Fraction Video Tutorials Available Now!! Dismiss

• ## A Challenge for Older Students

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
• Prime Climb Round 4: here

These are lessons inspired by Dan Finkel - Math for Love

I was first introduced to Dan Finkel and Prime Climb at a staff meeting one year ago through the video Five Principles of Extraordinary Math Teaching:

I was very curious what my grade three students could tell me about the color circles because through our daily math routine we learn skip counting, factors, multiples and divisibility connected to our school day number.

Here are the colored circles I put up for my Grade 3 students to explore.

My students engaged in this activity about 6 times within a 6 month period and I wrote 4 blogs to share the learning connected to the exploration time. If you haven’t read my blogs, see the links at the top of this post.

Many of my students found the first round quite challenging. There was definitely a productive struggle, but NOBODY was frustrated! When I put up the colored circles for round 2 many cheered and commented how much fun they were having. Our goal was to color in numbers up to 50 by the end of June, however, time ran out. I can tell you they were very disappointed!

The intent of this blog post is to share the learning of  Grade 4, 5 and 6 students who I’ve engaged in Prime Climb through recent subbing experiences.  I am fortunate that teachers who I sub for give me the freedom to plan the math.

Prior to subbing in a Grade 4 classroom I was told the focus of their learning was multiplication and division so without hesitation, I chose Prime Climb. For students to be efficient with these operations they need to have a conceptual understanding of factors, multiples and divisibility of numbers.

I put the colored circles up 1 to 20 and asked the students to look carefully at the circles, numbers and colors.

Here is what the students shared:

• All even numbers are orange
• Some circles are fractions-half, thirds

I did ask why certain circles were shared as fractions but they weren’t able to explain and that is okay. This would be something they would need to make the connections to through further exploration.

• 7 and 14 are purple cause you can count by 7 to get to 14 and 14 is also orange because it is even
• 5 is blue and 15 is ½ blue and ½ green because 3 x 5 = 15
• 3, 6, 9, 12, 15, 18 all have green because you are counting by 3
• Numbers you can count by 5 have blue
• If you double each number 2 to 9 those numbers have the same colors. Eg. double 2 is 4 so the circle with 4 also has orange, if you double 6 it is 12 so 12 has green, etc
• 11, 13, 15, 17 are red because there are not any numbers (2, 3, 4, 5, 6, 7, 8 or 9) you can count by to get to them
• Numbers you can count by 6 to are also green because you can count by 3

I didn’t ask but I should have asked if all numbers divisible by 6 can be counted to by 3? This would have been a great opportunity to see if a conjecture could have been formulated.

I asked why 1 would be gray and nobody was able to explain this to me.

After this discussion, I provided each student with the new blank color circle chart that Dan Finkel has provided which is awesome!

Prime Climb Color Chart

I left the sheets and an explanation of the activity for the teacher hoping she will learn more about Prime Climb for the benefit of students to move into a ‘deep’ phase learning.

Further opportunity for exploration would enhance their learning and understanding of multiplication and division as well as benefit the understanding of fractions.

A few days later, I taught in a Gr. 5 / 6 class as well as a Grade 5 class. Here is their thinking:

• The colors are based on multiples. Multiples of 2 are orange, multiples of 3 are green, multiples of 5 are blue, multiples of 7 are purple
• 12 is both orange and green because 2 x 6 = 12, 12 is also green because it is divisible by 3
• 12 is shared into thirds because 4 has two orange parts and 3 is one whole green and 3 x 4 = 12
• 7 is full purple but 14 is half purple and half orange because 2 x 7 = 14
• Multiples of 10 have blue and orange because they are divisible by 2 and 5
• 5 x 3 = 15 so 15 is both blue and green
• 1 is clear (white, gray) and all circles have some white because you can count by 1 to all numbers
• 2 is full orange, double 2 is 4 so circle has two parts, double 4 is 8 and 8 has 3 parts and double 8 is 16 and 16 has 4 parts. So as you double the number the circle has one more part. It goes whole, ½, ⅓, ¼.
• Red are prime numbers
• 10 has two parts cause 5 and 2 are factors and 20 has 3 parts (2 orange and 1 blue) because the two orange represent 4 and the one blue represents 5 and 5 x 4 = 20

Some very interesting thinking was happening. A productive struggle was going on which I think is critical and awesome. A few of the Grade 5 students were asking for ‘answers’ but my reply was, “with further exploration time you will come up with the answers (and conjectures) yourself.”

The educational assistant in the classroom asked the students if there was any pattern or reason for how the circle was colored. For example, if the top half was orange and the bottom half blue could these colors be reversed? Something to ponder.

The teachers of the two classes are going to continue this exploration and purchase the game so I am excited to hear (and hopefully get to see) how the learning grows.

Below you will find a few pictures of the students’ work. I found it interesting how many of the upper elementary students chose to color the numbers out of sequence whereas my Grade 3 students worked in sequence.

The pictures demonstrate the students' recognition of patterns (eg. all even numbers needed orange, all numbers divisible by 5 needed blue). As students were coloring I was moving around the class asking questions to see if any conjectures were being formulated. I didn’t hear any, but deep thinking was happening and I believe if these students are provided further opportunity to explore and continue to color in the chart conjectures will come.

