Math for Enthusiastic First Graders

The basic challenge as I see it is that a young child who is enthusiastic about math is generally enthusiastic about numbers. The natural response to such enthusiasm from any reasonable adult is to give that kid more numbers. But this very quickly puts the kid on the path towards accelerating through the school curriculum, making it harder for them to have a good time at school and narrowing their mathematical experiences.

And the other part of the problem is the kid’s maturity and independence. If you have a teen who is enthusiastic about math, that’s fine, they’ll do their schoolwork and then noodle around on Numberphile or Desmos. But younger children frequently lack the ability to direct their own mathematical lives. They want an adult to lead.

If you put those two pieces together — number-centricity and the need for adult guidance — it makes it very hard to give a young child what they want without accelerating them through the K-5 math curriculum.

I maintain that it’s worth trying to avoid as much of the school curriculum at home as you can. If you can get beyond the fun of “hey my six-year old is doing ten-year old math” you’re at risk of making it harder for schools to keep your kid happy. And while I’m not opposed to straight-up acceleration or grade-skipping in some cases, in general I think it’s best to avoid these with very young children, who however advanced they are in math often have totally age-typical emotional needs. (This is why, for all my love of Beast Academy, I don’t typically recommend it to parents.)

I’m trying to speak generally here, but without being super-weird about it I’ve encountered this both at school and at home. My oldest son is very enthusiastic about numbers, and I think it’s in his best interests to not race ahead too far of his classmates. And at school I have tried to express the perspective above to parents of my 3rd and 4th Graders, to mixed success. (Thankfully, most parents get it.)

OK, but all this leads to a question: what do you do with a kid who wants to talk about numbers during breakfast? and on the subway? and over dinner? and after dinner? and at many other times, especially when he’s bored and moody?

It’s in that spirit that I share some of the things that we’ve been talking about in my own home lately. Some of these I proposed to my oldest. Others he came up with on his own. I’ll add that in my teaching life I sometimes encounter accelerated students who totally lack the ability to ask questions independently. That doesn’t make sense to me, since often as kids become more independent they can get more out of their grade-level classes.

Anyway, here are some of the number-centric ideas my son and I have been talking about lately.

Switcheroo Numbers: Take a two-digit number, and swap its digits. 35 and 53. 72 and 27. 19 and 91. Then take the difference of those numbers. You will find that all those numbers are _________________. Why is that? Does it work with three-digit numbers? Four-digit numbers? Consider representing these numbers as sums — 30 + 5 and 50 + 3 — to play with some of these ideas.

Writing a Scratch Program for Counting Factors: My son is most eager to talk about factors, so we often end up drifting towards factor math. Together we made a Scratch program that finds all the factors of a number. I let him tinker around with it and he was able to show some indpenedence there, which I was gratified to see — coding is something unrelated to K-5 math that he can do outside of school. Together we added a factor counter to this program, so he could try to “set records” and find numbers with the most number of factors.

Factor/Number Fraction: Conversations about which numbers had a lot of factors led to discussions about the relationship of the number to its factors. My son loved talking about perfect, abundant, and deficient numbers. I, on the other hand, find this stuff dull as dirt, so I proposed a different question. 3 has two factors, so its factor to number fraction is 2 out of 3, or 2/3. 4 has three factors, so its fraction is 3/4. What numbers have the largest of these sorts of fractions?

Building Bigger Factors: This one also came from me, I thought he would like seeing how you could use multiplication to find factors of bigger numbers. For instance, the factors of 10 are 1, 2, 5, and 10. The factors of 9 are 1, 3 and 9. What are the factors of 9 x 10 = 90? How many factors does it have? And (if this is new territory for you) you might discover that there is a pattern that seems to hold for some, but not all numbers. (For instance, compare the factors of 9 x 10 and 5 x 10.)

Triangular Square Numbers: I don’t know if my boy got into triangular numbers from Marilyn Burns’ books or from Numberblocks, the TV show. But it might have been Numberblocks, with their annoyingly catchy “step squad” chant. You know the one I’m talking about.

That’s the one.

Anyway, are any triangular numbers also square? That made for a few days of fun.

The Difference Between Nearby Multiplications: I don’t know why, a lot of my son’s math has to do with finding the difference between slightly different things. For instance, he came into the kitchen the other day eager to share a “trick.” Start with two numbers, like 2 and 7. Make 2 x 8 and 3 x 7. Find the difference between them. He thought there was a pattern but it took him a few false before he figured out what was going on, and then it was interesting to try to explain what was happening and why.

The Sum of Consecutive Powers: This one came from him, I think. Or maybe it was a book he was reading? I’m not sure. Either way, add consecutive powers of 2, like 4 + 8 = 12 or 32 + 64 = 96. His claim was that these numbers are always divisible by 3. Really cool, right? Try it with other bases besides 2. Why is this happening? Fun times.

The “One and a Half” Sequence: This one is very much kid-math, but I think it’s interesting and it occupied my son for a while. He spent a few days (or weeks? what is time?) looking for numbers that were “one and a halfs.” For example, for him 3 is a one and a half (because it is 2 and half of 2) but 4 is not. The full sequence of “one and a half” numbers for him was: 3, 6, 9, … right! He noticed that these were multiples of 3. And then what about “one and a third” numbers? “One and a fourth”? And so forth. An interesting regularity for him to notice and a challenging one for him to try to explain.

