A little while ago, Tracy Zager wrote a fantastic post about the criteria she uses to evaluate math games. Within it, she recommended a program called DreamBox. It looked promising, so I installed it on my school iPad and signed up for a two week trial.
My son J and I just spent an hour playing with the games today, and I am very impressed. This is exactly the sort of math program that I have been waiting for years to see.
I am so sick and tired of games that only care about “math facts.” I want a program that deals with math patterns, math representations, and math concepts. DreamBox, at least in the early levels, is full of the sort of math representations that I see in the elementary classrooms I envy. J was working with ten frames and rekenreks, which are fun to play with even digitally. He wanted to play each tutorial twice just because he liked sliding the rekenrek beads back and forth or changing the composition of the ten frame.
There are even activities for the youngest learners, those who don’t even have a firm grasp on their numbers from 1-10. There is an early game, “More or Less,” where the game displays two quantities of dots and asks the student either “Which is more?” or “Which is less?” This game was great for J because he didn’t know the word “less.” I said it meant not as many which worked well for him until he got to a 1 vs. 2 problem and said “They’re both less!” I have learned a lot about talking math with my kids from Christopher Danielson, so I was elated to hear him say that. It started a fun little conversation about less and more.
Below, I have put a couple of 2-3 minute recordings of J playing through some other games. I have decided to write-up a detailed commentary of the math that I think is going on in each video. So consider this your chance to jump ship.
Check out DreamBox! I have no connection to this company other than I think they made a cool app. It’s got a free trial for 2 weeks and then costs $13 a month. That’s a pretty steep price. I’m still not sure if I’m going to pay it. We’ll see how the next 12 days go...
Ok, now down here let’s get reeeeeel mathy. Remember, I am not an expert in this stuff. I am a middle school math teacher, but I have a hobbyist’s enthusiasm for early childhood math. So what I write is not a firm account of what happened in the video. It’s what I think is going on, and what I notice about that. So in the comments, please add your thoughts!
In this game, J has to replicate the top rekenrek using a double rekenrek (these are the abacuses with the red and white beads. I just looked up the name today, so don’t feel bad. I still don’t know how to pronounce it).
The thing is, he must use beads from both the top and bottom rows. I think the purpose of this activity is to get kids to move from a visual understanding of matching towards the idea of equivalence; that is to say, the idea that 2 and 3 are equivalent to 5. So that’s what J is working on.
So on the first problem, which I missed, J immediately yelled “2!” when he saw the two red beads. That means J can recognize the quantity 2 without counting the beads. This is known as “subitizing” and it’s a huge early math skill. It depends on the quantity and the arrangement of the objects. Look at the two pictures below. One you can look at and immediately know how many dots there are. The other, you can’t. It’s because the 5 dots are a smaller number and the dots are arranged in a very familiar pattern. The other one you actually have to count because there are more dots and they are scattered across the page.
J can subitize 1, 2, 3, and sometimes 4. At this point, he hasn’t had to count any of the beads individually so far. So when he sees an amount he doesn’t recognize, he is in uncharted territory and asks me whether he can count. He counts up to 5, moves 4 across, asks me for help which I refuse to give, and then moves the fifth bead across. J then recounts his beads, gets 5, and celebrates by swinging the iPad around.
Notice that he hasn’t touched the green button yet. The program hasn’t told J that he is right. J already knows for himself that he is right. That is what a conceptual understanding looks like.
Things I did that I hate:
“Now make 3 in a different way” - I did this because I already know (or think) that J can subitize 3, and I wanted him to focus on the mechanics of the game. But I should have said “Now make the next one in a different way.” Let him do the work. No shortcuts, kid.
“Ooh, this is a hard one, J.” - This is such a tough habit to break. I’ve been a teacher for 6 years, and I’ve been trying to break this habit for 5.9 years.
It’s so understandable. You want to build up kids’ confidence, so you see a challenging-but-solvable problem and you say “ooh, this one’s hard!” so that once they get it right, they feel smart. But the thing is, how do I know whether this problem is hard? J is going to try the problem in a second anyway. He can decide on his own whether the problem is hard or not. And if you ask him his opinion afterward, he might even tell you why.
