Number Talks - week 3

After reflecting on our first couple weeks of Number Talks, this week we decided to make some changes. First, we thought it would be helpful to try doing them in small groups during table tops to help those that had trouble focusing. This time, I made sure to only briefly show the cards as my teaching partner was already doing so. To refocus on numbers versus pictures we also changed the question from “What do you see?” to “How many dots?”.  Since we started them in November we decided to review the first four strings of cards for the number three. We recorded their answers, and finally, we discussed them.  The results were very interesting.

Since I am the French teacher I was asking them in French "Combien de points?" I recorded their answers in the charts below, noting if they replied in English. We started the Number Talk sitting at a table but I found it distracting and moved the group onto the carpet. Unfortunately I can't recall if we moved for string 2 or string 3.

String 1 – Number 3
Student	Card A	Card B	Card C	Card D
A	1	2	3 (English) I encouraged in French "1, 2, 3"	1, 2, 3 (Counted but card wasn`t in front of her)
B	1	No answer (didn`t see card?)	3 (English)	3 (English)
C	5 (English)	2 (English)	5 (English)	12 (English)
D	No Answer	No answer	(showed 3 fingers)	(showed 3 fingers)

String 2 – Number 3
Student	Card A	Card B	Card C	Card D
A	2	3 fingers	3 (English)	3 (English)
B	2 fingers	3 fingers	3 fingers (encouraged to speak… 3 English)	3
C	2	10 (English)	8 (English)	8 (English)
D	2	No answer	No answer	No answer

I recorded their answers for strings 1 and 2 but found it unnecessary to continue as that was enough data to give us a sense of their thinking. Both Students A & B seem to have a strong sense of the number of dots but still must count from one in French.  found they qucly became unnterested for strngs 3 and 4 as t was not challengng enough so  ased them to show dfferent ways wth ther fngers. Student D was very hesitant and shy to answer but began to hold up three fingers. She may have been inimidated working with the others or speaking in French. I would like to redo the Number Talk on-one-on with Student D. Student C gave random answers and will clearly need extra assistance and practice to develop his number sense. This student`s thinking is revealed below during our discusson.

Here is an account of our conversation after completing the Number Talk:

Teacher: What do you notice about the cards?
Student A: They`re all the same.
Student C: No.
Teacher: How are they the same?
Student A: They`re all sets of 3.
Teacher: Does everyone agree with student A?
Students B & D: Nodded.
Student C: They`re not all the same to me.
Teacher: (Spread out the cards) Look at the cards. How are they different?
Student B: They`re in different ways.
Student C: Different ways.
Teacher: How are they different to you (student D)?
Student C: Mumbled.
Teacher: Asked to repeat but he was shy and needed encouragement.
Studen C: One is down and one is up.

Possible Next Steps:

• Further discuss with English Partner on Monday to see her results.
• Continue with small group Number Talks to assess students and make groupings for future learning
• Read through Kindergarten chapter on Number Talks for more ideas
• Some possible suggestions from my English partner who was concerned they are just expectng the same answer over and over:
• move onto number 4
• Mix the cards up from 1-4

Proportional Reasoning

This past month has been extremely busy and therefore my teaching partners and I decided to put a pause on our Number Talks until we have time to coordinate some next steps for our class in the New Year. In the meantime, we have joined our school board's Collaborative Inquiry in Mathematics and began another math journey into something called proportional reasoning.

I was first introduced to proportional reasoning in my Math AQ course this year, and at first glance, I found the concept seemed very logical and easy to understand since it just seemed to be a part of everyday life. I knew I had never heard the term before but it seemed as if we had always been teaching it anyway. My impression was that it was about problem-solving and as long as we provided students with real-life problems then they would develop their proportional reasoning. 

My thinking:
"What does this have to do with proportional reasoning?

Prior to our first session for the collaborative inquiry, we were asked to provide students with a task from a list and to bring in some student work samples. The tasks were all aimed at helping students develop proportional reasoning. We chose one about showing the number 10 in various ways since it seemed to be the most basic task. We began representing the number 5 for the first week and then 10 the following week. I was very unsure how this related to proportional reasoning, since I thought proportional reasoning only had to do with presenting real-life problems.

A few weeks later, as several of the primary teachers from my school board gathered for our first Collaborative Math Inquiry session, I realised proportional reasoning was not so easy to explain and not just developed by giving students real-life problems. As we discussed the idea and our understandings of it and read some supporting documents, I started to confuse it with the idea of subitizing. But after a full day's workshop with some very interesting discussions, videos, and attempts at solving problems on our own, I have a much clearer picture of the concept.  

In my own words, proportional reasoning is a way of thinking about problems and comparing amounts and their relationships in relative versus absolute terms. It involves solving real-life problems and the capacity to subitize is a precursor to looking at those real-life problems using proportional reasoning. Therefore, developing students' ability to subitize actually is a way of proportional reasoning in and of itself. And the task we chose - representing the number 10 in various ways -  also can involve subitizing, thus the connection with proportional reasoning. 

We were also given an excellent resource to aide in our understanding of proportional reasoning. I highly recommend this monograph: Paying Attention to Proportional Reasoning: Support Document for Paying Attention to Mathematical Education (2012). December has been an incredibly busy month and amid all the items on my to-do lists, reading that resource, I must say, was not my number one priority. I was quite frustrated by some unexpected financial “to-do’s” that suddenly were thrown at me in December; but to my surprise, my financial “to-do’s” not only involved many mathematical operations but an incredible amount of proportional reasoning. When dealing with anything financial, it is obvious how mathematics plays a role but I started to realise proportional reasoning seems to permeate more areas in my life than I had previously thought.

