In the Count I Trust!

Until I began working on the Elementary Mathematics Professional Learning Opportunities (EMPL) project, I had no idea what "Trust the Count" meant nor did I understand how much it underpinned everything in mathematics. This blog post will focus on this idea - what is it? - how do I know if students trust the count?- what do I do if they don't trust the count?

The EMPL working group set one of the big ideas for Additive Thinking as
"Once students trust “the count”, they can flexibly manipulate numbers in order to make solving problems easier by

• using Parts and Wholes
• decomposing / Recomposing
• partitioning
• compensating
• using Constant Difference"

Additive Thinking is about being flexible with numbers. This big idea says that before they can be flexible with numbers, they must trust the count.  

What is Trust the Count? 

In order to explore this idea, you are going to engage in a little formative assessment. Don't worry. It's not going in your gradebook!

Look at the image below. How many dots are there? How do you know?

Now, click on the video below and watch the image change.

How many dots are there now? How do you know?

Click on the video below and watch it change again. 

How many dots are there now? How do you know?

Let's take a moment to assess your responses.

For the first image, you might have counted out every dot. Maybe you saw the 5 dots on a dice in the center and one on each side. Or, did you see 1-2-1-2-1? Or, 3-1-3? Or, something else? How you figured out that there are 7 dots in the first image is irrelevant. 

After you clicked and saw the first and second animations, how did you answer? Did you know it was still 7 dots or did you recount the dots? Someone who recounts all of the dots does NOT trust the count. They have not yet internalized the idea that the quantity stays the same if all you did was move them around.

This is a simple test that you can use when your students are struggling with operations. Use a quantity of items that they have not yet subitized - 9 is a good number for this. If they recount the dots, then stop everything you are doing and focus on helping build the idea of trusting the count. Operations on numbers will not make sense if they don't trust the count.

Students who don't trust the count are most likely missing mastery of the following grade 1 Alberta outcomes:

• Number outcome 2: Subitize (recognize at a glance) and name familiar arrangements of 1 to 10 objects or dots.
  
• Number outcome 3: Demonstrate an understanding of counting by

• indicating that the last number said identifies "how many"
• showing that any set has only one count
• using counting-on
• using parts or equal groups to count sets

• Demonstrate an understanding of conservation of number

So what do I do to help students trust the count?

These activities should be started in kindergarten when students are learning what numbers are all about and then continue on throughout the grade levels. Begin with the lowest level that makes sense for the student you are working with.

Activity 1:

Step 1: Place some objects in front of the student. Ask "how many are there?" Remember, it doesn't matter if they subitize or count each item. This step isn't about trusting the count. Ask "how did you know?" Did they count each one? Did they see groups of 2 and skip count? etc. Their skills here let you know if they are seeing the dots as individual pieces or can visualize grouping them together.

Step 2: Move the objects around the table. Ask, "how many are there now?" If they count each one, that's ok for now. Ask, "how do you know?"

Step 3: Move the objects around the table. Ask, "did I add any?" No. Ask, "did I remove any?" No. Ask, "how many are there now?" If they count each one, that's ok for now. Ask, "how do you know?"

Step 4: Move the objects around the table.  Ask, "did I add any?" No. Ask, "did I remove any?" No. Ask, "Before checking, can you guess how many there are now?" It's ok if they are wrong or unsure of the answer. Say, "Let's check." Give them time to check. Ask, "how many are there now?" If they counted each one, that's ok for now. Ask, "how do you know?"

Repeat Step 4 several times but not too many times in a row. Spend about 5 minutes on this activity. Spending too long on it won't benefit them. It is better to spend smaller chunks of time on it but do it often. Don't change the quantity during a session, especially during these early stages.

Don't be surprised when (not if) at the end of a mini session, they are confidently repeating the number but then the next day, they are reverting back to counting each one several times. Building this idea takes time. Do...not...rush...this! I can not stress this enough.

Once students have done this activity with you a few times, you can have them work in partners, spreading the items out and pushing them tightly together but never changing the quantity.

Focus on building sense of very small numbers - 2 through 9 - before moving to larger numbers. Seriously, we rush this piece because we think students are getting it when they aren't there yet or that we have to get so far so fast. You have all the time in the world. Go slow.

CAUTION! I once had a teacher tell me that she did this activity with a student. She placed some objects in front of the student, had them figure out how many there were and then told the student to close his eyes! She rearranged the items, told the student to open his eyes and say how many there were. This is NOT about trusting the count. This is about trusting the TEACHER. They must KNOW that you didn't add anything or remove anything otherwise and they can only do that if they see you move the items.

Activity 2:

The situation: The student is subitizing all of the sides of a dice. They see six dots and say 6 without counting. They see 4 dots and say 4. When you show them a six and a four and ask them "how many are there in all?", they start counting each dot 1-2-3-4-5-6   7-8-9-10.  

A different situation: You give them a word problem - There were 7 hot dogs ordered on Monday and 8 hot dogs ordered on Tuesday. How many hot dogs were ordered in all? They draw out the 7 hot dogs and then the 8 hot dogs and then count all - 1-2-3-4-5-6-7   8-9-10-11-12-13-14-15. This is a counting all strategy.

