Guided Math ~ Chapter 7: Building Mathematical Proficiency

Being mathematically proficient is more than just knowing how to do something. It is an attitude, a way of thinking (pg 86).

This chapter goes into more detail about mathematical proficiency which I briefly mentioned in my post on Chapter 5.

5 Components of Mathematical Proficiency:

Conceptual Understanding:
Students need to understand math on a conceptual level. In guided groups, students learn about the concept using manipulatives, scaffolds, and tools. It is much more than just knowing how to do something.

Procedural Fluency:
Students need to be able to "do" math and think flexibly about math using different methods: written procedures, mental math, calculator, and manipulatives. In guided math groups, math discourse ensues while students work on procedures.

Strategic Competence:
Students need to be able to solve problems and represent their thinking. There is more to math than just getting the answer. In guided math groups, the focus is on students identifying/using different "pathways" for solving problems.

Adaptive Reasoning:
Students need to be able to explain and justify their work/thinking. In guided math groups, math discussions occur where students reason, listen, interact, and make connections. Students are encouraged to use more than one way to solve a problem.

Mathematical Disposition:
Students need a disposition that will allow them to be confident in math and are willing to persevere and appreciate math while reflecting and monitoring their own learning. Mathematical disposition is about ways of thinking, doing, being, and seeing math. In guided math groups, learning can be scaffolded to foster confidence and success. Perseverance can be nurtured when students are given time in guided math groups to wrestle with problems. Small guided groups create a safe environment for students to take risks.

Sometimes my students need help seeing themselves as mathematicians. I spend a lot of time focusing on this at the start of the year, and I revisit this notion when I see their perseverance start to wane. As a visual reminder, I have posted my Mathematician Posters on a bulletin board and made a bookmark of the mini posters for students to keep in their math books/folders.

Getting students to model their thinking using different representations is not always easy. Many times my students are so proud to be the first to get the answer! This, of course, is not what I focus on. I work hard to try and change this mindset. Some students have their own way of doing/showing their math. In guided math groups, modeling different methods and having students see other students share different methods can help build stamina toward this goal. 

Conceptual understanding is the core of my instruction. Many of my students come to me with a strong foundation in math. My goal is to fill the holes with conceptual understanding. Why is that answer correct? What is that algorithm really showing you? I want my students to be able to justify and explain conceptually why something works/does not work. One time, I gave a problem to a group of students that did not have a solution. Sure, students could "do" the problem using the steps in the procedure we learned and get an answer, but the solution would not make sense. One student returned the next day and explained why the problem would not work conceptually. He was right! The rest of the group solved the problem using the steps of the procedure and got an "answer." Unfortunately, this answer did not work in the context of the problem. We had a long discussion about "doing" math and "conceptually understanding" math. It made students realize math is much more than just doing the steps to get an answer to a problem!!

When problem solving, I have found that my students lean towards showing math with numbers. That's what they feel most comfortable with! Pictures are sometimes used. When they get stuck, I often encourage them to represent the problem in pictures. But, this is not something they do naturally, yet. Words. Written words. Oh, my. If they could avoid them in math...they would! When they have to...they do it...and they do it well. But this is not a go-to-method to represent their thinking. Definitely this is a work in progress to get students to model their thinking using numbers, words, and pictures.

Guided Math in Action ~ Chapter 6: Framework for Guided Math Lessons

Planning. Planning. Planning. In order to maximize learning during guided math, it is important to constantly monitor data and performance of students.

The key takeaway for me was right on pg 69 when Dr. Nicki Newton talks about teaching at the concrete, representational, and abstract levels. I have written about the C-R-A sequence of instruction in a previous post. Click the image to read that post.

When I plan for instruction whether I am working with elementary or middle school students, C-R-A has become a habit of mind. Sometimes it is a challenge, but it is definitely exciting to see the progression of learning by students.

The chapter goes on to talk about the framework for guided math lessons. The key components include:

• mini-lesson
• student practice
• share time 

Mini-Lesson
One thing I have found is keeping mini-lessons mini. Dr. Nicki Newton outlines that in presenting a mini-lesson you should (pg 70-72):

• hook the students into the lesson
• explain the focus of the lesson
• present specific learning expectations 
• model/demonstrate 
• check for understanding.

