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## Staff Development Conference – Las Vegas July 2017

Staff Development for Educators Conference, Las Vegas, Nevada, 2017

For the first time, I attended the SDE conference July 11 and 12th at the Palazzo/Venetian Hotel in Las Vegas. I enjoyed it and found it valuable. I always can walk away from a conference/workshop/session with at least one new idea, but what I found this conference did for me was reaffirm and confirm I am/have been doing great things for student and teachers learning.

I attended one morning keynote speaker and 7 sessions within the 2 days. I am going to provide you the highlights of each session as well as a wrap up of what messages I found to overlap in all sessions.

KEYNOTE SPEAKER: JO BOALER- MATHEMATICAL MINDSETS

Jo Boaler’s key message is that ALL students can learn mathematics. Mathematical ability is NOT heredity, nor is there smart math students and weak math students. Their success is the outcome of the teacher’s messages of encouragement and empowerment as well as the teacher’s instructional methods and practice.

SESSION 1: COMPUTATION: FROM CONCEPTUAL UNDERSTANDING TO PROCEDURAL UNDERSTANDING by Jana Hazekamp

• Number sense is developed, it can’t be taught.
• Visual learning of number is a necessity. Students need to visualize the decomposition and recomposition of number, first numbers to 5, then 10, then 20 and so on.
• It is important to introduce math words before symbols.

Although these two topics were presented separately, it is crucial to connect subtraction to addition when decomposing and recomposing number. This connection will enhance basic fact fluency which impacts multi-digit addition and subtraction.

Horizontal adding from left to right forces place value which is the foundation to developing number sense.

• Part part whole
• Friendlies
• Hundreds chart
• Compensation
• Open Number line

Subtraction in not just ‘take away’ and ‘minus’, it is about the ‘distance’ and ‘difference’ between numbers.

Strategies for subtraction:

• Part whole
• Place value (pull apart the value of each digit and/or two digits)
• Hundreds chart
• Open number line
• Compensation

SESSION 4: TAKE THE NUMB OUT OF NUMBER SENSE (K-2) by Eileen K. Ryan

There are 5 competencies of mathematics that create problem solvers

• Number sense
• Visualization
• Generalization
• Metacognition
• Communication

The key to developing number sense with our primary students is understanding number through the decomposing and composing of number.

SESSION 5: DEVELOPING MATHEMATICAL THINKERS(K-2) by Eliza Thomas

• Talk to your students, give them the opportunity to share their thinking.
• Pull out/see mathematics everywhere in our world.
• Vocabulary needs to be an instant piece of their repertoire because it develops mathematical thinking.
• Focusing on spatial awareness is also important in mathematics.

SESSION 6: FILLING IN KNOWLEDGE GAPS (Gr. 3-5) by Cassey Turner

• Students must develop number sense before being taught the algorithms.
• Decomposition and recomposition is crucial to all operations with all types of numbers.
• The Japanese believe division is the final step in teaching place value.
• Equivalent fraction knowledge is the foundation to all fraction learning.

SESSION 7: EARLY ALGEBRAIC THINKING by Kar Hwee Koh

Without an understanding of number-eg.  patterns, relationships, place value, regrouping, add, subtract, count on, problem solving- students will not be able to do algebra later on in the middle school and high school years.

WRAP UP/OVERLAP MESSAGES

• Empower students
• Engage students
• Encourage communication by asking the how and the why– can use the words , convince me, prove it, how do you know?
• Number sense is a MUST to develop in order to enhance mathematical problem solving in our students
• Do not teach traditional algorithms until later elementary years (Gr. 4-6) when number sense is developed.
• Speed is not always better, nor does it mean the student is smarter.
• Productive struggle is crucial to students growth and learning. It pushes them to think further and harder.
• Number sense is the understanding of number and number relationships; and the ability to use number flexibly, creatively and efficiently.
• The equal symbol is about ‘balance’ NOT computation
• Contextualize number equations for all operations. This means have students tell stories connected to the equations.
• The unknown in equations has to be in every place holder, not just at the end for the answer.  Eg. 5 + 5 = x ,  14- x = 8,  a x 8 = 40,  x = 25 divided by 5

Please do not hesitate to ask questions. As I continue on my journey, I am always learning and growing. I am so excited in having the time this Fall to pursue my passion in helping students, teachers and parents. Together, we can make a difference and improve the mathematics learning for children of all ages and throughout the entire world.

