Have you ever had a student that when you put up 4 fingers and asked them “How many fingers do I have?” they had to count each finger one-by-one? That happens way too often in early elementary classrooms. Those children can count, but they are not subitizing and without being able to subitize a child will never move past counting on their fingers to add and subtract.
This last post of my Fact Fluency Series is going to delve heavily into why & how you should be implementing subitizing into your classroom to build students’ number sense, which in turn builds their flexibility and helps them be truly fluent with their facts.
I shared how we should be focusing on building 4 Types of Facts instead of teaching facts as isolated pieces of knowledge kids are supposed to memorize, in the first post in this series,
In the second post I detailed out the three pieces kids really need to be truly fluent plus the 4 Number Relationships that help build the most important piece (flexibility).
I want to start off this third post with a definition of subitizing. I first learned about subitizing from Doug Clements’ article, but he defines it as “instantly seeing how many.”
Now, I’m going to talk about how subitizing helps build the 4 Number Relationships mentioned in post #2 and activities you can do in your classroom to build your kiddos’ ability to subitize. BUT, one word of warning….I have no research to back me up on this warning, just my experience….if you work with kids who come from home lives that have no structure, no predictability, no stability, I have found that those kids struggle with subitizing. It takes them longer to ‘trust’ that every time I have all the fingers up on one hand it is FIVE. They don’t believe that every time I push over all the red beads on the rekenrek that it is FIVE (you might be trickin’ me, Miss). When a child has nothing that is reliable in their life, why would they think they can rely on a ten frame to always be five when the top row is filled up????? Now, this is NOT to say they can’t subitizing…they can, it just takes them longer to trust the tools and images that other kids will quickly be okay with. Okay, done with my warning….on to the good stuff.
First I’m going to share a couple quotes that I LOVE, but tend to shock some people:
Counting is important, but it does NOT build any relationships about a number. In fact, it doesn’t even help kids to see how a number relates to a QUANTITY!!! Think about a kid counting out 6 blocks, they ‘tag’ each block with a number…but what does that kid really know about ‘6’?
Most people respond to that question with “They know it is 6 things.” But do they???? A quick way to tell is after they count the 6 items, ask the child to show you ‘four.’ A LOT of kids will point to the 4th item, NOT show you four things:
This really hit home to me when I was doing a training for a group of administrators. I gave them all a Number Path and asked them to circle “six.” Almost everyone in the room did this:
That is the digit ‘6’ but not the quantity of 6. If all we focus on with our kiddos is that “six” is ‘6’ and looks like this:
then when they start doing 6 + 5, their only strategies are either memorization or using these images of the quantities:
which results in kids counting one-by-one to solve.
Instead, when you take the time to build subitizing in your students using structured images, like ten frames and rekenreks (a.k.a MathRacks), they could count to solve….but they also get the opportunity to build number relationships that will lead into POWERFUL strategies.
Right now, maybe you already see some strategies kids could use when trying to figure out how many are in the ten frames or how many beads are “in play” (for those not familiar with a rekenrek, the beads on the left side are ‘in play’)….but honestly it isn’t for YOU to see. I want the kids to see those strategies and not have you tell them what to see and how to figure out the total (not that YOU do, but for some it’s easy to just tell the kids to move 4 from the bottom ten frame to fill the top ten frame).
They will only be able to see it for themselves if they have built the 4 Number Relationships. So, lets take a look at how subitizing can build the 4 Number Relationships (which build number sense) and the by-product of building that number sense is that kids get more flexible with their use of numbers and they come up with those strategies on their own.
#1: Spatial Relationships is recognizing how many without counting by seeing the visual pattern.
My favorite activities include just providing kiddos with lots of visuals. Create Dot Plates using paper plates with dots on them, create PowerPoints with images that flash in and animate out, create Subitizing Cards using 3×5 cards and dot stickers, do number talks and show an image and let the kids tell you what kinds of groups they see. The important thing is to make sure kids can see groups…don’t just put dots in a line or randomly placed. Grouping allows kids to subitizing, otherwise they will go back to counting one-by-one.
#2: One/Two More and Less, this is not the ability to count on two or count back two, but instead knowing which numbers are one more or two less than any given number.
