From K to 8: Math

FROM K TO 8

We sat down with a select group of teachers to talk about what math learning looks like over the K-8 curricular arc. Kindergarten teacher Jennifer Southern, 3rd grade teacher Eileen O’Hare, 4th grade teacher Donna Young, and 6th & 7th grade teacher Beth Bonzell offer their thoughts and expertise on the development of math skills and mindsets from kindergarten through 8th grade.

FROM K TO 8 math

We really emphasize “ being able to explain your thinking, showing more than one way to get to an answer, and reviewing learned skills so they obtain a deeper understanding of the concepts through constant practice.

What is the basic philosophy in your math class?

Jennifer (Kindergarten teacher): Our basic philosophy aligns with the Bridges math program, which is a hands-on way to teach math. Through the use of visual models and games, students develop good number sense and a conceptual understanding of math.

Eileen (3rd grade teacher): In 3rd grade, we really emphasize being able to explain your thinking, showing more than one way

to get an answer, and reviewing learned skills so they obtain a deeper understanding of the concepts through constant practice. Though we use some visuals, it shifts in 3rd grade to more problem solving and practicing. We also supplement with Reflex math and other visual aids.

Donna (4th grade teacher): Our general philosophy in 4th grade math is that everybody can do math.

well, I want the students to develop their mathematical thinking skills. I want them to be mathematicians and have fun with math.

What strategies and/or activities do you use to boost engagement and interactivity in your math class?

Jennifer: For every concept that we’re teaching, we practice during center games, which are hands-on and partner work activities. This week in class, for example, we’re focusing on presidents, U.S. symbols, and money. Students are learning to identify coins and their value, and how to exchange coins for exact value.

Eileen: In 3rd grade, we start with concrete concepts. For example, if we’re learning place value, students are introduced to manipulatives. And then we’ll use those same manipulatives for addition and subtraction of multi-digit numbers with regrouping, where they have to actually see how to regroup by trading in the 10s for 1s or the 100s for 10s and so on. And then they bring that new knowledge to paper in the context of an equation.

Donna: We don’t play a lot of games, but we do use hands-on materials that help students learn strategies for breaking a problem down. Generally, when kids

take the time to slow down and use their materials, they can overcome their initial block. One of the biggest things we try to shift is their belief that if you can do it in your head, it means you’re good at math. It’s actually the exact opposite—if you can show every step of your work on paper, that means you’re good at math. The students who are struggling the most just don’t believe in themselves, so our activities provide them with strategies to really build that confidence.

Beth: Whenever I can, I include a game in the curriculum and try to connect it to real life situations. I also avoid long lectures wherever I can because I want students to talk about math with each other. For example, in 6th grade, we play a game called Integer Roll to practice their integer operations. It’s kind of a dry subject, so I created a game board that has a bunch of different positive and negative numbers on it, and I have three dice. Two have negative numbers on it, and one has positive numbers on it. They roll the dice and get to pick two of the numbers to do whatever operation they want to try to get an answer that’s on the board. They try to get four in a row and beat the other person. It’s a game and it’s competitive, so they’re more likely to talk about what they’re doing. I also like to do hands-on activities when we can; recently 7th grade students built scale models of pyramids to better understand proportions and scaling.

What are the developmental stages that you encounter with your students? What do you see kids experiencing regarding math each year?

Jennifer: Whether it’s math or writing, and based upon their preschool experience, they’re all coming in at different levels. And so through whole-group, small group, and center group, we meet each child where they’re at. With the Bridges program, there are different components—teacher-led lessons, number corner, and work spaces—through which we do individual and flexible group work to teach different target skills.

Eileen: By the time they get to 3rd grade, students’ skills are a little more similar, but there’s always a range of ability, and we differentiate to support these varying abilities. We are all working on the same concepts and skills throughout the year, but with different levels of ability. One of the supplemental materials I use is an activity on the iPad called IXL, which is a great way to differentiate.

Donna: In 4th grade, students seem to have decided whether they’re good at math or not, which is something we work hard to shift. And, like I mentioned before, they also believe that if you can do math in your head, you’re good at it. They don’t yet understand the value of showing your work clearly and neatly, and that’s one of the biggest goals of this grade level. One of the biggest challenges is that quick is good. Fast is good. It’s usually not. Fourth grade is an important year when habits develop, not just in math but across the board. And if you can really hit home with good habits like highlighting, showing all your work, and labeling your units, then it lasts.

Beth: The biggest thing that I see in 6th and 7th grades is students’ mindset—their beliefs about who they are as mathematicians and their ability to do math. There’s something about math that makes students want to label themselves as good at math or not good at math. Sometimes their negative beliefs about math are based on having trouble with fractions or multiplication facts or certain concepts. So that’s always a battle, to change that mindset. Students at this age level also tend to compare themselves to each other. They’re really concerned with how people see them or if someone else is getting to the answer faster. I want my students to see math as an adventure—to see what we can discover and find. But it’s hard sometimes with

this group because they’re so concerned with their im- age and their grades. They’re not yet naturally viewing their education as a journey.

What are the learning outcomes from your class as they move to the next grade?

Jennifer: In kindergarten, to name a few learning outcomes, students are identifying and writing their numbers to 40, counting to 100 by 1s, 5s, and 10s, and by 2s to 20. They are identifying and representing numbers and comparing numbers with greater than, less than, or equal to. They are adding and subtract- ing within 10 and identifying 2- and 3-dimensional shapes and comparing their attributes. They are telling time to the hour and focusing on measurement. Stu- dents are graphing, learning about money, and identi- fying coins and exchanging them for the same value.