These are just the little number investigations that have landed at home. There are other things to recommend: Prime Climb, Math for Smarty Pants, The I Hate Mathematics Book are the biggest winners in our house at the moment. And of course it partly depends on your child’s taste.

But for parents of enthusiastic young children, I think the most important work at home is to broaden those tastes. Because even if what we love most is calculation and numbers, mathematics is big. Kids, especially those self-professed lovers of math, deserve the chance to explore it.

More YouCubed Research That is Difficult to Explain

Along with however many people are on YouCubed’s email list, I recently received this message from Jo Boaler:

Hello youcubians,

Three years ago we reached out to everyone who had taken Teaching Mindset Mathematics, a workshop we offer to share the curriculum, ideas and teaching practices from our 2015 summer mathematics camp for middle school students. We asked if they were planning to host a youcubed-inspired summer camp in 2019, and if so, whether they would be willing to participate in a study of student outcomes and teacher change. We heard from many people who were interested in engaging with us and ultimately enrolled 10 different camps and their school districts into our study, sponsored by the Gates Foundation.
The demonstration camp that youcubed hosted in 2015 had resulted in significant improvements in student mathematics understanding and mindsets, and the purpose of the new study was to learn whether this could be replicated by teachers across the US in a variety of settings. Today, we are very excited to announce that the results of this research have been published in the journal Frontiers in Education. I would encourage everyone to read the full study, which can be found here.

So, a published study funded by the Gates Foundation. I clicked and opened up the paper.

Look, you know what’s coming at this point, right? We’ve been down this road before. I’m going to show you that the paper is a mess. And who cares? Nobody. People who care about research, including a great number of math education professors, do not have respect for Boaler’s output. YouCubed’s popularity stems I think from consultants, PD leaders, and from teachers themselves, not from the math education research community. Frankly, I think YouCubed would be just as popular if they never performed any research whatsoever. It’s totally inessential. As am I. As is this post. What am I doing with myself? How am I choosing to spend my life? That’s how I always end up feeling when I write about YouCubed.

Anyway, back to the paper, this is what they did:

A matched comparison analysis was employed to assess the effect of the approach on students’ achievement. School districts provided a variety of achievement measures of both participant and non-participant students (GPA and MARS scores, before and after camp participation; and a baseline math standardized test score), and a battery of control variables (race, ethnicity, gender, free and reduced-price lunch status, English learner status, and special education status). 

OK basically the research team couldn’t run an experiment where they randomly place kids in these camps, so they did a very reasonable next-best thing: they created a control group, of sorts, by matching each child in their camp with a child with similar stats who didn’t go to the camp. So if Michael is a camper and has a GPA of 2.5 and a MARS score of whatever, etc., they tried to find a student Jackson who was not a camper but with the same GPA and MARS score. If Michael (and all the other campers) suddenly are performing much better after the camp than the “control” group then maybe the camp is responsible for it. Or maybe it’s some subtle selection bias.

This is all totally reasonable but in the end totally irrelevant for the issues I’m going to point out. I’m including it because, why not?

By the way, the details of this matching and a quantitative comparison of the groups are not detailed in the paper. Normally a paper would do that, but again that’s not so important in light of what I’m about to say.

OK, so we get to the part of the paper where we’re seeing how campers did compared to their control group. First issue is that in this table it looks like the “non-participants” in the matched group had a higher GPA after the camp than the campers.

The body of the paper says, “This analysis showed that at the end of the first term or semester back at school, the students who attended the youcubed summer camp achieved a significantly higher mathematics GPA (p < 0.01, n = 2,417).” Now this is the exact opposite of what the table says, which is a problem, because either the table is wrong or the sentence is false. And, honestly, I assume that the table is wrong and the entire thing is mislabeled. OK.

The paper goes on: “On average, students who attended camp had a math GPA that was 0.16 points higher than similar non-attendees (i.e., students from the same district and grade and who had a similar baseline math GPA and test score) (Table 4).”

This is a problem because the table also does not say this. The difference between a GPA of 2.679 and 2.622 is 0.057 points. That is not 0.16 points. What am I doing, providing this level of scrutiny to this paper, i.e. basic scrutiny? Shouldn’t I be writing a book? Doing math? Reading something? Was I put on this Earth to fact check these papers, providing infinitely more care than the authors themselves apparently did?

Anyway, presumably the issue is that the body of the paper misinterpreted its own table. The 0.16 is the effect size, provided (and clearly labeled) on the right side of the table, not the GPA.

And now a bit more from the paper: “In addition, compared to control students, camp participants were 6 percentage points more likely to receive a grade of B or higher, and 5 percentage points less likely to receive a grade of D or lower (Table 4).”

This is not right, again I think they must be rounding up the effect sizes and reporting them as a percentage change. Silly stuff, quite honestly, but none of it matters at all, I don’t know.

By the way, there is no supplemental information posted, despite this statement.

So, I don’t know. I wrote to corresponding author a month ago but haven’t heard back, I’m sure he’s busy and I’m not sure what he would say.

It’s absurd that this was published as is, and it obviously doesn’t speak well for YouCubed or Boaler that this was pushed through. I…I am unsure what any of this means. Like, yes, it’s bad. But in a way it’s so bad that it doesn’t really touch her core areas of advocacy. I wish the research she was attempting was better because then it would actually be about something. Instead it’s about nothing besides YouCubed and Boaler herself, who has managed to become the popular voice for math education research in this country while showing complete disdain for the work of research itself.

I need to find a better hobby.