What do you do? Let's consider the dice showing six dots and four dots. Point to the six and ask "how many?" 6. Cover the six with your hand and ask, "how many?" If a student is unsure, lift your hand so they can see it. Make sure they understand that it's not a trick. You are not sneakily turning the dice while they can't see it. Cover the dice and ask "how many?" again. Repeat this, covering and uncovering until they confidently know that there is six under there. Again, be patient. The key word in the phrase "trust the count" is "trust". Trust always takes time to build. When they are confidently saying six, point to the first dot on the four and wait patiently. More often then not, they will say "one". Point to your hand covering the six and ask, "how many?" six. Point to the dot and say "sssssssssssssssssssssssseeeeeeeeeeeeeeeeeeeeeeeevvvvvvvvvvvvvvvvvvvvveeeeeeeeeeeeeeeeennnnnnnnnnn". Seriously, drag it out. If they say "seven" at any point while you are doing this stop and say "seven". If they don't, that's ok. They are probably looking at you like you have grown two heads but that's because they aren't sure where you are going with this. If they don't say 'seven" when you have dragged out saying it, just say "seven" yourself. The goal is for them to figure it out themselves but you may need to help a bit but not too much. You can point to your hand covering the six again and ask, "how many?" and then point to the first dot on the four and wait for a couple of seconds and, if needed, say "seven". Now, point to the second dot, wait a moment and then stretch out the word eight. Repeat this process until all four dots have been accounted for. Ask, "how many altogether?" Repeat the count if needed.  Roll the dice again, and repeat this process.

This is called "Counting on". Students start with the total of one number (6) and then continue adding on from there.  6    -7-8-9-10. This is NOT an additive thinking strategy. It is a counting strategy but it will help move them towards additive thinking.

This is an activity students can do with partners once they understand it. Give them two dice and a cup. One student rolls the dice. The other student names them. The first student covers up one dice and the other student figures out how many. The first student checks to confirm.

You can also place a number of objects on the desk with a container that is big enough to cover many of them. Cover 5 of them and say, "There are five under the container. How many are there altogether?" When the student successfully counts on, uncover the container and check. Cover a different amount but don't change the total quantity. They might count on. They might add the two numbers. This helps build the understanding that you can separate a quantity into different parts but you still have the same quantity (trust the count). It also builds toward mastery of different addition facts. If a student is working on 9, and they see it as 1+8, 2+7, 3+6, 4+5, 5+4, 6+3, 7+2, 8+1 over and over again, they will add those facts to their basic fact toolbox. 

A student completing the hot dog question might still draw out 7 hot dogs and 8 hot dogs but, if they have mastered counting on, they will say 7    -8-9-10-11-12-13-14-15  OR 8   -9-10-11-12-13-14-15.  If they are still counting all, point to the first 7 (or 8 - whichever they drew first) and ask if you have to count all of these hot dogs again. Draw a circle around all of them and write the number 7 below it. Cover up the image of the hot dogs but let the 7 show. Ask, "How many hot dogs are hidden here?" 7. Point to the 8th hot dog and repeat the same process completed above with the dice.

Teachers will often tell me that they tell students to put the first number in their head and then keep counting. This is great. Visualization is an extremely important skill. I do this AFTER they have had lots of time to understand it first with manipulatives.  I find that this makes the transfer to visualization easier if they already have something to picture in their head.

Activity 3:
Here is one more mini assessment for students working with numbers higher than 20. Have them build 24 with base ten blocks using two tens and 4 ones. Then have them trade/exchange a ten for ten ones. Ask them what their new number is. You might be surprised how many might have to recheck. They've learned to trade; most do it automatically. However, they don't necessarily understand that the total value hasn't changed.

In a later blog post, I will be speaking a lot about place value and how to dig deeply into it. Alberta teachers, please remember that place value is currently introduced into the curriculum in grade 2. This is NOT an activity for grade 1 students.

A Support Document

The EMPL group created a document (just scroll down a bit after following the link) explaining "trust the count", some misconceptions about number sense, and everything I have outlined above plus more. It can be printed and sent home to parents if you wish.

Source

Trusting the count is something we have internalized ourselves and we may be surprised by how many students come into classrooms without this understanding. For me personally, I know that I was playing board games and card games all the time before I even started Kindergarten. I was already subitizing. I could count on. These activities had me playing with numbers all the time. Now, I watch students come into school who have grown up on devices instead of on physical games. Many of them are coming in never having played a board game or pulled out a deck of cards. I can't control what happens before they get to me. I make sure they get all of those experiences in my class. We play lots of board games, card games and dice games. I try to make sure we spend at least 5 minutes every day on a game.

Self Reflection question for you: What strategies have you used to support students who do not trust the count?

Turning "Mad Minutes" on their head!

Warning: This post will push buttons and I am ok with that. Every time I share this story during a workshop, I have push back from one or two people. That's ok. Everyone is at a different stage in their learning based on their own experiences. I am simply sharing my experiences and my observations with the students I have worked with. Some of you may question whether or not I expect my students to memorize their basic facts. The Alberta Curriculum requires students to recall their facts.


When I was in elementary school, I was great at memorizing. I could learn facts very quickly. People told me that I must be smart because I had memorized so much and so easily! I was very proud of this ability.

My teachers would give us Mad Minutes. These both thrilled me and terrified me. I knew my facts so I was pretty confident. How fast could I answer all the questions this time? At the same time, I was always terrified that I would make a mistake. If I made a mistake, nobody would think I was smart any more.

In my current role, I get to speak to a lot of adults. Some have a loving relationship with math. Others hate it. When I ask them why they hate math, more often than not, I hear horror stories regarding Mad Minutes. This is not the only reason they hate math but often it was the beginning of the downfall. Mad Minutes had taught some of them that you could only be "good" at math if you were fast and if you could memorize. We know that this is not true. When I am thinking deeply about math, I am not fast. I am making connections to other ideas. I am looking for nuances. I am analyzing. I am trying to find patterns. This can not be done quickly. I spend a lot of time talking with adults about the misconceptions that math is all about being fast and a good memorizer.