My favorite activities are pretty much the same as the ones I do for Spatial Relationships with one little twist….I will show the kiddos a visual and then ask them to show me “One More Than what I showed” or “Two Less Than what I showed” etc. This gives them a chance to see the original quantity and then what happens to that amount when we add One More Than or we make it Two Less Than. I also love doing Number Strings that start with an amount and the next image I show has “one more” and then the next has “one more” and so on. Here is an image of a completed string of images I have given kids, the goal is talk about how the images relate:
#3: Benchmarks of 5 & 10 are so important because 10 plays such an important role in our number system (and two 5s make up 10), students must know how numbers relate to 5 and 10.
My favorite activities involve the use of ten frames and rekenreks. Both of these tools/visuals are built upon a 5 & 10 structure. It allows kids the opportunity to see how a number relates to 5 and to 10. Do any of the Spatial Relationships activities, but just use ten frames or rekenreks as your visuals instead of dot patterns. When quantities are arranged in a ten frame in this manner, the structure of the ten frame highlights its relationship to 5 and to 10.
My favorite activities are again all the ones mentioned in the Spatial Relationships, but I like to use visuals that are intentionally highlighting the decomposition of a quantity into different parts OR even starting to use random placements so that the kiddos have to find their own parts. Take for instance the image below, for a LONG time this type of ten frame placement drove me nuts….but what I’ve come to realize is that this type of placement allows for the KIDS to find their own parts instead of always only seeing six as 5 on the top row and 1 on the bottom. Try this image out with your kiddos, I bet you get a lot of different ways that they can group the dots to determine the total amount.
I hope throughout this series that I’ve pushed your thinking about the way we teach “the facts” to our young kiddos. Most of them aren’t ready for the abstract symbols of 5 + 7 on a worksheet, but when you show them a visual of a 5 and 7 in a ten frame it’s amazing how they can determine the total!!! If you haven’t been doing Subitizing or spending time building your kiddos’ number sense I sure hope you take the time to make it a priority. Laying the foundation of number sense HAS to happen or else we are just building a mathematical house for kids that with the slightest storm will come crashing down.
If you are still wanting more, I created a 7 minute video in which I describe WHAT subitizing is, WHY it’s so important to do in your classroom, AND HOW you can incorporate it into your daily routine. Click the button below and I’ll email you a link to a video.
Have you ever seen a 5th grade student count on their fingers to add 7 + 8? Or a 6th grader doing repeated addition on their paper to solve 9 x 7? If you haven’t, I can guarantee you’ve heard the stories from the upper grades teachers during staff meetings as they complain that “Kids don’t know their facts!!!!” Yet, every primary grade teacher spends a TON of time trying to help kids develop their fact fluency. So, why are so many kids still not fluent with their facts?!?!?!?
In Part 1 of this blog post I discussed how the old way of teaching kids to learn isolated facts should be retired and in its place should be the idea that facts are related AND that certain facts come easier than others. In this 2nd part I will lay out the 4 Number Relationships that kids need to have around ALL numbers to develop their number sense and become flexible thinkers with their facts…and all other math concepts!!!
Let’s first get clear on what FLUENT actually means. Susan Jo Russell’s article outlines three parts to being fluent:
Many students have one or two of these, but few have all three. Kids who are counting manipulatives or fingers to solve 4 + 3 are usually very ACCURATE, but not EFFICIENT or FLEXIBLE. Kids who have memorized their facts are ACCURATE and EFFICIENT, but not FLEXIBLE.
We all want kids have fact fluency, yet most people think that just means accurate and efficient. However, here is the rub…if you focus on building a student’s accuracy & efficiency they will NOT become fluent. The key piece to getting kids to become fluent with their facts is FLEXIBILITY. We all have had those kids in our classroom that just “get” numbers. They have FLEXIBILITY with numbers that allows them to work so effortlessly in mathematics. So, what do those kids have that other kids who struggle with mathematics don’t have????
Educators throw that term around all the time, but what exactly makes up number sense? The most quoted quote about number sense says that number sense is “…good intuition about numbers and their relationships” (Howden, 1989). But, I gotta say, I don’t really like that quote. It gives me no ideas for how or what to actually teach to my students.