One of the grade 6 boys said to his partner, "Let’s color all numbers divisible by 11 red because 11 is red."

I was thrilled to hear the response of his partner, “No, we can’t do that because 22, 44, 66, 88 are even numbers and 55 is divisible by 5."

Through my classroom experiences, Prime Climb engages students and allows them the opportunity to explore. The learning that emerges is deep, meaningful and rich which is of benefit to understanding number. In turn, this deeper understanding of number will strengthen their conceptual understanding of skip counting, multiples, factors, divisibility, and the operations of whole numbers as well as fractional numbers.

I recommend that you purchase this game for your class or for home, to play as a family. If you can’t find it at a Chapters near you then you can purchase it using this link:

http://primeclimbgame.com/

If you have purchased this game, or choose to purchase this game,  and/or engage your students just using the circles you can retrieve here (link), please comment to this blog and share with us what learning is happening.

I would love to hear from you and I most definitely know Dan Finkel himself would love to hear any feedback!

# CBE's \$2M Move to Hire Math Coaches and Specialists

I am writing this in response to the CBC news article regarding Calgary Board of Education putting \$2 million dollars into hiring 25 math coaches. (October 11 http://www.cbc.ca/news/canada/calgary/cbe-math-coaches-calgary-schools-tests-teachers-specialists-1.4349500)

I will say good on you CBE! There has been talk for a few years regarding the mathematical crisis in Alberta. I will emphasize TALK because it seems very little action has been/is being taken to help our teachers and students.

The need for numeracy coaches/specialists have been in the forefront of the latest research.

Literacy seems to always take the front seat and Numeracy the back seat and I am just baffled with this as we want our students to be both numerate and literate.

The focus of this article is to put the focus on Grade 6 and 9 to improve test scores. These struggles and challenges are not quick fixes nor is the negative attitude towards mathematics going to be a quick and easy fix when most likely these students in Grade 6 and 9 have been ‘hating’ math since elementary school.

I have been a teacher for 25 years and I have obtained my Masters in elementary math from the University of Alberta in 2012. I have taken a leave of absence Sept. to Dec. to pursue my passion. That passion is to help change the much needed way in ‘how’ we teach mathematics starting in the primary grades.

With this said, I am hoping these 25 coaches are not hired as a bandaid solution which will be short term with the focus being ONLY on test scores These Grade 6 and 9 students are hating math and struggling because they do not have the foundation of number sense which is key to understanding mathematics. To add, these students have probably been sitting at desks learning math as memorized rules and procedures and then working through textbooks and worksheets to practice.

Students at all levels need to be engaged in active, rich, and meaningful tasks that allow for conceptual understanding and procedural fluency. To add, they need to be in classroom environment where they can see the beauty and creativity in mathematics. If all this is in place beginning in the primary grades our students will not only be highly successful in mathematics by Grade 6, they will ACTUALLY LOVE MATH!

No bandaids or quick fixes! So CBE, take the lead and invest this money wisely to change the culture of how mathematics is taught. I can guarantee you if you put coaches into your elementary grades, you will not need to fix problems at the Grade 6 and 9 level!

Yours in education,

Darlene Kusick

What are your thoughts about this move that CBE is taking? Do you feel that these numeracy coaches will make a larger impact at upper elementary versus lower elementary? Do you feel a coach would help at your school for your students and teachers?

# What Does A “High” Mathematics Student Look Like and Does This Mean They Have a Growth Mindset about Mathematics?

• ## How do you know they are ‘high’ in their mathematics achievement/learning?

When talking with teachers I often hear about their ‘high’ students in mathematics.

In conversation with these teachers I ask, how do you know they are ‘high’ in their mathematics achievement/learning? The majority of responses I get is that these students scored well on a paper and pencil assessment or did well on a worksheet or questions from a textbook.  When I ask to see these assessments (or this work) often they are simply asking for answers ONLY, not an understanding of how they got to the answer.  Another common response is that these ‘high’ achievers can complete number work with speed.

Here are questions I would like to ask:

1. Does getting the right answers mean that students have made and/or are making meaningful connections?
2. Does getting the right answers mean students are able to show and/or explain their thinking of how they got to this answer?
3. Does getting the right answers mean students have a growth mindset towards mathematics and are able to look at mistakes (if made) as opportunities for their brain to grow and make meaningful connections?
4. Is speed in completing a number task(s) mean students have conceptual understanding and procedural fluency?

Both Jo Boaler in her book, Mathematical Mindsets, and Mike Flynn in his book,  Beyond Answers address this. These two books are absolutely excellent educational materials to read.

Jo Boaler comments in her book that there seems to be a societal belief that those who can calculate number work quickly are both a ‘true’ and ‘smart’ math person.  She also talks about her connections with MANY mathematicians and she defines these individuals as not fast math thinkers, but rather as slow, careful and deep thinkers.

Within her book she also talks about students who have been called ‘smart’ in math tend to dislike math and more than often, do not continue with math after high school. These ‘smart’ kids tend to have a fixed mindset and are usually crushed when their answers are not right.

With this said, it is of the utmost importance to change our misconceptions that math is a subject that is about calculating numbers fast, following rules and procedures learned and memorized through traditional methods of teaching.

Let me turn your attention to Mike Flynn’s viewpoint. I was first introduced to his work through a free webinar a colleague had sent to me back in the Spring. After watching this webinar I ordered his book, Beyond Answers.