Let's take a quick peek into the past...
When I taught grade 3, I knew I wanted to do something different. I didn't want to put the time pressure on students but I still wanted to use Mad Minutes. So, my first change to Mad Minutes occurred when I displayed a "count up" timer on my board. My students all started with papers that displayed 10 questions on the 2 times table. They all started when I started the timer. When they finished, they would look at the timer and record the time they had finished at.  I collected all the papers and marked them. The next day, we repeated this process...and so on and so on. When a student answered all 10 questions correctly in 30 seconds, I upgraded them and gave them the 3 times table questions. Eventually students were working on all different levels of facts so I put many copies of each of the facts into separate folders so they would come in from recess and grab their applicable mad minute, wait for me to tell them to start, write it, rinse and repeat.

I was very proud of the change that I had made. It was still about memorizing the facts. It was still about getting faster. I felt that my change had taken away the stigma though of "fast and memorization = good math student".

Fast forward 10 years. I give a "mad minute" and mark my students work. Suzy does all the questions and gets them all right. What does that tell me? Suzy knows her facts. Did I already know that? Yep or Probably.  Jill does all the questions and gets them all wrong. What does that tell me? Think about it for a moment. Did you say that Jill doesn't know her facts? That might be true. It's also possible that Jill's dog died last night or her parents had a fight or she's mad at you or... There are many reasons that she might have received the mark she did. What about Patty who finished half the questions but got them all right? Take a moment to think about her. Is it because she needed more time to finish? Was she counting on her fingers? Was she working merrily along and then saw a snowflake out the window? I had an epiphany. I realized that those mad minutes weren't giving me any information about my students that I probably didn't already know OR I had to go talk to them anyways to get more information.

Before I share how I turned mad minutes on their head, let me share some concerns that are always shared with me when I talk about changing mad minutes.

Supporters of mad minutes will tell me that students need to learn to deal with pressure and mad minutes teach them that. I agree that we all end up working under pressure in our lives and knowing how to do that effectively is an important skill. I'm not sure that it's something that an 8 year old needs to learn under these circumstances. I can find better ways to help them learn that.

Supporters of mad minutes will tell me that mad minutes motivate students to learn their facts. They want to beat their score. Yes, this is true for some students. Is some students good enough for me in my classroom? Nope. I want more for all my students.

Some teachers will tell me about the changes they've made either to how they frame the activity or how they implement it. I love hearing about those changes!

Do I still want my students to master their basic facts? Yes! It's part of the Alberta curriculum that students master those basic facts. It has always been in the curriculum. It was never removed, despite what some people claim. All you have to do is go to the grade 3-5 curriculum and look for those multiplication facts. The clarifications introduced in 2014 more explicitly stated the need for students to recall the facts. Interestingly enough, even though mastery and recall are specifically stated in the curriculum, some will argue that mastery and recall is not the same thing as memorization.

What is mastery? They are able to recall their facts when needed AND they understand those facts and what they mean. What is recall? Recall occurs when students commit the facts to memory and retrieve them when needed. (Source)

I think of it this way...everybody has two toolboxes. One contains the "basic facts" that you have mastered. 3x5 = 15, 4x2 = 8, etc. You are able to retrieve these facts without thinking about it or figuring them out. You just know.

Your other toolbox is your strategy toolbox. This box contains all the strategies that your understand and use. It may include the traditional algorithm, doubling, knowing the distributive property, etc.

Now think of these two students: Joey has a very large basic fact toolbox. He's in grade 3 but has memorized everything up to 12x12. However, his strategy toolbox is very small.

Steven, on the other hand, has a very small basic fact toolbox. He is still working on committing the grade 3 requirements to memory. However, his strategy toolbox has several well used, well understood strategies that he accesses whenever he needs to.

Which student will be more successful?  Steven. He has the strategies mastered to help him figure out anything he doesn't know. Joey will often struggle to figure out what he doesn't have memorized because he doesn't have a large strategy toolbox.

Where do I want my students to be at? I want them to have a healthy strategy toolbox AND a healthy basic fact toolbox.

Ok ok. Enough background. This isn't a groundbreaking, earth shattering idea. It is a small change that I made.

I'm going to use a simple addition mad minute. I apologize that all of the questions are vertical. I typically like to have about 50/50 vertical/horizontal layouts. I figured for this blog post example, it would work.

Everything written below in normal text is something that I say during the activity. If it is in italics, then it is something I do or notice.
--------------------------------------------------------------------------------------This week, we have been working on facts of 10. Can someone tell me what a fact of ten is?
Students explain that two numbers that add to ten would be considered a fact of ten.

Can someone give me an example?
7+3, 2+8

I display just the top row of questions on the SmartBoard for students to see.

I have placed some questions on the board. If it is a fact of ten, we are going to draw a rectangle around it. Is the first question a fact of ten? Yes. Let's draw a rectangle around it.
I draw a big enough rectangle that it leaves me some space inside to answer the question but I do NOT answer it.

Let's look at the next question. Is it a fact of ten? No. Do I draw a rectangle around it? No. What about the next one? No.

I am going to hand out your sheet. Please keep it face down on your desk for now.
When they are all handed out...
When I say go, you will turn over your sheet and find as many questions as you can that are facts of ten. What are you going to do when you find one? Draw a rectangle around it.
I am only going to give you one minute though. You might not find them all and that's ok as long as you keep looking for the entire minute. Remember, do NOT answer anything! Just draw rectangles. Ready? Go.

Students have one minute.

Stop! Now, I am going to give you one minute to answer JUST the questions you drew rectangles around. If you don't finish, that's ok as long as you keep going until your time is up. If you finish before the time is up, you can look for any you  might have missed.  Go.

Students have one minute to answer the questions.

Let's go through the questions. Is 9+1 a fact of ten? Yes we already drew a rectangle around it.

How much is 9+1? 10. I write 10.

What about 9+5? No.