Never fear, I found the answer in my favorite book, which I now call my math bible; Teaching Student Centered Mathematics. The authors (Van de Walle, Lovin, Karp, & Bay-Williams) outline 4 Number Relationships that kids should develop for numbers up to 20. For this blog post I will keep with numbers up to 20, but really, kids should develop all 4 of these relationships with any number (even fractions and decimals).
These 4 relationships are WHAT makes up number sense. If you spend time really developing these fully with your students, you will get kids like this little boy who are ACCURATE, EFFICIENT, and FLEXIBLE with their facts.
If you didn’t watch that video, WATCH IT! It’s only 30 seconds and I’m going to dissect that kiddos thinking and relate it to the 4 Number Relationships (so it helps if you’ve watched the video) :).
He totally used the Benchmarks of 5 & 10 because he knew that if he could make one of those numbers a 10 it would make the problem easier. In order to make one of the numbers a 10, he had to understand the concept of Part-Part-Whole because he had to “pop off the 1” out of the 6….he understood that he could break that whole (aka 6) into friendlier parts. When he did pop off the 1, he instantly knew One Less would be 5.
So far we saw him use 3 of those number relationships, what about Spatial Relationships??? Most of the time you don’t actually ‘see’ kids using spatial relationships because it is the foundational piece that allows kids to build all the other 3 relationships. So, here is an example of a visual of 9 + 6 that does NOT build spatial relationships to help develop the other relationships:
All four number relationships are essential to helping your kiddos develop fact fluency…not just fact memorization. Even with multiplication facts. Think back to that kid who does repeated addition to solve 9 x 7….if children understand that 10 is a powerful benchmark, that 9 is just one less than 10, and they have built spatial relationships and part-part-whole understandings using arrays or area models, then we see lots of kids who can tell you that 9 x 7 is just like having 10 x 7 but you have to subtract 1 x 7 from your answer.
The coolest thing to me is that once these relationships are built, children develop strategies for their facts that are not tricks…they are real strategies that last well beyond their “facts.” Take a look at these strings of problems to see how a strategy kids develop for their “fact” can become super powerful as they go further into mathematics.
Last summer I left the job that I had for 7 years and have slowly been transitioning out of it (they kept bringing me back in :). With that job I was always busy during the summers doing trainings for teachers, but this summer and most of last summer I’ve gotten to be home hanging out with my 4 kids (YEAH! and EXHAUSTING!). To be honest, I started writing this post a year ago, but never finished. Thanks to @JamieDunc3 for pushing me to actually finish it. 🙂
One thing I’ve noticed being home more with the kids is that I probably do some mathematical things with my children that others do not (don’t ask me about my literacy endeavors with them). Or maybe you do them, but don’t realize the powerful mathematical foundations you are building with them. What I realized as I began writing this post was that these 5 things are not just for parents, they apply equally to the early elementary classroom and to preschool classrooms.
So, here are my top 5 Ways to Build Math Minds, that I do with my personal children:
Most of the time we think of making kids count sets of objects, which is important, but I’ve found so much power in just counting NOTHING. We live in a small town and so if we want to go anywhere it’s a long drive (like 45 minutes to get to a pool for swimming lessons). On our drives there is lots of counting, but we often don’t have objects to count (on our drive to swim lessons we were lucky to see 10 vehicles along the way). So, we just count to count. They count to see how high they can count, they count to a specific number, and they count to 60 for every minute we have left until we reach our destination as we get closer. I’ve noticed how much the kids are discovering the patterns. They have been able to transfer the 0-9 into the decades of 10-90 and on into the hundreds. However, my five year old, still battles the transition into the next decade and says sequences like “Twenty-eight, twenty-nine, twenty-ten, twenty-eleven.” It’s actually really cute and I LOVE that he is getting that connection to the counting sequence.
Counting isn’t just for young kids, either. All kids should be counting. Have you ever had kids count by 2/3, 0.6, or 2x+4? Or, how about instead of just counting by 2 starting at 37, try counting by 2 minutes starting at 12:17pm. As kids count they begin to notice patterns and math is built on seeing & understanding patterns (see #2 below).
