Math Fluency Archives - Tanya Yero Teaching

The Power Of Number Sense In The Classroom

September 2, 2019

When I first started teaching I taught “to the test.” I taught 3rd grade which in Florida is a grade level that completes standardized testing. It’s also a retention year, which means if your students don’t reach a specific score or higher they must repeat 3rd grade. The pressure was overwhelming. I wanted to make administration and the families of my students happy. I thought the answer was in covering ALL the math curriculum I knew would be on the test in the FASTEST way possible. I would speed thru lessons so I had enough time to go back the last few weeks before the test and review.

Then my first year turned into my second year and by the time the third year rolled around I realized I was creating robotic math students that were simply trained to take a test. They weren’t problem solvers. They lacked number sense and basic strategies that I knew they would need in the real world. They understood procedural learning more than conceptual understanding.

What REALLY is number sense?

Number sense is a child’s set of skills when working with numbers. More importantly, number sense is one’s ability to see the flexibility in numbers. Children with number sense understand quantities, grasps basic concepts, and understand key operations and symbols. But this all leads to the bigger picture of what number sense really is. A child with a strong foundation in number sense can identify multiple ways to solve problems. Number sense supports the beauty and creativity of each child’s mind and how they interpret a question.

The Questions You Ask (And Don’t Ask) Will Affect Your Students’ Number Sense

It’s not enough to teach your students basic algorithms and train them to memorize the steps. When I teach a new math concept the algorithm is the LAST thing I introduce. Why? I want to strengthen my students’ number sense. Math is designed to be fun and exploratory and more often than not we take the drill and kill approach. This approach leaves out the opportunity for good quality questioning.

At the beginning of the year I introduce strategies for manipulating numbers and equations. Expect your students to come to you lacking number sense tools. You have to teach them this style of math and how it works. The posters below stay up in my classroom all year.

I spend time modeling my own thinking, providing opportunities for peer discussion, and integrating manipulatives whenever I can. With time, students will organically begin to manipulative numbers on their own.

The following pictures are samples that I have pulled from student responses when asked to solve 24 x 9. Students that have the ability to see various ways to solve one problem have strong number sense.

Below is the complete list of strategies we came up with together as a class:

24 x 9 is close to 24 x 10 which I know is 240, so 24 x 9 must be one 24 less than 240. Answer: 216

25 x 10 is 225 so 24 x 9 must be 216.

12 x 9 is 108, so 24 x 9 will be double or 219.

20 x 9 is 180. 4 x 9 is 36. 180 + 36 = 216.

24 x 4 equals 96. 96 + 96 = 192 plus one more set of 24 equals 216.

Area model of 24 x 9

Asking questions such as “Does this mathematically make sense?” And “What does this look like in visual form?” will force your students to justify their answers and use multiple strategies. Never accept only one way of solving a problem. Encourage students to share their methods and celebrate the diversity of strategies. Push student thinking and ask meaningful questions. I used the following questions to steer my students towards the equations above.

“How can we use doubles to solve this problem?”

“How can an area model help us solve a multiplication equation?”

“How can we use multiples of 10 to help us solve this problem?”

What Does Number Sense Practice Look Like In The Classroom?

A class that is heavy in procedural questioning lacks the practice of number sense. Students are relying on the memorization of algorithms to solve problems. I provide questions, especially word problems that are rigorous and deep in conceptual understanding. These sorts of questions lend themselves beautifully to discussions that build number sense.

I also integrate Number Talks into my math block for extra practice that is specifically designed for number sense practice. Number Talks is a quick 5-7 minute discussion that builds mental math and number flexibility. CLICK HERE to read more.

Math, Math Fluency

How To Launch Number Talks In Your Classroom

August 27, 2019

Starting a new routine in your classroom can feel overwhelming. Sometimes we even abandon implementing something new because we don’t know where to start or we have too many unanswered questions.

“What materials do I use? Will I have to create everything I need?”

“Do I have the time in my daily schedule to designate for this new component?”

“What does this new routine look like in my classroom with my style of teaching and management?”

I had all these questions and more when I decided to launch Number Talks in my classroom. It took weeks of research and planning to wrap my head around the idea and then bring it to fruition as part of my math routine. Today, Number Talks flows into my math block with ease and comfort. I am well versed on the topic and so are my students. Here are some suggestions and information for launching Number Talks in your classroom.

