Number Talks Week 2 - What exactly is subitizing?

In the last couple weeks of our Number Talks, students have shared many different ideas about how they view the pips (black dots). The goal in my mind at the moment has been primarily to develop a sense of comfort for all students to share their ideas in our classroom and create an encouraging community. At the moment, there is no right or wrong answer since we are simply asking the question “What do you see?”. Beginning with this open-ended question is important so that students understand that everyone visualizes numbers differently.

Last week I found this question invigorated students and I was eager to allow many students a chance to share; but this slowed down the process and students had trouble focusing. This past week I limited the amount to 3 students sharing for each card and this helped us keep our attention. Our students began sharing really wonderful descriptions about the pictures they see within the dots; it was interesting to watch our focus shift from last week with numerical descriptions related to subtraction to creative connects to objects in the real world.

What do you see?

While it is very important to encourage students to think creatively and create this positive atmosphere where students aren't afraid to share their ideas, after our class on Number Sense and having read the article on Subitizing: What is it? Why Teach it?, by Douglas H, Clements, I have a deeper understanding of the goal of our Number Talks. This article reminded me that subitizing means “instantly seeing how many”. I learned that perceptual subitizing means being able to recognize a number without using any mathematical thinking and conceptual subitizing means being able to recognize a number as both the whole and the unit (i.e. You can roll a die and see 2 groups of 3 or one group of 6.)

These skills are important to develop prior to learning more complex addition and subtraction procedures. And as Clements suggests: “Young children may use perceptual subitizing to make units for counting and to build their initial ideas of cardinality.” I had to review the concept of cardinality from the Kindergarten Math Continuum from our school board: “Identify the quantity of items in a collection where the last counting word tells how many (e.g. ‘1, 2, 3, 4… there are 4’)”. Eventually, students begin to conceptually subitize through these counting and patterning abilities and will “discover essential properties of number, such as conservation and compensation… [and] develop such capabilities as unitizing, counting on, and composing and decomposing numbers as well as their understanding of arithmetic and place value” (Clements). Thus, through our Number Talks, our goal is to develop students’ ability to conceptually subitize, laying the foundation for future understanding of strategies for addition and subtraction.

Clements notes specific arrangements of sets that make subitizing easier for children to learn. “Children usually find rectangular arrangements easiest, followed by linear, circular and scrambled”(Clements). It is also important to note he says that preschoolers aged 2-4 cannot conceptually subitize and that for sets of 1-4, no arrangement is easier than any other, but linear is easiest when the set is larger than 4. This means that my 4-year-old kindergarten students are likely to be counting each item but eventually by the end their second year of kindergarten, students will begin to conceptually subitize.

Clements also suggested there are several number activities that will help students develop the ability to subitize. Below are my three favourite activities.

Activity 1

Give each child cards with zero through 10 dots in different arrangements. Have students spread the cards in front of then. Then announce a number. Students find the matching card as fast as possible and hold it up. Have them use different sets of cards, with different arrangements, on different days. Later, hold up a written numeral as their cue. (Clements and Callahan 1986).

Activity 2

Place various arrangements of dots on a large sheet of poster board. With students gathered around you, point out one of the groups as studenst say its number as fast as possible. Hold the poster board in a different orientation each time you play. (Clements 1999).

Activity 3

My kindergartners’ favorite numeral-writing activities involve auditory rhythms. They scatter around the classroom on the floor with individual chalkboards. I walk around the room, then stop and make a number of sounds, such as ringing a bell there times. They write the numeral 3 on their chalkboards and hold them up. (Clements 1999).

After watching recordings of our Number Talks, I realised I have been displaying the dot cards the entire time and allowing lots of time for reflection while my teaching partner has been flashing them briefly.  In order to bring back the focus onto subitizing, “instantly seeing how many”, we will need to change the questions slightly, and ensure we are both presenting the cards briefly.

My Next Steps:

• Show the cards briefly and then hide them.
• Change the question from “What do you see?” to “How many dots?”
• Record their answers.
• Discuss their answers. If they saw 3, ask “How do you see 3?”
• Find more Table Top Activities and games specifically related to subitizing

Here is a Subitizing Pinterest Board full of great activities: 

These changes in our Number Talks will help students focus on quantity so their answers should be more numerical. It will also help them realise there is a correct answer when it comes to counting an amount, just as there is a correct way to put together a puzzle, but we may have different strategies for figuring out which pieces go together, some more efficient than others. We will give them a chance to share their methods of visualizing the dots and we can continue to celebrate our differing thought processes, creating a safe atmosphere for all students to share their ideas. The emphasis is not on having the correct answer but uncovering different ways to figure out the puzzle and eventually analyzing which way we feel is most efficient depending on the situation.