• ## Skip Counting and Factorization are Important

If you do not remember what Prime Climb is or you are a new subscriber or viewer then please go back and read the blogs:

• Prime Climb Round 1: here
• Prime Climb Round 2: here
• Prime Climb Round 3: here

These are lessons inspired by Dan Finkel - Math for Love:

I have to say this fourth round of Prime Climb has been interesting and exciting because of the students continued growth in the understanding of factors and divisibility.

As mentioned in previous blogs my Grade 3 students have been working with skip counting, factors and divisibility based on the number of school days they have been at school.

It is important you keep in mind this learning began on Day 1 of school and we have not missed a day. It starts with understanding of what number patterns we can count by to that number beginning at 0. For example, when we started at 1 we could count by 1. One group of 1 is 1. Then for 2 we could count by 1 and by 2. There are 2 groups of 1 in 2 and 1 group of 2 in 2.

This learning continued every day and from the start we had to do the skip counting one number at a time with the support of a 100’s chart and/or number line to determine the factors and sometimes it was trial and error. When I say trial and error students may provide numbers in sequence as factor. For example, at Day 20 students may say we can count by 1, by 2, by 3, by 4, by 5, by 6. We would skip count to verify and this is surface level learning.

Then the deep learning began and by the time we got to 30 many students realized that in 10 there are 2 fives so in 30 there are 6 fives and in 10 there are 5 twos so in 30 there are 15 twos. They also knew that 3 tens are 30 so then 10 threes are 30. They began to connect that factors come in pairs and are opposite one another in the amount of groups. This of course, lends itself to the commutative property. If 10 x 3 = 30 , then 3 x 10 = 30.

It has been an incredible learning journey for my students and it was wonderful watching students’ surface level understanding of factors and divisibility move into deep and transfer learning.

You may ask what benefits does skip counting and factorization have to students’ success in mathematics?

• Skip counting is the pre-learning to multiplication and division.
• Factorization allows flexibility with teen digit and multi-digit multiplication and division.
• Both skip counting and factorization is of benefit to working with fractions which is connected to decimals.

I ask you to take a look at your curriculum, whether it is Kindergarten or Grade 6 and ask yourself where and how this learning is important to start and how it progresses and connects to other concepts/strands.

Now let’s get back to the work of my students. I have included pictures and some short video clips. Remember there are students who are still grappling with this, but it is productive struggle. You will specifically hear this productive struggle in a short video where a young boy is trying to explain why he colored 27 in the way he did, based on fractions. He knew how to color 27 based on the factorization and patterns of the Prime Climb circles, however, he wasn’t quite certain about the addition of his fractions. But again, think of the learning that is taking place. I didn’t correct him because I do plan for this learning to continue as we move into May and June.

Here are the pictures as well as the videos. I hope they are of value to your learning. If you haven’t yet tried Prime Climb, give it a try with your students and let us know what happens by adding a comment to the Blog. I would absolutely love to hear about the learning in your classrooms.

Prime Climb is a game created by Dan Finkel (Math4love) and it can now be purchased in Canada.

## Wanted 92 -- Deeper Learning Demonstration

In my WANTED 65  blog, I stated that through our daily math routine we focus on many different number concepts to enhance number sense. Without number sense, students lack the ability to think, reason and problem solve with flexibility

On Monday, February 6th, I had my Grade 3 students engage in this activity for the second time with their Grade 2 buddies. My intent was to identify if my students were developing a deeper understanding of the number concepts we engage in daily. To add, I wanted to see if there was transfer learning occurring since DAY 65 of our school year. Growth should be visible after one month of learning.

You may ask, did I see growth? I did, and of course, not all of my 26 students are at a deeper understanding. But I have to say that many have further developed their number sense.

Here are a few examples of what I heard and saw:

• I had a student tell me 9 decades and 2 years, and from that, another student built on that student’s thought and said 8 years less than a century.
• I also had a few others show 92 using various coins other than dimes and pennies.
• A couple of my students made reference to the number line work we do and said that 92 comes to the left of 93 but to the right of 91.

I was pleased with the outcome and the students enjoyed the activity once again. Our grade 2 buddy teacher came to me after and said her class came back so excited and told her all about the poster activity. I plan on doing this activity again around DAY 120.

You continue to hear me using the language: surface learning, deep learning, and transfer learning. These terms come from the book, Visible Learning For Mathematics, written by John Hattie, Douglas Fisher, and Nancy Frey. (Read this post: Procedural Fluency for more on this). Once again, I recommend this as an educational resource to read.

Here are some pictures to show the students work for WANTED 92. I hope you will give it try it with your students.

