“I am not good at math.”

“ I’m not a math person.”

“I am missing the math gene.”

“I don’t have a math brain.” 

“Math is just hard for me.” 

How many of these have you heard from students?  These are all fixed mindset statements.  

A fixed mindset is a belief that a person has a predetermined amount of intelligence, skills, or talents. Alternatively, a growth mindset is a belief that a person’s intelligence and/or talents can be grown or developed with persistence, effort, and a focus on learning.

Children in kindergarten are all about the growth mindset. They will keep trying and struggling to do something, just because they can. Usually somewhere around third grade children start thinking they can’t, or things are just too hard.  They begin to think of themselves as either smart or not smart (I refuse to use the ‘s’ word). 

This guest blog post is written by Pam LeSeure. See more from Pam about Guided Math Conferences HERE and the power of Guided Math HERE.

A growth mindset is important to all aspects of the classroom, but today we’ll focus on the math classroom. I begin the school year by asking students how they feel about themselves as mathematicians. Most say they are not mathematicians. That’s the first thing I squash like a bug. We are ALL mathematicians, scientists, revolutionaries…you get the idea.  

As soon as I hear the fixed mindset comments about themselves I start repeating the word “yet.” 

“I just don’t understand fractions…yet.”

“I can’t multiply…yet.” 

“I don’t know my basic facts…yet.” 

Then I read the book The Power of Yet. I even made a shirt that says it in case they think I am joking.

We post the idea of “yet” all around the room and refer to it often. But that isn’t enough to get them to buy into the idea that there is value in struggle and that mistakes are okay. 

There is another saying that doesn’t always have the positive effect we want: “Mistakes are proof we are learning.” I can’t remember one student that didn’t roll their eyes when I would say that…until I started really living the growth mindset life.  

There is real brain evidence that says teaching students that their work needs to be correct every time is detrimental. Instead, we should be teaching them to take risks, struggle, and fail.

We can use real-life problems to foster this. Here’s an example from my own experience.

Each Wednesday my daughter and I would stop at Starbucks on our way to school. One day the kids asked how much money I was spending on “just coffee” each week. So I asked the class to calculate how much I was spending. A couple of the quicker students asked what each coffee cost and how many times I stopped. Simple multiplication. Easy enough. These students were all about speed.

But the belief that the “smart” students get the answer the fastest is a fixed mindset. 

Depth is so much more important.

Then a student asked if we always got the same size. This was a great question that proved deeper thinking…and yes we both always got Ventis. 

The next question was if we only stop on Wednesdays. Yes, we only stopped on the Wednesdays that we had school.

Light bulb!

Someone asked for a calendar. The next person asked if we only got coffee each week or did we add a treat. Someone asked if the prices had changed at all.  This simple comment on my coffee spending habits continued for the entire math block. I never considered stopping them or telling them we would have to pick this back up the next day. 

This is a prime example of a growth mindset. These students never said, “This is getting too hard,” or “There is no way we can figure this out.” They kept going. The struggle and perseverance they showed that day was simply amazing. My only regret was I didn’t have a camera rolling to get a video. 

Related: 5 Steps to Teaching Growth Mindset in Math Class

Another day, a fifth-grade student decided that he was going to create a new strategy for 3 by 2 multiplication. He was a student that typically understood things easily and figured this would be a simple task. Each day he would come to me with a different idea. Each time his strategy would fail. Although he was frustrated, he never gave up.  This struggle was real, for sure. Finally, after many attempts, he said he didn’t think he could create an original idea. I asked him if he was giving up. He knew that I didn’t tolerate giving up, so he said, “NO! Never.  For now, I am going to stick with what we have, but eventually, I will discover a new strategy and will come to teach it to your students.  I just hope you haven’t retired by then.”  

It is so important to foster this growth mindset in our students. But how?

• We can become flexible in our teaching. The Starbucks coffee lesson was not in the lesson plan. But it proved to be so much more than just about the coffee. It allowed students to persevere and use math in a way that was relevant to their real-life experience.
• We can “notice” when students are using a growth mindset, and point it out to them. For example, if you see a student struggling but still working hard, you might say, “I have noticed that you never gave up on that activity, even though you faced some challenges.”
• We can be good role models. As teachers, it’s important for us to model a growth mindset too. We can show students that we are learning too, right along with them. We can talk about our struggles and how we’ve overcome them.

The ability to persevere as a child will ultimately lead to the ability to persevere in life as an adult. Let’s give kids as many opportunities to do this as we can!

The post Reflections on the Power of Yet and Growth Mindset in the Math Classroom appeared first on Shelley Gray.