I know this doesn’t sound like it would apply to the classroom, but hear me out. Since the time my oldest was three we have been paying him a quarter for each chore he did (and we continue to do it with all of our children). That may seem ‘rich’ to some, but I had some goals in mind that using quarters helped me achieve but you could do this with dimes or nickels, whatever you want. Here are a few things I noticed and focused in on with him.
So, in a classroom you can do the same thing by implementing a classroom store. There are all kinds of posts out on the internet about teachers who have implemented systems in their classrooms where students earn money for doing things in the room. Then you can have them do the exchanges mentioned above, but also they can purchase items from their classroom store. At home, I have my kids save up their money for toys or electronics they want. This gives you lots of opportunities to work on addition & subtraction along with learning about money.
Now, if you have followed me for any amount of time I hope you have seen what a big fan of games I am (check out my free Evergreen Games!!). I grew up in a game playing family and any chance we have we play games with our kids. Any card game, board game, dice game, or dominoes builds powerful math ideas. Often when you play games kids may not attend to the mathematics involved. So, of course I’m always talking to my kids about it. For example, when we are playing Chutes & Ladders, I will ask them questions like “How many do you have to roll to make it to that ladder?” and “What do you NOT want to roll?” when they are close to a slide.
As much as I love playing games and a lot of them do have math inherently part of them, I just can’t help myself and I’m always asking my kids questions that pull out even more mathematics or just get them attending to the math so that when I’m not right there playing with them, they have basically learned how to notice the math in situations they may not have noticed before.
The consistency of routines builds the expectation of relying on patterns. I remember all the books I read when my first child was born that talked about the power of developing a routine for your baby. The same holds true throughout their lives. These routines establish predictability. When you trust that predictability, you can then use it to create order in your life. The same is true in mathematics. The Standards of Mathematical Practice also acknowledge the power of structure (Math Practice 7) and repeated reasoning (Math Practice 8). Children need to be able to see the structure of mathematics and figure out how to use that structure to make sense of patterns they notice. This is what I’ll be working on with my 5 year old who gets the pattern of the numbers (eight, nine, ten, eleven…) but isn’t applying it correctly to the counting sequence as it grows higher (Twenty-eight, Twenty-nine, Twenty-ten…). They notice patterns, but often don’t understand how it applies.
The more we establish routines and patterns in our children’s lives the easier math becomes for them because math is All About That Pattern (not the Bass).
I have asked my kids “How do you know?” for so long that I don’t even have to ask them anymore. As soon as they figure out an answer they tell me their thinking process…they just know it’s an expectation. Or, instead, they just think out loud to begin with. Below is a video of my oldest (7.5 YO) that I took just the other day. I was sitting in the dining room and he was in the kitchen preparing a treat we like to make. I heard him count “6,12,24,30.” I asked him to repeat it and he said the exact same thing again. I thought he was trying to count by 6s but doing it incorrectly, so I went into the kitchen and asked him to show me what he was thinking (Look at the picture below to see if you can figure it out, then watch the video). Now, I was busy in the dining room working on stuff and I could have just said “No, Bud, it goes 6, 12, 18, 24.” But stopping to hear his thinking No.1 it was an awesome opportunity for me to see his cool thinking and No.2 I didn’t dismiss his thinking as wrong and correct him which inadvertently tells kids to stay on the “right track” and not think creatively with numbers.
With young children it is my belief (and research backs me up…Kamii, 1999; Clements & Sarama, 2004….just to name a few) that children should experience mathematics in a play-based way along with lots of visuals and discussion about what they notice mathematically in their world.
What are your thoughts? Do you have a favorite way to Build Math Minds in young children???
Welcome to my piece of the Balancing The Equation blog hop book study! If you are just joining in, the blog hop is almost over…however, there are links at the bottom of this post to everything so that you can go back to the posts about the other chapters. Make sure you head over and watch the replay of the webinar I did yesterday with Matt Larson, one of the authors of Balancing the Equation. The webinar was FANTASTIC and there were tons of comments from the live attendees like “This webinar should be seen by ALL educators!!” So after you read this blog post, go watch the webinar.