3+3? No.
Keep going until you have gone through all the questions.
--------------------------------------------------------------------------------------
Another week, I would give them the same sheet, either a fresh copy or this used copy. Maybe we've been working on doubles (ex. 3+3, 4+4). This time we will circle the doubles questions for 1 minute and then answer for 2 minutes. I only gave 1 minute on the "facts of 10" activity because all of the answers were 10. All other strategies will need more time so I give them 2 minutes of working time instead of 1 minute.

This process helps students look for questions that specific strategies are most useful for. They are quickly analyzing a question before solving it. They aren't panicking about getting all the questions done, resulting in the silly mistakes that often occur. 

As I am typing this, I am thinking about extensions that I could introduce. So here is a new one for you...if a student finishes before the 2 minutes is up and they have checked to make sure they have found all questions, they can create and answer their own questions based on the strategy they are working on. For example, we might be working on doubles addition to 9+9. The student might extend to 10+10, 11+11, etc on their paper.

When I am finished sharing this idea with teachers, I still have some "yeah butters". This is one of the areas that some people are unwilling to change in. All I ask them to do is try it with students. See what happens. Listen to the conversations about their thinking. Check their anxiety levels. If the teacher is willing to try it 3 or 4 times with a positive mindset, I feel confident that it will be a win for the teacher and for the students.

Reflection question for you: How do you assess students' mastery of basic facts? What do you like about it? Dislike?

How I use "Number Talks" in the classroom

Number Talks have become quite popular in the math class recently. You can purchase several books by Sherry Parish (NOT an affiliate link) that teach you how to run them or read many blog posts about them.

A friend of mine tweeted about his experience running a Number Talk and through follow-up conversations, we realized that I run mine a little differently than the format shared in the books.  Although I talk briefly about my process in my Parent Night blog post "What are these stupid strategies kids are learning days? Why can't they just learn the #@$#* way I learned it?", I thought I would focus a single blog post on it so that I can go into my process in more detail.

How many of you have students who are effective at communicating their mathematical thinking? They won't be amazing at communicating their thinking in the beginning (unless they have been practicing this a lot in previous years) and that's ok. This is a skill that students develop over time. They must be provided with opportunities to practice it in meaningful settings. 

You can use number talks at any point in students' learning and understanding. Maybe you are introducing 3 digit addition and want to see what strategies they have for 2 digit addition. Maybe you have been exploring 3 digit addition for a while and want to see what they will do if given a 3 digit plus 4 digit number.

I typically have students complete their work using as much mental math as possible, although they know they can always access manipulatives if they need it or use a mini whiteboard to record their thinking as they work through it. However, I have noticed that people often use different strategies when given paper and pencil versus solving it mentally. When I ask adults to solve an addition question in their head, they often use alternate strategies but admit that if I had said to use paper and pencil, they would have used the traditional algorithm instead.

Everything written below in normal text is something that I say during the process. If it is in italics, then it is something I do or notice.

------------------------------------------

Step 1: Post a meaningful question for students to solve on the board so that all can see it. This might be a "How many dots are there?" or "Add 324 + 493" or pretty much anything else that you would like to focus on. 

I am going to give you 2 minutes to work on figuring it out. It's only 2 minutes though so will everyone finish? (No.) Is that ok? (Yes as long as you keep working on it the entire time.) If you figure it out and still have time left, see if you can confirm your answer by solving it a different way.

While you are thinking, place a closed fist in front of your chest. That will tell me you are thinking. When you have solved it one way, put your thumb up. If you have time, solve it a different way. If you do, put another finger up.

Step 2: Students work. My "2 minutes" are up when I see that almost every student has their thumb up.

Turn to your partner and explain to them how you solved the question on the board. If you didn't finish, that's ok. Share as far as you got. I should hear things like "First I did this. Then I did this."

Students have time to share their strategies with their partner. During this time, I wander around the classroom listening for a variety of strategies that I would like to have shared during the group sharing time. I will let students know that I think they have explained an interesting strategy and will ask them to share it with the rest of the class. I try to focus on the students who may not typically share in class and the students who don't often feel success.

Step 3:  Class Sharing. During this time, I will call upon students to share strategies, starting with the least efficient / easiest strategy first and then moving onto more complex strategies. I will ask students who want to share their strategies, and of course the students whom I talked with earlier always put their hands up because they know I am going to call on them anyways. 

Who would like to share how they solved this question or how their partner solved this question? If you didn't have enough time to finish, that's ok. You can still share your thinking as far as you got.

I phrase the first statement that way so that students know they have permission to talk about the strategy that their partner used. I started saying it that way because, especially in the beginning, I would have students who didn't / couldn't solve it on their own until after they talked to their partner. This gave them permission to share even if it wasn't their own original idea. In fact, after students have had experience with Number Talks, I will require them to explain the strategy their partner used rather than their own strategy. This really helps them focus on listening to understand and to ask clarifying question. 

I phrase the second statement this way because it's important that students realize that it's not about solving it quickly. Even if you didn't finish, you still have ideas and strategies you can share.

As students are explaining their strategy, I am asking them clarifying questions, ensuring that they are being very specific and clear about their thinking. For example, if they are solving 24 + 95 and they say 3 + 9 = 12, I will ask them - Where did the 3 came from? They will tell me that it's from 1 + 2. I will ask them - is that really a 9? They will tell me - No, that's 9 tens - or - No, that's 90. As they are explaining their strategy, I am writing it out the board. This helps students see the thinking represented symbolically.

Step 4: After a student has finished explaining their strategy:

Who else solved it the same way?

Students are going to say - I solved it that way too - so you might as well ask. This also shows this student that other people thought about the question the same way they did.

Step 5: Does it make sense to others?

Let's take a moment to think about that strategy. If that strategy made complete sense to you, give me a thumbs up. If that strategy made no sense to you, give me a thumbs down. If that strategy makes some sense to you, give me a thumb in the middle. It's ok to give me a thumbs up, thumbs down, or thumb in the middle.

