Coordinate System Study

Students will use this template as they continue to study the coordinate system today.

I started the year with coordinate plane standards for many reasons. First the standards are typically easily learned by all. In opposition to much of the number work we do, work that students meet with a range of interest and skill, coordinate grids is a new learning point that they are generally excited about. Therefore to it makes a good standard to start the year with since it helps me to build an invested learning community of students and teachers. Further, coordinate plane standards provide some nice opportunity for "floor to ceiling" exploration and learning this provides students with a tool they can use all year as they learn multiple concepts in math.

Last week we reviewed all the standards in this area as students learned about the reasons for the coordinate system, made coordinate grid pictures, wrote and plotted math stories, and created algebraic equations to demonstrate the relationship graphed on the coordinate plane.

There are many ways that I could choose to approach today's lesson, and I've decided to do the following. We'll review the coordinate system language as we review a Ping Pong story that we'll graph together. We'll also work together to graph the relationship when X = Y times 3. I'll use the story related to the fact that one pound of apples cost three dollars. While I don't like to teach from the front of the room too often, I think the students need this kind of review before tomorrow's coordinate system assessment. Then later in the year, I'll ask students to do what I had originally planned for today which is to write their own math story, chart the numbers, graph the points, and describe the algebraic relationship using variables and/or words.

There are many ways to teach each lesson, and it's up to the teacher to lead the path, a path that will result in best learning and engagement. Onward.

What Do You Do When a Child Arrives at a Grade Level "Behind" the Others with Regard to the Standards?

A great resource for math teaching and learning.

After reviewing a host of student data, I'm left with the question, What is the best way to teach students who are one or more years "behind" with regard to the math curriculum? I used quotation marks for the word, behind, as it suggests a negative label. We all learn in different ways and time frames, so I want to be careful with that term, yet I also know that when students come in far from the grade-level knowledge expectations, the learning community often thinks of those children as "behind" the others.

To answer my own question, I will create Boaler's "floor to ceiling" learning experiences that have what I've always called the 1-2-3 approach where 1 is review, 2 is grade level, and 3 is enrichment. I always open all levels to all students. Willingham in his book, Why Don't Students Like School, affirms the opportunity that exists when students review material that they "know" since there's always more depth to be gained by that review. I know that to be true because I've taught similar standards for years and each time I teach the standards anew I make more connections and learn more.

So with this 1-2-3 or "floor-to-ceiling" approach what's important and how does it work?

It works like this. Students enter the experience at level 1 with a review. For some the review will be quick and pointed, and for others the review stage will represent new learning. Level two is the grade-level norm or standard, and the place that we hope most students will master with the support of educators, classmates, and family members. The third level is enrichment and it's open to all. I find that when I open the enrichment level to all, it's amazing to see who reaches since it's never just those students you'd expect to reach. Often a child who struggles in one area of math may reach in another. That's why I'm not a fan of permanent grouping especially in the early years. I like the flexible grouping we use during RTI related to specific skills, knowledge, and concept goals.

Another great book when it comes
to matching learning with how the
brain works.

With the "floor-to-ceiling" or 1-2-3 approach, what's important are the following notes (and probably more):

• Introduce everyone to the big idea in an intriguing way. 
• Scaffold the experience so there are levels 1-2-3 and all levels are open to all.
• Leave the experience "loose-tight" with plenty of room for student choice and voice.
• Make time and space for discussion, share, and response.
• Use teamwork when possible and be strategic about how you create groups. Use a variety of group types so that students are working with like-partners, unlike-partners, mixed-ability groups, like-ability groups, interest-based groups, and more. As you employ various types of grouping, observe which groups inspire, engage, motivate, and forward student learning. 
• Use strengths/interest-based content and approaches as much as possibly and introduce many ways to learn and let students try out all the various ways to learn too. 

We can't predict how a child one or more years away from the curriculum grade-level standards will learn and grow. Sometimes when a learning experience is truly engaging, a child will leap ahead and demonstrate significant growth. Sometimes students will do this even if the experience isn't engaging as they reach a point of cognitive shift that helps them to merge and consolidate ideas with greater ease and strength. 

There are also times when a student remains stuck with a concept, knowledge, or skill. Give it the good try in multiple ways, and then take a break and return to that learning point later. 

Learning is not linear.

When teaching students who are behind when it comes to standards, think broadly and not narrowly. Make engagement the number one goal. Challenge yourself to work with that child in ways that helps him/her LOVE learning math. Next talk with the child and find out how he/she likes to learn. Work as the child's "servant" in learning. Tell them you are there to help them learn and you need their help in that regard. Assess regularly with the child and reflect too. Ask with the child, "What's working? and What's not working?" Work together to make optimal change. Teach the child how to be their own best advocate and learning manager. Explicitly let all children know that we're all on our own learning paths and those paths wiggle and waver as we learn more. 

How do you teach students who are behind when it comes to standards learning? Please share as I want to broaden my understanding of this as much as possible as I now we can grow our expertise in this area.

How Do You Keep Up with the Research?

Boaler's research points to the advantage of collaboration for students, and
it can be inferred that this collaboration works for educator development too. 

I received an invitation to sponsor a learning event within my school system. It was a clearly outlined invitation, the kind of invite I support when it comes to creating dynamic learning organizations. Hence, I had no choice, but to respond positively by submitting a presentation topic.

I thought a lot about what I wanted to present and lead. I knew I wanted it to connect to my "word," the acronym that will lead my work this year: TOME - Teamwork, Observation/Reflection, Math w/STEAM, and Everyday Learning, and I knew that I wanted to choose a topic that would contribute towards creating a useful, valuable math learning community.