In the webinar and in his book as well, he speaks to the fact that teachers of primary students need to help young children ‘contextualize’ problems. What this means is that if we are to write a mathematical equation, eg. 3 + 2 = 5, students need to be able to make connections to these symbols through storytelling. A story for this equation, might be, there were 2 birds on a branch. 3 more came to join them and together there are 5 birds on the branch.

When I work with my grade three students I always have them tell me stories for all the operations, addition, subtraction, multiplication and division. I will share a story with you about my Gr. 4 colleague who shared the webinar with me. After watching the webinar she went back to her classroom and gave a basic fact multiplication equation to her students and asked them to give her a story.

She was in disbelief that many could not do this, yet they could provide the correct answer. This, undoubtedly, demonstrated to my colleague. that a deep understanding of the process of multiplication was not understood. If students are challenged with understanding multiplication, then division will also be a challenge, as well as fractions, factors, divisibility.

If students are challenged with understanding multiplication, then division will also be a challenge, as well as fractions, factors, divisibility.

If you are one who is doing timed basic facts drill and practice and/or teaching the steps of the traditional algorithms, do your students have a deep understanding of the operations or are they simply ‘fast’ at calculating and/or have memorized the facts and/or the steps of the algorithms?

I will also share another story with you that I recently experienced in a Grade 5/6 classroom. I asked the students how they knew 15 was divisible by 5. Many were unable to explain this to me. I kept pressing for understanding through guiding questions and finally one student responded with 15 divided by 5 is 3, three 5’s make 15.

I then asked them if 5 and 3 are factors of 15 and if there were any other factors. One of the students told me 7 and 8 so I asked the class, if I times/multiply 7 and 8 will that leave me with a product of 15?

As I looked at the students I could tell there was much confusion. So I asked what are factors and what happens if we multiply two factors, such as 3 and 5? Finally a student was able to tell me that 1 and 15 were also factors of 15.

Perhaps you are not seeing my connection to Mike Flynn’s view with ‘contextualization’. These grade 5/6 students were not contextualizing the equation of 15 divided by 3 = 5.

With all this to contemplate, I ask you to reflect on your assessment of your students’ mathematical ability with the four questions I put forth at the beginning of this blog.  I would love to hear your thoughts, comments and questions with regards to this.

Let’s engage in a conversation! This, I believe will help you with your own mindset around the teaching and learning of mathematics. And if teachers have a growth mindset and understand the need to change the way mathematics is taught, we can most definitely help ALL children make connections, think logically, interpret data, use space and work flexibly and creatively with number.

# Why Incorporate Money and Thermometer into our Mathematics Lessons

• ## Examples demonstrating the need to teach money and thermometer

Despite the fact that the learning of money is minimal in the Alberta curriculum; and thermometer(temperature) completely removed, I strongly believe it is of the utmost importance to add it to our students’ math learning at ALL grade levels.

I have included money and temperature into my Grade 3  math lessons on a daily basis. And because of it, the development of number sense and connection to other outcomes has been incredible!

When writing my Kindergarten through Gr. 6 Math For Success teacher guide books I didn’t hesitate to incorporate both money and thermometer (temperature). I am most definitely including it into my Gr. 7 book, which has been in pilot mode.  Math For Success Guidebooks

I often have been questioned through the years, why I have included money and temperature into the teacher guide books I have written if it is not in curriculum? I have also heard from many teachers since it is not in curriculum they will not incorporate it into their students’ math learning.

I will put forth the following questions:

• Is money not number?
• Is money not connected to place value, operations,  fractions, decimals, percentages?
• Is it not an important life skill that is needed for daily life and decision making?
• Is thermometer not a visual representation of a number line that is vertical rather than horizontal?
• Is it also not part of our daily life with regards to weather and decisions that would need to be made in relation to temperature.

I think sometimes, teachers become too focused on the ‘what’ that is to be taught rather than the ‘how’. I also think that teachers are looking at mathematics as parts/pieces that are taught in an assembly line manner rather than seeing it as an interconnected entity.

BOTH money and thermometer are not only visual representations to develop numerical knowledge; but also, in my opinion are mathematical tools to help students develop number sense; and from that many outcomes are learned and meaningful connections made.

# Money

FREQUENT QUESTION ASKED REGARDING THE PENNY:

If the penny has been discontinued why are you including it?

It is connected to place value and place value in mathematics is the foundation in helping develop number sense.

# How it is Used in Teaching

• Skip counting by 1’s (pennies), 5’s (nickels), 10’s (dimes), 25’s (quarters)- connect it to the hundreds chart or a number line and that will help with the understanding of addition, factors, multiplication, divisibility
• From skip counting will come the understanding of multiples because if you can count by 1’s and 10’s, connection to counting by 100 (loonies), 1000 (10 dollar bill) is made
• Connection to place value which is the foundation of developing number sense
• Decimal numbers
• Fractional numbers - proper, improper and mixed
• Percentages
• Pattern

# Real Life Stories Relating the Importance of Incorporating Money

## Example 1

At the beginning of July I went to get a coffee from McDonalds. The lady who served me was close to 40 years of age, if not older. I ordered a large coffee which is \$1.75. I handed her a \$5.00 bill but for some reason she punched into the register \$20.00. Of course it showed to give me back \$18.25.