Having students self assess this way gives you an opportunity to see how students feel about this particular strategy. Does it make sense to most students? Who doesn't it make sense to? When students see a strategy for the first time, you might see a lot of thumbs down. The next time they hear it, you will probably see some more thumbs in the middle. The more they see it or hear it, the more their thumbs will move.

Step 6: Repeat with the other strategies. Don't do too many strategies, though. Focus on about three. Any more than three and students tend to lose focus.

Step 7: Reinforce the idea that there are many ways to solve a single question.

We just heard 3 ways to solve this question. Which way was correct? (They all are). Take a moment to think about the one that makes the most sense to you today. Wait a few seconds. Turn to your partner and tell them which one makes the most sense for you today and tell them why. Give students a minute or two at most to share.

------------------------------------------

Number talks shouldn't take up more than 10 minutes in the class. I wouldn't do these every day but I like using them about 2-3 times per week, regardless of the grade level.  They have so many benefits to them that the time invested is well spent. Every student shares their thinking with their partner so every single student is practicing communicating their thinking. They are seeing their thinking represented symbolically. They are listening for understanding to others' strategies. They are seeing strategies move from less complex to more complex. They are reinforcing the idea that there are many strategies for solving the exact same question.

Reflection Question for you: Have you ever used a process similar to Number Talks in your classroom? What benefits have you seen? Are there parts of the process you have struggled with?

I would love to hear your thoughts in the comment section!

Homework and "My Favorite No"

Let's step back in time to 10 years ago. I am teaching high school math. Every day, I diligently teach the class, assign practice questions and then assign homework which I would check the next day for completion. 0 for incomplete. 5 for partially complete. 10 for complete. This mark was not based on understanding. It was based on compliance. What did I notice? My compliant students were completing it. The students who were struggling or who weren't engaged, didn't complete it. Writing a mark in my book changed nothing. Their compliance was affecting their grades since I was following the typically standard "Homework counts for 10% of the grade" designation.

I quickly realized that I was completely and utterly wasting my time. This did nothing to help me understand what my students knew and didn't know.  So I changed. At the end of class, I gave students questions out of the book but told them to do as many as they felt they needed to do in order to demonstrate understanding. I wasn't checking it though. I created videos for each of those questions where I solved them and posted them online for students to view if needed. If they were stuck, they had the video they could look at. I encouraged them to google for help as well. Anything they were still stuck on, they could still come to me for help.

So, if I removed homework checks from my daily routine and deleted it from my gradebook, was there something I added? Absolutely. At the beginning of each class, I handed out a "mini quiz" with one question from the previous day on it. They had about 5 minutes to answer the question. At the end of the class, while students were working, I went through the mini quizzes, calling students up as I marked each one to return it, having a quick conversation or "mini lesson" to fix understanding. If I felt that enough students were struggling, I planned to rework the lesson for the next day.  These mini quizzes were worth 5 marks each. Students could come and rewrite mini quizzes over lunch hour or during their end of class work time if they felt they understood it better. I was writing a lot of new mini quizzes.

One day, I saw a video entitled "My Favorite No". This teacher started out by giving a one question quiz (just like I did) but here's where she differs. She immediately, in front of all those students, went through the quizzes and sorted them into "Yes" (the answer is correct) and "No"  (the answer is incorrect) piles, saying "Yes" or "No" as she went. When she reached the end of the quizzes, she went back through the "No" pile looking for her favorite "No".  She looked for a common misunderstanding / error that students predictably had.  Rewriting the students' work (to protect their identity), she told students she had found her "Favorite No". Yes, there is a mistake but there is good math happening in it as well. Then, she asked students to find all the good in the work. Students shared all the things that this student had done correctly. Then, she asked them to find the error. Once the error was identified, she asked them to help her fix it.  

I love this process. Instead of waiting until the end of the class to go through mini quizzes individually with students, errors and misunderstandings are addressed immediately. Students who made mistakes are seeing it solved correctly...not at the end of class but right away. Students are celebrating mistakes in class, looking for the good work that occurred and looking for errors that need to be fixed.

For me, this was powerful and a game changer. It allows me to immediately address misconceptions. I can refocus my lesson for the day if need be. The time at the end of class became usable for other support. Students weren't spending their lunch hours in my classroom rewriting mini quizzes and I wasn't spending my own time marking them.

Did I enter marks into my grade-book? No. This wasn't about compliance. This wasn't about compiling numbers to show that students had done work. This was about finding and addressing mathematical misunderstandings. Did I spend time looking through each paper to identify students with specific issues that needed to be addressed? Absolutely, but I only really needed to deal with students with major misunderstandings as most errors were dealt with during the classroom discussion.

I love how one little change can make a huge difference in the culture of the classroom. 

Reflection Question for you: What do you do to build a classroom culture that allows for students to respectfully discuss errors that have been made?

What are these stupid strategies kids are learning days? Why can't they just learn the #@$#* way I learned it?

Wow! This turned into a huge post. I apologize! You may want to break this reading up into several sittings.  Sorry! I have included the slides that I share so you could read the slide and then, if interested, read the details below it and if not interested, just skip ahead to the next slide. 

I chose to start my blog reawakening with this question because I get this question a lot from parents...not all parents but many. It's imperative that we have parents on our side when we are teaching in a way that is often different that the way they learned. If parents understand that these changes are good, then they are more open to them.


In order to help with this, I hold parent nights so that parents can come and learn more about the pedagogy of our curriculum.  They are encouraged to bring in their children with them. (Preferably no more than one child per parent) Some of these sessions take place where I work (CARC) and others have taken place within a school that has arranged for me to hold it there.  Typically, I set each session to be grade specific or a very small span of grades. This helps narrow the examples and activities that are provided and to focus of our conversations.