 That is a lot . . . which goes back to the need to plan carefully and thoroughly. 

Student Practice:
This is the exciting part of the lesson for me. I get to interact and have math conversations with my students. One key component of this time is to ask effective questions that foster student thinking. It is a time to observe students in the moment and get a front-and-center perspective on how they engage with the math. It is a time to record observations that can help drive future instruction. This time never seems like it is enough time!

Share Time:
This is the time when synthesis of the lesson occurs. It also is the time that I have to consciously ensure happens before the end of math class. The debrief is so important for student learning. It brings everyone back together to restate the goal or focus.

Dr. Newton recommends having some sort of planning sheet. Honestly, I have not found a planning sheet that works for me. I have found in the past when I have used a template, my planning becomes too contrived and I feel like I am just filling in the template because I am supposed to. I do have a checklist of points to keep in mind, and I plan from there.

It is one thing to plan a great lesson, it is another to spend time reflecting on the lesson. I like the questions on pg 76 to help keep a pulse on how students are progressing.

• Are students learning and independently applying the concepts, strategies, and skills?
• Do students transfer the learning to their daily/independent work?
• Are students developing fluency and flexibility of numbers and thinking?
• Are students able to explain and model their thinking?

I think Dr. Newton sums it up best when she states, "Guided math lessons follow a particular protocol. You just don't pull some students together and work with them randomly (75)." On the teacher's part, guided math lessons take planning, forethought, and using best practices for teaching.  On the student's part, guided math lessons allow for active engagement where exploration, conversation, questions, and demonstration of understanding occur.

Guided Math in Action ~ Chapter 5: Balanced Assessment

When we hear the word "assessment," so many things pop into our heads. This statement on pg 51 was powerful: "Balanced assessment means that we continually look at the whole student in various ways." With the focus being on the whole child, it is important to have various assessment tools.

In this chapter, Dr. Nicki Newton references the five elements of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and mathematical disposition. (pg 52) I definitely want to work on creating a better balance of these five elements in future assessments. I picked one question as a trigger for each element using the list from pg 53.

CONCEPTUAL UNDERSTANDING: Does the student understand this concept?

PROCEDURAL FLUENCY: Can the student do the math?

STRATEGIC COMPETENCE: Does the student use different and efficient strategies when solving problems?

ADAPTIVE REASONING: Can the student talk about the concept?

MATHEMATICAL DISPOSITION: Does the student monitor his/her own learning?

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The chapter then focuses on pre-assessment, ongoing assessment, and evaluative assessment.

Pre-assessment:
I really liked how Dr. Nicki Newton flipped the way she launched the lesson on division. As a whole-group pre-assessment, Dr. Newton probed the students to get a pulse of where the students were at by asking the following questions (pg 55):

• What is division?

• Who can give me some examples of when we use it?

• What are the tricky parts? 

The whole-group discussion can activate prior learning for students. Sometimes students just need a trigger to activate and help them remember what they already know. In the past, I have allowed students to quickly preview a concept before a pre-assessment so it's not a cold topic. It has given me a more authentic read for those students who really may have some understanding of the concept, but may just need a quick review. I keep students like Miguel (pg 41) in mind. The data from the pre-assessments is then used to help group students for guided math and to help drive instruction.

Ongoing Assessment:
Ongoing assessment helps keep a pulse on where students are at so we can take them where they need to go (pg 9).

Observations, graphic organizers, and conferences are a few of Dr. Nicki Newton's suggestions for ongoing assessment. In using these, adjustments can be made to the instructional plan for students when needed. I like the idea of using a Teacher Observation Sheet when monitoring and keeping anecdotal notes from guided math sessions (pg 61). The parts that I want to add to my observations during the coming school year are:

• What does the talk sound like?

• What are keywords and phrases being used?

•  Who is not talking? (This is important to notice and track so I can help these students be more comfortable in taking an active role during guided math sessions.)