I told her I gave her a five, not a 20. She became nervous and I tried to explain to her how much change she needs to give me. She fiddled with the money and proceeded to hand me \$15.75 cents. |I was confused as to how she was trying to calculate this. I told her again, I gave her five dollars and she needs to give me \$3.25 back because it is 25 cents to 2 dollars (175 + 25 is 200) and 2 plus 3 is 5. (200 + 300 is 500)

She became absolutely rattled and said to me, I will just believe what you tell me and she handed me the \$3.25.

I’VE HAD MANY OF THESE TYPES OF EXPERIENCES OVER THE PAST FEW YEARS, AS I AM SURE YOU HAVE AS WELL!

Now, let me give you another story with a very different outcome. It was middle of June and our Gr. 4 students were selling cotton candy as a fund raiser. A Gr. 3 student in my room ordered 2 bags and the cost was \$3.50. My educational assistant was standing near the children when the exchange was taking place. My young student (age 8) handed the gr. 4 girls a \$5.00 bill. The change given back was less than it should have been. She was able to tell them, without hesitation, as well as with confidence, that what they gave back was incorrect. She also explained to them what they owe her and why.

My educational assistant came to me right away to tell me what had happened and it demonstrated for me the importance of learning money and how it is connected to developing overall number sense!

## Example 2

I went to a store at the beginning of September and there was a 10% discount on all products. The clerk was in her mid to late 60’s and she is a very intelligent woman who was a school teacher years ago and has ran a business for 25-30 years.

My total came to \$117.50 and she figured something out on paper and asked me if 10% of \$117.50 was \$7.50? We were so busy chatting about a variety of things that my mind was not focused on what she was calculating; and I have dealt with this business for years, so of course I trusted her.

I left the store and on my way home I started to think, how could that be? 10% of \$100.00 is \$10.00 and 10% of \$10.00 is \$1.00 and 10% of \$7.50 is 75 cents, so therefore, 10% of \$117.50 is \$11.75. Or 10% of 11 750 is 1175 but written as a money notation and connecting to place value that would be \$11.75.

I turned around and went back and explained my thinking and was reimbursed the shortfall of \$4.25.

I am confident that some of my grade 3 students by the end of the year would have been more than capable of figuring this out because of the work we engaged in with money on a daily basis through the math routine in my Math For Success teacher guide books.

# Real Life Situations Involving the Need to Understand Money

• Young children earn money for chores/allowances and they may be saving to buy something. Therefore, they would be able to figure out how many chores and the length of time it would take them to earn the amount they need. (or perhaps they are collecting can/bottles to take to the Bottle Depot and keeping track to a specific value)
• Young children may be given money for a birthday gift or allowance and want to go shopping. They need to know if they have enough money to purchase an item(s) of choice?
• Children as they grow into young adults need to learn about budgeting their money/earnings.
• When borrowing money there are interest rates on lines of credit, visa cards, mortgages, loans
• If there are sales at stores they need to determine if it is a savings and what is the better buy (eg. two for \$5.00 or one for \$2.75)
• If there is a discount on sales items and/or extra discounts on already discounted items, is it really a savings and if so how much?
• They are setting up a lemonade stand to earn a specific amount of money and figuring out that if they sold an x amount of cups of lemonade they would make an x amount of dollars
• If wanting to make investments understanding of money is needed
• Making wise and solid economical decisions in life
• If wanting to start and/or run a business, figuring out prices so materials/products/wages can be paid, but also a profit can be earned
• Job decision (Eg. Do I take this job that will pay x amount per hour and work 4 hours daily or do I take this other job that will pay x amount per hour but I work 8 hours 3 times weekly?)

I WILL STOP THERE BUT THE LIST IS ENDLESS AND IT APPLIES TO INDIVIDUALS AT EVERY AGE!!

# WHAT ELEMENTARY KIDS SAY ABOUT LEARNING MONEY

I asked if I could buy a \$6 loaf of bread with \$5 and we said no! I asked why they said you don't have enough money, so the discussion went to value of money:

• We need to know the value
• It is important to know the value of money

#### Karen Bouliane - Gr. 1 Class

I always ask my kids how important is \$ to them...?  They always respond it's very important to them because of all the things they can buy with it.  I then ask then isn't it important to count it and know how much change you should get so you aren't short changed?

Connections, connections connections...!!!!!!!

#### Mary Schatz - Grade 4 Teacher

• So we can get food for our body to make us live(survive) and be alive longer
• Be a banker
• Buy food for people who are too old to go the the store
• Make money when people shop
• To buy stuff to grow

#### Diane Waldie - Kindergarten Class

• You need to know how to pay for stuff
• You have to earn money to buy things you need (house, clothes, car)
• To buy water, groceries and important things to stay healthy
• It shows you how to use money

#### Vanessa Hardy - Gr. 1 Class

• You need to buy stuff, so you need to know how much it cost.
• You need to know how much money you have before you can buy something
• You are learning about the value of money (money and coins)
• When you are grown up, you need to know how to pay your bills.
• You need to understand the money to avoid making mistakes when paying for things and getting change.