Here is the link to the Google Slides I use/adapt. You have my permission to use/adapt this presentation to meet your needs!

Let's see how a typical parent night runs.

Everything written below in normal text is something that I say during the workshop. If it is in italics, then it is something I do or notice.

--------------------------------------------------------------------------------------

The parents arrive, some with their children and some without. They all have different expressions on their faces when they come in. Some are excited to be here. They are ready to learn and to have conversations around math. Others look like they are ready to bolt. It is obvious that they have not built a positive relationship with math. Others look like they are here to challenge everything. They don't like the way that their child is being taught and have come prepared to put an end to that. The rest of the parents are somewhere in between.

During our time together, we will be looking at math in a variety of ways. Everything we do follows the philosophy of the current Alberta Mathematics Program of Studies. However, your teacher may be teaching strategies that are different that the ones we do today.

We are going to explore strategies that may be new to you. When you see them or hear them, keep an open mind. Ask yourself, do they make sense to you? Have you ever seen them before?

We are going to have an opportunity to make connections between what you learned in school and how your child is experiencing math now.

At the end of tonight, I hope that you feel like you have a better understanding of the philosophy of the Mathematics curriculum.

We only have 1+ hr together which means that our time together is just a teaser. We could spend days and days on these ideas.

Each of these questions was posted on the screen one at a time so that they could focus their discussion.

Question 1: Parents, turn to your child and ask them this question.  Listen carefully to their responses. If you don’t have a child here tonight, write this question down so that you can ask them later.

Question 2: Parents, tell your child one thing you think THEY are really good at in math.

Question 3: Parents, ask your child “What is one thing you feel you really struggle with in math?” Again, listen carefully to their responses.

After conversations: You may or may not have been surprised about some of the answers shared during this conversation.  Often, we have different beliefs about our strengths and areas for growth. Have this conversation several times throughout the year.  Expect the answers to change over time.

These two questions show up separately.
Question 1: Parents, do not answer this question out loud. When you were in school, what was your least favorite subject?

I talk with a lot of people around the world and often ask them this question. What do you think most would say? Pause Unless they are a high school math teacher, university math professor, engineer or in another math related jobs, many adults (but not all) have told me that math was their least favorite subject. Why?

Many people were shown algorithms and procedures and told to follow them blindly. "When dividing fractions, invert and multiply. Don't ask why." Often when asked how the algorithm works, they were told just to follow it. Some people had no issues with the algorithm. They could follow it. Many who could follow it also understood it while others followed it blindly but could get the answer. However, many people need or want to understand how it works. Without that understanding, they couldn't follow the algorithm.

People often identify their memorization skills as the reason they are good or bad at math. If you have all your basic facts memorized, you were often perceived as good at math. If you didn't have them all memorized, well...you obviously weren't good at math.

How fast were you at finding answers? Could you do it in your head? If you thought slower than others, if it took you more time to find an answer, if you needed paper to solve it, you were often perceived as being bad at math. Speed became a division at math.

If you were good at running algorithms, following procedures and memorizing basic facts, you were successful in math which made you feel like you were good at math. If you weren't good memorizing, you probably felt you were bad at math.

Question 2: What area in math do you think people often identify as being a huge issue? Algebra is often identified as the biggest issue for many people. Even though we use it all the time, people don't see it expressed in variable form very often so they assume they are not using it. Algebra in symbolic form can be intimidating. People have told me that they don't understand how there can be letters in math, or why the value of those letters change. They "learned" algebra by jumping to abstract concepts too quickly rather than exploring it concretely first.

It is through our experiences with math that makes us feel that we are either good at it or not good at it.

If, in your head, you said math was your least favorite subject or if you feel like you are bad at math, the good news is that you were NEVER bad at math.   You just weren't taught the way you needed to be taught in order to understand it. And that's what we are trying to do in schools. Teach math in such a way that it makes sense to them rather than through rote memorization. 

There are three things teachers want you know about their mathematics classes. The first goal is for your child to love and understand mathematics.  Both goals need to be attained for future success. If a kid understand math but never loves it, they won’t think about all the opportunities they may have in their future that could involve mathematics.  For those that love math but don’t understand it, there may come a time that they dislike math due to their limited understanding. And we want kids to keep as many options open as possible and the reality at this point is that some students fear of mathematics prevents them from being as successful as they can be or pursuing certain areas of study

Secondly, you are going to see a lot of concrete manipulatives in your child's class. Tools such as base ten blocks, integer tiles, unifix cubes, cuisenaire rods, and other tools help students build a concrete understanding which allows them to make connections to abstract representations. The use of manipulatives or other materials does not mean that the mathematical understanding is any less, and often it means the opposite.  If a student can understand an algorithm AND explain that with the use of manipulatives, they have a stronger foundational understanding of the concept.

Math class is no longer sitting in rows, working on worksheets. It is an active learning process. They will not be talked to for a period of time and then sent home with a ton of homework to practice what they heard. They will be doing activities and group work along with some individual work. They will build mathematical ideas using manipulatives. They will talk to people about their understandings and strategies. Some of the activities may spill into their outside of school life and will hopefully engage family or friends.

I hear a lot of people complain about the "New" math. The math has not changed. They still need to learn to add, subtract, multiply and divide. They will still learn math in school. However, the WAY your child is taught math will not be the same as the way you were taught.  And change isn’t a bad thing.  There will still be times that some things seem familiar, but there are other times that things will look or sound different.  What we hope you will do is encourage your child to work on mathematics, to ask your child to explain what they are doing and why, to involve your child in as many ways as you can think of that involve mathematical thinking (ie grocery bills, filling up with gas, counting out money) and tell your child that they are capable and can be a successful mathematical thinker.