I decided to propose a 15-hour learning event that would help colleagues and I to keep up on the latest research and literature related to math teaching. In addition, I chose to include creating websites and "hosting conversations" information and practice within the learning event. Hence educators who choose to sign up will engage in the following activities:

• create a learning/teaching website, 
• read, study, and reflect on new research related to math education, 
• engage in collaborative conversation that leads to wise action ("hosting conversations")
• readily apply the research to their current math teaching
• reflect and share the application of research to their teaching and its results

I believe this learning process will help the teachers who sign up and me when it comes to updating and deepening our mathematics teaching in many positive ways.

I've come to realize that the best presentations I offer are presentations I'm curious about and invested in--presentations that serve to create conversation, encourage collaboration, and result in positive student learning.

How does your system support collegial share and professional learning? How do you contribute to these efforts? When do these efforts result in meaningful learning and share?

I wonder if any or many will sign up for my class? One obstacle to this is that teachers' schedules are very tight leaving little room for extra study. Yet, one strength is that teachers can earn a salary increment credit and Professional Development Points (PDPs). Also, it will provide educators with evidence for their evaluation goals and provide them with a collegial team with whom to develop their math teaching and learning.

Let's see what happens.

Math/STEAM Year: Classroom Set-Up

Initial plans are set for the new school year, and now it's time to get in there and set up the classroom. Last year I changed the room around a lot as I got used to a room focused mainly on math and STEAM study. I ended up with the set-up above, a set-up that served the many ways we teach and learn well.

Important areas of consideration include the following:

STEAM Supplies

I have a good collection of STEAM (science, tech, engineering, art, and math) supplies. I also have lots of containers and shelving to host the supplies. The job before school starts will be to organize those supplies in a way that makes it easy for students to find what they are looking for. I plan to use zip-lock bags for many supplies, and then I plan to put those zip-lock bags into draws and baskets for easy access. This list of supplies is a starting point if you're just beginning to create your STEAM Center.

Maker Math Supplies

I have a good collection of math tools and supplies including graph paper, rulers, scissors, colored pencils, calculators, protractors, tiles, grid white boards, and more. I will place those in the front of the room for easy access to support our math exploration, investigation, and project work.

Individual Supplies

The students who are in my homeroom will store their supplies in student desks (we still have the old fashion desks, but if I were choosing I'd order tables and chairs with wheels for flexible set up). Students who come to my class for STEAM and math, each have a colorful drawer to store their individual notebooks and baggy of supplies. 

Presentation Space

The presentation space includes the document camera, cart, white board, and student desks. We'll use this area to present learning, foster class discussions and debates, and learn together.

Rug and Book Space

This will be a cozy space for reading and working together. The space will include cozy bean bag chairs and other comfy furniture. There's also a book shelf and baskets of books.

Coach's Corner

There's a space in the back corner of the room with a big desk, white board, shelving, and a rug. This is where the teaching assistant will sit and it will be known as the coaching corner--a place for students to receive extra support and help. 

Tables

Often students are invited to choose their work space. The tables will provide a space for students to work on projects together or to spread out when they need more space.

Check-In

The check-in space will give students a space to sign in each morning, read important notices, and collect important supplies and information.

Tech Cabinets

I have a number of small tech cabinets that store the tech equipment we have. I'll place those in the back near the coaching corner for easy teacher/student access.

Playground

Fortunately the classroom is located next to the playground. We'll often use the playground for our investigation, exploration, creative, and energized learning activities. 

Signage

There will be many signs to encourage students' positive mindsets and choices. Signage will include:

• List of names
• Schedule
• Growth mindset reminders
• Inspiring quotes from notable individuals
• Notes outlining routines

Students will create a lot of signage throughout the year to guide our collective efforts and learning. 

The plan is set and time has been set aside to create the space. I want to make it as inviting and welcoming as possible to help every child begin the year in a positive way. 

Merge Learning Mindsets/Behavior Lessons with Content Focus

Particularly at the beginning of the school year, it's essential to merge learning-to-learn mindset and behavior lessons with content focus. This integration helps students to gain confidence and skill as a learner as well as acquire the math content knowledge, concept, and skill.

The best way to do this is to have a ready list of learning-to-learn behavior/mindset lessons and match those lessons to the daily focus. For example I'll match a focus on collaboration with an initial number review and assessment during the Math Team T-Shirt lesson and the Global Cardboard Challenge project.

I'll weave more of the lessons from my Learning Mindsets and Behaviors website into other math lessons as well. Last year I did this regularly, and every time I focused on a learning to learn mindset and/or behavior lesson, I saw students sit up straighter with bright eyes as these lessons served to empower them and let them know that I knew everyone in the class was capable of success.

In a sense, the learning mindset and behavior lessons are catchy introductions that motivate the best of what students can do. In many cases, these lessons dispel long held myths about learning that students have learned at home and in the broader culture, myths such as some people can learn and others cannot.

To build this empowerment out event more, after I've exhausted my learning behavior and mindset lessons, I'll weave in current events, notable experts from all cultures, races, lifestyles, and religions, and new content and discovery. These kinds of introductions also serve to empower and engage students while making math meaningful. Valuable learning experiences always include a strong rationale as well as ways to make the learning brain-friendly and successful.

Please feel free to use the lessons I've created on my Learning Mindsets and Behaviors website--blended lessons that invite students into the learning. Also if you have other great empowering learning experience additions, please let me know. Motivating, engaging, and welcoming students to learning is as important as the content, concept, and skill learning--it takes both areas of focus to reach successful teaching and long term investment and learning success.

This little film is one example of an introduction I share at the start of the year--it's amazing to see how many students view this as new knowledge when they watch it and we discuss what it means.

Successful Math Community: Explicit Routines

It's important to visualize the math classroom and determine the routines that matter. Then it's important to explicitly teach and practice those routines.

Many elementary school teachers are masters at routine--they establish good routines at the start of the year and stick to those routines all year long. Others find this more of a challenge. A routine-oriented teacher is not better than the other, but the truth is that good routines create greater freedom and time for deep, meaningful, and joyful learning.