#### Elena Reynolds - Gr. 2/3 Class

• When we get older we will have to do taxes.
• When we want to buy something we need to know we have enough money.
• So we don’t get ripped off.
• So we do not take long when paying.
• Budgeting
• To save
• For working at a store (power off, tips)
• So we do not go broke
• Invest
• You may not have an adult with you
• To be independent

# How it is Used in Teaching

• Vertical number line
• Sequencing of number
• Magnitude of number
• Intervals
• Difference between (subtraction)
• Negative and positive integers

# Real Life Situations Involving the Need to Understand Thermometer

• How to dress for the weather
• Would you run through the water sprinkler outside in a temperature of -5 degrees celsius?
• Boiling and freezing temperatures for the purpose of cooking, building (certain materials contract and expand as temperatures drop and rise)  pouring concrete, putting out bedding plants, gardening, hatching baby chickens

I HOPE THESE REAL LIFE EXAMPLES CLEARLY DEMONSTRATE THE IMPORTANCE OF WHY TO INCORPORATE THERMOMETER

• Ask your students why it is important for them to learn money and share the responses with us.
• Incorporate the learning of money into your classroom and share the learning that came from it?

# Numeracy Specialists and Numeracy Coaches: Does this Impact Teacher and Student Learning?

• ## Should districts have a Numeracy Specialist hired at the District level?

This is a very short blog asking you the importance of having numeracy specialists at the District level and Numeracy coaches at the elementary and middle school level.

I have read several articles within the NCTM Children’s Mathematical Magazines over the past two years that speak to the importance of numeracy coaches at the lower and upper elementary levels. Being elementary teachers means we are expected to teach ALL subject areas.

This, without a doubt, requires a lot of work and time to fully understand the front matter as well as the content to be taught for every subject. Not only do teachers have to learn the ‘WHAT’ to teach, they also need to be engaged in  a variety of professional learning opportunities to assist them in learning ‘HOW’ to help students learn and engage in meaningful and enjoyable lessons.

I recently read an article why Quebec has the highest math scores within Canada and one of the reasons is that they have math specialists teaching math from Gr. 7 through to Gr. 12. Another reason is that continual professional learning opportunities for teachers at every grade level is provided;  and their teachers are ‘TROOPERS’ who are willing to learn and grow for the benefit of their students’ learning in mathematics.

Jeannine Ellis, a teacher and  principal at Iron River School in Bonneyville, AB, sent me her Masters capping paper and in it she stated that it is because of having a Numeracy Specialist hired at the District level, who is solely responsible for working with schools to improve mathematics learning, their mathematics instruction has and continues to improve.

Having a Numeracy Specialist hired at the District level, who is solely responsible for working with schools to improve mathematics learning, their mathematics instruction has and continues to improve.

I know for myself, as well, in the past, I was provided time in my school to work as a math lead with my colleagues. There are so many ways in which a math lead/coach can provide help:

• 1 to 1 mentorship
• Classroom modelling
• Team teaching
• Observation of me teaching
• Small group sessions.

This was so successful that 2 years ago it grew to almost half the staff (14 teachers) wanting to be a part of this mentorship opportunity. Unfortunately, this practice did not continue and professional learning opportunities in mathematics became minimal.

Without me continuing as a math lead within my school, I believe it affected the mindset of our teachers, which in turn, impacted instructional methods. What tends to happen, is teachers fall back into old patterns of teaching because it is what they know and how they were taught.

In the district in which I teach, there isn't a numeracy specialist hired at the district level, but rather curriculum leaders who are responsible for all subject areas. Is it possible for someone to be highly skilled and highly knowledgeable in all subject areas for various grade levels?

I would love to hear from you about this topic. Let’s start a discussion with the following questions:

• Where do you teach?
• Is there a numeracy specialist at the District level? If so has it made a positive impact?
• Is there a numeracy coach in your school? If so has it made a positive impact?
• Do we need specialists either teaching students right from an early grade or what grade should there be specialists?
• Are your teachers/colleagues engaged in various professional learning opportunities in the area of mathematics?
• How frequently is your staff engaged in learning opportunities for mathematics?

• ## Do we want our students and children to learn to LOVE mathematics and to see the beauty and creativity in what they are doing?

In 2007 a new mathematics curriculum was mandated. The implementation from K-12 was staggered. The main goal of this change was to reduce content at the elementary level with the intent that teachers have time to delve deeply into the number concepts which empower our young students to develop number sense and gain conceptual understanding and procedural fluency.

Students were to engage in their learning and to think creatively and out of the box. Teachers’ instruction was to enhance their students’ ability to explain and show their thinking using multiple representations. This, of course, was much different than the traditional delivery of instruction where teachers taught and students performed by mimicking rules and/or following memorized procedures and steps.

This, of course, was much different than the traditional delivery of instruction where teachers taught and students performed by mimicking rules and/or following memorized procedures and steps.

Within a couple of years of the implementation of the elementary curriculum, there were rumblings from many Alberta parents. The perception of this ‘new math’ was referred to as ‘discovery math’ and, unfortunately, parents felt it was failing their children. The messages in the news indicated that parents thought their children were left to learn on their own without any teacher instruction.

With the rumblings came a petition from parents, demanding the Minister of Education change the curriculum back to the teaching of traditional algorithms and memorization of basic facts. Unfortunately, without any discussion, research  or follow up with educators, the Minister of Education bowed to  public pressure and amended the curriculum so that traditional algorithms and memorized facts were back. Within this amendment is a small blurb stating students will learn conceptually.

