number sense trajectory Archives - Shelley Gray

More or Less? How to Strengthen Comparison Thinking Through Games

Shelley Gray — Tue, 19 Apr 2022 17:05:33 +0000

“Who wants more!?” Growing up in an Italian household, this was a common phrase that was heard at the kitchen table. I can just picture my father checking in on all the kids to make sure we had enough food to make us full. The concept of more or less is learned at a very young age. Think of infant children. Aren’t adults constantly asking the child, “Do you want more?” Because of this, children have an unconscious sense of a change in number within the first few months of life. This change could be with very small collections such as 1 or 2 objects, or with larger collections, such as double the number of objects.

Comparison In Early Grades

When students enter the classroom in Kindergarten, the idea of comparing a collection of objects still stays the same. Teachers provide students with two groups of objects to compare and determine which has more and which has less. Teachers make sure it is obvious which is more or less, so that students do not need to count to compare. This is because the focus is on developing the vocabulary of “more” or “less” and then transitioning that language to “greater than” and “less than” further in their learning.

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.

Things to Remember

There are a few important notes to remember about the early stages of comparison thinking. When you first start talking about comparisons, make sure to provide an abundance of opportunities to compare obvious amounts in order to help support the new vocabulary. Obvious amounts mean more than just one or two more. Your child needs to really SEE that one group of objects is more or less than the other. Furthermore, it’s also important to note that this idea of comparison is grounded in concrete representations. Let your students play with two different sets of objects and use prompting questions to elicit a conversation of more or less. It’s okay if they’re unsure at first! Multiple exposures will continually reinforce the vocabulary.

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Games to Practice Comparison

Games are a great way to engage young learners, and the idea of comparison can easily be practiced during a math game! Below are three simple games you can play with young learners to practice the concept of more or less. These games will also reinforce the concept of subitizing, since all you need to play is a deck of subitizing cards!

Comparison Game #1: More or Less?

Two players decide if they are playing more or less before starting the game.
Players split the deck and each player takes the top card from their pile and places it face up. Whichever card is more/less wins both cards.
The player with the most points/cards wins the game.

DIFFERENTIATION IDEA: If your students struggle with the pictorial representations, have them recreate the image using concrete objects!

Comparison Game #2: More or Less War

Players split the deck and each player flips over the top card from their pile.
The first player to say the quantity that is less keeps both cards as points.
The player with the most points/cards wins the game.

DIFFERENTIATION IDEA: Playing this game with older students? Use multiplication subitizing cards instead of five or ten frames!

Comparison Game #3: In Betweens

Two players split the deck in half and each place one card down face up. These are the two target numbers. If they are equal, redraw.
Players take turns flipping over their top card. If the card is in-between the two target cards they keep that card as a point. If it matches a target card or is outside the range, it goes to the discard pile.
Players continue alternating turns until all the cards have been played.
The player with the most points/cards wins the game.

The Number Sense Trajectory

If you’re interested in finding out more about the early number sense trajectory, here are some other posts that will be helpful!

The Power of Early Number Sense: How to Support Curiosity in K-2

How to Investigate Subitizing Through Games to Spark Early Number Sense

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 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.

The Power of Early Number Sense: How to Support Curiosity in K-2

Shelley Gray — Thu, 17 Feb 2022 17:37:09 +0000

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.

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

The post The Power of Early Number Sense: How to Support Curiosity in K-2 appeared first on Shelley Gray.