mathematical Tasks, and CCSS Mathematical Practices) to designing effective lessons and units using the ACT Framework.  ACT Lesson plans and units can be evaluated using the Guidelines for developing an Instructional Unit.  Teachers evaluated the Algebra 2 Unit 4 using the rubrics and focused on the following: 

·        Student toolbox:  What tools do your students use in your classroom? 

o   Are they given opportunities to add new tools to their toolbox? 

·        Essential Questions: How do these develop over a unit and lesson?

o   How are the probing questions used daily? 

·        Assessment strategies:  What embedded assessments are used throughout the unit? 

o   Are scoring rubrics used? 

o   Are there a variety of assessments used throughout the unit?  What DOK is addressed? 

·        Engagement strategies:  How are the students engaged in the content? 

o   Is there a variety of instructional strategies used throughout the unit? 

o   Is the content relevant and meaningful? 

o   What critical thinking skills are developed? 

You can find the PowerPoint from the session: at http://www.slideshare.net/motoncheff/day-2-act-training-2

As teams continue their work with CCSS and developing the student mathematical practices, the ACT framework will provide guidelines for creating rich units of mathematics instruction.

On a side note:  The Math IL's book study this year is "5 Practices for Orchestrating Productive Math Discussions" (Smith and Stein, 2011).   The Task Analysis guide used during the ACT training is also presented in the book to as a model for reviewing tasks to ensure that the task will supports student thinking and reasoning.  This is also a way to match tasks in the classroom with the expected student learning outcomes. 

My next blog will be about how our PLCs are planning for creating these rich mathematical tasks. 

Lesson Planning for REIL

As Algebra and Geometry Teachers are working on developing ACT units, I thought I would share some tips for writing lesson plans.  I have included terms from the REIL evaluation so you can see the connection between the evaluation instrument and purposeful lesson design. 

Below is the ACT Lesson plan format and in each area are tips for addressing that section of the Lesson Plan.  The tips are blue and the examples are in black. 

ACT Lesson Plan Template

Learning Target: As a result of class today, students will be able to - In this section, list the specific learning targets that you want the students to be able to do by the end of the class.  You may have some targets that are taught over a 2-3 period lesson.  For longer lesson, you will still have daily learning targets.  This is also where you can list your "sub-objectives".  You will also reference the CCSS standard.  When choosing the learning target, how does the learning target connect to the bigger focus of the unit? And how or why is this particular lesson relevant and meaningful to students? Example Learning Target:  You will be able to graph linear functions in slope-intercept form, with and without technology (main objective)  (You need to task analyze to list sub-objectives.) You will solve equation for y.  (sub-objective) You will identify the slope and the y-intercept in an equation written in slope-intercept form. (sub-objective) You will use a graphing calculator to graph a linear function.  (sub-objective)  CCSS: F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Assessment:  How will I know that the student met the learning target?  What evidence will you use to determine the level of student learning of the target? How will students’ self assess and monitor their own learning?  This is where you list what the embedded checking for understanding activities are in order to assess if the students are meeting the stated learning targets (objective) and sub-objectives.   Checking for understanding list  from Metro Tech
Probing questions:  What are the questions that are going to be asked throughout the lesson to guide students’ discovery of the concepts? This is where you anticipate student misconceptions and write clarifying questions that represent all three DOK levels.
Vocabulary What vocabulary will be needed for today’s lesson?
Mathematical Practice:  What student mathematical practice will be developed during this lesson?  Which specific practice will you be developing?  For more specific description of each practice click here, Mathematical Practices. ·       Make sense of problems and persevere in solving them. ·       Reason abstractly and quantitatively. ·       Construct viable arguments and critique the reasoning of others. ·       Model with mathematics. ·       Use appropriate tools strategically. ·       Attend to precision. ·       Look for and make use of structure. ·       Look for and express regularity in repeated reasoning
Activity	What will the teacher be doing?	What will the students be doing? How will students be actively engaged in the lesson?
Beginning of class routine (Bell-work, Warm-up Activity, Tapping into Prior knowledge)	What prior knowledge is needed for today's lesson?  When will I address questions from independent practice (homework)?	This is what the students do from the moment they walk in your classroom.  This is where you list the “HOW” students are engaged.
Activities 1, 2, 3, or more	What are the student learning experiences needed to develop student understanding of the learning concept?  How do we break the learning into manageable chunks of information?  Is this a Procedural (modeling the product) or a Conceptual Knowledge (open-ended with inquiry) lesson?  What research-based strategies from the Educators tool box (instructional practices) are going to be used?  This section is to list the teacher actions needed to facilitate the learning.  List specifics that will address what mathematical tools will be used. If students aren’t meeting the learning target or sub-objectives, what differentiation will be provided?
Closure/Checking for Understanding of the Learning Goal	What will you do to assess if the students met the learning target for the day?	Students will take action on the learning target for the day.  They should be able to tell you what the learning target was and how it connects to the entire unit.