### Try it with your students

I can’t wait to hear how it goes and for you to ask me any questions that may come up as you work with your students. What numbers are you using? I will be helping anyone interested in extending their learning through videos and webinars in the weeks to come.

• ## Cardinality - knowing the number of elements in a given set

Through the many years of learning and working with subitizing, it is evident that finger patterns play an essential role in helping young children decompose and compose numbers 1 to 10.

As students move into the deep understanding of decomposing (breaking apart) and composing (putting together) numbers 1 to 10, arithmetic fluency begins: first with the basic facts and then with the teens-digit numbers. Once connections are made with the teens-digit numbers, students move into greater number(multi-digit) arithmetic with ease.

### This understanding of decomposing and composing number will become evident as students engage in the learning of the basic fact strategies.

For example, let’s look at 2+3. First, before students have cardinality, they will find answers through a ‘count all’ strategy. Once cardinality is developed, students will use a ‘count on’ strategy. For the ‘count on’ strategy, the sum will not change whether the count starts with either addend (2 or 3).

If you are uncertain as to what cardinality means it is knowing the number of elements in a given set(group).

For example, if I ask a student to use counters (manipulatives) to show me a set (group) of 8. He counts out 8. Then I ask him, "How many do you have in the set(group)?"

If he can tell me 8 without hesitation he has cardinality. If he has to do a ‘count all’ to tell me there is 8, then he doesn’t have cardinality. And research states that without cardinality, it is a challenge to develop number sense.

As deep learning occurs, the transfer will be made to the doubles plus 1 strategy.

With this, a student would say:

I know 2+3 =5 because 2+2=4 + 1 is 5.

What he or she did is decompose 3 as 2 and 1.

This, in turn,  will lead to the doubles plus 2 and other efficient strategies. I am focusing on addition, but this understanding is crucial to all operations (addition, subtraction, multiplication and division).

Without the work of finger patterns, efficient strategies are less likely to be understood or may take a longer period of time. In turn, our students will either remain using less efficient strategies of ‘count all’ and ‘count on’. They may require support from counting on their fingers, a number line or a hundreds chart; and/or they will memorize the facts without any understanding of the how and why.

I have heard from many upper elementary and middle school teachers that they still have students counting on their fingers. This indicates that students have not gained a deep learning of decomposing and composing number.

To help you understand the strategies I am referring to I recommend you listen to Jo Boaler’s Youtube titled, Jo Boaler’s Clip on Number Sense:

### When deep and transfer learning is happening with numbers 1 to 10, students demonstrate fluency with multi-digit numbers.

For example, a student who decomposes and composes numbers 1 to 10 would explain finding the sum of 37 and 25 as such:

I will take 5 away from the 7 leaving 2 ones and I will add the 5 to 25. Now I have 30 and 32 which is 62 or I can take the 5 from 25 and break it apart as a 2 and 3, then add 3 to 37 which is 40 and 22 more is 62.

When you are working with students on greater number fluency you can work with number strings based on basic facts. A number string would look like 5+7, 15+7, 25+7, 35+7… if a student was thinking:

I know 5+7 is 12 because I think of 7 as 5 and 2 so double 5 is 10 + 2 is 12, then 15+7 is 22 because I think of 7 as 5 and 2 so 15+5 is 20 + 2 = 22.

Number strings can be used for subtraction as well. It may look like 12-8, 22-8, 32-8… if a student was thinking:

I know 12-8 is 4 because I think of the 8 as 6 and 2 so 12-2 is 10 and take away 6 more is 4, then 22-8 is 14 because I think of 8 as 6 and 2 so 22-2 is 20 take away 6 is 14 (or a student may think of the 6 as 5 and 1 so 20-5 is 15 take away 1 is 14).

If you are wanting to learn more about number strings Cathy Fosnot’s book, Young Mathematicians At Work...Constructing Number Sense, Addition, and Subtraction is a rich resource.

The above examples require prior learning of finger patterns. Finger patterns can be easily incorporated into one of the research based subitizing activities known as Dot Flash. I have engaged my Grade 3 students with finger patterns from the beginning of September, and will continue to the end of June. It is benefitting their understanding of teens-digit and multi-digit adding and subtracting, as well as with the multiplication and division basic facts.

I did the Dot Flash activity with my class on Friday, Feb. 3rd. First, we did the activity Dot Flash which I have explained in my subitizing blog. You can also find this activity in my professional capping paper I have written and published: The Benefits of Subitizing: Helping Early Childhood Educators.