What routines matter with regard to your math classroom?

Transitions
Since our students move from classroom to classroom, making the time to teach transitions is integral to using time well. My colleagues did a great job with this last year, and I want to listen to their strategies so I can follow their lead. Good transitions save time and prepare everyone for quality learning.

Respectful, Targeted Use of Supplies
Since students move from place to place, they each have a drawer in the math classroom for their math notebooks and baggy full of supplies. It's important to make sure that everyone has the needed supplies including a grid notebook, pencil sharpener, colored pencils, and fine line marker at the start of the year. It's also important that shared supplies such as scissors, rulers, protractors, and manipulatives are easy to access, use, and put away. Early year introduction to supply locations, care, use, and clean-up sets the stage for a successful year.

Learning Experience Routines
Generally the learning experiences will follow a basic routine each day. In my class, students generally have an introduction, time to study, learn, make, or practice, and a closing comment or share. It's important to explicitly teach the routine so students know what to expect. You can ask students to monitor the routine too so that most of the time is their time to learn, create, share, and practice.

Speaking and Listening
One of the greatest challenges of classroom life is that you're typically working with large numbers of students at a time, that's why it's important to set speaking and listening rules and rationale up front. Typically, these rules lead speaking and listening:

• Raise your hand if you'd like to speak.
• Keep your hand down if someone else is speaking and then raise your hand when they are done if you'd like to say something.
• Speak with a strong voice so everyone can hear. It's advantageous to have a microphone available for the soft spoken students.
• Don't side talk when someone else is speaking.
• Listen carefully to what someone is saying, and if you want to add, but afraid you'll forget, write down your thoughts in your notebook so you're ready to share when the time comes.

Clean-Up

It's good to explicitly talk about and practice clean-up routines and expectations. Large numbers of students can make a big mess and it's important that they know how to clean it up.

Ask Questions, Seek Help

Students need to advocate for themselves, and it's important that we teach students how to do this. Generally I let students know that they can write me a note, seek me out when available, or email me if they have questions or need help. If it's an emergency, I tell them to interrupt me right away, and for leaving the room for the restroom or other reasons, we have a hand signal so it doesn't take too much time from the learning.

Organization

Teach students how to organize their supplies in varying work spaces and for a diversity of projects. For many students, organization is not natural and to provide students with organization tips can really help them succeed. 

Rules, Policies, and Protocols

It's important to review the "Never Go There" rules--rules that have to do with safety, rules such as how to use scissors, never throw things, don't use bad words, no name calling, and safe use of appropriate tools on the computer. You can ask students to add to this list. I tell students that it's always good to have a "Never Do That or Go There" list so that they don't do the things in life they can't reverse--actions that truly disrupt and hurt others.

Then we talk about general policies and protocols--what to do. Students will readily come up with this list. With the upcoming election in the news, this can be matched with the creation of a class constitution which is an effective way to learn about our government and govern the class too. 

These are some important routines to explicitly teach during the first six weeks of classroom life. Ruth Charney's book, Teaching Children to Care, further outlines important attributes of school life--elements that invite all children to a year of successful learning. 

I'm sure I'll add to this list in days to come, and in the meantime, if you have anything you would add, please let me know. 

Making Teaching/Learning Goals Explicit

Too often goal setting doesn't result in good work. This is often true because the goal setting does not identify the success criteria and tracking processes upfront.

To set a goal and move towards that goal with careful thought and attention is an amazing process. To honestly identify the goal's movement from start to finish provides information that helps us to reach the goal with strength, and reaching a goal with strength means that we're doing the work we set out to do.

As I think about my teaching/learning goals, goals that are mandated by the Massachusetts' Educator Evaluation system, I want to clearly identify the success criteria and tracking process.

Student Learning Goal
Students will progress in the following areas:

• Identified fifth grade math standards
• Identified learning-to-learn mindsets and behavior with a focus on collaboration, character, perseverance, organization, empathy, confidence, self-awareness, and self-advocacy

Success Criteria/Tracking
Formal and informal formative and summative assessments will demonstrate growth. Assessments will include the following:

• Math Progress: Content, Concept, Skill Assessments, Project Work, and Oral/Written Reflections.
• Learning-to-Learn Mindsets and Behavior Assessments: Observation, conversation, reflection, project work.
• With acknowledgement of new ESSA requirements, tracking will identify growth for all students including the categories of economically disadvantaged, English Language Learners, Special Needs, and Historically Underserved Cultural/Racial Groups.

Teaching/Learning Path

The path to meet these goals will include the following elements:

• Explicit introduction, discussion, and goal setting related to each goal.
• Transparent, inclusive student-teacher project work, practice, reflection, revision, and assessments as part of the overall learning/teaching process
• Open, honest communication with the learning team related to teach goal and the activity related to that goal
• Regular parent/student meetings and communication related to individual coaching for each goal. 
• Extra support where needed to support children's needs related to the goals above. 

I look forward to explicitly sharing the goals, success criteria, and tracking processes above with students as we learn. I will not explicitly discuss the ESSA goal categories with the whole class to avoid any kind of judgement, prejudice, or name calling. However, I will pay attention to those categories as I teach and look for ways that I can build greater cultural proficiency to make good growth for all students including those specifically identified by ESSA. 

A tight focus as noted makes the teaching/learning year more interesting and focused on what I can do for and with children. We know that good progress mostly depends on the relationships we make with students and the time and attention we invest in each child. Our good work matters in this regard. 

Early Year Conversations: Architect Your Learning

Students need to understand that they are the Architects of their Learning.

We all know the familiar adage, "If you give a man a fish, he eats for a day, but if you teach a man to fish, he eats for his whole life." We know that it's essential that students learn-to-learn so that they become successful lifelong learners. This is critical to living well in a world of ever changing resources, situation, problems, and opportunities.