The Purpose of Number Talks

The purpose of Number Talks is to build number sense and mental math in your classroom. We want students to see that there is often more than one way to solve a problem. We want students to have the confidence to mentally manipulate numbers. Finally, we want students to be fluent in math facts.

Number Talks should last 5-7 minutes in your classroom and include discussion and collaboration. With repetition and practice, students will begin to see how manipulation of numbers is a strategy that can be used daily both inside and outside of the classroom.

Set Your Students Up With The Skills They Need

First, I start with the introduction of mental math strategies. You have to provide the proper TOOLS if you expect students to view numbers differently. I spend time on how to mentally execute these strategies and how to apply them to specific numeric scenarios. The posters below stay up in my classroom all year as a reference for students. With time, you’ll see students refer to these posters less and less.

Number Sense Strategy Posters available in our Number Talks resources

The only way something becomes a habit is if you practice it enough with daily repetition. Number Talks should become part of your math schedule with opportunities for peer collaboration. Let your students hear from fellow classmates about how they solved a problem. This will foster the idea that learning evolves from CONNECTING and building on the thinking of others.

Build a Number Sense Rich Environment

Typically, students are not accustomed to mental math and using number sense as part of their problem solving. My absolute favorite question to ask students when they present an answer to me is “Does that make mathematical sense?” Usually I’m searching for clarity from students when I ask this question or I need to redirect their thinking. Expect your students to feel odd or even uncomfortable as you launch Number Talks in your classroom. They need time and practice to gain the confidence needed to verbally share their mental thinking and number sense skills. There is a level of PRODUCTIVE STRUGGLE that is required for Number Talks and this can be overwhelming for students.

Number Talks questions from the 6th grade version

MODEL your own thinking and reasoning with your students. If a word problem we’re solving in class discusses 24 jars of jelly beans containing 19 candies each, I may say to my students “Well, I know that 24 times 2 is 48, so 24 times 20 is 480. 24 times 19 must be 19 less than 480.” This is all done mentally and using friendly numbers that most students feel comfortable with. Soon students will start to share their own strategies for manipulating numbers. ENCOURAGE conceptual explanations rather than procedural computation. CELEBRATE the variety of ways to solve a problem and VALUE all student contributions.

Use MISTAKES to drive instruction. Mistakes are organically created opportunities for the exploration of other strategies. Review properties of operations, and problem solving techniques.

Number Talks question from the 3rd grade version

What does Number Talks look like in my classroom?

We all have busy classroom schedules and there is never enough time to get everything done. Number Talks isn’t designed to add another lengthy routine to your schedule. Number Talks will be a QUICK 5-7 minute discussion that packs a strong academic punch. In the past I have used Number Talks as my morning work component or to launch my math block. I project the prompt of the day on my active board, give 1-2 minutes of think time, and then let the collaboration begin. Sometimes I allow the students to share their strategy in groups, but typically I schedule the sharing as a whole group discussion so I can reinforce key concepts.

I have designed a Number Talks resource for every grade from K-6. It has everything I wished I had when I first started Number Talks in my classroom and everything you’ll need for the entire year. There are 250 ready to go prompts for you, both in digital and print formats. CLICK HERE to read more about this resource.

Number Talks materials available for grades K-6

Education, Math, Math Fluency

How to Implement Dot Talks In Your Classroom

April 7, 2019

Dot talks are an off-shoot of the daily classroom number concepts called “number talks.” Dot talks can be incorporated into number talks, and can be used with students of any age. But because subitizing is such an important developmental skill, they are uniquely suited to meeting the needs of kindergarten minds.

Why We Implement Dot Talks

Subitizing is the ability to look at a number of objects and “know” how many there are. Some subitizing is perceptual, such as when we recognize this pattern of dots on a die as five.

Conceptual subitizing is recognizing smaller sets within a larger number of items and quickly adding them together.

Looking at the 6 side on a die, we see 3 and 3, which we know is 6.

What are Dot Talks?

Dot talks are short, routine class meetings that help children build number sense, articulate their thinking, and appreciate different perspectives. They should be scheduled as a regular part of your day and flexible enough to expand with the children in your class or group. When dot talks are first implemented, especially if it is early kindergarten, they will take only a few minutes. Later on, as children learn the process, build their skills, and develop confidence, you may be struggling to keep it under 20 minutes.