However, this has not and is not happening, and students as young as Grade 1 and 2 are memorizing facts and procedures to add and subtract. By the time they enter Grade 3 and 4, number sense has not been developed and mathematics becomes a struggle and a subject they hate.

However, this has not and is not happening, and students as young as Grade 1 and 2 are memorizing facts and procedures to add and subtract. By the time they enter Grade 3 and 4, number sense has not been developed and mathematics becomes a struggle and a subject they hate.

Since the 2007 curriculum change, mathematics learning and instruction has declined in Alberta.

However, is this issue a result of the ‘new math’ or is the issue a result of not fully understanding HOW to teach mathematics for conceptual understanding, as well as WHY it is so important to teach for that conceptual understanding?

• Do we want students to go through their school years performing math through rules and memorized procedures?
• Or, do we want students to go through their school years being able to explain the whys and the hows, and to be able to think, reason and problem solve?
• Do we want our students and children to learn to LOVE mathematics and to see the beauty and creativity in what they are doing?

What is needed in a 21st century employee? Being able to calculate numbers by relying on rules and procedures or being able to think critically, reason, problem solve, interpret data and ask questions when something doesn’t make sense?

What is needed in a 21st century employee?

I almost failed high school math and it took everything I had to finish my Statistics and Math Curriculum and Instruction course in my Undergraduate Studies. In the first three years of my teaching career I taught Gr. 1, Gr. 4 and Gr. 5. Interestingly, although the way in which I was taught math in school didn’t help me succeed, that is how I taught my students.

I did my best but didn’t feel it was the best for my students. In my 4th year I was assigned to a Gr 2 classroom and I determined, more than ever to change my method of instruction and my own mindset towards mathematics. If I felt unequipped in my own ability and lacked passion, what were my students feeling and thinking? I began to play and the curriculum became my bible. I had to know and understand the curriculum I was required to teach. I became very excited because the children were so engaged AND they were having fun learning math! One thing led to another, and I have written and self-published Kindergarten through Gr. 6 Teacher Guide Books titled Math For Success. I also have provided workshops for teachers throughout Alberta and I have obtained my Masters in Elementary Mathematics from the University of Alberta in 2012.

Mathematics is my passion and I want to help change the mindset and improve teachers’, parents’ and students’ understanding. I made the decision at the end of June to take a 4 month leave of absence from my teaching position to be available to mentor teachers to provide students with a more complete understanding of mathematical concepts. My hope is not only to help improve mathematics learning but to ignite a passion in all stakeholders involved in this journey. We have all heard the phrase, it takes a village to raise a child. It is my thinking it will take a village to change the mindset and instruction around mathematics. Developing true partnerships among the school, home and community will help to engage and excite our young learners AND lead to improved results!

Mathematics is my passion and I want to help change the mindset and improve teachers’, parents’ and students’ understanding.

Mathematics instruction is most definitely the key to students’ success.  However, before learning, understanding and implementing rich instructional methods and tasks, you must ask yourselves the following questions:

1. What is your mathematical mindset?
2. Do you believe ALL children can succeed at high levels?
3. Do you think mathematics is just for those with a mathematical mind or who have inherited the ability to do math?
4. Do you see mathematics as a performance subject in which students memorize rules and procedures or do you see the beauty and creativity in how mathematics can be learned and demonstrated?
5. Do you understand what it means to be ‘numerate’ or what ‘numeracy’ means?
6. Do you know how to develop number sense?
7. Do you understand what it means when mathematical literature (educational books, articles, curriculum) state that students are to demonstrate a conceptual understanding or procedural fluency?

I would love to hear your thoughts/ideas with regards to these questions. I am also passionate about helping make this MUCH NEEDED CHANGE in the area of mathematics. With that said, I am hosting a public meeting on Thursday, September 28th at Ecole Olds Elementary School in Mary Schatz’s room from 7:00-8:00 pm. Her room is on the East side of the school. If you come to the East doors for 6:45 pm someone will be there to greet you and let you in. The meeting will start promptly at 7:00 pm.

This meeting is open to anyone who is interested. If you are not in the area and would love to take part, this meeting will be shared via a Google Hangout. To take part in the Google hangout please use the link provided to sign up:

https://www.eventbrite.com/e/mathematical-crisis-webinar-tickets-37865027316

Do you want to see a MUCH NEEDED CHANGE in the area of mathematics? Please attend this public meeting on Thursday, September 28th at Ecole Olds Elementary School in Mary Schatz’s room from 7:00-8:00 pm.

Share this with a friend and let's all work together to make a change in math education in Alberta. This meeting is open to anyone who is interested. If you would like to take part but are unable to attend in person, this meeting will be shared via a Google Hangout. To take part in the Google hangout please use the link provided to sign up:

https://www.eventbrite.com/e/mathematical-crisis-webinar-tickets-37865027316

# What does differentiation of instruction mean?

Differentiation of instruction means that the teacher is able to engage ALL students within THE SAME lesson. The lesson being taught should incorporate specific tasks that allows for multiple entry points.

Jo Boaler in her book, Mathematical Mindsets refers to these tasks as being low floor and high ceiling tasks. If you are looking for a rich educational resource to read this is one I would most definitely recommend.