My part of the blog hop is the 2nd half of Chapter 4, which is all about how to stop the pendulum swing and find an Equilibrium Position….which, being a Recovering Traditionalist, is right up my alley!
“Let us teach mathematics the honest way by teaching both skills and understanding.”
-Hung-Hsi Wu, Professor Emeritus of Mathematics, UC-Berkeley
Pages 75-85 of Chapter 4 has four main ideas: Perseverance, Practice, Feedback, and the Use of Technology. I’d like to focus in on Perseverance and Practice.
One thing I think that helps build perseverance is ensuring that the tasks we provide for kids allow for productive struggle and help develop kids’ growth mindsets around mathematics. Balancing the Equation does a nice job of explaining each of these but here are two more resources:
Both perseverance and growth mindset are, at their heart, pushing kids to become more than they are right now by getting them to a place where they are slightly uncomfortable and they might potentially fail. Now this is super hard to do as a parent and as a teacher because we want our kids to succeed. So, to illustrate the point I’m going to pull a line from the movie Kung Fu Panda 3 where the Kung Fu Panda (Po) is talking with his master:
Po: “I don’t know why you ever thought I could teach that class”
Master Shifu: “Oh I knew you couldn’t”
Po: “What!?!?! You set me up to fail? Why!?!”
Master Shifu: “If you only do what you can do, you will never be more than you are now.”
The practice section of the chapter does a good job of giving parents and teachers ideas of what practice in class and homework should look like. I’m honestly not a fan of homework and during the webinar, but Matt explained it perfectly when someone asked a question about the use of homework and reminded me to keep a balance. He said:
Anyone who is good at whatever it is that they do practices. Learning mathematics is no different. Students need practice. Now, that practice needs to be appropriate. Practice should be based on understanding. It doesn’t need to be lengthy…Has there, in some cases, been way too much homework? Absolutely. Has there been inappropriate homework? Absolutely. But we also can’t throw all of that out because someone tells us now that it’s inappropriate.
When he said that it hit me that homework is another area that the pendulum swings one way and then the other. Some people assign a ton of homework, some assign none. It’s time to stop the pendulum swing and find a balance of using homework appropriately. In the primary grades there may be no homework or the homework is a math game. But kids do need practice if we want them to get better and I can also see how practice can help build perseverance and the growth mindset.
A lot of times when we swing towards helping kids develop their conceptual understanding we might only do one or two problems during math time. This can be wonderful for helping develop their understanding and their perseverance…however, kids aren’t getting a lot of practice. So, one of the challenges is determining when to have kids persevere and when to be practicing.
My belief is that when the concept is new, then we should be spending more time doing rich, interesting tasks that allow for productive struggle ( don’t forget about that talk by Robert Kaplinsky) and building of conceptual understandings. Once kids start gaining familiarity with a concept and have a base of that conceptual understanding then we can move into purposeful practice. For example, kindergarten kiddos should NOT be doing timed tests and worksheets full of addition and subtraction problems. They should be modeling and acting out problems that they encounter in their lives that require them to add & subtract. On the other hand, if you have 6th grade kiddos who still don’t know their math facts AND you know that teachers in previous years have been building the conceptual understanding and they still don’t have it…I think there comes a time when we just need to focus on practice.
These three items come from a presentation I do on Family Math Nights for local schools and the message is for parents, but is equally important for how we “help” in the classroom.
If you want some ideas for things to do in a Family Math Night at your school, download my PDF of 3 recommended set-ups (and resources) for a Family Math Night.
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I get emails a lot from teachers asking me what I think of this program, or that app, etc. Well, two times in one week I got asked about the math game Prodigy Math. I hadn’t heard of it, so after the second question I decided to look in to it, especially because the second person was at a school that uses Dreambox (which I LOVE) and their district is looking into using Prodigy instead.
I went in and spent 20 minutes in there playing as a student because I have no experience with the program. So here are my thoughts…but again only after 20 minutes of exposure…so I’m sure I’ll get heat on this, but I’m posting it anyway because what I saw in 20 minutes is so much like every program out there and I wanted to give you all some of the things I saw from the kid side.