How to Teach the Doubles Facts So Your Students Actually Understand

July 15, 2022 1 Comment

What are the doubles facts in math? How do you teach them? Why are the doubles addition facts important? By the time you finish reading

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Fun Multiplication Games for 3rd and 4th Grade (help them master the facts)

July 12, 2022 No Comments

How can you make learning multiplication fun while also allowing your students to master the multiplication facts? WITH GAMES! Multiplication games are an effective way

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How to Teach Math Effectively Using the Concrete Representational Abstract Model

July 16, 2020 7 Comments

WHAT IS THE CONCRETE REPRESENTATIONAL ABSTRACT MODEL? The CRA Model is an instructional approach for teaching math. It consists of three phases: Concrete Representational Abstract

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The post Three Easy Ways to Transform Scary Mad Minute Drills into Something Purposeful appeared first on Shelley Gray.

Congratulations! You made it to summer break! Summer is not only the time for rest and relaxation but also for reflecting on the previous school year. Your students have worked hard all year long to grow their math skills, and in order to prevent summer slide it is helpful for them to continue to work with basic operations and other skills they’ve learned.

Students need a break too, and many don’t touch a math problem or activity over the summer. But if we can give families ways to incorporate relevant and fun math, there is a better chance that we can prevent the summer slide.

Here are some fun math activities and resources to try over the summer:

Baking and Cooking

Baking or cooking are wonderful ways to incorporate fractions into real life and help kids understand how they are used in the real world. When baking with children, be sure to include discussions such as:

• What if we were doubling this recipe? How much {insert ingredient} would we need then?
• What if we had no full cup/half-cup/quarter cup? What is another way that we could make this amount? (for example, if the recipe calls for one-half cup of flour, we could make that same amount using two one-fourth cups.)

Human Clock 

This is a fun outside activity to do with students at school, but can also be done at home with families. The focus on physical movement makes it a favorite for kids!

You will need: 

• Concrete area (larger area) 
• Sidewalk Chalk 
• Two children 

Instructions: 

1. Gather materials and find ample concrete space outside. 
2. Draw a large clock on the sidewalk with chalk. It needs to be large enough to fit two children in it. Label the clock like a regular analog clock with each hour number 1-12. 
3. Tell two children that one will represent the hour hand and one will represent the minute hand of the clock. 
4. Give the children a specific time to make on the clock. Example: 1:15
5. Have the students lay down on the sidewalk and make the correct time. 

Crayon Resist Math Mystery

If your child/student loves art, this might be the activity for them to try! You will need blank white paper, watercolor paints, paintbrushes, and white crayons. 

1. First, write some math equations on paper with a white crayon. 
2. Have kids use watercolor paint to reveal the first part of the equation slowly.
3. The child then says the answer, and self-checks by painting over it to reveal the answer.
4. Once finished, they can hang their pieces up to display their beautiful math art! 

Rounding Chalk Number Line 

Like the Human Clock, you will need a large concrete area and sidewalk chalk. Students will draw a number line from 0-100, or they can draw a number line counting by 100’s from 0-1,000 (100, 200, 300, 400, 500, etc.) 

Instructions  

1. Draw a number line on the concrete area. 
2. Have students label the number line. They can mark it by tens or hundreds. 
3. State a number and have the child identify what number it will round to. Example: 67, rounded to the nearest ten is 70. 
4. The child can explain why they rounded that number to the specific ten or hundred. 
5. Give the child more examples so they can practice rounding and have them explain how they found the number. 

Working with Money

If you are traveling this summer, or even going out for a meal, involve your children in money decisions. Some examples include:

• If you go out for ice cream, have your child estimate the total cost using the prices on the menu.
• If going on a road trip, have your child calculate an estimated cost of fuel using current fuel costs and the distance being travelled.
• When traveling, rather than buying your child a certain number of items, give them a specific amount of spending money to use as they wish. This will help them learn the value of money and how to budget.

Math Games

Games are a wonderful way to review math facts in an engaging way that feels fun, and many can be played with just dice or a deck of cards.

Some fun games that lend themselves well to family play are Salute, Knock off the Clock, Target Number, High Low, and Bump. 

Summer is all about having fun in the sun, but it is also a great time to review math skills and concepts so your child can avoid the summer slide when they begin school in the fall! 

Would you like all of these ideas in printable form to hand out to your families before the end of the school year? Download and print here!

The post Six Activities to Use at Home to Help Prevent the Summer Slide appeared first on Shelley Gray.

The post Three Types of Tasks You Should Add to Your Math Routine, Tomorrow! appeared first on Shelley Gray.

Still having a hard time picturing how a five-year-old can determine more or less without counting? Here are some strategies students may use to compare in the kindergarten classroom. These strategies are explained in detail in the book The Common Core Mathematics Companion: The Standards Decoded, Grades K-2 by Linda M. Gojak & Ruth Harbin Miles.

The post More or Less? How to Strengthen Comparison Thinking Through Games appeared first on Shelley Gray.