Evaluative Assessment:
This section's key takeaway is to review the results from an evaluative assessment with the class and individual students. Assessments should be more than a grade; students need to look deeper at their performance beyond the grade they received. I like how Dr. Nicki Newton talks about helping students to concretize their learning before going on to the next concept by having them express what they learned, understand, and still need clarification on (pg 64).

I would like to incorporate more performance assessments. I feel they are more comprehensive and help students see how to apply math. Due to time and finding quality performance tasks, this is not always easy. But it doesn't hurt to set a goal, right?!

Two for Tuesday at Pam's Place

That's My Rule: Candy Machines with In/Out Boxes has two activities. Each activity has 10 cards with different function machines, otherwise known as Candy Machines in these activities. The cards target addition and subtraction to 20.

Candy Machine ~ What’s My Rule? is an activity designed to have students look for patterns and then identify the rule for the In/Out Boxes. 10 cards included. Math tools might be helpful for some students.

Candy Machine ~ Working or Broken? is an activity students really enjoy, especially when a Candy Machine is broken! For this set of cards, students have to determine if the pattern or rule for the Candy Machine is followed for each set of numbers in the In/Out Boxes. More than one set of numbers might make a Candy Machine broken. 10 cards included. Math tools might be helpful for some students.

This is a DEAL. Check out these Mathematician Posters. I am participating in two online book studies this summer: Teaching Numeracy, 9 Critical Habits to Ignite Mathematical Thinking by Margie Pearse and K. M. Walton at Math Coach's Corner and Guided Math in Action by Dr. Nicki Newton at Adventures in Guided Math and have math on the mind. Don't we want all our students to see themselves as mathematicians?! These mini posters could be that visual reminder ;)!

 

Students as Mathematicians Mini Posters can be used to create a bulletin board to remind students they are mathematicians. Pass out the mini posters to students and have them role play what being a mathematician looks like and sounds like in the classroom. Keep a set of mini posters available during small group instruction to point out positive math behaviors or to support learners. Use the posters to anchor in good habits of mind when learning math. 

Guided Math in Action ~ Chapters 3 & 4: Managing the Math Workshop and Forming Guided Math Groups

Welcome back! Today I am going to discuss chapters 3 & 4 from Guided Math in Action as part of a book study hosted by Adventures in Guided Math.

Chapter 3: Managing the Math Workshop

I liked how the chapter started. Start slowly. Start calmly. Start immediately. Just start! (pg 29) The whole management piece can be daunting.  Routines. Procedures. Expectations. Setting the stage and adhering to the expectations are key. I keep thinking back to guided reading. If similar routines, procedures, and expectations can be found in both reading and math, the transition for students might be easier.

After reading the chapter, I keep thinking back to "hot topic centers." (pg 31) It's not to say I don't think about "hot topics" or the need to recycle and review "hot topics." But I want to keep monitoring and adding to it throughout the year. What "hot topics" come to mind for your students?

One key component of math workshop is having supplies and math tools readily available. In looking at the Figure on pg 36, I really like the varied representations of math tools. I feel it is so important that students have multiple ways to view math. It reminded me of a lesson where students represented decimals to the hundredths place using base-ten blocks, a centiwheel, and money.

All math tools are kept in a community location for easy student access. Organizing manipulatives in student packs makes for more efficient use. The Standard for Mathematical Practice #5 states: Use appropriate tools strategically. After instruction, I allow students to choose the tools they feel will be most useful when practicing new concepts or solving problems. One thing I am sure to do is to introduce and model the correct use and functionality of math tools for students first before they are added to the community location.

Management of guided math takes time, patience, and PLANNING.

Chapter 4: Forming Guided Math Groups

This is the best chapter so far. Small group instruction is key to helping us meet students where they are so we can take them where they need to go (pg 9). While reading the beginning of this chapter what resonated with me was the statement that it is very important for teachers to work with small guided math groups at ALL levels, not just the lowest-performing (pg 41). I know this is not always easy to do, however, if we are going to take our students where they need to go, ALL students deserve the benefits of small group instruction. Dr. Nicki's story about the first grader, Miguel, was a gentle reminder to those students who may already know the current lesson and probably need something more. Math workshop seems like it can afford opportunities for ALL students to complete tasks at their readiness levels in order to move their learning along.