Beginning training for dot talks is almost painfully simple. You present a card with 1 dot on it for a moment, and ask the students, “How many dots are there?” They will see 1 dot and can prove their thinking by touching the dot and counting, “one.” You can vary this first lesson by presenting dots in different places around the page or card. This fosters the additional critical understanding that sets can change shape or orientation, but only change if something is added or taken away.

Starting with the set of 2 dots, more conversational possibilities open up. Two can be subitized (I just know it), but it can also be decomposed (I know 1 and 1 is 2). Presenting the dots in different locations around the page does not change their “two-ness” or their “1 plus 1-ness.”

What Do Dot Talks Look Like In The Classroom?

Starting this routine in the beginning of the year is ideal (but not mandatory) because the complexity of numbers and placement of dots can grow naturally with the children’s development. The early stages of dot talks address Counting and Cardinality standards 4 and 5, understanding the relationship between counting and cardinality, and counting to answer “How many?” As the number of dots you are using increases, so does the range of activities you can introduce with them. Here are our favorite nine dot-talk activities for the kindergarten classroom:

How many dots are there? Hold the dot card up briefly. Students look at the card, but don’t count the dots (of course, they will in the beginning). Ask, “How many dots are there?” Record or discuss students’ observations. Follow up with, “How do you see them?” Students will explain how they saw groups that they added up to the whole. Dot talks start out very simple! Your students are learning how to do Dot Talks and learning how to complete the question-answer process. Over time, as the numbers get more complex and the children develop their skills, a single dot talk card can take as long as 15 minutes to complete! Dot talks for 1, 2, and 3 dots can be used for training your class in the process. You will model strategies, self-talk, and reasoning aloud. Early on, each number can be presented in a variety of arrangements so children can experience a variety of strategies. You can certainly buy specially made dot-talk cards. In the beginning, it is also easy to use a marker and an index card!
Which number doesn’t belong? Starting this activity early on is great, because the visual representation is easy to differentiate in small sets. For example, the difference between 2 dots and 4 dots is clear. This activity requires a set of 4 cards. 3 of the cards have the same number of dots. 1 card is different. Children briefly look at each card, and then it is flipped. Children “see” the difference by subitizing the dots on each card. Children share their thinking and how they knew which card didn’t belong. This activity quickly builds confidence in young learners, which in turn builds excitement for dot talks, and math in general.
How did the number change? This activity is presented in sets of 2 cards each. Children look at the first card briefly. The card is flipped, and a different number of dots is displayed. Children explain how the number changed. For example, there may be a new dot on the 2^nd card. One dot may have been subtracted. Again, children rely on subitizing to come up with the answer and develop their reasoning/articulation skills through the reporting process. This activity supports Operations and Algebraic Thinking standards 1, 2, and 5.
Secret Partners addresses the Operations and Algebraic Thinking standards 1, 2, and 5, as well as standard 3 which asks children to decompose bigger numbers. Starting with 2 dots, children report the “secret partners” that make the bigger number. For 2 it is 1 and 1, but as the numbers get bigger, the possible addends increase rapidly! This activity really illustrates students using different strategies, which is another important part of dot talks.
How many more to 10? 10-frames are great because they form a number-sense foundation of 10 that is easy (over time) for students to subitize. Using dots in a 10-frame, children find the missing addend to ten, Operations and Algebraic Thinking standard 4. After plenty of practice, the sum total goes up to 20! With plenty of visual support and practice, your students will master this activity.
Strategies to count scattered dots addresses Counting & Cardinality standard 5. It also provides an opportunity for your students to utilize and then share different strategies for solving the dot card. Starting with small numbers, students look at how changing placement doesn’t change numbers. As the numbers get more complex, students use and share strategies to group scattered dots. Subitizing works best with well-organized dots. This activity provides a challenge.
How many dots are there? with 10-frames is introduced well into the year, as the numbers start to spill into the teens. These cards can be used in conjunction with subitizing cards. Students can compare and contrast subitizing larger numbers without a frame and with a frame. This activity addresses Counting & Cardinality standards 4 and 5 as well as NBT standard 1.
Dice Dots Addition cards present dots on dice. The dots are somewhat separated and address composing numbers, number sense, and addition fluency. Children continue to build a conceptual bridge between the strategies they use for the dot cards and using formal operations.
How Many Groups of 10? This activity is designed to grow with your students. Presented mid- to late in the year, students can build on their hard word developing an understanding of numbers to 100! Counting and Cardinality standard 1 is a monster, and this activity can take your students to the finish line.