Showing students that math is a flexible subject can be powerful. So many kids (and adults) believe that math is all about getting right answers. But it’s not! When we can show students that there are many ways to solve a problem and that we can use flexible thinking to think in different ways, we empower our students.

We allow them to see math as a creative subject and give them the opportunity to see math in their own way.

In this post, Shareen Hando, a teacher from Winnipeg, Manitoba, shares the many different ways she uses Math Conversations in her classroom to empower her students – helping them to see themselves as mathematicians and share their thinking in creative ways.

My Math Background

I come from a background of experiencing math anxiety, timed tests, and focusing on mental math and fluency rather than problem-solving – I share this commonality with many colleagues. Because of my past experience, I was passionate about finding methods and research-based practices to build mathematical mindsets and problem-solving practices with my students, and sharing math ideas in creative ways. 

Related: 13 Simple and Quick Alternatives to Timed Math Fact Tests

Encouraging Risk-Taking, Flexible Thinking, and Confidence with Problem-Solving

I started the year with an emphasis on, and celebration of, every student’s ideas – correct or not. As part of our daily number routine, reluctant sharers knew that every morning, every student had an opportunity to share. They could see that other students might have made a mistake, but those students were recognized for trying and explaining their reasoning.

The focus was not on the right answer, but instead on the thinking and the courage to try.

Constant daily exposure helped students begin taking risks and feel comfortable sharing ideas without the pressure of the teacher calling on them. My goal was for students to feel comfortable sharing and to respond with a comment or question that kept students thinking. Slowly but surely, I began to hear the discourse in my classroom with phrases like

“I’d like to add on to…”

“I agree because…”

“I like how you solved it.”

“I disagree. How about…”

I began seeing math talk signals being used in other subjects and the phrases we use during play.

Resources that Help Me Promote Flexible Thinking and Risk-Taking in Mathematics

I found Shelley Gray resources during a time of hybrid learning. Teaching both in-person and online was already a near-impossible task – throw in creating engaging digital and paper activities and it was, well… is there a word for worse than impossible? There were not enough hours in the day.

In this digital space we have so many amazing websites to find tasks such as counting collection images, which one doesn’t belong (WODB), subitizing images, and story problems – but wouldn’t it be nice to have ALL of these diverse rich tasks in one place, catered to your students’ ability levels?

I aligned immediately with the philosophy and ideas that Shelley incorporates into her resources. I jumped right in, and both students and I LOVED her Math Conversations. The next year, I began exploring different ways I could incorporate Math Conversations into my combined classroom.

How I Adapt and Use Math Conversations in My Classroom

Math Warm-Ups

We pull up a math conversation slide and right after the morning meeting my students are ready for a minds-on thinking task. We establish our math talk symbols, reinforce that mistakes help us grow and we are off! 

Related: Do You Need Help With Number Talks? Here’s the Quick Start Guide

Activating Prior Knowledge Before Introducing a New Concept

If we are starting a new unit, I will scroll to find a Which One Doesn’t Belong (WODB) to spark curiosity and activate what students already know. 

Sharing Ideas for In-Person and Online Learners Via Flipgrid

Work smarter, not harder as my colleagues would say! I was able to have rich conversations with students during hybrid learning by posting a slide of the Math Conversations through Flipgrid. In class, we would have a whole class math talk, and then share student responses on Flipgrid; online learners were able to add their own ideas to the same math conversation, and view their friends’ responses afterwards! It was so powerful! It made my students feel connected, and they felt like they had a responsibility to share and teach their friends at home. 

Displaying Math Tasks and Student Thinking in the Classroom

Recently, I have started making 4-5 monthly math tasks that center around concepts and goals we are focusing on in numeracy. I pulled slides from Shelley’s Math Conversations to showcase and display students’ learning. We started with projecting the slide on the screen, then I would record students’ thinking with their initials and have a math talk.

Students love seeing their work on the board! If students agreed, I would also add their initials beside one solution.

I also used the Math Conversation slides to create bulletin board displays where students could share their flexible thinking. I love displaying students’ thinking because even those who are more reluctant to publicly share are still given a chance to write down their answers to the math task. They are given a choice if they want to write their name on the back or front of the card.

Math Conversations have been a huge hit in our classroom. During our math instruction, it has been amazing to see students reflect back on our bulletin board and make connections when we are practicing skills such as two-digit addition or equality. Showing students’ thinking allows them to feel not only more confident in sharing ideas, but also to see themselves as mathematicians. 

Ready to Start Math Conversations to Foster Strategic, Flexible Thinking in Your Classroom?

Each Math Conversations package includes 200 ready-made slides. There is NO prep required – just display the slide and start your math talk! See all the details HERE.

The post How To Promote Flexible Thinking and Risk-Taking in Math Class appeared first on Shelley Gray.