# EXAMPLE OF LOW CEILING/HIGH FLOOR PROBLEM

You went to Walmart with your mom and saw a game you would like to buy for \$20.00. Mom said you had to earn the money by helping out with the dog.

Mom's deal was this: each time you help with the dog she will pay you 2 quarters. You told dad when he came home from work and he told you he would match the two quarters with 5 dimes.

• When you have earned \$20.00 how many quarters and how many dimes would you have?
• How many chores do you need to do?
• You must work in small groups and represent your answer in different ways.

## You may ask how is it this a low floor and high ceiling task?

• Some students may take out paper or plastic coins and lay out ALL coins and COUNT ALL until they reach a sum of \$20.00. Then they will count all the repetitions of the pattern core/unit to determine the number of chores to be done to purchase their game.
• Other students might show the unit/core that is 2 quarters and 5 dimes and know that is a total of \$1.00 so therefore 20 chores would have to be done to earn \$20.00 because 20 groups of 1 is 20 or 1 x 20 is 20
• Other students might show/draw 10 repetitions in sequence to \$10.00 and just realize 10 and 10 is 20 so they just need to double it.

Can you think of another solution/pathway a student may take depending if they are at a surface, deep or transfer level of learning with patterns, money, counting by 25’s, 10’s, 50’s, as well as their overall number sense?

You may ask where the terminology of surface, deep and transfer learning comes from. This is from a book I read, Visible Learning for Mathematics, by John Hattie, Douglas Fisher and Nancy Frey. This is another excellent resource to get your hands on!

## Another way to differentiate is to provide a question and then build on it by asking, if you found that solution, what if…  or are you able to start at and….

Example 1:

Your class is discussing number patterns or divisibility and you ask your students to skip count. For example, we are skip counting by 2. I would say your count can start a 0, 20, 78, 134, 546. (record these choices on the board)

Once students start their work, I walk around because sometimes students will make the choice that is too difficult or too easy and I will quietly encourage them to try another.

Example 2 - Problem:

My natural path doctor prescribed a special capsule for me to take. The recommended dosage on the bottle is to take 2 every morning with breakfast. I opened the bottle this morning and took 2. What date would I take my last two capsules or how many days would it take until the bottle is empty?

Again I walk around, engaging with my students, and if some students are finishing  and others need more time or are needing my guidance, I will extend this problem with the following question:

However, my doctor said if I wanted to give my body a good kick start, I could take 3 capsules daily. If I did take 3 this morning what date would I finish the bottle of 30 and would it be more or less days than if I took 2 daily and why?

I could further extend this problem by asking,

I was recommended to take these capsules for 3 months. If I took 2 a day, how many bottles would I need? If I took 3 a day how many bottles would I need?

If the cost of one bottle is \$15.75 including GST, what would the total cost be if I took 2 a day? 3 a day? Would the cost be greater or less if I took 2 a day or 3 a day? Why?

THIS REQUIRES YOU TO BE ON YOUR TOES. IT IS QUICK AND EASY TO INCORPORATE BUT YOU NEED TO BE THINKING. I EITHER JUST WRITE KEY WORDS ON THE BOARD OR YOU CAN HAVE THIS TYPED UP.

I MAY STOP BEFORE ALL SOLUTIONS ARE FOUND, BUT IT GIVES TIME FOR ME TO WATCH AND LISTEN TO MY STUDENTS.

IT ALSO GIVES A LITTLE EXTRA TIME FOR THOSE WHO MAY ONLY FINISH THE FIRST QUESTION AND THIS IS PERFECTLY OKAY. ALL STUDENTS ARE ENGAGED AND SOME OF MY STUDENTS EVEN COME UP WITH MORE QUESTIONS OR THOUGHTS/CONNECTIONS RELATING TO WHAT WAS ASKED!!

Throughout my Math For Success teacher guide books, at every grade level K-6, you will find these types of problems!

Day By Day Math Guide

# How does/should a teacher approach this in his/her classroom?

I am going to begin by saying how this SHOULD NOT be approached and that is by homogeneous or ability grouping students. I am so very disheartened to hear of teachers and/or schools (districts) making the choice to group students according to ability.

The talk is that the reasoning behind this decision is based on the RTI model in which students with challenges are to be provided targeted instruction.

I have read the book, Simplifying Response to Intervention, Four Essential Guiding Principles by Austin Buffum, Mike Mattos and Chris Weber,  and I will differ in opinion that the founders of RTI are moving schools in the direction of ability grouping with insurmountable research that proves the negative effects of it.

Is it perhaps being interpreted to meet the ideas of specific schools and individual teachers? No matter what the thinking is and I am confident that these decisions are being made in the best interest of our students, BUT it still comes down to grouping students into low, medium and high levels of ability.

In my Grade 5 and 6 years of school,  I was ability grouped for L.Arts. There was the A, B and C groups and I was placed into the B group. Behind backs and in corners students talked and everyone knew, despite the positive talk of the adults, who was in which group.

That stigma stayed with me right through to college. I failed my first English writing assignment in College and was absolutely devastated. I went to the Learning Centre to help improve my writing skills and without any challenges, I was successful. I completed my Masters in Elementary Mathematics and throughout my Masters program I was writing at an academia level and achieving scores of A and A+. With that said, I will put forth the following question, was I really a ‘B’ level student or could have I excelled if provided rich instructional tasks in a heterogeneous class?

