Teen numbers in English are STRANGE. I mean, look at these weirdos in the first half of the numbers:
11 – eleven (not one teen)
12 – twelve (not two teen)
13 – thirteen (not three teen)
14 – fourteen
15– fifteen (not five teen)
16 – sixteen
17 – seventeen
18 – eighteen
19 – nineteen
When we start teaching these numbers to our youngest learners, we often take for granted the strangeness of the names of these numbers.We are used to the names and counting with them…we don’t even question or notice how strange they are. The first half of them don’t even follow a logical pattern! In addition, I often hear videos being played where the singer is saying “Teen numbers start with a 1!” Actually, teen numbers are written with a 1 first yes, but that one in the teen number does NOT have a value of “one”.
Teen numbers are the basis of place value. I still work with third graders who will see 10+6 and start counting on their fingers to figure it out! How they don’t know that shortcut always shocks me. I think somewhere we failed them in understanding the value of the 1 in the teen number.
Beginning to learn about place value for our early learners needs to include some key things. We have to spend a lot of time letting them play with the numbers, get comfortable with different representations of those numbers, including finding the ten that is hiding in the number. In many state standards teen numbers are expected to be thought of/written as a group of 10 and some more ones. YES, this is key! In the photo below you’ll see some of these various representations with fingers and numbers. In future posts, I’ll show some other representations and tools that work AWESOME.
In addition concrete tools should also be incorporated as much as possible! Here we are using units of one and ten to make it match the fingers. Very powerful for them to connect a tool from their body (fingers) to a more abstract tool (base ten blocks).
Never skimp on those teen numbers kindergarten teachers! Future grade levels depend on you to explore these concepts deeply with your students.
Flashback to 4th grade when my math teacher asked me to subtract mixed numbers, my palms were sweaty, and I felt like I was looking at the demonstration cross eyed WITH brain fog. If you are a loyal follower of my blog, you might know about my past traumatic experience with fractions. It was not pretty, and the last thing I want is for any student of mine to feel this way.
I created the video below to try to show regrouping with visual models. I can’t stress enough how visual models have changed and revolutionized my teaching. It almost never fails that when students struggle, I’ve forgotten to include a visual to help them “see” the math. I am not alone, check out Dr. Jo Boaler, Berkeley Everett and many others who truly understand how we can deeply teach mathematics through visual representations.
The video below will speak for itself, but if you don’t have time to watch it (it’s less than five minutes), the idea is that you can regroup whole numbers into fractions just like students do with base ten models. It’s SO MUCH FUN when students get to learn this way with you. Please check it out and let me know what you think!
I love to be sneaky with using vocabulary in the classroom. One of the ways that I like to embed it is through game playing. For example, instead of using two color counters, we pull out the pattern blocks. Instead of saying “I want blue!”, I tell the students that I would like them to sound more like a fancy mathematician. Instead you must say, “I would like to play with the trapezoid, please.” It causes a laugh with the kids, but even better it starts to get them to use those words more regularly and with purpose. Plus they love to say the word trapezoid! There is a nice benefit to getting these words in their mind early, they’ll start to connect that vocabulary to real world examples of lines, points, segments and other geometry terms.
There is a natural reason to determine the difference between the two tools in the photo below, and to examine how the rhombus and trapezoid are different so that they know which one to pick. This can start as early as the primary levels! Most pattern block sets come with hexagons, triangles, rhombuses, parallelograms, and trapezoids.
Should we allow students to use their fingers when learning and solving math facts? I’m going with an emphatic YES…with one condition. They have to be using them efficiently. For example, are they solving 10 – 4 by putting up one finger at a time to show ten fingers, and then removing one at a time? That might not be the best finger usage. However, all 10 fingers up and putting four down in one “poof” makes TOTAL sense mathematically. We are letting students make sense of quantity and structure numbers when we do this.
There is abundant research out there about finger use, and the deep connections it makes for our student’s fact fluency knowledge.
The best part? Students feel SO confident with learning subtraction facts, which is usually much harder to teach and learn than addition.