Subitize comes from the Latin word “subit(us)”, which means “to appear suddenly.”  In my last post, we talked about the Early Number Sense Trajectory that was created from the work of John Van de Walle, Doug Clements, and Julie Sarama. According to the trajectory, subitizing is the first mathematical stage a learner experiences. Students who can subitize are able to visually recognize a quantity of five or less. However, there are multiple levels of subitizing, and recognizing the difference between 1, 2, or 3 objects happens as early as within the first few months of infancy! 

It’s important to note that when students call out these number names, they are only matching the number name to a visual pattern. In other words, students do not know what those number names mean as a quantity yet. For instance, suppose I show the following image to a young learner:

The learner yells “3!” Yes, the young mathematician knows the number name of the visual pattern but when asked to count how many, the learner may respond “1,3,5.” This is known as perceptual subitizing: the learner can state the number name of a visual pattern but cannot explain why. At this stage of learning, that is okay! We are working with learners who are beginning with perceptual subitizing.

The next stage is conceptual subitizing. This is a learner who can explain why a number name matches a visual pattern and recognizes subgroups that can be combined together to compose a whole. So for the image above, the student may explain that it’s 3 because 2 and 1 make 3.

With that said, there are many ways to support young learners at home during this stage of learning, and what better way to talk about math than through games! Below are three simple games you can play to practice the concept of subitizing. All you need is a deck of subitizing cards (find them for free down below)!

Subitizing Game #1: How Many?

• Two players split the deck and take turns flashing the top card from their pile to their playing partner for 1-3 seconds. 
•  If the player correctly identifies the quantity on the card they keep the card as a point. Otherwise, it goes to the discard pile.  
• The player with the most points/cards wins the game.

Subitizing Game #2: Memory

• Place all the cards face down and players alternate flipping over 2 cards to find a match. 
• If a player finds a match, they collect the cards as their points. The player with the most points/cards wins the game.

Note: You will have to make two copies of the subitizing cards to ensure that each card has a match. Mounting on colored paper or copying on cardstock may also be necessary so students can’t see through the paper.

Subitizing Game #3: How Many? War

• All the cards are kept in a single pile and one card is flipped over. 
• The first player to say the quantity on the card gets to keep the card as a point.
• The player with the most points/cards wins the game.

Related: Subitizing for Multiplication: Building Number Sense and Multiplicative Reasoning

Differentiation

There are many ways to differentiate these games to meet your learner where he/she is at.  If you have a young learner in Pre-K or Kindergarten, subitizing cards within 5, like five frame cards, would be more appropriate. For first and second graders, subitizing cards within 10 and 20, like ten frame and double ten frame cards, are a great fit! 

Do you have any subitizing games that you love? Share in the comments below! I’ll be sharing some more games for the next stages in the Early Number Sense Trajectory, so stay tuned! Always remember, we are better together.

Resources to Help

Grab and Go

Remember: Depending on what you are using the subitizing decks for, you may need to make multiple copies.

Five Frame Cards

Ten Frame Cards

Double Ten Frame Cards

K-2 Learning Trajectory Document

Deep Dive

Learning and Teaching with Learning Trajectories- This website will provide a wealth of information on specific learning trajectories for learners of all ages!

Learning and Teaching Early Math: The Learning Trajectories Approach by Clements & Sarama

The post How to Investigate Subitizing Through Games to Spark Early Number Sense appeared first on Shelley Gray.

Rounding numbers is a tricky concept for many kids. It can be somewhat abstract and a difficult concept to grasp. If there is one thing I’ve learned over the years, it’s that no one strategy works for every student. Concepts often need to be taught and reinforced in multiple ways to reach every student in the classroom.

When it comes to rounding numbers, I believe in a conceptual approach where students are taught to find the proximity of the number being rounded to the benchmark numbers and midpoints. This is one way to ensure that students truly understand what rounding means – rather than simply memorizing a rhyme or the steps to a procedure.

Related: Not Sure How to Teach Rounding Numbers? Make it Hands On! (Try This)

However, I also understand that students need to see concepts in different ways. In this post, my guest writer Katie has given a breakdown of how she uses Rounding Mountain in her classroom in order to help students visualize the rounding process. This is a visual that may give some students the “a-ha” moment with rounding, and we can use it to connect their new understanding to rounding on a number line.

Rounding mountain can be a helpful visual for students; however, remember to use this to support your students’ understanding of benchmark numbers and midpoints, not to replace it.

If you’ve taught elementary-level math, then you probably have taught some form of rounding. Rounding is important in elementary math because it helps our students build their sense of place value.  

When you round a number, you determine which benchmark number (ten, hundred, thousand, hundred thousand, million, etc.) a number is closest to. 

I like to teach my students different strategies when I review or introduce this topic to them. One of the most popular ways that my students have enjoyed learning and using has been the rounding mountain. 

What is the rounding mountain and why is it beneficial? 