Two questions are posted on the screen for everyone in the room. These are adjusted based on the grade level. Choose one question to solve in your head. Do not use paper or a calculating device other than your brain. Hold a closed fist in front your chest while you are thinking. When you have solved it, put up your thumb. If you have time and can think of another way to solve it, put up another finger.

I give enough time so that almost everyone has at least one thumb up If both of you have your thumbs up, please explain to the other person exactly how you figured out the answer. I should hear things like, "First I did this and then I did that."  Parents, you will go second. If you are still thinking, go ahead and finish thinking before sharing. 

Participants are given time to share their thinking. When it looks like everyone has had a chance to share their thinking...  Let's share your thinking with the rest of the group. Who would like to explain the strategy they used or their partner used to solve one of the questions on the board?

Someone shares their thinking. I ask questions to clarify their thinking for others and make sure that steps aren't skipped. Throughout their explanation, I am recording their thinking on the board.  Did anyone else do it the same way? Cool. If that makes sense to you give me a thumbs up. If that makes no sense to you, give me a thumbs down. If that makes some sense to you, give me a thumbs in the middle. Who did it a different way?

The traditional algorithm will always be shared by a parent. Sometimes they are embarrassed that they used it. Others are proud. Is it ok for students to use the traditional/standard algorithm in school?Yes! There is absolutely nothing wrong with using the traditional/standard algorithm. It is absolutely a strategy that works.

We run through several examples So, we have heard many different ways to solve those two questions. Who's strategy is correct?

Participants always respond that all of them are correct. If you only walk away with one understanding from this night, I want it to be this - there are many ways to add numbers. The "grade 3 outcome" says that students must be able to add "3 digit" numbers. It doesn't say how they must do it, just that they must be able to do it. They must master a strategy for adding numbers but what they master may or may not be different than the one we are most familiar with and that's ok.

There are many different strategies for adding, subtracting, multiplying and dividing.  There are the traditional methods we learned in school but there are also many other strategies students are exposed to like the ones shared today. You may find your child using strategies that are very different from the ones you learned and that’s ok. We want children to understand that there are many strategies but…

They don’t need to master all of them for a particular concept.

There is no right or wrong strategy as long as the strategy is efficient and effective for your child. It might not be the most efficient strategy available but it may be the most efficient one for them at their level of understanding at this moment in time. With experience, students will develop more efficient strategies.


So what does it look like when a student has mastered a strategy?  If a student has mastered a strategy, then for that student, the strategy is efficient, effective and explainable. If it is efficient, the student can use it to find the correct answer in a reasonable amount of time. Although there is no time limit, I think about a minute would be a fair time estimate. When a student is first introduced to a new strategy, it might take them 10 minutes to use it to answer a question and that's ok. They just haven't mastered it yet. Through more experience with a particular strategy, they will usually become more efficient at it. What is efficient for a grade 2 student to add numbers will look very different than what a grade 6 student may use. Efficiency increases over time and experience.  If a student has mastered a strategy, then for that student, the strategy is effective, it must get you the right answer every single time. Sometimes students make up a strategy that flukes into the right answer but doesn't work in any other situation. So, it must be effective every single time. If a student has mastered a strategy, then for that student, the strategy is explainable. They have to be able to explain why the strategy works, why they chose that strategy to use for this particular question and whether it will work for really big numbers and really little numbers.

We have to remember that many of us were taught math focusing on learning the traditional/standard algorithm. If we felt we were successful in math, we often were able to learn those algorithms easily and feel comfortable using them. We have had many years of experience working with that algorithm. If you were one of the adults who, in your head, said that you don't feel like you were good at math in school, you may have said the algorithms and procedures didn't make sense to you. Alberta Education dictates the outcomes we must teach, such as "adding three digit numbers", but they do not tell us how to teach those strategies. That has always been up to the professional judgement of teachers. Nor does it dictate which strategies students must use. That is up to the student.

At this point, we engage with a math activity that is specific to the grade level of students who are attending. For example, if it's grade 5, I will show them how to use base ten blocks to build a conceptual understanding of 2 digit by 2 digit multiplication then work through representing that as an array model. I will be sharing these activities in later blog posts.

So what can you do at home to help your child see more success in math? Play games! Board games, dice games, card games, etc. It’s better to play for a short amount of time but more often than playing for a long time once in a while.

It's important to understand the difference between a game and a practice activity. "Games" that don't require any kind of strategy are simply tools for practicing basic facts. For example, roll two dice and multiply them together to get an answer fits in here. These aren't games. These are pounding a nail in over and over until the nail is embedded in the wood. Children need some of this but it doesn't need to be the main focus in order for them to master those facts.  

A game requires some sort of strategy. Being strategic will help them build deeper mathematical thinking. They need many opportunities with this.

So let's take a look at some examples of games that we have played in the past.

 The card game war.  Is it a practicing activity or a thinking game?

War is definitely a practicing activity. There is absolutely no strategy involved. You simply flip over cards and figure out who is larger. It's about practicing comparing numbers. One person doesn't even have to think if they rely on the other person to decide who is bigger each time. So how do we turn War from a practicing activity to a strategy game"?

• Divide a deck of cards so that each player receives one suit of cards.  (ie. You get all of the Clubs) Place the rest of the cards back.
• Players fan out their cards so they can see them but the other player can not.
• Players each choose one card to play and place it face down in front of them.
• When both players are ready, they flip their cards.  Highest card wins. If there is a tie, players replace the card back in their hand.
• Cards that have been claimed go into a discard pile.  Not back in their hand.

Participants are given a few minutes to play "War...with a Twist".

Variations:

• Player A takes all the red cards. Player B takes all the red cards. Each player chooses two cards for each round.  Decide at the beginning of the game which operation will be applied.

Variations:

• Player A takes one black suit AND one red suit. Player B takes the other black suit and red suit.
• Black cards are positive numbers.
• Red cards are negative numbers.

