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
Writing numbers correctly is often a struggle for young kids. Most kids can say the numbers but when they go to write them down the way we write numbers doesn’t align with how we say them. This causes confusion in our students and is one of the reasons students lack place value. In this post I will share with you how my 1st grade daughter struggled with writing numbers (she would write 501 as 1500 and as 5001) and how I helped her in one night, while I was cooking diner (aka…I really didn’t do much, it was all because of the tool I had her use).
So, this all came about just this week. My mom was watching my two school aged children when they got home from school. When I got home, my daughter, Sierra, couldn’t wait to show me all the numbers she had written in her journal while she was home with Grandma. She handed me her journal and as I took a look she said, “I know that I was doing it wrong to start with and then I changed it to the right way towards the end.” This is what I saw:
I am used to seeing kids write numbers similar to her “correct way”…she was trying to write 504, but wrote it 5004…How many of you have seen that in your classroom???
However, the way she knew was incorrect, I had to ask her about. She said “Well, I put 400 then put a 1 in front.” So in that first picture she has 401 written as 1400, 501 as 1500, 801 as 1800 and so on. And seriously it was on and on…she filled up almost a whole page before I got home, all the while writing the numbers incorrectly (even when she thought she had changed to the “correct way”).
So, how did I help her? I pulled out my favorite tool for helping kids understand Place Value…Place Value Cards! These are also known as Hide Zero Cards or a variation is known as Arrow Cards.
The image above shows her modeling 13 using the cards, but it wasn’t easy for her to start out with. She was fine showing 1-10, but when she got to 11 she wanted two of the 1 cards. I asked her “What is a 1 and 1?” Her response was “Eleven.” So we talked for a minute about what really makes 11, how it isn’t made from 1 and 1, because when you put 1 and 1 together it makes 2. She caught on very quickly as the cards helped her to see 11 as 10 and 1. Then, while I was cooking dinner, she continued making numbers, writing them in her journal (correctly this time), all the while building her place value:
The cool part about the cards is that in the upper left corner is the VALUE of each digit. So when she puts the 90 and the 1 together she sees it as 91, but can also see it as 90 and 1. She continued on building numbers correctly until 110, and this is what she put together:
It looks correct, right??? But, let’s zoom in a little closer to see how she actually made that number:
Now when this happened my husband said she was right…I said she was wrong. We got into a little debate about it :). I see his side that, symbolically, she is showing 110 correctly, BUT she isn’t showing me understanding of place value…she used the 100 and the 1 to show 110. Often we get wrapped up in helping kids to write numbers in “standard form” but when they do write numbers in that form, they loose the value of the digits. I am such a big fan of these cards because children get to see the standard form of a number but they ALSO get to build their place value understandings. Here is my daughter explaining how she showed 153:
Now, I know she is using the value written at the top to help her say the numbers…but that’s what you want. In the early grades we need to build that solid foundation of the value of the numbers so that when we move to just the standard form students know the value of each numeral. I love that these cards help kids see the value while also seeing how to write the number in standard form. I’ve already heard my daughter getting better at reading larger numbers. Like 134…it’s not 1, 3, 4 anymore when she says the number…it has turned into 100-30-4…YEAH!!!
And just an FYI, I did not make her sit down and go until 153. She just kept doing it, all night. We had to make her put the cards away during diner. 🙂
Leave me a comment and let me know if you’ve used Place Value Cards before. How did they work for your students? Do you have other ways to help kids learn to write numbers which also help them understand the numbers??
Yesterday, during a PD session, I was asked my thoughts on timed tests. With all the talk about growth versus fixed mindsets, the topic of timed tests for facts has become a popular topic. My thoughts may not match up with the popular thought of the moment, but I’m going to post it anyway. (If you don’t know about growth versus fixed mindsets check out this quick synopsis.)
Our education system is famous for its pendulum swings and math education is right in the thick of some now. One is the debate about timed tests. For quite awhile the pendulum has been over on the side of using timed tests daily. Now, that pendulum is swinging to never using timed tests. (See this article and this article.)
While I’m not a big fan of timed tests, especially the way they are often used in classrooms, I am a firm believer in moderation…in everything in my life. For example, I have a massive sweet tooth. If I try to go without candy, a few days later I’ll end up eating 5 candy bars in one day. Instead, of going back and forth between the two extremes (no candy to 5 in a day), I find it easier to allow myself a candy bar now and then when appropriate. And that is where I stand on timed tests. If we go to the extreme of no timed tests I think we are going to see it not work and the backlash to it will swing us back to daily drill & kill. Instead, lets help teachers to learn the difference between using timed tests to TEACH versus using timed tests to ASSESS.
Teachers who do timed tests every day as their “fact practice” are using the tests to teach. However, by using them in that way teachers are facilitating a fixed mindset in their students; “I must be bad at math because I’m not good at these tests,” or “I must be good at math because I’m fast at these tests.”
Teachers who use timed tests every once in a while to assess how well their students are progressing with their understanding and development could actually use the tests as an opportunity to develop growth mindsets in their students. What if every once in a while (maybe once a month, maybe every two weeks??) we assessed our students to see how many facts they can get correct in a minute….but instead of telling them they didn’t get enough correct in that minute, what if we had them keep a graph every time so they could chart their progress? Or try having a set number of facts (ones that particular child has been working on building their understanding of) and time them to see how long they take to do them all…then later assess those same facts again to see if they can get through the set faster than they did previously. Wouldn’t this help them see where they are and encourage them to grow in their understanding and fluency?
I am intrigued, yet concerned with the movement to do away with timed tests. I’ve seen the damage they can do to kids when they are pressured to perform, especially in front of the whole class (I’m sure we all have those memories of standing at the chalkboard). However, just because teachers in the past might have taken timed tests to the extreme by using them daily and in ways that put extreme pressure on students, does that mean we shouldn’t encourage kids to get faster at math?
I am a firm believer in the power of building a solid foundational understanding that helps children think flexibly about mathematics. I encourage teachers daily to use number talks, put mathematics in context, and go deep with their students to develop a conceptual understanding. I complain all the time about how Common Core has put the introduction to multiplication & division starting in 3rd grade and by the end of that year they need to know the facts from memory, hell, I even wrote a book to help teachers develop students’ flexibility with numbers as a lead into developing fluency with addition and not once in there did I ever encourage the use of timed tests. But I still think there is a time and place to encourage kids to become faster with math (notice I said faster, not FAST), because I’ve worked with school districts who only emphasized the conceptual understanding of math and did not spend time helping develop procedural fluency. I see that without both, our students can often struggle to work with deeper tasks because they can’t work fluently with the “menial math” in order to get to the deeper understandings.
What do you think? Could we actually use timed tests in a way that helps develop a growth mindset in our students and helps them to see how they are growing and becoming more fluent in their math understandings instead of the way we all remember timed tests being used?