One of the things I have learned when forming groups is I not only need to allow data and teacher observation to guide my decision ... but also student voice. Student voice through reflection and conversation. The groups need to remain fluid. I think building a sense of community early on in the math classroom helps students to understand that "fair is not always equal." Highly capable students are not highly capable in everything; struggling students have areas where they shine. Honoring where students are ready to learn is important to me, yet sometimes it is not always easy to find the "right fit" on the "first try."

Record Keeping. I jot down little notes in the moment when working with students so those thoughts and observations are not lost. Because I don't want to take away from the interaction with students, I keep my notes concise but try to target key points. Sometimes after reflecting about the lesson, I might have an afterthought and will jot down any additional ideas on a sticky note if there is no room left. One record-keeping sheet that I have used is the one below. Click on the image and grab the FREEBIE if it is something you can use.

Guided Math in Action: Chapters 1 & 2: Perseverance and A Numeracy-Rich Classroom Environment

 

Welcome to the Guided Math in Action Book Study hosted by Adventures in Guided Math. Today's discussion is going to focus on Chapters 1 and 2.

While reading the first few chapters, one of the key points that resonated with me was the goal of  Guided Math (GM) to meet students where they are so you can take them where they need to go (pg 9). For so long, this has been the goal/emphasis in reading that I am glad to see this is now shifting to math.

Another quote that stuck with me is from pg 11, "It means that if Johnny doesn't get it after you've tried to teach him three different ways, then you try a fourth." We talk about students showing perseverance, and sometimes, as teachers, we too, have to persevere in finding ways to reach all our students. It is the idea of filling a toolbox with ideas and strategies to pick, choose, and revisit as many times as necessary.

Chapter 1 ~ Guided Math: An Introduction

To help foster perseverance in the classroom, I have a bulletin board that addresses the 8 Standards of Mathematical Practice. Perseverance is covered in Standard 1: Make sense of problems and persevere in solving them. As a class, we talk about the Standards of Mathematical Practice, and we anchor back into the practice when students do not show perseverance. It seems simple, but the visual reminder helps students remember what is expected. You can see where these mini posters were downloaded from if you click on the image.

As a class, we also talk about ways to overcome being "stuck" in math. Another strategy I use to foster perseverance is Need a Hand? Try This! Click here to read a previous post. Basically, there are tips on each hand that I encourage students to try before the infamous saying, "I need help." or "I don't get this." or "I can't do this." I encourage students to TRY THREE before ASKING ME.

Chapter 2 ~ Guided Math in a Numerate Environment

I keep a numeracy-rich classroom environment to help develop strong mathematicians. I have found it is better to start the year off simple and then add to the room as learning occurs and needs are recognized.

•  I like to use each new poster or anchor chart as part of a learning experience so students can make the connection and see value in it. I have been known to tell my students, "Take a short field trip to the back of the room and reread that poster. Come back and let me know what you learned/remembered." They *LOVE* to take a field trip and often come back grinning with the answer. 
• Math time should not be a quiet time. Looking on pg 17, I loved seeing all those -ing verbs that help to foster a numeracy-rich environment. For too long math was thought of as computing. Period. Posing open-ended problems with multiple solutions or multiple entry points of learning can foster rich mathematical discussions where students have to connect, explain, listen, and prove. The Tell Me All You Can Routine (pg 23) not only fosters fluency of thinking, but perseverance to tell ALL you can...and not stop at just one or two ideas.  
• I have a space where math manipulatives and math resource books are available for student use. Students know where they are; students know they can use them when needed. Something I was left to think about was when it mentioned on pg 23 that centers are brought to students to save instructional time. So much of math workshop is creating routines to maximize learning. 
• You have to love all the wonderful math literature that is available nowadays. No better way to launch a lesson than by reading a quick book about math...whether it is fiction or nonfiction. One book I love to launch the year with is Math Curse by John Scieszka and Lane Smith. It's a great book to make students realize that math is ALL around them! They live in a numeracy-rich environment. They just have to notice!!

Two for Tuesday!