Eileen: We also want them to understand how to differentiate in word problems and know what to do when they encounter a multi-step word problem. Also, to understand how to read and write basic fractions, show their thinking, and solve problems in more than one way. By the end of 3rd grade, we aime to have students going into 4th grade have their basic addition/subtraction facts automatic and accurate using the Reflex math program. We want them to have a pretty good handle on multiplication facts—not necessarily mastery but to know most of them. We also want students to know at least two strategies for multi-digit addition and subtraction, and one of those strategies should be the standard algorithm.

Donna: In general, they should have learned how to use a highlighter to isolate facts and really understand what’s being asked. We hope they also learn the value of showing their work, that quick is not the way to go, and that there’s always more than one way to arrive at the same answer. Also by the end of 4th grade, we want most kids to become math confident.

Beth: There are the obvious math skills like being able to do arithmetic and write and solve an equation by the end of 7th grade. But more importantly, I want them to have the tools and confidence to know how to figure out new types of math problems they encoun- ter. I tell them a lot, if you don’t remember how to do it, that’s okay, because you can probably figure it out. Fourth grade is an “ important year when habits develop, not just in math but across the board. And if you can really hit home with good habits like highlighting, showing all your work, and labeling your units, then it lasts.

So much of mathematics is logical, and I want them to have the confidence and skill base to figure things out when they don’t remember. I also want them to know how to use their resources because there are so many math resources out there. That way, they can be successful no matter what math class they’re put into.

What’s the most exciting thing you’re learning or developing as a math teacher?

Eileen: The Bridges program has opened up a new way to look at strategies that I didn’t learn when I was learning math. As a kid, I learned standard algorithm for everything, and I didn’t always learn why and how things worked. So, knowing that I’m teaching kids the foundation of how multiplication works, how a standard algorithm makes sense, using a number line for multi-digit subtraction, and breaking numbers apart, that’s pretty exciting to me.

Donna: As a teacher, you think you have it all figured out and no matter what, a student will come up with another way to do something that you haven’t thought of. That’s always pretty cool. Also, I feel like our school does a super good job of challenging strong learners. I also put a lot of energy into really figuring out the best and most productive way to work with the kids who are struggling.

Beth: Right now, I’m really focused on trying to find other ways to assess students and determine if they are developing the skills that I’m teaching. For example, I’m not calling their tests “tests,” they’re called checkins—because that’s what I’m doing—I’m checking in to see what they know. And if they don’t do well, then they can retake it. This way, it’s viewed as more continual and not final or set in stone. So much of math is built on previous knowledge. We’re always learning. I go online, read articles, and dive into my own research as a lifelong learner. Youcubed by the Stanford

Graduate School of Education is one online resource I use. Particularly Jo Boaler, a Professor of Mathematics Education at Stanford University and a faculty director of Youcubed, has been a great resource for me. She focuses on the brain science behind learning math and talks a lot about mindset and how that affects learning.

How do you use technology to teach math?

Jennifer: I model everything first through ImageMate on the smartboard. So when they go back to their seats, they’re confident. And then I have a lot of interactive games from Lakeshore Learning, an online resource, to teach different number sense like identification of numbers, number sequence, and comparing numbers and quantities.

Eileen: We use the iPads with Reflex math to practice basic facts. We also use the iPad to play math games such as numbers league, math bingo, and pop math. I’ve recently implemented IXL this year, which is an online learning tool, because it’s a great way to differentiate. And when you get an incorrect answer, it explains why. I’m also always using the smartboard. Having the visual image up there is huge because the kids are so engaged with it.

Donna: I don’t know if I could teach math without the Smartboard. Because it’s interactive, they can come up and engage with it and it just helps it to be clear visually and in an engaging way.

Beth: It’s hard in some areas in math because typing math is harder than writing it by hand. But there are resources that can help with data, like spreadsheets. There’s a website or an app called Desmos, a free online graphing calculator. Students can enter data and it’ll graph it for them. It’s very versatile and helps them visually see the information, which is important, and it’s faster than drawing by hand. I also use games online. What are the benefits you see students getting from learning math together in a collaborative, discussion-based community of learners?

Jennifer: It makes learning real and engaging. And it also teaches more than just math skills. It teaches those collaboration skills—that sportsmanship. We have a saying in class, “It doesn’t matter who wins or loses, it’s how you play the game. You’re going to win some, you’re going to lose some. But you’re a good sport. We say good luck before, we say good game after.”

Eileen: They learn from each other. They are doing partner work and explaining their thinking to each other. If students are explaining their thinking and it doesn’t make sense, then they talk about that and they show each other why it doesn’t make sense. I just think collaboration and kids being able to hear each other and see each other’s work is important. And you’re a team, you’re a community.

Donna: Talking about math and explaining what you’re doing is crucial to learning. If you can explain it, it means you’re really understanding it. Also, the kids who generally like to do individual learning like Khan Academy can be pretty rigid in their thinking. And conversation allows them to see methods and hear ideas that they would never have thought of themselves. There are multiple approaches and nobody has all the best ideas individually. You’ve got to get them from each other.

Beth: To me, working together in math is huge because students need to discuss it with each other. For me, talking to my peers when I’m trying to learn or develop something is so much more powerful and the learning is deeper. I love it when students get in math “arguments.” There’s so much math learning happening in that math argument and that can’t just happen in an individual setting. You need someone to banter with.