At the start of the school year, it's advantageous to engage students in a number of conversations about what's important when we think of learning success.

One concept that's important can be titled, "Architect Your Learning."

Tony Wagner spells this concept out well in his talk below:

You may start this conversation with your students with this question, "Who is responsible for your learning and what does that look like?

Let students talk. List the names of the people they feel are important for their learning and what that looks like under each name.

Then say, many are talking about the big idea: Architect Your Learning. What does that title mean to you? Define the word architect:

Then show students this short video:

Ask students to share any new ideas or revisions they have for the "Architect Your Learning" chart you've made.

Then share the fishing adage with students and emphasize, that students are truly the people who are in charge of their own learning. It doesn't matter if the student is young or old, a girl or a boy, or from any country, culture, religion, or race in the world--the first and most important person responsible for a student's learning is the student. 

After that give students some time to think about that by writing, drawing, or quietly talking with the statement:

As the Architect of my Learning I . . . .

Make some time to share reflections. Keep students' reflections in their online or offline portfolios for later review, revision, and conversation. Also post the videos above on the student website and in class newsletters for the whole learning team including students, family members, educators, administrators, and school community members to view and revisit in order to set the stage for a year of successful teaching and learning for every child. 

This is one of many integral lessons to teach students how to learn well. More lessons like this can be found on this Learning Mindset and Behavior website. 

Eleven: Design Math Learning for Student Engagement and Success

Good learning design impacts student engagement, empowerment, and education in math and all
other disciplines.

How does a teacher design learning well?

It's important to develop a learning design process. Most educators prefer the backend design method, a process brought to life in education by Wiggins and McTighe in their book, Understanding by Design. John Hattie further exemplifies this approach in his wonderful book about how to teach well, Visible Learning for Teachers, Maximizing Impact on Learning. 

Over many years, I've crafted a design process that I use to design units of study. It is a successful approach that takes into account research from the books and expertise mentioned above as well as many experiences and professional learning events I've engaged in.

As I continue to write this math book/blog, I'll utilize the design information, approach, and resources I outline in the presentation below. As you think about your math units of study, I recommend that you also use some or all of the information below to organize and implement good design to reach and teach children well.

Ten: Establish Math Learning/Teaching Norms, Rules, and Protocols

What norms and protocols create a great math teaching/learning community?

How do you create those norms, protocols, or rules?

I suggest that early in the year, perhaps after you give the first social skills, effort, and attitude survey, you and the class talk together about what makes a great classroom community. Preface the discussion with a short rationale such as students who learn and know more generally have more choices and opportunity for success and happiness. To know math well means that you'll be well equipped to problem solve, make good decisions, and discern facts and figures--this is all good when it comes to living well. That's why it's important to me that everyone learn as much as possible this year, and together we can make that happen by creating positive guidelines for our teaching/learning community.

There are a lot of ways that you and your students can create a guiding list of rules, protocols, and norms. One good way is the following:

• Pose the questions: What makes a great math teaching/learning classroom community and what doesn't make a great math teaching/learning classroom community? Typically thinking about the positive and negative together elicits more ideas.
• Next have small teams of students discuss this and write down their ideas.
• After that gather the class and let teams share their ideas one by one. As each team shares an ideas, other teams should cross that idea off their lists if its the same.
• In the end, review the list with the class. Ask students if they noticed anything missing. 
• In the end, if there are other norms, rules, and/or protocols you'd like to include, share those.
• Make a clean copy of the list yourself or ask a student(s) to do it. Hang the list up where you can easily refer to it throughout the year. Make revisions and additions as needed. 
• From time to time stop to talk about the list and as a group evaluate the math teaching/learning community? 

Generally when students help to create the norms, protocols, and rules, everyone takes takes it more seriously. This is one good way to start the year. 

Nine: Math Collegiality

Collegiality is a key ingredient to teaching well in any discipline today. I am definitely not an expert in this area, but I do regard this area with respect and a desire to learn more.

What makes collegiality an essential ingredient to teaching well?

First, none of us as educators can do all or be all. The job is enormous and intense, and every day you meet a new challenge, one you didn't expect or perhaps, haven't encountered before. That's when collaboration helps. You can talk with your colleagues to brainstorm and troubleshoot, and generally, together you meet a good resolve.

Next, you can't know all things. That's why our shared teaching model is so dynamic. Each of us has chosen areas to focus on with depth, and we direct our professional learning and leadership in those areas. For example my colleagues have focused in on reading and writing. They lead and I follow in those areas. I lead the math and they follow. That doesn't mean we don't share our expertise or time in the areas that we don't lead, but it does mean that we don't have to teach or stay on top of all content areas as our colleagues will do that for us.

And, as colleagues, we have a variety of life experiences, personalities, learning preferences, and family/friend groups. Acknowledging those differences and maximizing the positive attributes of our diversity supports good teaching and learning too. For example, if I'm having difficulty with a student that my colleague relates well to, she or he can guide me to teach that child well. Or, if I've learned something from my years of experience that my colleague may not know, I may share that. And, on the other hand, my colleague's youth and enthusiasm has the potential to spark my creativity and energy as well. In summary, together we do better.

In truth, no teacher is the best or worst, we all have talents, experience, knowledge, and skill to bring to the job, and our strength is in our TEAM: Together Everyone Achieves More.

Good collegiality relies on good character, and good character sets the stage for mentoring and modeling what it means to teach and learn well for each other and for our students too. The attributes of good character are the foundation of teaching and learning well:

Respect: Polite tone and language, put yourself in their shoes, kindness
Responsibility: Do the job and do it with care on time
Perseverance: Don't give up. Keep pushing forward. Take risks, make mistakes and learn from them.
Thankfulness: Show gratitude, recognize what others do, be grateful.
Kindness/Courtesy: "Please, thank you, help, excuse me, can I help, what do you need. . ."
Self Control: Patience, a good routine, taking time to see the whole picture, good strategy/process
Honesty: Tell the truth
Cooperation: work with others, listen to their point of view, make time to make decisions together.
Tolerance/Acceptance: I'm only one in the sea of people and ideas, we all have a right to be here.
Friendship: Take care of each other, do what you can for one another.
Courage: Don't be afraid to speak up, ask questions, and stand up for what is right and good.
Humor: Have a sense of humor that helps as life is going to take many twists and turns.