The rounding mountain is a visualization tool that can help your students understand whether a number should round up or round down. As you can see in the example above, 41, 42, 43, and 44 would round down to 40. 45, 46, 47, 48, and 49 would round up to 50.

My students have always enjoyed this strategy because it allows them to “practice climbing up and down the mountain.” They enjoy making it a game or a competition. The more ways we can visualize and practice a concept, the better!

Related: Quick and Easy Place Value Activities (Try These to Boost Number Sense!)

How to Use the Rounding Mountain

1. Pass out a chart or allow your students to use an anchor chart on a stand or whiteboard. I like to laminate my anchor charts so they can be used year after year. 
2. Decide which place you are rounding to, for example, the nearest ten.
3. Give your students a number. In this example, we will use 43. Have them identify the benchmark numbers. These are the next lowest ten and the next greatest ten. In this case the benchmark numbers are 40 and 50. Students can use sticky notes to place these benchmarks on either side of the chart, or write with dry-erase markers. 
4. If we are rounding 43 to the nearest ten, we will underline the number in the tens place. Then we will circle the number to the right (in this case, the 3). 
5. The circled number will determine whether we round down or round up. In this case, we round down to 40.

This can easily be used as a partner activity to encourage collaboration and teamwork. Have students take turns and double-check each others’ rounding.

IMPORTANT: Make sure your students understand WHY they are underlining the number in the tens place, and circling the number to the right. We want to ensure that students understand rather than follow a series of steps with no real understanding. Ask questions like, “Why does the circled number determine if we round up or down?” or “What are some options the circled digit could be if we rounded up to 50?”

Helping Students Make Connections

One of the most important aspects of helping students develop a deep understanding in math is to help them make connections between concepts. If you decide to use the Rounding Mountain visual in your classroom, ensure that you are helping students make connections to a number line. Here are some ideas:

• Ask, “How is our Rounding Mountain the same as a number line? How is it different?”
• Discuss with students: “Look at our Rounding Mountain. Now look at this number line. What do you notice? What do you wonder?”
• Refer to the numbers at the base of the mountain (ex. 40, 50) as benchmark numbers, and the number at the peak of the mountain as the midpoint.
• If you are working with rounding to the nearest hundred, thousand, or ten thousand, ask: “Is there a way we could use the rounding mountain to round to these other place values?”

Free Rounding Mountain Printables

If you’re interested in using Rounding Mountain as an activity to supplement your students’ understanding of rounding numbers, here are some free printables that might be useful for you!

More Rounding Resources

The rounding mountain activity above can easily be connected to rounding on a number line. When students round on a number line, they use their knowledge of benchmark numbers and midpoints to find which ten (or hundred, thousand, etc.) a number is closest to.

Here are some other resources that will be helpful as you teach your students how to round on a number line:

• Rounding to the Nearest Ten (within 100) worksheets
• Rounding to the Nearest Ten and Hundred (within 1,000) worksheets
• Rounding to the Nearest Ten, Hundred, Thousand (within 10,000) worksheets
• Rounding: Digital Mystery Pictures

The post How to Round Numbers Using Rounding Mountain (Plus Free Printables!) appeared first on Shelley Gray.

Assessment is not a favorite word for teachers or students. I remember back in the ancient times (the late 70s) when we would have a quiz about halfway through a unit, and we would have the end-of-unit test at the end. The memories that stuck with me the most were from fifth grade – when I was the student, not the teacher. Mr. Fouts would assign us the test at the end of the chapter – not just the even or odd questions, but all 50 of them. I can still remember the anxiety I would feel. 

But back then, it wasn’t called anxiety. It was called, “If you had paid attention, you would know this.” 

Assessments have evolved and come in many different forms now, but let’s be honest, they are not anyone’s favorite part of teaching. When I began teaching, we didn’t perform assessments to guide instruction – well, not officially. Guided math changed my entire mindset about teaching math, and assessments are part of this shift. The students reaped the benefits.

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This blog post is part of a Guided Math series by guest writer Pam LeSeure. See more from Pam about Guided Math Conferences HERE and the power of Guided Math HERE.

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Assessment to Guide Instruction

Formative and summative assessments are an essential part of guided math groups. The five strategies for assessment as discussed in Math Workshops in Action by Dr. Nicki Newton are: 

• clarifying and sharing learning intentions
• engineering effective classroom discussions and other learning tasks that elicit evidence of student understanding 
• providing feedback that moves learners forward 
• activating students as instructional resources for one another
• activating students as the owners of their learning

By using a few fundamental assessment techniques, I put these five strategies into practice in my classroom.

Tried and True Assessment Techniques for Guided Math Groups

Guided math groups allow for quick face-to-face formative assessments that provide you with valuable information to drive your future instruction. Here are some of my favorite ways to assess during Guided Math groups.

