It’s hard to believe that we are nearing the end of the school year. Whether you get out in May or June, it has certainly been yet another school year “that was like no other.” Many of us were back to full days of in-person instruction with our students, and I know I’ve realized that students are so much better in a classroom than they were staring at a screen for virtual instruction!
Of course, we had bumps in the road as we navigated going back to in-person instruction. One of the biggest things I’ve noticed is that we have to think about how many full years of school our students have actually had. If you teach second grade, this was probably your students’ first full year of school, as their kindergarten and first grade years were certainly not normal! Certainly our fifth grade students also have large gaps since they didn’t get the kind of instruction we would have liked for their third and fourth grade year, but I’ve definitely seen the profound effects of gaps in learning as I work in early childhood classrooms.
Before the Intervention Window Closes
I always talk about how the early intervention window closes at the end of third grade, which means that if we’re able to shore up a student’s reading skills and help them develop numeracy and number sense skills before then, those foundational skills will stick with them as they grow. That doesn’t mean that students in fourth or fifth grade, or even at the end of their third grade year, can’t grasp those concepts, it just takes much longer at that point to make the same amount of growth.
In the face of this staggering amount of learning loss, I hear some conversations about just sweeping it under the rug and continuing on. It’s such a struggle for educators. So many of the classroom teachers I talk to have said they have something like 18 out of 24 students on reading plans – literacy levels are much lower than we expected coming back this year. But I’m also looking at the math side, and students are just as low, if not lower!
Obviously, we can’t slow down our math classes to make sure there is 100% accuracy for all students in every area. We do have to move on, but why can’t we have students on differentiated levels to meet them where they are and address some of this learning loss, just like we do in reading?
Targeting Numeracy Skills
As I work with students, I’ve seen that the lack of numeracy skills is profoundly affecting their number sense. This is a really key component that needs to be addressed in our conversations as educators.
I’ve noticed that students’ conservation skills, to 5, 10 and 20, are not strong. I see first graders relying on the “pound and count on” method of putting the larger number in their head and then counting on their fingers. Sadly, I’m also seeing second graders that only know how to count on. One student I talked to, when given a problem like 27 + 32, looked at the hundreds chart on his name tag, started with 27 and counted up 32. We talked about why this strategy might not be the most efficient, and the student actually landed on the wrong number, but had no idea whether or not he was even close because there was absolutely no number sense.
In this blog, I want to focus on peeling back the layers to see where our students are struggling. We’ve taken many of our Molding Math Mindsets schools on this same journey of looking back to see what students know before we move on. Do they have conservation to 5 in a vertical, horizontal, and dice arrangement? Do they have conservation to 10? Have we done a good job of looking at part-part-total and helping students connect it to a 10-frame? Can they see part-part-total on a linear Counting Buddy or on the first row of a rekenrek? Do they have conservation to 20, not just on the surface but in-depth. Can they instantly recognize a number from a double 10-frame, a linear Counting Buddy Sr. and a rekenrek?
Students need hundreds and hundreds of experiences with numbers on these different levels to really understand them. But I’ve noticed that we tend to check the box, and then skip over to adding or learning D.C. Instead, I’m advocating for a close look at the layers in between to make sure students have a solid foundation of number sense.
Conservation to 5
This is not just flashing a number and subitizing – “How many do you see? Ok, let’s move on.” It has to be “How many do you see? How do you know?” If a student can recognize a four on a 5-frame, great! But we are missing a valuable opportunity if we don’t also talk about the ideas and principles of part-part-total, number bonds, and all the other pieces that we want students to have in number sense. If they say, “I know that two and two makes four,” build it with them! Put red counters and two yellow counters on the table so they can see the two parts of four. Get out a number bond and write the four at the top and the two and the two in the spokes so students can see it’s made up of two parts. (If you need a number bond template, download one from our website! You can also order three-section buffet plates on Amazon that work really well.) Make it physical by clapping on one side of your body and saying “two” and then clapping on the other side and saying “two” and then clasping your hands together in front of your body and saying “let’s put it together! It makes four!” Repetitive? Maybe. But when students are learning conservation to 5, it is very important not to just fly through subitizing and “tell me how many you see.” We want kids to be able to really understand the concept a little bit deeper, so in this first layer of conservation to 5, we want to spend a lot of time looking at part-part-total.
Our Deck o’ Dots games provide a lot of great opportunities to practice conservation to 5 and part-part-total. Pull out just the 5-frame cards and have students look at how many they see on a card and just talk about that. Play Deck o’ Dots Duel with the 5-frame cards, or Same Less More, or even Deck o’ Dots Difference. The Five Frame Shake game on our website is also another all-time favorite that helps students get additional practice in this area.