But if I start the problem by saying “this one’s hard,” then I’ve already created an artificial sense of difficulty that colors his perception of the problem before he even starts to solve it. This is especially harmful when a child struggles to understand a problem that you’ve called “easy.” Every level of teacher does this, and it always hurts the kids. We start with easy and by middle school we say obvious and in college we say intuitive, but it all means the same thing. Which is “you better get this right pretty quickly or you’re dumb."
It’s better just to let the problem and J decide together whether or not it’s hard.
Things I did that I like:
“I don’t know. Try it.” - This is my standard answer when he asks for confirmation. J is more than 50% confident in his answer, but he’s looking to me to tell him he;s right. I don’t want to do that. I want him to get used to finding possible answers, thinking it through, and testing the ones he feels fairly confident will work. I want him to get comfortable with being uncomfortable.
All the parts where I wasn’t talking - Shut up and get out of the way, Dad. That silence is the sound of someone’s brain growing. Don’t interrupt it.
This video was from the next game we played. In this game, you are supposed to make your ten frame match the given frame exactly and then click on the numeral that matches the frame. J is pretty familiar with the numerals from 1-10, but this is one of several games that helps to strengthen that connection. The numbers are laid out in order so you can count up to the appropriate numeral if you don’t recognize it by sight.
Since we had just been working on equivalence, I think J is just trying to match the quantity. I think he didn’t realize that his ten frame had to look exactly the same as the frame above it. So that’s what I’m explaining to him at the beginning.
First thing: J can subitize 10 in a ten frame! That is news to me, and very impressive considering he just saw ten frames for the first time today. Unless they’re using them in his pre-school class already, which I doubt, this means he has learned a new idea from DreamBox already. Also, he knows that moving the 10 dots at once is more efficient than moving them 1-by-1.
J decides the next problem, nine, is a tough one without me even saying anything! See, he does have his own internal sense of difficulty. So he counts the nine dots, sings a little adorable Nine Song, and starts to build his nine in a very efficient manner, by starting with the 5-dot piece and then getting the 3-dot piece. Then he concludes he’s done. I try to ask a question without giving away that he’s made a mistake.
So he counts his frame, gets eight, goes back and counts the original frame, has a little I’m-a-three-year-old moment, and just gets a bit discombobulated. So I ask him to put down the number he thinks it is, and he picks 9. I find this interesting since he didn’t get nine for either of his most recent counts. But the correct answer was still in his brain somewhere. If you go back and look, just after the Nine Song is a moment when his finger hovers over the numerals. I think he was about to pick 9 then, so he made that connection before he even started building the ten frame.
I am in love with this second video. For one thing. I did a much better job of getting out of J’s way and letting him reason through the problem.
Secondly, I love that J made mistakes in this video. He sees two similar ten frames as being the same, and he doesn’t properly count the original frame on his second attempt. The concept of touching one object per number when counting, which is called one-to-one correspondence, is something that J is decent at but still messes up occasionally. When he gets off track in counting, he starts singing the numbers in sequence but is no longer connecting them to the dots.
AND THAT IS OKAY. He is a three year old and working on a new skill. I don’t need to stop him and correct him every time I see him make that sort of mistake. The program will catch J’s mistake and over time he will become a more precise counter. I love mistakes. It means that kids are working at the edge of their abilities. And I don’t always correct J, any more than I correct every little grammar mistake he makes. J used “he” and “she” interchangeably for almost a year before he realized that they connoted gender. I never corrected him for months and months. I just used the pronouns correctly in my own speech and figured he’d figure it out when he had a firm grasp of “boys” and “girls” as an underlying concept. And you know what? Almost a year later, he was ready.
What I care about is that the program found his mistake and asked him to try again. This is one of Tracy’s 3 big criteria for math games, and it’s spot on.
So right after I shut off the video, J picked up a 5-dot piece and then filled in the second row one-by-one until he got to nine. Then he said “It’s easier to do ones” and on the next one, which was a seven, he filled in all seven dots by ones. He shifted to a less efficient strategy that was within his comfort zone. Again, I am fine with this. Over time, he will shift back to using the larger pieces once he finds single-dot pieces too time-consuming to use. I’m not going to force it.
So J, like every other child in the entire world, isn’t a perfectly elegant problem solver yet. But he’s a little bit better than he was at the beginning of the day. Credit where due: DreamBox has put together a very promising app. I am really looking forward to the next 12 days.