Do you have a great lesson that your team is working on?  What are you and your teams’ challenges as writing these lessons?  Share your responses in the Post section.   

Mathematical Practice #5: What tools do your students use?

To what extent are you integrating technology as a TOOL in your math classroom?  How are you preparing students for the 21st century?  Did you know that students can go to wolphram alpha to type in a math problem and it "spits out"  a variety of information?  The screen shot below is an example:

The key to using this website with students is not just so they can find an answer...it is so students learn to discriminate which information is needed for their purpose.  Imagine assigning a math concept that students can explore using Wolphram.  As they are considering the results and making new meaning of the content, students can share their learning with classmates via facebook, twitter, edmodo, blogging, or through other social networking. This is just one of many examples of how teachers can teach students to use appropriate mathematical tools in the classroom.  Mathematical Practice #5 states,

"Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software." (CCSS, 2010) 

As we are plan for teaching students to consider which tool to choose, teachers have to decide which tool to use with students and how the tool will support student learning of the mathematical concept.  Below are some guiding questions:

• How will this tool develop student understanding of the content/standard? 
• What tool can assist with displaying student solutions?  
• Will the tool present dynamic and multiple mathematical representations?
• Can I use the tool to collect and disseminate formative assessment data?
• How can I use the tool to enhance student collaborative learning?  (Common Core Mathematics in a PLC: 9-12, In press)

Here are some resources for mathematical tools that PUHSD Math teachers have readily available. 

Graphing Calculators

Math Tools website resources
Future of Math Technology tools
Cabri Jr. 
TI-Navigator Tutorials
Math Bits (Tip for TI-84 and TI-Nspires)

SMART board lessons
Library of Virtual Manipulatives

GeoSketchpad
Resources
Sample lessons for GSP

What tools are using in your classroom?  Post your best tool idea. 

Reflect on the progress we are making and CELEBRATE!

As we are going through our first year of implementation of the CCSS within the ACT Framework, I plan on sharing some of the “great news” as teams of teachers focus on rigorous expectations of student learning.  PUHSD has amazing mathematics teachers and I want to “shout from the rooftop” and share some of the exciting stories.  Below are just a few examples from the first few weeks of school. 

Story from a Pre-Calculus teacher at Maryvale High School:

“I just wanted to take a moment to share some good news from Phoenix College.  I recently took my 3 pre-calculus classes to the computer lab and a representative from PC came to administer the accuplacer which is the college entrance exam.  I was able to get 79 students cleared to take the test and just received the official results yesterday.  Of the 79 students that took the test, 20 of them actually tested into Calculus (MAT220, which also allows them to take my class), 51 tested into the appropriate Pre-calculus class (MAT187) and just 8 didn’t make the cut.  Of the 8 that did not make it, the largest margin was missing it by only 3 points!

These results make me very happy that we, as a department, are moving in the right direction.  An interesting note is that these students came through our math program where we implement the use of the calculator daily.  The reason this is interesting is because no calculators are allowed on this test.  I feel we are doing a much better job implementing higher reasoning skills that can only be accomplished by using the technology we all have available to us and appreciate all the hard work that has gone into increasing our students’ level of understanding.”