The first real-life situation that comes to my mind in which proportional reasoning is involved would be comparing prices at the grocery store. Often we must not only consider the prices and quantities but in order to properly compare we must use proportional reasoning to decide whether the prices are even comparable and multiplicative thinking in order to calculate how much the item is relative to the other. Our comparison must be based on similar quantities. Comparing prices is an example of an authentic situation that occurs more regularly in our adult lives. Other examples include: using recipes and cutting them in half/doubling them, dividing up pizza among a group of friends, and calculating the temperature from Fahrenheit to Celsius or vice versa. In order to prepare students to solve these everyday problems, it is necessary to help students learn how to use proportional reasoning and provide them with these types of authentic problems at their own level of understanding and interests.

I just had to share this funny  article  I found while searching for
some images about sharing pizza!

After dealing with my finances, I finally found some time to look over the document and contemplate just how complex proportional reasoning can be. “Proportional reasoning is a complex way of thinking and its development is more web-like than linear. Students do not think through an identical concept in exactly the same way so there are myriad possibilities at play when developing the ability to reason proportionally” (p. 4 Paying Attention to Proportional Reasoning.) It is very interesting to note how I was feeling when tackling my own real-life problem that involved proportional reasoning. Mostly, I was frustrated that it was consuming an incredible amount of my time when I needed to put that time into writing this reflection. The task was also very complex. I had to review and compare amounts and make many calculations to verify if they were correct. On top of all this, I had to explain my thinking and I realised that was one of the most difficult parts. I think it was so difficult because I was trying to explain my proportional reasoning which involved many numbers and operations and as stated before, there is not one way of thinking about a particular problem. My explanation may not make sense to others. Thus we must allow students to present multiple ways of thinking about problems just as we are doing through our Number Talks.

I must note here, before moving on, a memory from my childhood. When I was about 10 years old I can recall having some math homework that was a bit tricky and my dad was attempting to help me out and trying to convince me to try to use his method. I was very much convinced I needed to use the formula and strategies my teacher taught me in class and had so much trouble looking at the problem from another perspective. I understand now that my dad was trying to get me to look at the big picture and use proportional reasoning. I clearly was not in the mindset that math problems could be solved in different ways and not able to apply proportional reasoning in solving my math homework. Since the concept of proportional reasoning seems like a relatively new term, looking back on my mathematical education from the perspective of an educator, I now see how it is very important for students’ math education from an early age. The Paying Attention to Proportional Reasoning document states: “Although the Ontario curriculum documents for mathematics do not reference the term proportional relationships until Grade 5, activities in the primary grades support the development of proportional reasoning” (p. 3). Since proportional reasoning is so complex, it is not explicitly in our Ontario Math curriculum until the later grades; however as the document addresses, it is important to understand it as an educator and provide students with opportunities to develop it, even at a young age.

The Paying Attention to Proportional Reasoning support document divided the complexity of our thinking into different concepts which we use in a “web-like nature”: understanding rational numbers, multiplicative reasoning, relative thinking, understanding quantities and change, spatial reasoning, measuring, linear models, area, volume, unitizing, comparing quantities and change, scaling up and down, and partitioning (p.4). The one type of thinking that drew my attention was unitizing and spatial  reasoning. As the document notes: “Unitizing is the basis for multiplication and our place value system which requires us to see ten units as one ten and one hundred units as ten tens… this is complex since unitizing ten things as one thing almost negates children<s original understanding of number… Spatial visualization allows the student to understand the unit as equal intervals of distance. This is exactly what we were working on in both our Number Talks and in the task involving representing 10 in different ways.

Originally, when I joined the Collaborative Math Inquiry, I thought that the tasks would take us in a different direction than our Number Talks. But as I embark on my own math learning journey it is exciting to develop a greater understanding of the big picture and

see the interconnections among the mathematical concepts I am teaching. I am really eager to be a part of this school board inquiry and further explore how to develop primary students’ proportional reasoning as well as improve their mindset, communication and number sense skills.

Other Proportional Reasoning Resources:

Another ministry document that has samples of questions and other suggested resources:
Continuum and Connections: Big Ideas and Proportional Reasoning K-12

A colleague of mine was also searching for Kindergarten resources related to proportional reasoning and found the following document. I have not yet had a chance to read it but it looks quite useful. The document is long but you will find a section in the contents on proportional reasoning. 

http://www.sde.ct.gov/sde/lib/sde/pdf/curriculum/math/K.pdf

During the first session of our Collaborative Math Inquiry, each Grade team got together to discuss what area we thought the students needed most attention/improvement.  Our Kindergarten Team was able to group our ideas into three categories which we prioritized below:

1) Mindset

2) Communication

3) Number Sense

I found these resources for mindset and communication on a document we got at the first session:

Student Interaction in the Math Classroom: Stealing Ideas or Building Understanding (What Works Research into Practice, January 2007)

Communication in the Mathematics Classroom (Capacity Building Series Special Edition #13, September 2010).

Maximizing Student Mathematical Learning in the Early Years ((Capacity Building Series Special Edition #22, September 2011).

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!