I used to hate teaching math when I began my education career. It was scary! I didn’t know what I was doing and had ZERO confidence. 18 years later, you’ll find me traveling across the country teaching other teachers the best ways to teach math. My number one goal in starting this blog is to help YOU love to teach math. However, blogging isn’t the main form of learning these days. I find myself going to social media to learn so many things. So…I am (very nervously) trying a new idea where I do things way beyond my skill and comfort level:
I have started an instagram account (a professional one), you’ll find it by clicking here: @beyondtmath
My goal is to release as many teaching tips as I can using reels or video posts.
I want every video to be under 60 seconds, and something you can try immediately.
I plan to try to do these weekly.
It feels worrisome to put myself out there, but I’m going to be posting all those tips that I tell teachers and I hear back: “I wish someone had told me this!” or “I love that idea!” or “I am SO trying that!” or “That finally makes sense to me!”
My first post is about how to develop number sense with cubes. I hope you’ll venture over and find me! (Please, please, please!)
Flashcards are the worst for practicing fact fluency. I’m just going to say it. They are JUST the worst. Actually maybe timed tests are the worst, but we will tackle that another day. Talk about taking all the joy and fun out of mathematics. They only work for students who can memorize, and while you might be thinking “Hey! They worked great for me!” You were probably one of the lucky ones. Even if they DO work for SOME kids, is that the best way to practice math facts?
My favorite, favorite, favorite thing is to have students play with multiplication first. Multiplication fact practice doesn’t have to be devoid of all fun. You start out by purchasing really great things that are high interest for children, and then find a way to help them put them into groups. We use fake eggs and egg cartons, pop its to show rows with pressed in dots, penguins on icebergs and the fan favorite among every student I’ve ever done this with…is the mini shopping carts with groceries.
As long as the items can be put into groups, they can represent multiplication. Now in this case you can actually pair flashcards with these items. Pick a flashcard, then build it. This will help your students make some meaning out of the number sentences. The more students build their multiplication facts, the faster they learn the meaning of multiplication conceptually. The faster they learn the conceptual part, the easier time they have visualizing multiplication. The easier time they can visualize multiplication facts, the faster they can learn strategies for skip counting or doubling groups. It snowballs quite quickly! I’m a huge proponent of games and activities like this vs. time tests and flashcards. This does a lot of great things for students:
It reduces anxiety big time and adds in play, which research shows allow you to be a better problem solver.
Students start to look for shortcuts (our brains are wired this way), and they will begin to skip count rather than count by ones to find the total.
It normalizes the use of tools for ALL students.
Eventually these items won’t be needed, as they will eventually become tired of building the items and will come up with other ways to solve multiplication facts.
This is an awesome thing to work on at your teacher table, or to have as an ongoing station for students to play around with during independent work time in your math workshop.
The switch from working with addition/subtraction to multiplication/division for students can be pretty jarring. With multiplication, everything they have previously known about abstract symbols suddenly changes from straight up just adding what you see to adding groups or rows. While many students can successfully use multiplication, I will often see the writing of number sentences to be full of small errors. I’ve found the best way to clear up those errors is to use really engaging REAL WORLD multiplication examples.
For instance, in the image below, you can see five lily pads with 3 frogs on each lily pad. The number sentence that matches this story could be 5 x 3, because there are 5 groups of 3 frogs. I love to help them make sense of the symbols and numbers by simply labeling that number sentence (see pic below), and then having them work through several examples.
You can have them practice this skill in a few ways:
Write the number sentence and have them build it. Do it over and over, until they get used to the first number being how many group they need.
Build it for them and have them write a matching number sentence.
Give them a number sentence, build it incorrectly and then have them find your error. (THEY LOVE THIS!)
The more practice they get with these concrete items, the faster it will settle in, and the more confident your student will feel!
Today while working with some third graders in my intervention group, I had that moment that I know you all have as a teacher. When you pose a story to be solved (I’m tired of calling them problems…is it ok to call them stories to be solved?), and the students just start panic crunching numbers. Like throwing numbers around and adding up every number they hear in the story.
Today it was simple story, one that I know these three could probably picture in their minds, or at the very least draw:
“There are two trees, each one has 15 cherries. How many cherries are there altogether?”