Perimeter, Area, Volume...Oh, my! includes Poke and Check cards along with three anchor mats for perimeter, area, volume. The Poke and Check cards can be used in a center or hung up around the room for students to walk-the-room and review these concepts. I also have used the cards where students use markers to put on each paint splat and then check with the answer key. So many different ways to use these cards.

Along with the Poke and Check cards are three anchor mats. I *LOVE* using these anchor mats with my students. I use them in many different ways. I use anchor mats before learning as a preassessment, during learning to add information, and/or after learning as a summative piece. What I really like about these anchor mats is students use them as a resource throughout the year. It makes them accountable for their own learning when they "get stuck." I encourage students to fill out the anchor mats using pictures, symbols, and words. Students can fill out the sections using their prior knowledge, as well as information learned during instruction, from picture books, from math resource books, or from online resources. This tool allows students to conceptually develop a working definition/ understanding of perimeter, area, and volume.

 

Calling all Bill Peet book lovers!! Check out this product that helps students take a closer look at author, Bill Peet. Add a little twist to a traditional author study with this download. Although this author study is based on the author and books of Bill Peet, it can be adapted and used with other authors!

Bill Peet's books are known for their figurative language. Have students go on a scavenger hunt through his books and find examples of similes, alliteration, and personification to name a few. Have students complete  a Five Star Rating where students evaluate one of Bill Peet's books and then defend their reasoning.  A friendly letter writing activity is included where students choose to write a letter from the point of view of one character to another character from another book. Students can complete The Important Thing summary about what they learned about Bill Peet and his books. Every time you read a Bill Peet book, it seems like you find something new/different.

Wisdom Can Begin with Wonder

Imagine. Explore. Learn. Grow. Start a lesson with "Have you ever wondered..." and pique your students' interest. Use technology and interesting topics/images to hook and engage your students. Something you oughta know for your classroom is Wonderopolis.

Wonderopolis is one site that can help to address CCSS: Reading Informational Text. Students can use text, videos, and pictures to help answer the guiding questions. Find some wonders that might work with your units of study. Might you find a topic that relates to a unit of study in science? Or find a topic that connects to a story in reading? Wisdom can begin with wonder.

Wonderopolis is a great way to engage students. Each day there is a new wonder...posed as a question. To begin have students predict answers to the wonder question of the day. These questions are just like the questions students often ask, yet we don't always know the answer to. Do you know why batteries are different sizes? Click here to find out.

And...look at all the great resources you will find to engage students for each wonder.  The text and the short video guide students to finding an answer to the wonder question. For each wonder, students will find Wonder Words that students can use for vocabulary work. Students can complete a Try It Out section at home to extend learning. What a great way to foster the home/school connection. The Still Wondering section allows children to explore the wonder through a different context. Wonder What's Next piques students' interest in upcoming wonders.

Here are some ways I have used this website in class.

• Have students explore the wonders and then answer the question in their reading journal.
• Project a wonder with a doc camera for students to read. The wonder can be read to younger students as they follow along. Pass out something similar to wonder stems to each student for them to complete and share during the discussion. Then students can work on finding evidence from the text to answer the wonder question after the discussion occurs. Click on the image to grab a few wonder stems.

• Have students watch the video and write a one sentence summary.
• Read the text and have students write three new facts they learned. 

Check out this Wonder Mat where students can record their ideas after investigating a WONDER. Click on the image to grab the FREEBIE.

 The possibilities are endless!

Another way to foster wonder is to show unusual photos that might elicit student curiosity. Consider these possible sources for finding unusual pictures: Pinterest, Flickr, and Google Images.  What do you wonder about when you see the image below? Post an image like this and have students generate questions. Use this as an opportunity to teach students how to write rich questions.

 

Wisdom can begin with wonder. I combined all my activities in a product: Dare to Wonder. Check out the preview to see some of the ways I incorporate these ideas in the classroom.

Two for Tuesday

Check out Let's Have a Ball in Math! You will find three activities that target problem solving, computation, divisibility, and order of operations. Some of the tasks allow for more than one right answer...and some may require students to demonstrate some perseverance.