Again, if you were to read Jo Boaler’s book, Mathematical Mindsets, which I highly recommend, she cites countless research from various countries, in which ALL students who were homogeneously grouped according to level of ability in math scored lower on testing than students who had rich instruction in heterogeneous classes. She also said even the students in the the so called ‘high level’ classes held negative attitudes towards math and didn’t go further into math studies after high school or if they did move on they dropped math after their first year of college or university.

I have taught at a grade 3 level for quite some time and it is so sad to count how many students over the 25 years have come into my class telling me they hate math or they are not good at math. HOW CAN THIS BE AT AGE 7 AND 8?

We want to help ALL STUDENTS develop a mathematical mindset through RICH, HIGH LEVEL INSTRUCTION; SENDING THE MESSAGE WE BELIEVE IN THEM; AND CELEBRATING THEIR MISTAKES!

Mathematical ability is NOT hereditary, nor is it for ONLY those with a math brain. ALL our students can learn mathematics and that is the mindset we as teachers need to have as we teach to our class that includes children of all ability levels.

Are you finding it challenging to meet the levels of all your students in your classroom through one lesson? I can help you understand how to differentiate instruction in your classroom.

One of my Workshop Options to help you meet your Teacher Professional Growth Plan requirements will specifically focus on Differentiation of Instruction. This can also be a part of a school wide professional development workshop. I can also offer this session to a group of parents if there is interest.

If you are a subscriber you will receive this information in the monthly newsletter that will be emailed September 20th. If you are aware of any of your colleagues and/or parents who would love to take part in this learning opportunity please share.

# How have you or how are you starting your math class this Fall?

• Paper and pencil assessment
• Teach a lesson and have students practice by completing a worksheet or questions out of a textbook or workbook
• Printables to make math fun. Eg. Coloring in numbers on a hundreds chart to create a picture
• Games
• Or have you even started yet because you are doing ‘get to know you’ activities

I start engaging my students in math from Day 1 of school with the following intentions:

• Setting classroom norms regarding expectations of what math class will look like
• Getting them excited about math by making it active and alive
• Checking in on their understanding of number as well as a few other concepts.

I understand if your district or school has given clear direction to complete an assessment with the purpose to identify students requiring targeted instruction. However, can you also include ‘informal’ activities (assessment) in which you engage your students in a way they don’t even realize they’re doing math?

I know from many years of experience that if you can get your students hooked right from the get go, you will have them eating out of the palm of your hands and success will follow.

I have provided you with a few different number activities to  engage your students. Whatever level you are teaching, each and every one of the  activities can be modified or adapted.

# Activity One

## Ask students where numbers are important to them in their lives?

• Birthday, age, house number, street number or number of family members
• A lock code or password
• Amount of money in their piggy bank or bank account
• Earning a specific amount of money to purchase an item

Write this out on a piece of chart paper and post in your classroom

# Activity Two

Eg. Five (5)

• 3+2
• Number of fingers on one hand
• Number of toes on one foot
• Half of 10
• The value of a nickel
• Years in one half of one decade
• Comes after or to the right of 4
• Comes before or to the left of 6

# Activity Three

## Put a few numbers up and ask, What can you tell me about these numbers?

#### 9, 7, 13, 3, 33, 35, 21, 5

• All are odd numbers
• 3 is a factor of 33
• 3 groups of 7 is 21
• 5 is 1/7 of 35
• 33 and 9 are multiples of 3

# Activity Four

## Write a few numbers and ask, "Which of the numbers does not belong and why?"

#### 28, 20, 10, 16, 24, 36

Possible answer: 10 doesn’t belong because it is a multiple of 2 but not a multiple of 4

# Activity Five

## Can you determine my secret number from the following clues:

Eg. My secret number is:

• A two digit number
• Greater than 35 but less than 48
• Not odd
• A multiple of 2, 3, 6 and 7
• The difference between 100 and 58

Eg. My secret number is:

• Less than 5
• More than 2
• A number I can count by 1’s and 2’s to get to when I start counting at 0

# Activity Six

## Give students a hundreds chart and ask the following questions

• Is the sum of 1 and 99, 100?
• Is the sum of 2 and 98, 100?
• Is the sum of 3 and 97, 100?
• Is the sum of 4 and 96, 100?
• Is the sum of 5 and 95, 100?
• Are you noticing anything with the addends?
• Without writing all the word sentences or equations out, is there a way to determine how many two addend equations have a sum of 100?
• Can a relationship/connection be made to subtraction? Eg. Is the difference between 100 and 1, 99? Is the difference between 100 and 2, 98? Is the difference between 100 and 3, 97? Is the difference between 100 and 4, 96? Is the difference between 100 and 5, 95?
• What are you noticing? Is there a pattern?

## The number of activities to engage our students in is endless.

Do you want your young students to be excited and engaged in mathematics class?

OR do you want your students bored and disliking math?

Give one of the above activities a try with your students and comment to this blog.

Tell us what you did and how you changed or modified it to your grade level. Tell us how your students reacted. Tell us what the outcome was. Did it give you insight about your students’ number sense?

# I know from many years of experience that if you can get your students hooked right from the get go, you will have them eating out of the palm of your hands and success will follow.

## 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.