After flashing the collection of dots 3 times, I went around the room letting each of my students whisper in my ear how many dots they saw. Then, I asked them to show me on their fingers how many dots they saw in the collection. The pictures below show some of the finger patterns shown by students.

I asked each of those students to tell the parts of 8 represented by their finger patterns. The picture below shows you how I recorded the parts.

Using the correct vocabulary is important so as I was recording the students’ thinking I said,

You have just decomposed(broke apart) 8 into different parts.

Have we decomposed 8 into all the possible parts?

Very quickly, I heard no and students began providing me with more answers. The picture below shows their thinking:

### Try it with your students

If you teach Gr. 4, 5 or 6 you can do this with numbers beyond 10.

What I do with my Grade 3 students come April, May, and June is I have them pair up or get into groups of 3 or 4 to show me the parts of a number greater than 10. They have a lot of fun with this and I have even been asked if they can take their socks off and use their toes. That we do not do, of course!

I will also be hosting a webinar to help you further understand the research-based activity known as Dot Flash on Feb. 22nd from 7:30 to 8:00 pm. I hope you can join me for this!

• ## Does Memorization Work?

My principal passed me a book called Visible Learning For Mathematics written by John Hattie, Douglas Fisher and Nancy Frey.

Apparently, John Hattie is coming to the forefront in mathematics learning. Is he the one mathematical guru to focus on?

Through my Masters studies and ongoing reading of various books and articles over the past few years, there are a few gurus who make reference to each other regularly. Presently, I have read approximately half of this book and it is very interesting and so true. If you are looking for professional learning I most definitely recommend this as an educational read.

His thoughts are similar to that of Cathy Fosnot's, in that we are putting students at a detriment by teaching algorithms without understanding. He refers to this understanding, the ability to explain the how and why when doing the work, as procedural fluency.

If you had the opportunity to view the video, Deep Understanding Versus Memorization (below), my colleague, Mary Schatz and I spoke of the importance of students developing a strong number sense before introducing the traditional algorithms. The Clarifications of the Alberta Curriculum that came out last spring indicated that teachers are to teach a variety of strategies and then each student chooses the strategy (or strategies) in which their understanding lies.

The question I pose is:

Are educators aware of the different strategies to teach and do they know the ways in which to teach these strategies for understanding?

Why I ask this is because of two recent scenarios: One that I experienced and another I heard from my colleague, Mary Schatz. As I said in the video, I was having a conversation with a grade two teacher back in October and she was already introducing the algorithm for adding to her grade 2 students. I teach grade 3, and I have not yet introduced the algorithm. I won’t until they have had the opportunity to learn other strategies that will help further develop their number sense.

The day after I interviewed Mary, she emailed me about a teacher expressing her frustration in helping her students learn the traditional algorithm. Her comment was that she was repeatedly showing them the steps again and again but they just couldn’t get it! They kept forgetting to carry the regrouping from the ones to the tens place value.  In conversing over email about this, Mary asked how I would provide this teacher help. My reply was that those students need further work to enhance their number sense (understanding that a digit in a given place value has a specific meaning or value, eg. 85 is 8 tens and 5 ones or 80+5, NOT 8+5 ) and until they do the steps of the algorithm can be demonstrated hundreds of times and they still won’t get it. Well,  perhaps they may memorize the steps eventually, but there won’t be any procedural fluency in which they can explain the how and why. Mary’s final response was that sometimes we need to go back before we can go forward. So very true!

Here are pictures of two strategies in which my grade 3 students have learned up to the present time in the school year. I also have included 2 short video clips.  It will allow you to hear a student explain their thinking using these two strategies.

### Try it with your students

I am unsure as to what grade you are teaching and where you are at, but take a moment to reflect and carefully observe as well as listen to your students. It doesn’t matter if you are simply working on the single digit basic facts or with multi-digit numbers.

Do your students have procedural fluency? Can they explain the how and the why? Take this back to your classroom and please share with us the outcome and/or ask questions for clarification.

• ## Recap to Recall Lessons and Link to the New

I had my students go at Prime Climb (Review Prime Climb Number Activity here) for the third round on Monday, January 23rd, 2017. It was absolutely fascinating to listen to their thinking since the 2nd round (Second round here). It is important to keep in mind that my students are engaged daily in activity and discussion to further deepen their understanding of number.

To start the lesson today I put up the colored circles from Dan Finkel:

I said to the students,

Let’s recap what we have learned about these circles previously in connection to our learning of number in our daily routine.