Related: The One Transformative Change I Made to My Math Instruction to Boost Understanding

Brief interviews are quick check-ins to assess and see how students feel about the concept in general. These interviews can help you plan for lessons in mathematical mindset. During a brief interview ask questions like:

• “Which strategy did you use?”
• “Can you model this strategy for me in another way?”
• “Why did you choose this strategy?”
• “How are you feeling about this strategy?”

An exit ticket is another simple, but effective method of formative assessment. Include one or two problems that allow the students to show what they know. Exit tickets can be pencil-paper, digital, or dry erase board. 

More To Learn

During my conferences with students, I felt confident that I was gaining information that would help me place them in the correct group, and I could then plan my lessons accordingly. But I knew there was more to learn. 

Involving students in the assessment process was another piece of the puzzle.

When students are involved in the process, they understand the expectations, take ownership over what is happening, and are motivated to take greater responsibility for their learning. Setting personal goals and reflecting on those goals is also part of the process.  

Sharing Time as Guided Math Assessments

Sharing Time is a way to involve students in the assessment process; it is also an opportunity for quick check-ins. During a quick check-in, the whole class has a conversation and reflects on the day. We summarize, discuss, question, and reflect. This is also a time for students to share where they are on their learning path, which leads to conversations about mindsets and goal setting. During Sharing Time, we talk about the entries. Each day has a time for reflection where students journal their thoughts, questions, etc.

Math Journals as Guided Math Assessments

I am all about journaling. I journal constantly and I teach the importance of journaling to my students. Our math journals are full of practice, reflections, self-assessments, plans, etc. These journals are probably the most insightful formative assessments I have ever collected, and the best part is that they don’t feel like an assessment. 

Randomly I will ask students to share something from their journals. This could be a simple reflection from a lesson or a question they had during the day’s math block. We add these to our class journal. 

Like these journal prompts? Download them for FREE here!

The class journal has taken many forms, including chart paper displayed around the room, notebooks broken into quarters, and a digital version. Although my personal favorite was chart paper, my students loved taking our journals digital. 

The digital journals were accessible anytime, including at home, and they included graphics that were more appealing than my doodles. They could also easily add comments to the entries. 

Assessment has evolved so much since I was a student struggling through fifth-grade math, and I am thankful for that. The more we can use our assessment to guide instruction, and create opportunities for student involvement in assessment, the more effective our instruction can become.

What are your favorite ways to assess your students? I’d love to hear in the comments below!

The post Powerful Guided Math Assessments: One Teacher’s Experience appeared first on Shelley Gray.

Tell me if this has ever happened to you…you’re about to launch a lesson on fractions and all of your learners exclaim, “We did this last year!” 

What about this: you start that lesson on fractions anyway, and maybe a couple of people remember the terms numerator and denominator, but most of them have them mixed up, and even though they “did this” last year, no one can really explain what a fraction is, let alone solve problems that involve them? 

Chances are that these learners worked with fractions by completing a sequence of classroom activities, leading to a unit test last year and haven’t thought about them again until now. This type of long-range planning is called “massed” or “blocked” and it is linked with poor retention. Learning Mathematics through these units is, unfortunately, exactly how I was taught, so it is no wonder that many educators struggle today to feel confident in our teaching when we didn’t have a lot of opportunities for deep learning when we were young. If you are concerned with providing deep, long-lasting learning in Mathematics so that your learners become genuine and critical problem-solvers, you need to try spiralling your curriculum!

Serendipity and “Sticky” Learning

My first encounter with the word serendipity was in Grade 2. It was the title of a levelled reader that I borrowed from my classroom library (that was way too much for me) and I don’t think I even attempted to read it before returning it. But the cover, with its furrowed Queen of Hearts character and magical title, captivated me. 

Without using Google now, how would you define serendipity? Does it make you think of that John Cusack movie? Is it about romance, the flow of the universe, or when things just fall into place? I think I have the gist of it, but after all these years, I never learned the true meaning and have to look it up as I use it. 

My Hero’s Journey as a Mathematics educator has, as it would happen, been very serendipitous. I have come across resources and mentors on and off over the past eight years that have been very valuable to me, all while I wasn’t looking for them. This blog post is inspired by one such mentor, a principal I’ve never actually met, whose motto, “Try something new, no one will die,” has stuck with me ever since I heard it!

In 2016, Kristin Phillips delivered a TEDxTalk (one of my favourites!) on what she calls “sticky learning.” This is that deep, enduring understanding we are always aiming for but that rarely happens. Turns out there is an ideal approach to achieving it, and we have been doing it the opposite way, especially in Math class, for way too long. 

When we spiral curriculum, instead of “teaching” and testing in units, we create multiple opportunities throughout the school year for learners to explore key concepts and experiment with strategies and tools. Spiralling allows for learners to do this in a variety of contexts which later translates to them becoming not only problem-solvers but critical thinkers! Instead of falsely demonstrating their ability to problem-solve right at the end of the unit, but not retain the ability the next year on, choosing to apply the right stuff from their repertoire and in new and increasingly complex situations is something that can happen when learners get to see content more than once in the year. In the 21st century, guiding learners to grow into these kinds of problem-solvers is essential. 