Story from an Algebra Team: 

“We used the data from the Pre-test exam on D2SC to modify our pacing calendar.  We found that the students had specific strengths and weaknesses so we readjusted the calendar to spend more time on concepts that students didn’t understand”   

Thank you to the teams/teachers that are sharing their stories.  If you have a success story, take a moment to respond to this post.  Sometimes, great ideas stem from hearing what other teams are implementing on other campuses. 



CCSS and Assessments

PARCC recently released a draft of their model content framework for Mathematics.  PARCC states:

"The model content frameworks serve several purposes: They help identify the big ideas in the Common Core State Standards for each grade level; help determine the focus for the various PARCC assessment components; and will support the development of the PARCC assessment blueprints." 

As we begin to plan for the new assessments, what can math teams do to prepare for this transition?  At the Math Team Leader training last week, a portion of the agenda was focused on ways to use Formative Assessments to engage the learner in the process.  Traditional classrooms  assessment cycles consist of teachers giving tests or quizzes, then they grade the tests, return the results to the students, then start a new unit of instruction.  During this cycle, teachers may or may not know what specific math gaps exist in students, yet the students are not purposefully reflective about learning from their mistakes.  Dr. Kanold shared models on how to use summative assessment measures as a learning tool when "viewed formatively rather than evaluatively."  Below are some examples to reference when creating common assessments that engage students in being reflective about their current level of understanding. 


Pre-Calculus Exam:  Notice the GUS.  As students are completing the practice exam, students monitor their understanding of each concept.  When students receive the results from the practice exam (prior to the actual test), they keep track of each concept on the student self assessment form.  This provides the student specific feedback on each learning target they were supposed to master for the upcoming exam. 

Algebra 1. Chapter 5 example

Geometry Quiz example 

It was a fantastic kick-off for the 2011-2012 school year for the math team leaders.  If you want to read more about the training, you can follow Dr. Kanold's Blog at or you can see the newest blog by North HS Math department. 
Click on SLIDES to view the PowerPoint from the Team Leader training. 

E2020 and CCSS Mathematical Practice #5

Last year we piloted E2020 at 5 schools for mathematics concept recovery (Trevor Browne, Cesar Chavez, North, Camelback, and Bostrom). This Saturday, 25 more teachers attended the E2020 training.  The following teachers are resident experts and will be training the rest of their site over the next month. 

School	Teacher Leaders
Alhambra	Kris Cunningham	Kerstin Vikstrom	Marsha Stoddard	Donna Pickens
Metro Tech	Gwen Lee	Suns-Diamondback	Rashawn Gilmer	Jay Osegueda
Central	Dawn Davis	Jeffrey Johnson Greg Stephens	Lisa Spresser Dorothea Lynch	Lisa Sommer
Maryvale	Shana Rose	Surani Joshua Aracely Shepherd	Vicki Burkhart	Fernando Galvez
Carl Hayden	Steve Pusec	Tanya Arnold	Bioscience	Po-Pe Enrique
Cesar Chavez	Oscar Ramirez
Camelback	Becky Angus	Daphne Anderson Smith	Miranda Jackson

A second group of experts will be attending Saturday, August 28,  from Metro Tech, Carl Hayden, Fairfax, Franklin, South, Desiderata, and Homebound. 

E2020 is currently being used for concept recovery as a Tier 1 or Tier 2 intervention.  Students who are struggling with a specific concept can be assigned an online support unit to supplement the classroom.  At the end of the training, teacher leader teams created plans for implementation at their site.  Some sites will use this in the lab classrooms or as part of their class time.    As I was watching the presentation on the labs I discovered that one of the easiest way to implement E2020 as a Tier 1 intervention (intervention for all students) is to use the labs to Model Mathematics.  Below is an example lab: 

CCSS Mathematical Practice #5 is Use appropriate tools strategically.  Below is a snapshot of “student look-fors” for Mathematical Practice #5.  What student actions would be observed while participating in the virtual labs?  How can teacher teams plan to incorporate the labs (gizmos) as part of their students’ mathematical experience?  (The labs can be presented on SMART boards or on presentation system.)

To see the labs, login to E2020 and view Course Structure for your course.   

Below is a short video on the E2020 training, webistes to access E2020 and course activities within E2020. 

 If you need assistance with accessing E2020 please contact your Math IL.