Skill levels vary within the tasks so you can differentiate based on student readiness. Have a Ball Problem Solving has eight different task cards. Some of the cards allow for multiple answers. Problem solving and finding combinations can be found on some of the cards too. Beach Ball Round Up has two different number mats that students can use. The recording sheet asks students to create problems that satisfy certain conditions using the numbers found on the beach balls on the number mats. Guess and Check is a strategy my students used for some of these problems. Be on the Ball...And Hit the Target is an activity with students practicing order of operations.

These back to school activities can engage students as they settle into a new school year. The Gumball Glyph is a craftivity that can help students get to learn more about each other. The Summer Sub is a writing craftivity that allows students to reflect on their summer. It is also a great piece to share with parents to see what their children remembered most about their summer!

That's My Rule ~ Candy Machines with In/Out Boxes

 

That's My Rule: Candy Machines with In/Out Boxes has two activities. Each activity has 10 cards with different function machines, otherwise known as Candy Machines. The cards target addition and subtraction to 20.

Check out the full product by clicking on one of the images.

Fraction Menu & Thinker Keys

I love teaching fractions. And I LOVE Thinker Keys. Click here to read more about this framework by Tony Ryan.

Okay, now onto the post. Hope you can use this fraction menu with your students. And, I hope you consider trying Thinker Keys in your classroom.

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Thinking is essential in math. I am always looking for ways to push my math students' level of thinking beyond just calculating an answer. With Common Core, students are asked to show multiple representations and dig deeper into mathematical understandings. Using Tony Ryan's Thinker Keys, I designed this Learner Menu for students to use to push the level of conceptual understanding of fractions with some higher level thinking. The Learner Menu could be used after instruction on the key concepts has taken place. Notice how the criteria at the top of the menu asks students to demonstrate their mathematical thinking and understanding when completing the tasks. Also notice how there is not one right answer, and sometimes that answer is not right there.

 

Thinker Keys are a tool teachers can use to embed thinking into any lesson. They can be used as part of a Learner Menu as seen here. Thinker Keys can start a lesson. Thinker Keys can be used as part of small group work to encourage Math Talk. In 2005, Tony Ryan, a Learning Consultant from Australia, updated the Thinker Keys. The earlier version, used on this menu, helps foster creative thinking while focusing on a deeper understanding of the standards. If you want to learn more about the Thinker Keys, click the links below.

Earlier Version of the Thinker Keys

Multiplication Stories

Today I am sharing a Multiplication Stories worksheet. Once students have a conceptual understanding of multiplication, they then are ready to create their own multiplication "stories" using data from a table. But here is the twist. The answer has to fulfill certain criteria. For example, the answer has to be a product that is even or a product that is greater than 20 and less than 35. In this way, students have to think about the type of "story" they will create based on the criteria given.

GraphicsfromthePond, DigitalClassroomClipart,
MyCuteGraphics, TheLearningTree 

Each student's multiplication story will be different. Students then can share their math stories with a classmate and have them solve! This gives students an authentic audience to share their work. Just a different way to reinforce and recycle the skill of multiplication and mathematical vocabulary. Click on the image above to download your FREEBIE.

Summer Bloggin

Step "Write" into Narratives has a variety of activities and tools to use to support narrative writing in your classroom:

• two seasonal narrative prompts
• an emotion activity where students create a word bank to use in their narrative writing
• a narrative organizer with a student version and a model for teacher think aloud
• paragrids to use as a tool to help students monitor and assess writing for targeted goals
• a narrative color code checklist
• a narrative menu to use as independent work or to support narrative mini lessons
• a self-evaluation tool to help students gauge needs and progress on Narrative Menu activities.

Capture the Bushels is an activity that helps students review and become more proficient at identifying factors and multiples of numbers while mixing in some strategy play. The goal of this activity is to capture the most bushels. How? The first player to place the fourth apple card into a bushel captures that bushel.

You can select the bushels and apple cards students will use based on their readiness levels. A multiplication chart and a divisibility rules chart are included to support learners.

Vocabulary anchor mats are included for students to review the following terms: factor, multiple, prime, and composite. Students can complete these anchor mats in small groups before this activity or independently as a formative assessment.