How I Spiral Curriculum In My Math Lab

At my school, we organise learning by sessions that are anywhere from three to seven weeks in length. When I began to plan for a year of Math Lab, I first laid out a foundational theme for each session and then selected activities from a variety of strands (in Ontario we have strands and expectations, in the US these are called domains and standards). My intention was to “space” and “interleave” important concepts and skills again and again throughout the year, because, like the fractions anecdote above, I know that learners need more than two weeks with something for them to be able to really make sense of it. Here is a snapshot of what our first four sessions looked like:

As you can see, I have quite a bit of freedom in my planning and it is may be alternative to your situation. I work at a school with a learner-driven programme where our guided Math Lab time is limited to one hour per week (they have a 75-minute block every morning where they pursue Mathematics learning and other foundational skills independently at their own pace and level). With such a small window for me to guide, I selected problems that would touch on a variety of strands each session, with each successive session spiralling deeper. 

For example, let’s review the Spatial Sense and Measurement pathway we have followed so far. In Session 2, the learners took on open-ended problems like “The Broken Ruler” or “Mowing the Lawn” and uncovered concepts like how to use a ruler, and measure perimeter (and calculate area for the older group). Then, in Session 3, they had to recall this learning and extend it in new contexts like finding enough materials for “Framing Photos” or making Metric unit conversions in “Cutting Paper.” In Session 4, these skills were needed again, although this time in the context of a longer-term real-life based project such as working on a farm and designing new livestock pens or planning an agricultural conference venue with enough space for vendor booths. 

Next session we are moving into the theme of Entrepreneurship and our learning focus will spiral back to Financial Literacy. In Session 1, the learners explored basic Canadian currency and later in Session 4 started to make calculations with money amounts. Building and running our own businesses over the next several weeks will take us even deeper into finances, from evaluating and managing costs and revenue, to calculating taxes and interest. 

The case for spiralling the Mathematics curriculum with your learners is pressing but, I’m sure, also daunting. It is my hope that this blog has made it feel at least the tiniest bit more accessible, even mid-way through the year. Maybe you’re even experiencing a little serendipity of your own right now, like you needed a change but didn’t know what! 

Start small, perhaps by choosing just one skill or concept that you know your learners struggle with – how could you connect to this more often throughout the school year? Check out the resources below for more ideas and examples and please feel free to comment below and let us know how it goes! 

Resources

For the readers:

Kristin Phillips’ blog: https://educationbits.wordpress.com

Scientific American article on interleaved learning: https://www.scientificamerican.com/article/the-interleaving-effect-mixing-it-up-boosts-learning/

For the grab and goers:

OECTA Slides on Spiralling: http://www.oecta.on.ca/OECTA/media/pdfs/Professional%20Development/Web%20Experience%20Sessions/2017-18/Math%20Monday/Spiraling%20in%20Mathematics/we_mm_spiralingInMath_v1_nov20_2017.pdf

Elementary Long-Range Plans from Ontario Ministry: https://www.dcp.edu.gov.on.ca/en/learning/long-range-plans

Sample Secondary Long-Range Plan:

https://docs.google.com/spreadsheets/d/14G6_za6iifsNkx2i6kNvIlVpRcTcGkCHL3Y6WZroYPg/pubhtml?gid=835151258&single=true

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As a K-5 Math Consultant, I can’t tell you how many times I’ve heard a teacher say, “My students just don’t have number sense.” To be fair, I’ve said it during my teaching career as a special educator too. As teachers, we love to use this phrase when discussing math with our colleagues, our principal, and even with our students’ parents. We offer parents strategies to work with their child at home, and we provide Tier 1 interventions within the classroom to support this “missing” piece. But, as we discuss number sense in our math classrooms, there’s this ominous cloud that hangs over us: 

What does it really mean, to have number sense?

Take a moment and reflect on this – the meaning of number sense.

What does it look like in your classroom?

What do you imagine when you are asked to describe a student who has number sense?

It’s difficult to put into words. I have these discussions every day with educators and I still struggle with this.  I like to refer to the research from The National Council of Teachers of Mathematics to help guide number sense conversations.  Here is what the NCTM has to say about number sense.

“As students work with numbers, they gradually develop flexibility in thinking about numbers, which is a hallmark of number sense… Number sense develops as students understand the size of numbers, develop multiple ways of thinking about and representing numbers, use numbers as referents, and develop accurate perceptions about the effects of operations on numbers.” NCTM Principles and Standards for School Mathematics, 2000

Notice some of the key takeaways:

• flexible thinking
• multiple ways of thinking about and representing numbers
• accurate perceptions about the effects of operations on numbers.