Bump Games are something I have added to my classroom over the past few years. First, I'll teach you how to play and then you can decide if they are a practice activity or a strategy game.

Instructions: Player 1 rolls the two dice and add together. For example, I roll a 1 and a 3. What do I get? 4. This is Alien Bump - Multiplication by 6. What do you think I will multiply the 4 by? 6. What is 4 multiplied by 6? 24. I cover the alien beside the number 24 using a tile (or a coin, or a bingo chip - whatever you have at home.) Do not cover the number 24 otherwise you won't be able to see it. Make sure to put it on the alien.  My partner rolls. They get a 2 and a 5. She adds those together to get 7. Then she multiplies 7 by 6 to get 42. She places her different colored tile on the alien beside 42. We keep going back and forth one at a time.  What happens if I roll a 7? My partner has already covered the 42! I bump her chip off and give it back to her and then cover the alien beside the 42. What happens if I roll another 4? I'm already on the 24! This time, since I am already on the spot, I get to put my tile on top of the number 24, claiming this spot. Now that I have a tile on both parts of the spot (the alien and the number), it can't be bumped off! Play until someone uses all their tiles or the entire board is covered.

Give participants time to play.

There are many different Bump Games that you can play.

So, is a Bump Game a practice activity or a strategy game?

Bump Games do not require ANY strategy.  It’s complete luck. This is a good way to introduce the game however.  Once they are familiar with the process, then switch it up! Make it more challenging.

Variations:

• Add an extra die. Ex. When the game asks for TWO 6 sided dice, give them THREE 6 sided dice. 
• Roll all THREE dice.
• Player chooses which of the TWO dice he/she will use.

Variations:

• After rolling, player may add ,subtract, multiply or divide the two dice to get a number that is useful.

There are many games that you can play to help students practice basic facts but still allow for strategic choices.

Parents, it is extremely important that you never say "I am/was bad at math". Women, this is especially true for you. Research has shown that when a mother tells her daughter that "I was bad at math", the daughter's belief about their own ability drops and as a result, so does their achievement.

It is ok to say, "I don't understand this...yet" or "I didn't learn this way. Please help me understand."

Your child is going to come home using strategies that you may not have seen before. That's expected and it's ok if you don't understand them right away. Ask your child to explain it to you. Remember though, they may not have mastered it yet so be patient. Remember that they will often explore the strategies using simple examples and then move on to more complex questions. I don't know how many times I've seen a strategy presented on Facebook using an over simplified example. Adults get upset because you would never use that strategy on that particular example. Please keep in mind, we only do that when introducing a strategy. We are not saying it makes sense to use that strategy on such an over simplified example. However, it helps students explore the strategy before it they have a chance to use it with more appropriate questions. Try that strategy out yourself.  Remember, it might not be your most efficient strategy and that’s ok. If you are stuck, google it or ask your child's teacher to help you understand it. I will be creating blog spots exploring many different strategies and how they can be introduced, moving from simplified questions to more complex questions.

There are many opportunities to work with numbers at home. At this point, I share activities that are appropriate for the age level of the students in the session. Baking is a popular one for all age levels. Giving children two few forks to put out at supper and asking how many they are short. Counting toys. There are too many to list here.

My "dumping grounds" website contains some of the activities that I have created for different grade levels (based on Alberta Curriculum), strategy games and practice activities that you can access to help your child. Videos are slowly being added for many of the ideas as well. Feel free to use this to help you understand and explore math!

One final reminder: My goal is to help my students build deep, conceptual understanding, through concrete experiences and making connections to abstract representations. They are still expected to master basic facts. They will still learn how to add, subtract, multiply and divide. They are just not doing it through rote memorization, flash cards or mad minutes.

--------------------------------------------------------------------------------------
When the session is over, parents always come up to me to talk about their math experiences and how they connected with the math in this session.  Here are some comments that I always receive, regardless of the grade levels I focused on or whether they believed that they were good or bad at math.

"Why wasn't I taught this way? If I had been taught this way, I would have understood math!"

"When I came here tonight, I didn't understand why my son was drawing boxes to add. Now I understand." [This was after I showed them how to add using base ten blocks]

"I always thought I was cheating. I couldn't do math the way the teacher showed me so I had to figure out my own way of doing it. Now, I know I wasn't cheating."

"When you said that I was never bad at math...that I just wasn't taught the way I needed to be taught in order to understand, that opened the world to me. I'm not stupid! I can learn this."

"My child might use a strategy that's different than I do and that's ok!"

"When's the next Parent Night?I want to learn more math!"

Parent nights have evolved so that parents who have participated in this beginning "theoretical" portion come just for the new activities and parents who are new to the event come earlier to learn the "theoretical" part as well as all of the activities.

When parents have an opportunity to experience and understand what is going on in the classroom, they will support you. I believe that it's imperative that they have an opportunity to participate in a parent night the first few weeks of school. This will set the stage for the rest of the year. I encourage you to create a class website where students create videos explaining all of the strategies you explore. What does it look like to solve a question using concrete materials? What does it look like to solve it using each of the different strategies they have learned? What a great assessment tool for students as well? Do they understand their chosen strategy well enough to explain it to someone who has zero understanding?

Wow! Did you make it to the end of this post! Great job! That was a lot of reading. I won't be making all of my blog posts that long. I won't be able to keep up with my one post a week if I do that every time. :) 

If you're not too exhausted from all that reading, I have a Reflection Question for the you:  What opportunities have you provided to help parents understand the strategies that their children are exploring in class? I would love to hear about them in the comments below!

If you have any questions about how I ran parent nights that I didn't address, please feel free to ask! I will respond to all questions as best as I can. 

Have a fantastic day and I will see you in a week!

Sandi