Good collegiality demands also that we give each other space to be who we are and do the work that we need to do. Team profits from a regular schedule of time to work independently and time to collaborate. Collegiality is imperative when it comes to working well together, and this is a primary element of a successful math teaching/learning community.

Seven: Know the Math Content

You will be able to teach math well if you know the content well.

Too often elementary school teachers don't have a solid foundation of math concept, knowledge, and skill, and realistically, for many educators, it's been some time since they studied math with depth and it's also likely that when you learned math you may have learned it mainly by memorization rather than deep, conceptual study, the kind of study we utilize with students today.

There are many paths to learning the content well in preparation for the school year.

One of my favorite teaching/learning vehicles is Khan Academy. I recommend that you get to know this platform well and do all the exercises associated with the grade level that you will teach in the year ahead. Khan Academy utilizes the latest research with regard to teaching and learning math and develops student understanding in a blended way using mathematical language, models, real-world problems, and many exercises. You may complete the whole grade level section during the summer or plan to complete each unit section prior to teaching that unit. Khan Academy is a terrific resource for teachers and students everywhere when it comes to deep understanding of math concept, knowledge, and skill. Even better, Khan Academy is free and accessible anywhere that you have a computer and WIFI.

Another way to learn the standards and content is to take a course at a local university. When the new standards were announced a few years ago, I signed up for a course right away and it was an awesome, collaborative way to gain expertise with regard to the standards.

Further there are multiple standards-based platforms online that help you to gain a firm understanding of all the standards and how to teach those standards, and system resources offer support too.

To know the content well prepares you well for the teaching and learning ahead. Summer is a great time to shore up content knowledge and skill. Once you gain expertise in the grade-level standards, its important to grow your knowledge of the foundation skills, concept, and knowledge leading up to your grade level and the mathematical content that your grade-level knowledge leads to--the multiple grade-level standards beyond the grade you teach.

Six: Math Manipulatives, Tools, and Proceses

As you prepare for a terrific math teaching year ahead, affiliate with supportive organizations, establish a positive math teaching mindset, make time for a dynamic room set-up, create a website, build knowledge of expectations, and make plans to introduce math manipulatives and tools as well as the protocols for strategic use and practice.

Utilizing manipulatives help students to learn and apply math knowledge, concept, and skill.

To use math manipulatives, process, and tools well students need to know what they are and where they are located. They also need to understand the protocols associated with good use so that the tools, processes, and manipulatives are available when and where they need them.  You may couple the introduction of specific manipulatives with early year lessons, practice, and vocabulary review. Essentially beginning the year with these kinds of lessons lets students understand the playful, hands-on, and accessible discipline math is and provides them with engaging avenues for mathematical thinking, exploration, and study.

I've listed some of the tools, processes, and manipulatives I plan to use at the start of the year below. It's a list I'll continue to add to, and if you have suggestions for me, please let me know.

Grid White Boards and Pens

Students really enjoy learning math with grid white boards and pens. A great early year lesson is using the white boards to set up a coordinate grid, plot points, and create a picture. After that students can create another picture and set of coordinates to share with a friend. The most important protocol with regard to white boards is where to store them, how to clean them, and organization of the pens (and caps).

Rulers
A great way to learn about rulers is to make your own rulers. An early year lesson on using a ruler to make a ruler introduces students to the language of measurement as well as benchmark fractions and the fact that the ruler is a number line. Later students can practice drawing lines of specific lengths, making margins, learning to start at the 0 point, and later practice using the rulers to draw designs. Early learning about how to use the rulers and where to store them allows students to access rulers whenever needed during the year as they complete projects, make math models, and study numbers.

Tiles

Tiles are a great resource for all kinds of math teaching/learning, and at the start of the fifth grade year using tiles to review facts and arrays is a great exercise. Beginning with 24 tiles and asking students to make as many different arrays as possible is a great exercise for initial tile use and review of arrays, facts, area, and perimeter concept, knowledge, and skill.

Legos and K'nex
Building and creativity toys like Legos and K'nex offer positive math model making materials and free time play too that supports math teaching and learning.

Google Apps
Google Apps are terrific for math teaching/learning. Beginning the year with teaching students how to use Google table and draw to create math models is very helpful. Students in fifth grade will use Google apps to make number posters for numbers 0-100. The posters will include meaningful illustrations of the number, the numbers' factors and some multiples, number facts, expressions equivalent to the number and equations too. There's more information about this activity on this post.

Pattern Blocks
Early year introduction to the definition of a pattern and the properties/attributes of pattern blocks enables students to use these blocks all year to inform math study and project work as well as to have the lens of "pattern seeker" and "property/attribute identifier."

Tangrams
Tangrams are lots of fun to use and provide a great way for students to develop mathematical thinking and understanding of two-dimensional shapes and their properties. There are great online tangram exercises as well.

Origami Paper
Having a collection of origami paper, origami books, and online origami sites also supports math teaching and learning. In the past few years, we've had an origami expert come in to teach math with origami. Starting the year with a simple math origami activity introduces students to this wonderful math activity--an activity they can explore more at home and during their free time.

Math Books
Having a basket or book case filled with math related books is a great way to demonstrate to children that reading about math is a great way to learn math. There are lots and lots of great math books out there, so many that you could actually begin math class every day by reading a short excerpt from or entire math picture book. These books have been well organized and researched, so it's a great way to teach.