Here is what I heard:

• The circles with orange are numbers you can skip count by 2
• The circles with green are numbers you can skip count by 3
• The circles with blue are numbers you can skip count by 5
• The circles with purple are numbers you can skip count by 7
• The circles with red are prime numbers
• 2 and 4 are factors of 8 and that is why there is orange in those circles and 2x4 is 8

Why does the circle with the number 4 have 2 orange parts and why does the circle with the number 8 have 3 orange parts?

This took a little grappling which was a productive struggle. I referred my students to think about what they do daily with the school day number and divisibility. Then I had about 6 students who had made the connection. They said,

Because two 2’s is 4 (2 x 2 is 4) so that is why the circle with the number 4 has been shared in half. The circle with the number 8 has been shared into three orange parts because 2 parts represent 4 and 1 part represents 2 and 4 x 2 is 8.

One of my students put up his hand and asked,

3, 5 and 7 are also prime numbers so why aren’t they colored red?

After I asked him what he thought; the students grappled with this until one of my students replied,

The prime numbers begin with numbers greater than 10.

We also took the time to discuss why the circle with the number 1 was colored grey.

I put them with a partner and gave each one a piece of 11 by 17 paper with the numbers 20 to 40 on the bottom and space on the top to record their thinking.

They were given approximately 20 minutes and I wandered around listening to groups and asking questions when needed. It is important you know that not every one of my 26 students was fully grasping this.

For instance, I came across one pair who colored in 21 solid purple. I questioned their reasoning and they responded with, you can count by 7’s. I asked them to take out their hundreds chart and count by 7 to 21. Once they did this I asked, "So what can you tell me about this count?"

"Oh", was the reply. "You can count by 7’s three times."

"So", I asked, "Would the 21st circle be purple?"

Then they realized it needed to be green and purple.

Another example to share was a different pair who colored in 22 orange. I went through the same questioning for them to come to realize there were eleven groups of 2 in 22 so that circle needed to be red and orange.

As I engage with my students it allows me to gain insight into where they are at in their learning and this guides my teaching.

There were some students in my class rolling with this activity and demonstrating deep and transfer learning from our daily routine and various number concept activities we engage in regularly.

The pictures below show you this thinking.

The students love this activity and my plan and hope are to continue this until we reach 100.

Each time we do this activity, learning continues to grow and connections are being made.

I would love to purchase the Prime Climb game from Math4Love and it is now available to purchase and be shipped into Canada at Boardgame Bliss

### Try it with your students!

If you haven’t tried this yet with your students, give it a go - start here!

If you have, go for Round 2 and see how their thinking has developed.

For those of you in Kindergarten and Grade 1, you can work with the numbers 1 to 10. I am certain you’d be pleasantly surprised with the ability and reasoning our young students have. There is a productive struggle and it is perfectly ok for them to not know at first. Eventually, this surface level learning will lead to understanding.

## Wanted 65 -- A Change of Pace

My mathematics instruction is based on a daily routine which incorporates many number concepts. Although this routine is repetitive, I am mindful of my questioning and of the task which they are expected to engage in. Purposeful questioning as well as variety in the manner in which the tasks are approached will help support student learning and will undoubtedly push their thinking forward.

In my Grade 3 classroom, the daily routine tends to include the following number concepts:

• Place value (digit and value)
• Tally marks
• Even or odd
• Expanded notation
• Number word
• Comparing numbers
• Representing number differently using base ten blocks
• Writing number equations
• Money (value) and its connection to place value
• Money (value) represented in a hundreds grid and its connection to decimal numbers
• Divisibility which connects to skip counting, multiplication, division, understanding the patterns and relationships of number within a hundreds chart.

I engage students in so many various tasks to enhance their deep understanding of number concepts so the transfer of learning (connection) is made.

There are times throughout the year where students need a change of pace to show understanding of learning in a creative way. This is what the activity, WANTED 65 is about. I had the students complete this task the week before Christmas holidays when I know their brains just can’t absorb any more.

My students thoroughly enjoyed this activity and as you can see in the pictures, many of the concepts learned through our daily routine were included. I plan to do this activity again sometime between Day 90-100 as I would like to see further understanding in their learning.

I hope you take the time to have your students try this activity and then post a couple of examples onto my website.

Kindergarten and grade 1 students can write a little, but they can also show their understanding with manipulatives.