References:

Clements, Douglas H. (1999). Subitizing: What Is It? Why Teach It? The National Council of Teachers of Mathematics, Inc. www.nctm.org.

Kindergarten Math Continuum. Teacher Learning and Leadership Program. OCSB.

Number Talks - Week 1

This week we introduced Number Talks for the first time. It was exciting to see many students with their hands up ready to share their thoughts. Even more exciting, was seeing students that do not often share, raising their hands. 

I was eager to give many students the opportunity to speak, which slowed down my number talk, so I noticed many students started to lose focus until I switched to the next card. Since I was doing the Number Talk in French and students were learning to say “Je vois…” before answering, this also slowed down the process. I debated if it would be better in a small group but ultimately decided I need to limit how many students get to share before moving onto the next card.

This week, we were focusing on learning how to do Number Talks and not on documentation but on my second Number Talk I was eager to start documenting. Since answers were given out so quickly, it would be impossible to document them myself, so l asked my ECE to write down their responses. I showed them 4 different cards with 3 dots arranged in different ways. We noticed many interesting ideas and an interest in subtraction. Here are their responses:

Je vois...

• 2 dots
• Take away 1, gets 3, take away is 2
• Take 3 away, 2, bring back 3 it’s this.
• Teacher: “Interesting, I like how you used the words take away”.
• 1, then becomes 2,
  Teacher: “Interesting , you see 1 and then 2.”  I jumped in early here since she continued to speak after me saying: “then becomes 3.”
  Conclusion: She wasn't actually seeing two groups, she was counting.
•  Then becomes 3.
• This one is at the bottom half of it. (In reference to 3 dots in a line on the bottom of a square card whereas the other was in the middle.)
• Take 3 away, no more.
• 3 same line
• A diamond. (In reference to a line of 3 dots on a diagonal. I asked her to come and show me the diamond and she turned the square on an angle.)
• Take one away, will be 2.

Next Steps:

1. Speed up the number talk by limiting the number of students chosen to share.
2. The ECE will use the iPad mini to record the Number Talks in both English and French so we can compare them.
3. Documentation of student thinking through recordings. 

Kindergarten Number Talks

This year there is a focus on math in our school board so I decided to take the Primary Math Additional Qualification course. We have been talking about Number Sense in the early years and have had a chance to watch a few videos on communicating our mathematical thinking through something called "Number Talks".

For my inquiry assignment I decided to work with my colleagues and plan to implement these "Number Talks" into our Full-Day Kindergarten program for both English and French days. Today we met to begin planning. I wanted to share our plans in hopes that it might spark an interest among other Kindergarten teachers to check out this great resource and begin their own Number Talks!

Organizing our Number Talks:

Afternoon Table Top Activities (12:40-1:00pm)

o   Provide math games with materials to be presented during Number Talks

§  Ex. Dice, dominos, dot cards, Math books in the library

Afternoon Community Time  (1:00pm-1:15pm)

o   Number Talks – We will start by presenting dot cards and following the progression of cards from the book “Number Talks” by Sherry Parrish.

o   English Talk first, followed by repeating the same talk in French the next day.

Our Number Talk process:

1)    Present the question.

a)    Show a dot card briefly. *Not long enough for them to count the dots. Ask: How many dots do you see?

2)    Invite students to answer and ask: Does anybody else see them a different way?

3)    Students will use hand signals:

a)    Thumbs up means "I have an answer".

b)    Sign language for “the same” means they agree with another student’s answer.

4)    Students must use full sentences.

a)    Ex. "I see…"

b)    Ex. French “Je vois… + English explanation"

5)    Prompt students to explain their thinking and repeat their answers aloud.

a)    How do you see them?

b)    Show me how you found ___.

c)    Tell me what you see. (Ex. I see 4 dots on the bottom and 1 dot on top).

 

Next, our team watched the Number Talks Kindergarten videos and documented the teachers’ questions and prompts. We created a list of language we wish to use during Number Talks with dot cards, 10 frames, Rekenreks, and counting books.

As we begin this inquiry into Number Talks and reflect on their effect on students' mathematical development and ability to communicate their mathematical thinking, I will continue to post our findings in my blog. We will start our Number Talks next week using dot cards and look forward to this new adventure in our classroom!