Online Math Games, Songs, and Videos
Introducing students to online math games, songs, and videos, and making those sites easily accessible via your math website provides another enjoyable avenue for math learning and teaching.

Scissors
It may seem silly, but many students don't know how to use scissors well. An early year activity with paper folding and cutting can help students to study symmetry, right angles, and the properties of two-dimensional shapes. You can use this lesson to review scissor safety and protocols too.

Colored Pencils and Markers
Since the fifth grade year includes lots of math model making, students can practice making arrays using graph paper, colored pencils, and markers early in the year. This can be coupled with the tile lesson and will support students' model making all year. Students can learn how to use fine line markers to border edges, while using colored pencils to shared different model parts. Research has demonstrated that using color helps students to delineate concepts, parts, and ideas in math. An early year explicit lesson in this regard helps students to make precise models all year long.

Cubes
Learning to use cubes provides a great opportunity for students to learn volume. Our cubes became difficult to connect last year, so this year I'll likely use sugar cubes. Or I may use this as an opportunity to teach students how to create nets and then use those nets to make three-dimensional figures, and later use those three-dimensional cubes to learn and study volume. The more we can deconstruct concepts with manipulatives and project work, the better able we are to help students develop deep understanding of mathematical concepts, knowledge, and skill.

Base Ten Blocks and 3D Models
At fifth grade I prefer using online base ten blocks to review the base-ten system, but it's also good to have some base-ten blocks available for trading games to build understanding. Last year I used the 3D printer to create a 3D base ten model which was a terrific exercise, one that would be great for students if you have access to 3D printers.

Coding
Teaching students to code early in the year will support students' creation of animated math models. In the past, I've used Khan Academy coding and Scratch in this regard. Prior to the Base Ten Number System unit, I'll make some time to review coding with students so they can animate models that demonstrate the "behavior" of the base ten number system.

Paper
I have many types of paper in my classroom. Organizing the paper well and introducing the paper to students will help them to choose the right paper for the right task. I'll discuss paper types and we'll use lots of different kinds of paper at the start of the year so students know what to choose and how to use.

Pencils, Sharpeners, and Erasers
So often students' hands get tired because their pencils are dull. We'll talk about sharpening pencils regularly so you have a good point. We'll discuss when to use an eraser and when crossing out is just fine. The goal here is to know how to use the tools well so that most of the focus is on developing a strong mathematical skill, concept, and knowledge foundation.

Four: Create a Math Website

Create a math website to host all resources, ideas, and exercises. Weed and prune your website regularly to make sure it is easily accessible to the learning team: students, families, colleagues, administrators, community members and your PLN

Create a website to host all your math teaching/learning ideas and resources.

Create the website with the entire teaching/learning community in mind. Think of your website as a resource for students, families, colleagues, administrators, and community members.

Make your website as easy to navigate as possible. Think of Russian Nesting Dolls as you create the site and create learning paths on the site.

This is a link to my Magnificent Math website. I have some weeding and pruning to do before the school year starts, but this website has served as a terrific, evolving math/teaching resource over the past many years.

I used Google sites to make my website as we're a Google school and the platform works well with our heavy use of Google apps. There are many website platforms available, and what's important is that the platform is easy to use and access as you want to spend most of your time on math teaching and learning rather than trying to navigate a difficult platform.

Three: Know Your Math Progress Report

Creating 2D, 3D, and animated math models is an integral
part of teaching and learning math today. 

Taking a close look at the report card or progress report helps an educator to define the priorities that are important with regard to his/her teaching and the organization he or she works for. I analyzed our fifth grade report card below with regard to social development, work habits, math effort, and math skills, concept, and knowledge expectations. The analysis points to a large number of lessons that should occur early in the year and throughout the year so students are well aware of what's expected and how they can reach those expectations.

Jo Boaler, author of Mathematical Mindsets, is not a big fan of homework, however she recommends that reflective homework assignments can be very helpful and positive. To begin the year, I may include lessons related to social development, work habits, and math effort during each lesson. I also may ask students to reflect on these elements as part of their nightly homework to build capacity as well as practice and knowledge of reflective practice--a skill that's a valuable element of any learner's toolbox.

As a new or existing math teacher, take some time this summer to review and analyze your school's report card or progress report. Think about the standards students will be scored on. In systems like the one I work in scores are simply progress notes such as "progressing towards expectations, meets expectations, or exceeds expectations" and in other systems students are still scored with old-time traditional grades. I am a fan of standards-based reports as they foster a progressive, learner's mindset. The kind of mindset that demonstrates to learners that we're all on the continuum and our good efforts help us to move closer to our goals (standards).

Below I analyze each element of the report card that I have to grade in the year ahead. The analysis demonstrates to me a good order and process for teaching each concept. Not all required standards are included on our reports, but the standards with greatest priority are included.

Social Development

Perseveres when faced with challenges:
What does this look like? How does one persevere? When have you persevered in the past? To introduce this topic, I may present an open task and have students work on it with a focus on their needed perseverance. We'll reflect later on what that felt like, and how they motivated themselves and each other to keep going.

Uses positive strategies to resolve conflict:
We'll review the "Take the first step" process of conflict resolution which is to talk to the person you're having the conflict with. The second step is to get help. We'll brainstorm the types of conflicts that happen at school and the many ways one can resolve those conflicts.

Accepts responsibility for own actions:
What do you do when you make a mistake or do something wrong? How do you deal with that?

Follows school and classroom rules:
First we'll create a class constitution with our rules and protocols. Then we'll talk about ways one can work to follow those rules.

Plays cooperatively with peers:
What does it mean to play cooperatively? What does this look like?

Work Habits

Demonstrates initiative:
What is initiative? How does one demonstrate that in school?

Seeks help as needed:
Don't stay stuck! When you need help, seek it out. How can one do this?

Manages time well:
What does it mean to manage time? What strategies do you use to do this.