### Try it with your students

I can’t wait to hear how it goes and for you to ask me any questions that may come up as you work with your students. I will be helping anyone interested in extending their learning about subitizing through videos and webinars in the weeks to come. I also am aware of how precious the time of a teacher is and therefore, I will also have subitizing Smart Board files available for purchasing.

## Subitizing -- What is it? Why do we do it?

Many years ago, after attending a session at a math conference, I started flashing a collection of dots to my students. Although, they were able to tell me how many dots they saw altogether, I did not have a deep understanding of the power this activity had to enhance number sense for our young students. It was not until I was working to obtain my Masters that I unpacked my own mathematical understanding. I became so interested and intrigued that I chose subitizing as my final project to complete my Masters. Rather than just writing a paper, I wanted to provide teachers with a valuable resource so I wrote and published a professional resource for teachers titled, "The Benefits of Subitizing, Helping Early Childhood Educators". (click for more information)

After completion of my Masters, I led a group of teachers in my school through an AISI project. Our focus was to determine if subitizing daily in the early elementary years benefited students’ number sense identified in the research. We set up baselines and assessed throughout the year.

Our findings verified what the research states: if a teacher takes 5 minutes a day to engage their students in a subitizing activity, number sense is vastly enhanced. Therefore, it improves children's mathematical success.

The teacher group became so intrigued that we continued on with the project for another year. This project impacted our school enough to put money into a numeracy support project in which educational assistants worked with those students who were targeted as having challenges. The success rate was phenomenal, but unfortunately when budgets were cut the project had to be let go.

At every chance, whether it is with individual teachers or when I present to teachers at a conference or at a Professional Development Day, I incorporate  learning time for subitizing. No matter if I am working with teachers or students, they are completely engaged. My students love it and if we have to miss it they are not happy campers.

## What is subitizing?

For those of you unsure of what subitizing is, it is recognizing a collection of dots at a glance. It is also referred to as part-part-whole because within the larger collection are smaller collections that make the whole.

The research states that the collections be made from homogeneous dots, not fancy pictures of dogs, cats, etc. They do not state the reasons why but say it puts students’ learning at a detriment. There is also a sequence in which to organize and present the collections to our students.

We must first begin with arrays (matrixes), then move to linear arrangements, then dice arrangements and lastly scrambled arrangements. Research also states we could have some dots different colors to help children see those smaller collections within the larger collection.

Once you have a collection of dots created, either on a paper plate or on the Smart Board, you give the students 3 flashes… and it is a flash. If you show it for too long the students will do a ‘count all’ rather than visualizing and recognizing those arrangements at a glance.

The first flash is to VISUALIZE. WHAT DOES THAT ARRANGEMENT LOOK LIKE?

The second flash is to LOOK FOR SMALL GROUPS.

The third flash is to QUANTIFY (which means to add and find the sum or the total).

1. Visualize
2. Look for small groups
3. Quantify

There are no more than 3 flashes (this is research based), so I tell my students if they didn’t see the whole collection to quantify what they did see and then when they see the collection they can determine where they made an error.

DO NOT SHOW THE COLLECTION UNTIL STUDENTS HAVE SHARED THEIR TOTAL.

Also, I do not show the collection until I have 2-4 students share how they saw the PARTS OF THE WHOLE within the 3 flashes.

If I show the collection then students start making up other ways in which they saw it. It is important they share their thinking on what was seen on the flash.

For example, for the collection of dots below, one of my students saw the sum of 10 as the parts,  4, 2, 1 and 3. This student quantified these parts as 4 and 2 is 6. Add 1 more to 6 is 7 and I know 7 and 3 make a friendly 10.

The above example gives insight into  the researched based activity called Dot Flash.

The three pictures below show another researched based activity called Reproduce. It follows the same 3 steps as Dot Flash but the students reproduce the collection of dots.

They can’t start reproducing until after the second flash (this is so important because the first step of visualization is pertinent) and they must reproduce the collection exactly as they see it. I then extend the reproducing activity to help with the understanding of multiplication.

As I see understanding grow deeper I extend this learning of multiplication to using grid paper to show these arrays and we move into division. Subitizing lends itself to the learning of almost the entire primary curriculum. It is quite amazing and the best thing is that the students love it!

Subitizing can also be done with a collection of coins(money) which leads into a deeper understanding of fractions and decimals. This comes from one of the Cathy Fosnot books I read 3 years ago. It is fascinating and the concepts that can be learned from this is incredible…For example:

For money, the coins are always set up as an array/matrix and the coins must all be the same. (eg. all quarters, all dimes, etc)

You do this like the dot flash activity: The students get 3 flashes. The first flash is to visualize, the second flash is to look for small groups and the third flash is to quantify. However, they are to quantify two things, the number of coins and the value of the collection of coins.