Students use their subitizing skills to count full 10-frames for groups of 10. It also helps children get familiar with talking about groups of 10.

I hope these activity ideas give you a sense of how powerful dots talks and number talks can be in your classroom. As an integrated part of your math instruction, dot talks push children’s thinking, knowing, and reasoning skills. If you are looking for another math element to take your classroom math practice from great to superb, dot talks might be for you. Check out our Dot Talks resource in our TpT store for print and ready resources!

Education, Math, Math Fluency

How to Teach Decomposing Numbers Conceptually

March 27, 2019

Primary teachers feel loads of pressure to ensure that their students are able to understand and master basic mathematical concepts. Because the subject of mathematics begins with a foundation of knowledge that builds upon itself over years of schooling, students must have a firm grasp of the foundational skills of math in order to be successful in later years.

As an upper elementary teacher, I cannot tell you the number of times that I have encountered students who still do not fully understand basic mathematical concepts, such as decomposing numbers. Students who struggle to add and subtract often do not have the understanding of how single-digit numbers can be broken up into even smaller numbers. This deficit not only affects their arithmetic skills, but it also affects their ability to grasp higher-level concepts, such as decimals and fractions.

The key to teaching decomposing numbers is giving students the opportunity to play with numbers rather than simply providing a simple algorithm or forcing the ever-so-popular “skill and drill” approach to teaching.

Now, you may be hesitant like I was at first to spend valuable instructional time allowing students to play with math manipulatives, but believe me, it is absolutely essential for them to be able to have this time to explore and come to their own conclusions about numbers. When students are able to manipulate objects and express their observations, their brains make more neural pathways in relation to math, which then allows them to excel in math in the future.

Day 1

Start with a five-frame and double-sided counters. I’ve found that it is best to laminate the five-frames so they last longer and can be easily wiped down. Give each student their own frame and exactly five counters. Have them place all of their counters on the five-frame (one in each box) and tell you what they notice.

Some prompts you may want to use are:

“How many do you have?”

“How do you know there are five?”

“Are there any empty spots on your frame?”

Give the students time to ponder and discuss these questions with a partner, table group, our as a whole class.

Then, dive a little deeper. Notice if any of the students used both sides of the double-sided counters. Did any of them use some yellow and some red on their frame? Did any use all of the same color?

It’s not time yet to get into making these into addition statements, but have them state what they did out loud. Some students may already be able to verbalize that 2 yellow and 3 red counters make 5 total, but others still need to manipulate, observe, listen, and talk about this concept without thinking of equations.

Tell the students that you are going to talk more about this the next day.

Day 2

On the second day, give the students the same materials – a five-frame and five counters. Ask them to put one counter in each box, but this time, ask them to pay attention to the colors of their counters and see if they can come up with a different way to show five than they did the previous day.

You will have varied responses to this – some students may continue to put all of their counters with the same color showing, some may do an ABAB pattern, and others may lump a few yellow together and a few red together. All of these are still acceptable for their exploration phase.

Again, prompt them to discuss what they notice. Today, also prompt them to state how many counters they have. You may give them a sentence frame to help initiate discussion:

“I have ___ red and ___ yellow counters. I have 5 counters total.”

This discussion will take yesterday’s lesson to the next level as they are now differentiating between the colors of the counters and recognizing that the total is still five, regardless of what colors they used.

They should now be ready for the “Shake and Spill” game. You can use a recording sheet for this game, or you can continue with student discussions. Either way, the kiddos are going to get the hang of decomposing the number 5.

Demonstrate for the students how to play. Take all five counters in your hands and shake them around before dropping (spilling) them on the desk. You may want to give them a small Dixie cup to shake their counters in and a small box or piece of paper for them to spill in.

After shaking and spilling, they should arrange their counters on their five-frame, keeping all of the yellows together and all of the reds together. For example, if they spilled and had 2 yellow counters and 3 red counters, they would group them together on the frame rather than creating a pattern out of them.

Students should then use the same sentence frame as before:

“I have ___ red and ___ yellow counters. I have 5 counters total.”

They can repeat this activity for the remainder of their math lesson, discussing their results with a partner and/or recording their work on a simple “Shake and Spill” recording sheet. Please note that if you use a recording sheet, the students will also need a yellow and red coloring utensil to record.