I’ve been reading about the benefits of using vertical white board surfaces, to get students to do more thinking. One student wrote 2 + 15 = 17 with her partner. She was CONVINCED (and there was nothing I could say to change her mind), that because there were two trees and 15 cherries, those two numbers must be added together. The other student knew to add 15 and 15, but used an inefficient strategy of drawing each individual cherry on the two trees. I will address the inefficient strategy issue in another post, but today I really want to address the idea of students just crunching numbers without thinking.
I think for too long, students who struggle with math have been led down this road of not have the proper exposure to tools, drawings and then the abstract notation of mathematics. I see tools disappearing as early as first grade in classrooms, which we all KNOW are critical in child development. Not only are tools disappearing, but I see math programs that encourage abstract notation before connecting tools and drawings to that notation.
Take for example, 3 x 14 or 14 x 3. This student was able to build it with tiles (showing an understanding of area), then was able to take a shortcut and show an area model (without the tiles inside), and eventually use partial product notation to solve.
Somewhere along the line, students who struggle to problem solve missed these connections. They look at you with panic in their eyes as they try to figure out what you are asking and what they have to do. Then in pure desperation instead of solving, they just start mindlessly crunching numbers, not making any connections whatsoever.
Here are some ways that I intervene:
Make tools available, have them act it out first, then draw what they made. Don’t worry at this point if the notation is correct, can they make sense of the problem with tools? At first it might be helpful to provide the tool, then the next time have a few different ones to choose from. Start watching if they are choosing an appropriate tool.
Write a hint in a different color. Explain that the hint will lead them in a different direction, and that they need to “unthink” some part of what they did. In the case of the cherry trees, I drew an additional tree next to the one they had already drawn and wrote 15 under it. That little hint made all the difference.
Look for places they are actually making connections and compliment them on that. The student who wrote 2+15 was actually thinking about multiplication! I told her that, but then asked the two of them to back up and think about what that would mean if they multiplied those two numbers. Find SOMETHING that you like about what they did. Those compliments stick, and they remember it again next time and will use it.
Don’t give up on them, and have high expectations. I hear this said over and over again, “My ‘low’ students have a really hard time problem solving.” When you call them low, you will expect nothing but low performance.
Try not to give the answer at the end. Explain that problem solving is about thinking, not the answer. This will enrage them at first, but eventually they will start thinking instead of spitting out an answer. They will also see that they need to persevere while solving instead of waiting for the teacher to give the strategy and answer in the end.
Have them compare two strategies next to each other. Have the students look for similarities and difference.
Choose random partners whenever possible. This allows students to learn different strategies from others, and it keeps your groups fresh.
Students who struggle in mathematics can be excellent problem solvers. It’s so critical that we believe this, and that we continue to challenge them at high levels with the right supports.
The number one issue that I have with math tools, is that students need to be able to relate to it on their level. Base 10 blocks, while concrete, can be difficult to work with primary students without any context. Here is the context for today…please enjoy the delight your students experience as they pretend their blocks are candy! Base 10 Blocks have never been so exciting!
I can’t stop using visuals in elementary math. They are on every anchor chat, every lesson plan, every assignment if possible. It started last year when I noticed students were having trouble understanding place value until I made a visual 100 chart.
When students have a visual to connect mathematics to, it’s like something magical starts to happen. Students who excel want to make their own visuals, students who struggle start to understand…it’s truly remarkable.
You might be wondering more about what I mean by visuals. I’m going to introduce you to Berkeley Everett at Math Visuals. He is a K-5 Math Specialist out in California that has been working on making math come to life with visual animation. It’s truly remarkable the amount of hours he has put into this task, and it’s all FREE.
Need to learn to count in kindergarten? There’s a visual for that.
Need to see different types of division? There’s a visual for that.
Need to understand the concept behind compensation in addition? There’s a visual for that.
Need to work on different ways to represent two digit numbers using place value concepts? Theres a visual for that, too.
Go to this site and you’ll be lost for hours. Better yet, it will inspire you to create your own visuals on your math anchor charts. It will inspire your students to connect those very abstract math concepts to something that they can hold in their brain.
Thank you Berkeley, you’ve made me a better math teacher, and helped a whole lot of students at our school.