After the 3 flashes, I ask the students to share their thinking before putting down the screen because it is important they share what they saw on the flashes.

Following this discussion was the connection of their understanding of fractions we have learned from our daily routine as well as our number line work.

## Remember: All learning is CONNECTED and REVISITED Daily.

As you can see from the picture, I had a student come up to show his thinking of how he quantified the value of the coins as the fraction of one-half and that each one-half of the whole is 50 cents and two halves is the whole and has the value of \$1.00.

Another student said she saw the four quarters and counted 25, 50, 75, 100. I asked her to circle each count as a part of the whole and write the fraction and value for each count. You see this displayed in the next picture below. We counted each part as one-fourth, two-fourths, three-fourths, four-fourths and then I asked so if ¼ of the whole is 25 cents, what value is ¾ of the whole.

The next picture shows a student who circled ¾ of the whole and identified the value as 75 cents.

Another student piped up and said, "I know another way to write ¼ of the whole." The next picture shows that this student circled ¼ and wrote 25/100.

This then led me to introduce equivalent fractions. The next picture shows our discussion. I asked if 25/100 is the same as ¼ and I wrote this down. Then I wrote ½ and asked if there was another fraction to mean/show the same value and many of the students could tell me 50/100. Lastly, I wrote down ¾ and asked if there is another fraction to mean/show the same value and again I heard 75/100. Very briefly, I talked about these fractions showing equivalence.

As I have mentioned previously, subitizing can impact almost the entire primary curriculum and impact the later elementary years. The simple flashing of a collection of dots and/ or money can vastly enhance our students' number sense. The best part of this is that students love it and I hear it if we have to miss subitizing in our daily routine.

I have included a free Smart Notebook file with 10 slides so you can give it a try with your students. Click here to download the file: Subitizing Freebie (please note that you need Smart Notebook software to open this file. Contact me if you have questions.)

### Try it with your students

I can’t wait to hear how it goes and for you to ask me any questions that may come up as you work with your students. I will be helping anyone interested in extending their learning about subitizing through videos and webinars in the weeks to come. I also am aware of how precious the time of a teacher is and therefore, I will also have subitizing Smart Board files available for purchasing.

## Tiny Polka Dot Cards!

I also wanted to make you aware that Dan Finkel (MathforLove) has a new material for sale called Tiny Polka Dot Cards. I had told one of the kindergarten teachers at our school. When she checked it out on Amazon, you could only get it shipped within the States. However, there is a printable version for a cost of \$5.00.  She has shared this with me and looks it absolutely awesome! You would have to copy onto thicker paper and laminate but it would be worth it! Here is the link: Tiny Polka Dot Print and Play

• ## Building mental mathematics

Using the hundreds chart in your classroom helps build number sense. The hundreds chart, if understood, helps children develop the understanding of a relationship of numbers which helps with skip counting, factors, adding, subtracting, multiplying, dividing, even and odd, fractions, etc.

I have my grade 3 students use their  hundreds chart from Day 1 of school. They have a hundreds chart in their math duotang and then when they have marked up one they recycle it and get another. From Day 1 we use the hundreds chart to see what factors there are for every school day. From this we can see patterns and relationships between numbers.

For example:

• They become aware that 2 and 5 are factors of 10
• When determining how many twos in 44, they know that there are 5 twos in 10 so for every 10 they count by 5’s and at 40 that is 20  two’s
• From there they count on 21 groups of 2 at 42 and 22 groups of 2 at 44

Many now have gained the understanding of the factors for the multiples of 10 which helps find factors more efficiently.

For example:

• On Day 75 of school (January 11th, 2017),  many students could tell me they know there are 10 fives in 50
• From 50 to 75 is 25 so that is another 5 groups of 5
• There are 15 groups of 5 or 5 groups of 15 in 75

Through our daily work students have made the connection that when you move to the right within a row numbers increase by 1 and if you move to the left within a row numbers decrease by 1. To add, they understand that if you move down within a column numbers increase by 10 and if you move upwards within a column numbers decrease by 10. With this understanding we can add and subtract using a hundreds chart.

Following many different activities using a hundreds chart, we played a game using a die and a hundreds chart. I gave each student a hundreds chart and I had the die. I modelled how to play with my educational assistant and she showed and explained her thinking on the interactive hundreds chart on my Smartboard file.