Conservation to 10
At this point, we’re flashing 10-frames, looking at the Counting Buddy, using one of the great tools from DreamBox, and students might even be building on their own 10-frame mats. We’re asking questions like: How many do you see? What’s one more? What’s one less? How do you know? But are we spending quality time in this layer of conservation to 10 looking at part-part-total?
I would say this is one of the most critical areas where I am seeing learning loss in schools. Kids might be able to memorize that 5-frame and eventually transfer their knowledge to the top row of a rekenrek or a Counting Buddy Jr., but they aren’t fully understanding part-part-total. This is why I created my new Part-Part-Total cards which are designed to help students slow down and really examine the 10-frame that’s become so familiar.
Typically, we have missing addend/part-part-total cards where I show four on a dice and three on a dice and I’m hoping that you say seven. But, as I was doing this with students, I realized that kids were using their nose to count on, or counting on with their fingers. They didn’t really understand the concept of it.
Beginner Part-Part-Total/Missing Addend Cards
Clearly, we need to back it up a little bit within this layer, with what we are calling the beginner level of part-part-total. These cards were designed with actual digits for the total, but 10-frames for the parts. Let’s say the total is nine, but in one 10-frame, students will see five red counters, and the other, they’ll see four yellow counters. We can put a question mark card over the number nine and flash it to students. Can they see the two 10-frames and instantaneously tell you it’s nine? We’re expanding on what we’ve been doing with subitizing and helping students see that it is important to get a visual imprint in our brain to see those parts and the total.
Now can students do it with seven? Can they see four on one 10-frame and three on the other, and visualize pushing those counters together? Can they tell you they’ll end up with five at the top and two at the bottom to make seven?
I absolutely love using this set of cards with our students because it really gives them the understanding of how to see those part-part-total pieces. The cards show all kinds of different configurations so students can visualize moving over the counters to see what a 10-frame would look once it has been filled with the two parts. That’s really the essence of what we want when we’re talking about conservation to 10.
These cards are free for our Molding Math Mindset members, of course, or available for purchase and instant download in our store. Get them this summer so you can print, laminate, cut and have them ready for next year. I promise they’re going to be a great tool!
We can also take these cards a step further, and use them, not just for part-part-total, but also for missing addend. This is another important piece we look at while we’re developing conservation to 10.
Let’s go back to the example of nine, and a card where students see the digit nine, the five red and four yellow. Now, can I take the question mark card and put it on top of the five this time, with a statement like “I wish I had nine but I only have this many (the four yellow). How many more do I need?” Can students do this conservation process in the reverse? Can they think, “I have to have nine, so I know if I see four, the other part of it has to be five”?
Notice, in this level, I am not giving students an algorithm that says 5 + ⃞ = 9, because,99% of first graders will count out on their fingers, or put dots under that little cute box and they’ll go back and simultaneously count 1,2,3,4 and put it in the box. That is a procedure that students at this level are not ready for.
These cards would be great for a station, and all you have to do is print, laminate, and cut. There are lots of question mark cards included because those will, of course, go missing. You might also want to break up the pack into decks for students to use different ways or work on different totals. The cards could be used for an intervention. This could work for my special-ed students. This could work for my students in first grade, and many of my students in second grade that still don’t have this process down.
Beginner Number Bond Cards
Now once students have this idea of part-part-total, we experimented a lot this year to see if that was enough. Could students now go into conservation to 10 and understand it? And I decided I wanted to add one more layer to solidify the concept. I wondered if they could look at a number bond structure and see that they were looking at a part-part-total, or a total and a part with a missing part. Enter – my new number bond cards!
I’m super excited to use these in schools! Just like the part-part-total cards, these number bond cards have digits for the total, and then 10-frames for the spokes. Students might see a total of eight, but in one spoke of the number bond they’ll see two red counters on a 10-frame, and in the other 10-frame they’ll see six yellow counters, which will look familiar from the part-part-total cards they’ve been using.
In these number bonds, there’s always going to be one missing number. Maybe the actual digit is going to be missing and you’ll see two red in the 10-frame on one spoke and six yellow on the other spoke. Then the student needs to figure out the total of those two parts. We could reverse the process, and show the eight at the top, but only one spoke with two yellow on the 10-frame. Can students figure out how many would be in that hidden piece? A lot of kids can visualize that 10-frame. They know that eight would have five at the top and three at the bottom, and if two were gone, how many would that be? They know it will be six because they’re visually seeing it in their head.
Both of these new sets of cards can be used as a take home tool! They would make a great topic of a parent night, and then they could take home a set so parents can help students practice at home.
Obviously, once students get some of these concepts, we can move them into regular number bonds with digits. We have physical sets of regular number bond cards on our website, and you might have four of five of those in your Math Salad Bar. We also have a digital set of number bond cards with digits that you can download for home or school use, where students would be able to cut out their own set.