Number sense encompasses a multitude of mathematical ideas.

So, how can you support your students with numbers sense in a K-2 math classroom? Check out the Number Sense Trajectory below!

This progression was created from the work of John Van de Walle, Doug Clements, and Julie Sarama. Graham Fletcher also did a great video on the early number sense learning progression based on John Van de Walle, Doug Clements, and Julie Sarama’s work. It’s similar to other natural developmental progressions, such as a child first learning to crawl, then walk, run, skip, etc. Remember, in real life, some students may learn to walk before they crawl, so some of these concepts may come in a different progression. It all depends on the learner.

Before we dive into what each stage is, we must remember that number sense is not something that can be taught directly. As the NCTM definition states, students develop number sense as they begin to discover and understand the flexibility of numbers. Therefore this learning progression will help you understand what to look for when students first enter your classroom and how you can support students seeing numbers naturally.

It’s our job as teachers to provide students with opportunities to explore, visualize, and play with numbers.

Let’s dive in!

Subitizing

Flash this dot image to your students for 2-3 seconds and pose the question: “How many dots? “

Some students might count all to tell you how many, but some students may know instantly that it’s five dots. How do they know that? To be able to visually recognize a quantity of 5 or less is the power of subitizing. Kids start seeing objects as sets, or groups, and this helps students move past the 1:1 counting phase. The image above refers to our young learners, but as a teacher, I’m always wondering: How does this support number sense in the intermediate grades? Check out this dot image below:

Subitizing in the upper elementary grades can help elicit multiplicative thinking.

Related: Subitizing for Multiplication: Building Number Sense and Multiplicative Reasoning

Comparison

Children learn the concept of “more” at an early age: more food, more toys, more hugs, etc. This concept becomes generalized within their everyday life, and children are soon able to compare quantities by identifying which has more and which has less. This idea of comparison relates to concrete representations, like the one below. 

Students can VISUALLY see the difference. It’s okay if they cannot tell you exactly how many! Students don’t need to count the objects at this stage. The focus at this stage is visual recognition, which ties in with subitizing.

Counting

Think of the alphabet song – our young learners can sing along, but do they really know that there are letters embedded in that song? The same goes for rote counting, which is the procedure of counting without meaning and understanding. In kindergarten, students learn the counting sequence from 0 to 100. Teachers introduce counting tools like a number path and a hundred chart to support the counting sequence.

One-to-One Correspondence & Cardinality

After children learn the rote counting sequence, we begin to introduce the idea of 1:1 correspondence. This means students can connect one number with one object and then count them with understanding. We can build this idea not only with number matching, but also with object matching, like the image below. We want to elicit the idea that we can match 1 star to 1 cube.

1:1 correspondence may take time and we need to make sure we are giving students the opportunities to explore and practice. While children are learning about 1:1 correspondence, they are simultaneously exposed to the idea of cardinality: to tell how many things are in a set.  When counting a set of objects, the last word in the counting sequence names the quantity for that set.

Hierarchical Inclusion

Hierarchical Inclusion sounds like a complicated concept, but it’s something that children will discover when practicing counting and 1:1 correspondence. It means that numbers are nested inside of each other and that the number grows by one each count. For instance, 4 is inside 5 or 5 is the same as 4 + 1.

You can practice this concept by asking students to count out a set of 7 objects. Then, ask the students to use that set to show a smaller set of 5 objects. Instead of counting out a whole new set, students with hierarchical inclusion will know that they can take 5 counters from the 7 to make the new set. Concrete representations are extremely important at this level of learning.

Number Conservation

The big idea of number conservation is derived from hierarchical inclusion. Students at this level of learning begin to understand that there is a decomposition of numbers. Number conservation means that the number of objects remains the same when they are rearranged spatially. Let’s look at the example below. You may ask your students: Are these the same?

Yes! These are the same. Students relate this arrangement to 1:1 correspondence.  Now, show students the image below and ask: “Are these the same?”

Some students may say no, there are more lightning bolts. These students do not understand the concept of number conservation yet. Continue to expose students to objects in a variety of arrangements and continue to connect this idea to 1:1 correspondence.

Now What?

You may be thinking, “This is great information, Allison, but what does that look like as instruction in my classroom?” There are so many activities you can do to help support your students’ number sense in the math classroom! I’ll be sharing some activities you can do with just double ten frame cards for each phase of this learning progression soon. If you already have some activities/lessons, feel free to share in the comments below. We are better teachers together!

Grab and Go

K-2 Learning Trajectory Document

Graham Fletcher- The Progression of Early Number Sense and Counting (video)

Deep Dive

Learning and Teaching Early Math: The Learning Trajectories Approach by Clements & Sarama

Teaching Student Centered Mathematics PreK-2 by Van de Walle & Friends

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