The rules are:

1. You will have 10 rolls of the die and you must use all 10 rolls.
2. You can decide if the number that lands face up on the die when rolled will be given the value of ones or the value of tens.
3. The goal is to reach as close to 100 as possible or 100.
4. The work will be recorded within a Tens and Ones games chart (see the example below).

Our  score was -5 from 100 in the example below:

 Tens Ones 1 (first roll player chose 1 as 10) 6 (2nd roll, 6 ones was chosen so total is 16) 3 (total is 46) 1 (total is 56) 5  (total is 61) 1 (total is 71) 6  (total is 77) 3 (total is 80) 5 (total is 85) 1 Total is 95

I told the students they need to strategize throughout the 10 rolls, knowing if to choose the value to be ones or if to choose the value to be tens.

After the game they were asked to reflect on their decisions to determine if they would have strategized differently at a roll or rolls to reach a score of 100. If we reached a score greater than 100 before or on the 10th roll then we busted and they need to reflect on this as well. This builds mental mathematics as well as allows the students the opportunity to use the 100’s chart.

After we played together, they played independently:

Due to the students having challenges keeping the rows aligned I created a game sheet and have provided this for your classroom use.

### Try it with your students

If your students are in Kindergarten you can modify:

• Give each student a number line 1 to 10, a die and some sort of counters.
• Tell them they have 3 rolls.
• After each roll, they need to show the count of the die using the counters.
• Once they have completed the third roll they need to strategize to determine if the sum of all three counts is 10, less than 10 or greater than 10.
• They are not allowed to put the counters along the number line to see.
• What they then have to do is determine if they will choose the sum of  2 rolls or all 3 rolls for a sum of 10 or as close to 10 as possible.
• When this is done, they can lay the counters down along the number line to see if their thinking/reasoning was correct

If your students are in Gr. 1 perhaps you can use half of the hundreds grid, 1 to 50.

If your students are in Gr. 4, 5 or 6 you can use a hundreds chart to 200, 300 or 400 and they multiply the numbers rolled on the die.

Think about the modifications and/or variations that can be made to meet the needs of your students.

• ## Extending Students' Understanding

I revisited the colored circles 1 to 20 on Thursday, Dec. 1st. It is most definitely an activity in which rigor is evident, and as teachers , it is crucial we engage our students in activities that require rigor.

Review Prime Climb Number Activity here

I paired my students and gave each group a hundreds chart to work with. The colored circles were displayed on the Smartboard. The instructions were to color in the numbers 1 to 20 and to review with their partners what patterns and relationships they have identified in the first round, or perhaps they are recognizing new patterns and relationships this second round. As I saw students approaching the number 20 I stopped the class and asked them to use what knowledge they have gained for numbers 1 to 20 and use this understanding to color in the numbers 21 to 30.

As they were working I walked around the classroom listening and talking with the students. What I noticed the most, was that I had many of my students able to identify factors for numbers 21 to 30 but weren’t quite sure how to show the colors connected with the factors. For example for 21 I had 3 or 4 pairs of students color in 3 purple circles. When I asked why, the groups said because you can count by 7 three times. Whereas, they should have colored in a green circle and a purple for 21 to show 3 x 7.

Due to a lack of time, we were only able to discuss their learning for numbers 21 to 24. If you look at the hundreds charts (see pictures above), all students were able to color the number 22 orange to show groups of 2, but there were only a few groups who colored in red to show 11 groups of 2. All my students identified 23 as a prime number and colored it red. Again, for 24, many groups identified 24 as an even number so had colored in an orange circle. There were a couple of groups who colored in green and orange and when asked why, said they knew two, 12’s make 24 so the 12th circle was green and orange. This, I thought, was a deep meaningful connection. I will add, however, they were unsure how many orange circles to color in because 2 is orange and 4 is two orange because 2 x 2 is 4. I did try to extend their thinking by asking is 4 groups of 3 twenty-four, because nobody made the connection that 3 X 4 is not 24 and that 8 groups of 3 is 24. We skip counted by 3’s to 24 to see the 8 groups. I asked so how would we color in the 24th circle? I heard a lot of ‘hmm’s” from my students…so that means we have more work to do with these circles.

They were very engaged for the 20 minutes that we had spent on this and seem determined to figure it out. I collected their hundreds charts and this activity will be continued in the next 2 weeks. I shall keep you posted on my findings.

I am hoping our daily routine with finding factors for our school day number will continue to develop number sense which will benefit students to further make meaningful connections to the colored circles.