Works independently and productively:
What id independent work? When might a student be asked to work independently? What does "working productively" mean? What does that look like?

Transitions in a timely and appropriate manner:
What does transition mean? When do we transition? What is appropriate behavior for transition?

Follows directions, routines, and procedures:
First we need to establish routines and procedures and make those routines and procedures explicit. Similarly all directions need to be clear and explicit. So to follow directions, routines, and procedures, first students need to understand what they are. Next students need to brainstorm together the strategies they use to follow directions, routines, and procedures well.

Mathematics Effort

Participation:
What does optimal participation look like in the math classroom? How can we encourage one another to participate? What hinders student participation?

Asks for Clarification:
What does the word "clarification" mean? What are the ways that you might ask for clarification during class or after class? What happens if you ask, and you're still not clear?

Attends to Work:
What's expected with regard to "attending to work" and what does that look like in a classroom? Will this look different during different kinds of learning experiences?

Perseveres when Challenged:
What does perseverance look like in the math classroom? What behaviors are exhibited when you are persevering, and what activities are not exhibited in this regard?

Attends to precision while applying mathematical concepts:
Why is precision important in mathematics? What strategies can one use to be precise?

Math Concept, Knowledge, and Skill

Communicate mathematical thinking orally and in writing using math vocabulary:
Math vocabulary study is integral to math understanding. Every unit should begin with a review of the vocabulary. Vocabulary should be clearly visible in the classroom and accessible for at-home study too. Students should regularly express their mathematically thinking with written or spoken words. Creation of math scripts and videos support this work which ties in nicely with the Standards of Mathematical Practice.

Represent and interpret data using graphing skills:
Initially this is a great standard to meet when making early year infographics to help a class get to know one another. The line plot is the featured graph for fifth grade at this time. Later in the year this standard can be reviewed as you solve fraction problems and chart fraction/decimal data related to real-world problem solving.

Automatically recalls multiplication and division facts through 12:
What do we know about the numbers one through twelve? What factors and multiples can we identify related to these numbers? Which of these numbers are square numbers, composite numbers, prime numbers, or perfect numbers? Where are these numbers present in our culture, and what other names do we have for these numbers? How can we gain automaticity with facts? Why is this important? How do we do with this now, and how can we develop this skill?

Coordinate Grid Knowledge and Use: Able to graph points on the coordinate plane to solve problems.
Understand how to graph points, and then how to use graphing to determine numerical relationships. Student enjoy learning this and practicing it. This study can be embedded throughout the year as students analyze numerical relationships and solve problems.

Classifies two-dimensional figures based on their properties with particular attention to parallel, intersecting, and perpendicular lines:
Students enjoy this study which can be taught well using drawing, tangrams, geoblocks, and online games/venues.

Understand the concepts of volume and applies formulas to solve problems:
Since volume problems essentially include simple numbers, this is a great unit to introduce as you review basic facts. It's also good to review area and perimeter at this time before you get to area models of multiplication and division with larger numbers. Further this unit comes nicely after an initial review of two-dimension geometry since understanding those shapes and how to talk about them informs the volume, area, and perimeter unit.

Use positive and negative numbers to describe quantities:
This concept fits nicely into initial number talk at the start of the year when we look carefully at the numbers 1-12 and where those numbers fit on the number line. Again this can be reviewed during money-related word problems.

Writes and interprets numerical expressions using parentheses:
This is a great skill to teach as you teach students to use calculators effectively. This is also a great teaching goal to master when you are working with "easy numbers" and warming up math year skills. Though it is scheduled on our reports for Term Two, most teach this early in the year as it can be useful and practiced throughout the year as number concepts, knowledge, and skill become more sophisticated.

Generates, extends, and compares patterns and relationships:
Again this is scheduled for term two, but should be a focus of every math lesson with the following questions: What patterns do you see? How do the patterns compare? How might you extend the pattern? What relationships do you see? How would you define the relationship? Can you represent the pattern and/or relationship using a numerical or algebraic expression? Can you graph the relationship or pattern?

Understands the base ten place value system from millions to thousandths?
What are the parts of the place value system? How do these parts work together? Why was the base ten place value system invented? What value does it have in our culture? How do the values of numbers change as you move up and down the place value system? How do the values of numbers change when you add, subtract, multiply, or divide by base ten numbers?

Perform addition, subtraction, multiplication operations with multi-digit whole numbers?
Why do we use the word "operation" when we talk about addition, subtraction, and multiplication? Why do we add, subtract, and multiply? How can we perform those operations effectively and efficiently? Is there one way to do this? Is there a best way or does this depend on when and why we are performing the operation? What 2D, 3D, and animated models can we make to demonstrate these operations?

Perform division operation with multi-digit whole numbers?
What does it mean to divide? When would we use this operation? How can you divide one number by another? What words do we use to describe the numbers associated with division? What models can we make to depict division?

Performs operations with decimals to the hundredths:
When do we need or want to add, subtract, multiply, or divide decimal numbers? Why is this important? How can we perform these operations? Working with money and problem solving is an important art of this study.

Convert standard measurement units within a given measurement system:
This study fits nicely with new science standards that included working with matter. Many experiments can be done that find students converting measurements while exploring matter. This also alines well with base ten system study as students look at models of the base ten system and match those models to metric models.

Interpret a fraction as division of the numerator by the denominator:
1/2 = 1 divided by 2 which equals .5, both 1/2 and .5 are equivalent to half of the whole. Proper fractions like decimals demonstrate part of a whole. We can easily turn a fraction into a decimal by dividing the numerator by the denominator. Sometimes it's easier to work with fractions and sometimes it's easier to work with decimals. A good way to practice this skill is to begin by making a fraction, decimal, percent, and ratio equivalency chart. Later it's good to look at how this skill relates to meaningful fraction numbers with real-time problem solving and project work.