Be mindful that these sets are looking at digits, not quantities. If kids are struggling on number bonds that contain digits by counting on their fingers, they’re not instantaneously visualizing the parts, they may not be ready for the traditional number bond cards. Just step back to the beginner number bond cards, or back to the part-part-total cards until they’ve mastered those.
Advanced Part-Part-Total/Missing Addend Cards
As I’ve said before many times, students need hundreds of experiences with numbers in multiple modalities. Once students have mastered our beginner number bond cards, move on to the advanced set, which looks at part-part-total with dice or domino patterns instead of 10-frames. These cards are similar to our Understanding Combinations and Missing Parts Screener, but they are all organized in dice or domino patterns. Students might see a six as three and three. They recognize the double and know the total is six, but the threes are in a diagonal line like on a dice.
If the card had a six with a four and a two (both in dice patterns), you could cover up the six, flash students the card, and ask how many they see. You’ll know the students that are pausing and staring at the card and actually nose counting those numbers versus the students that are really understanding the idea of part-part-total. Again, if students are struggling with these cards, just go back in the layer to the part-part-total cards with 10-frames and give students more practice there before they try dice patterns again.
You can also use this advanced level for missing parts, by putting the question mark over one of the parts, showing the students the number, and asking if they can figure out the other part. If they know we need six as our total, and we covered the four, how many are under the question mark?
Again, you could take this further and helping students understand at a higher level by looking at blank number bonds, students can do all of these things by looking at it in a blank number bond, getting out the three-section plate to write out that number bond, or making the number bond on a number bond template.
Conservation to 20
We’ve done a LOT of work in the conservation to 10 layer. Is it time to move on to conservation to 20?? My answer to that might be no.
We might be working on conservation to 20 concepts in our numeracy talks. In Kindergarten, we’re obviously working on it in our teen numbers, but let’s hold on a second. Let’s not move students into subitizing a double 10-frame and getting them to memorize the rekenrek, etc. Doing some of these things is fine, but we really want students to take their understanding of conservation to 10 a step further. How?
D.C.’s Number Bond Race
With another new product – D.C.’s Number Bond Race! Everybody loves D.C.! In fact, everything we’ve been talking about in this blog is really the beginning parts of what DC does. Take Five-Frame Shake – We’re taking a 5-frame and using D.C.’s mallet to decompose and break it into two parts. The same thing is happening as we’re working with all these parts of 10. When I flash a nine on the 10-frame, and a student says they see a five at the top and a four at the bottom, that is D.C. hammering and breaking apart that number! So what better way to continue to practice than with a fun D.C. game! D.C.’s Number Bond Race is focused on missing addends. We’ll have an eight at the top and a five on the side. Students, in digit or number form, then have to fill in the other spoke of the number bond.
However, I do have a small disclaimer. If you’ve seen me present or listened to any of my blogs, you know that I do not believe that faster is smarter. So DO NOT put a big timer on the smartboard for D.C.’s Number Bond Race and see how fast your students can do it. That’s not the point! We just want them to be more comfortable with their number bonds.
Members, you can download this new game for free on our membership website, but it’s also available in our store.
There’s one more product that you might already have in your classroom that is a companion to D.C.’s Number Bond Race – the Two Out of Three Race. If you’ve never played, we have a great video that explains the whole game, but here I want to point out the difference between the two games as we’re talking about conservation to 10 and part-part-total.
Two Out of Three Race gives students a target number at the top of the page, and numbers in sets of three down the page. If they’re looking for seven, and their numbers are five, three and two, students will circle the five and the two, which make seven. Two Out of Three Race is great for part-part-total.
D.C.’s Number Bond Race focuses on the missing addend concept, like a number bond dash or similar game. Students would have seven at the top part of the number bond, and one part would be five, so they would have to write in the other part, which would be two.
Give Them Time
The conservation to 10 layer is DEEP! Students just need more time to understand all the facets of it, especially if we want them to be able to use D.C.’s strategy and decompose numbers and make a 10 later on.
We’ve already seen massive growth in our M3 schools by targeting conservation to 10 this year. We’ve accomplished much, but there is still much work to do. Remember, our K-3 students have experienced significant learning loss, but we are in the window of being able to create more than a year’s growth in a year’s time. This concept is vitally important to solidify before the end of their third grade year.
As the school year winds down, I think you’ll find these are super simple things we can implement in Kinder, first, and second grade, and even with third graders that still don’t have a handle on conservation to 10.
I hope these new products will help in your efforts to rebuild the foundation that’s crumbled over the last few years.
I would also love to know how things are going and what you’re seeing in your classroom with learning loss. Shoot me a message or find me on social media (Facebook or Instagram or Twitter) and let me know how you’ve seen these layers affect your students and how you’ve been working to set them up for success.
Next time, we’ll take this topic a little bit further next time and look at conservation to 20, and then connecting that to the universal skill that we want kids to eventually learn, which is D.C.’s strategy of making a 10 (or a friendly number or a decade).