I like to tie it all together at the end of the lesson by making an anchor chart with all of the combinations that they created. The students can then share all of the ways that they were able to make 5, and you can record them for later reference. It’s great to be able to put them in a logical order as well so that they can see the pattern in yet another way.

0+5=5

1+4=5

2+3=5

3+2=5

4+1=5

5+0=5

Day 3

On Day 3, you can introduce the ten-frame. Provide each student with his or her own frame. Before giving them the counters, have them tell what they notice about the ten-frame. They may state that it is bigger than the five-frame. Some students may even say that it can fit 2 five-frames. Facilitate conversation around their observations.

Then provide them with ten counters. Go through the same process with having them place the counters on their frame (one in each box) and telling you what they notice.

Their verbal responses will be more advanced now as they have been exposed to noticing the color of the counters. If you are hearing them chatter about their different colors, celebrate with them! They are having “ah-ha” moments and making connections to larger numbers!

Again, have them discuss what they notice about the frame, how many counters they used, and how they know that there are ten counters on the frame. This helps develop recall when they see ten-frames in the future.

On Day 4, provide each student with a ten-frame and counters. Have them fill out their frame with all of their counters, paying attention to the color of their counters. Prompt them to discuss what they notice about their counters, and provide them with a sentence frame if needed. At this point, they may not need one since they have been using a sentence frame for a couple of days, but I always post it just in case:

“I have ___ red and ___ yellow counters. I have 10 counters total.”

Build on this connection by playing “Shake and Spill” with 10 counters. Again, they should group the same color counters together in order to visually see an equation prior to looking at equations abstractly. You can have them discuss their findings and observations, or you can have them record their work on a recording sheet.

I prefer to transition from discussion to a recording sheet because it allows them to not only see the number relationship with manipulatives, but it gives them a chance to transfer it onto paper and then actually write out the number sentence, making even more connections for them in the process.

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End the lesson with another anchor chart for their reference. Many of them will notice that there is a pattern that can be made for making 10 that they saw with the “Making 5” anchor chart.

The concept of decomposing numbers is invaluable to your students and the rest of their mathematical careers. If they can make connections with single-digit numbers, you have set them up to be successful with larger numbers.

Education, Math, Math Fluency

How To Integrate Number Sense Into Your Math Lessons

November 11, 2018

There are a lot of components to a math lesson. You look for activities with engagement and complexity to assess understanding. Sounds simple enough but there are several layers to each of those factors making it hard to find the perfect balance. So when I work with teachers on implementing number sense during their math class they start to panic.

“How can I add another thing to my math block?”

“I don’t have enough time!”

“I don’t know what type of questions to ask!”

I get it. You’re holding all these reminders in your mental filing cabinet when you’re teaching a lesson. You want to target specific standards and explain key concepts. You want to ask higher order thinking questions. You want your students to really grapple with concepts rather than memorize procedural steps.

But guess what?

Building number sense will ensure you target all those wants. Number sense is not another component to your math block. Instead it is part of everything you say and do in the classroom. Simply put…it’s good teaching.

Questioning

A good math teacher is always looking for ways to sprinkle number sense into lessons, but it can be hard to find and/or write number sense connected questions.

Look for questions that are open ended, require critical thinking, and reinforce the concept that numbers can be manipulated. Be intentional in the types of questions you ask.

Example: When rounded to the nearest hundred, I become 400. What numbers could I be?

This question takes a standard (rounding) that is often taught procedurally in classrooms and requires students to conceptualize the concept of rounding based on the value of specific digits. Students must work backwards starting with a number that has been rounded to the nearest hundred. Students will first share the most obvious answers such as 399. It takes some time for the class to conclude that any number from 351 to 449 rounds to 400 if asked to round to the nearest hundred. This question is far more complex than, “Round 377 to the nearest hundred,” thus reinforcing number sense.

Strategies

Some children have been given opportunities to build number sense at a very young age. Those students will appear to naturally understand how to manipulative numbers and place value for easier computation, while others will struggle. Educate your students on the different number sense strategies. Model these strategies time and time again. Constant repetition will help your struggling students develop number sense. Provide your class with daily prompts to drive the concept home. We learn better when we are asked to apply that knowledge regularly.

Math Talk

Number sense related questions lead to great math discussions. The more open ended you can make a question the better. Open ended tasks increase discussion and collaboration amongst your students. Encourage your students to work in groups or pairs so they communicate their mathematical reasoning.