That is my first recommendation….when looking at an online program, go in and play it as a KID! Don’t trust the marketing the company puts out about how great their program is and how it meets the standards and uses models to help the kids solve problems. Go play it and see what it feels like/what’s required from the child.
So, here is what I saw:
As I was looking over my presentation for this week at NCTM and getting sidetracked by checking Tweets about NCSM (which I didn’t get to attend this year), I saw a few tweets about Steven Leinwand and Patsy Kanter’s presentation at NCSM and how well it connects to my presentation for NCTM (tomorrow, 4/14) and it sparked me to write this post about building addition and multiplication fact fluency.
I wrote a book a few years ago that included this addition fact chart and since then I also created one for multiplication:
I share these with teachers when I do math professional development trainings, but I’ve never written about them on here. The idea is that the old way of teaching kids to learn isolated facts should be retired and in its place should be the idea that facts are related AND that certain facts come easier than others. Thus there is really only 4 types of facts that students need to “learn” that then help them with all the other facts:
4 Types of Addition Facts:
Green: Make 10
Blue: 10 + something
Purple: Adding Zero
4 Types of Multiplication Facts:
Purple: Properties (x1 and x0)
If you focus heavily on those 4 types of facts PLUS building your students’ number sense so that they can use number relationships to help them derive the related facts to those 4 types of facts, teaching the “facts” becomes a whole lot easier. I.E. If a child knows 3 + 3 (an Orange fact) AND they know how numbers relate to each other, then 3 + 4 (lighter Orange fact) is a piece of cake. That is the case for all those lighter colored facts in both charts, if you know your x10 facts (Red fact), that can help you with your x9 and x8 facts (lighter Red facts)….but,only if you understand how the numbers relate to each other.
So, stay tuned for Part 2 of Fact Fluency where I will tell you the biggest mistake we make when trying to teach fact fluency…..and that I am saving until AFTER my presentation tomorrow or else you wouldn’t come to it. But for those unable to make it, I’ll share after the session.
Many of you know that I am a HUGE proponent of using math games to build your students’ mathematical minds. However, the biggest complaint is that math games take so much time to prepare and then you have to teach the new game to the kids before they can go off and play the game on their own in math centers. Then it’s just a cycle of rinse & repeat every time you want to introduce a new game. Well, in this post I will share with you a way to decrease the prep time for math games, both for you and for training the kids on the games.
The secret is a thing I call Evergreen Games…. I give my childhood years a little credit in the name here. I grew up working on my family’s tree farm, so for those of you not familiar, evergreen trees stay green all year round thus the name ever-green.
These games are Evergreen because they can be used with ‘ever-y’ 😉 math concept, plus, many can be used with any subject as well. The BONUS is that once you teach the general rules of the game you don’t have to re-teach the game, you can just swap out the concept. For example, Memory (finding two cards that ‘match’) is an evergreen game because you can play Memory with any content. I can make a deck that the kids have to find a card with a number and find a matching card that shows a visual representation of that number. Or I can make a deck where they have to ‘match’ two cards that add to 10. Or I can make a deck that shows a decomposed area model and the kids have to find the matching expression that shows the distributive property. Each time I swap out the content within the game, I don’t have to teach how to play a new game because they already know. I just have to explain what is considered a ‘match.’
The concepts you can ‘teach’ through Memory are endless and thus it makes the game Evergreen. There are 5 games that I feel are evergreen math games; Memory, Capture 4, Bump, I Have Who Has, and Difference To. I’ve created a PDF that gives you an explanation of each one PLUS three pre-made games for each type of game…yup, that’s 15 versions of these evergreen math games. Click the image below and I’ll email them to you.
Also, if you haven’t been following me over on Periscope, get over there!!!! It’s a fun way to interact as you get to watch me play math games with my personal children every Saturday. The videos are only available for 24 hours and then they disappear. So, go on over to Periscope and search for me as @BuildMathMinds. If you want to see my old Periscope videos, I do ‘katch’ them and you can see the math games I’ve played previously over at https://katch.me/BuildMathMinds