Add and subtract fractions with unlike denominators and solve word problems with addition and subtraction of fractions:
Use a signature story to describe and discuss this skill: "Amy and Paul were making cookies. They each had a table of ingredients. The cookies called for 3/4 cup of sugar. In the end, Paul had 1/8 cup of sugar let and Amy had 2/6 cups of sugar left. Together do they have enough sugar left to make one more batch of cookies?" How would we figure this out? What can we do to figure out if they have enough sugar left? Later students can make up their own stories and demonstrate this skill conceptually. Lots of fraction model wok supports this effort as well.

Applies and extends previous understandings of multiplication to multiply a fraction or whole number by a fraction and solves related real-world problems.
This work also benefits from signature stories. Begin with multiplying a whole number by a fraction: "Emily wanted to give every child 1/2 a piece of paper for the art project. There were twelve children. How many whole pieces of paper did Emily need? What if Emily wanted to give each child 1/3 piece of paper, then how many sheets of paper would she need?" Draw a model of this story. Solve the problem with a model and mathematically. Make up your own problem, solve with model and numbers. Present your problem to the class.

Next, work with multiplying a fraction by a fraction using the area model. Again use a signature story: "I bought a big rectangular cake for my dad's birthday. After the party I had 1/2 of the cake left. I wanted to give my brother 1/4 of the half cake. What portion of the original cake did I give to my brother?" Show how this problem looks with a model, then solve the problem mathematically. Have students make up their own problems to solve with the area model (similar to the cake), picture model, and using the fraction multiplication algorithm.

Applies and extends previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions:
Introduce with a story: "I was going to the fireworks with my family. I had three pieces of licorice, but there were eight people in the car. I decided to split the licorice into 1/4 pieces. If I did that, would I have enough licorice pieces for all people in the car?" How can we write this problem as an expression? What would this problem look like as a model? Do I have enough pieces if I divide 3 by 1/4? Or "I had 1/4 of a pizza left. I want to split it into 3 equal pieces. What size pieces will I end up with?" What does this problem look like as an expression and model? Students can make up their own fraction division problems to solve with friends and present to the class.

Understand what a mixed number is and solve related world problems:
Discuss mixed numbers. Brainstorm when we use mixed numbers in real-time. Look at the many ways we can write and/or model a mixed number. Add, subtract, multiply, and divide mixed numbers. Solve and create real-world mixed number problems.

Introduction: Teach Math Well

Throughout the summer, I am going to focus my writing and research on teaching math well. Mostly I will focus on teaching fifth grade math well, but many of the posts will relate to math teaching/learning for elementary school teachers everywhere.

Throughout the summer, I welcome teacher comments and ideas. I hope to use the writing to guide my teaching/learning work in the year ahead as my main goal of the year ahead is to teach all fifth graders math in successful, affirming, and growth-producing ways.

Two: Affiliate to Learn and Teach Math Well

To teach well, it's important that you belong to a supportive team within your teaching/learning organization and outside of it.

I recommend that you join the National Council of Teachers of Mathematics (NCTM) and your local math teaching/learning affiliate. I also recommend that you join Jo Boaler's youcubed at Stanford association for timely, research-based information and ideas.

These organizations will provide you with up-to-date research, learning opportunities, tools, and conferences that will support your positive teaching and learning.

In general, most school systems provide money so that educators can attend professional learning conferences. You'll want to learn about that. Some organizations also pay for professional periodicals and members in professional organizations too.

It can be overwhelming to receive all the information related to professional organizations, but if you make a pattern of reading one article a month or weekly review of the materials, it makes this professional learning pattern both productive and doable.

One: Teacher's Mindset

Are you a fifth grade teacher who wants to teach math well?

If so, begin with a successful teaching mindset.

What does that mean?

It means believing in the following tenets:

• Every child is capable of learning in meaningful, positive ways.
• Relationships matter. Develop strong relationships with every child.
• Build Team with families, students, colleagues, administrators, and community members--when we work together, we do better.
• Understand the teaching program expectations well--know the content.

What actions contribute to these tenets?

Signage

Find or make signage that reminds you, your students, and other visitors to the classroom that everyone is capable of learning in meaningful, positive ways. Position that signage in your classroom in a place where you and your students will see it as you teach.

Activities:

• Storytelling: Explain to students that there was a time when people thought some could learn and others could not. Tell students that myth has been proven untrue, and now we know that every child can learn and every child can learn well. The challenge is helping every child recognize the way they learn well and putting that effort, mindset, and tools in place. 

• Ask students to think about their own learning journey. Let children reflect and then present their math learning journey via writing a letter, making a timeline, creating a poster, drawing a cartoon or using another way to demonstrate their math learning to date. Give students time to share their "math autobiographies" with classmates. Respond to their work in writing. Save a copy of their work for later student reflection and additions. 
• Relate: Tell students your own math journey. Also explain to students that good learning depends on positive relationships. Ask students to think about the relationships they've had with teachers in the past, and then, with students, list the attributes that characterize positive teacher-student relationships. Keep that list posted in places where you and the students will see it daily.
• Communication: Establish regular (perhaps weekly) communication patterns for teacher to the learning team (families, students, colleagues, and administrators) and student-family/teacher. When everyone on the team knows what's happened, what is currently happening, and what's coming up, people feel like they are part of the team and people contribute to that team.
• Open Door Policy: Invite students, families, and colleagues to email you or set up appointments with you when there are issues, questions, or ideas to discuss. Typically when you do this along with regular communication, the meetings are greater at the start, but less as the year goes along because the learning team develops an efficient, targeted teaching/learning relationship.
• Study the Content: My posts in the days ahead will outline patterns of study that help a teacher teach well throughout the math year.

Successful teaching depends on a positive teacher mindset. Books like Mindset help teachers to strengthen a positive, proactive mindset for teaching well.

What else would you add with regard to developing a strong mindset for teaching well?