Talking Math with Your Kids week continues.
I worry about how much fraction work we do in early elementary grades, before lots of kids are really ready.
Griffin (seven) has been doing his second-grade math homework, which is to find of 4, and then of 8, and then of 20, etc. After several of these, he is to fill in the blanks: of ___ is ___. He begins with of 4 is 1.
Me: You already did that one; it was the first problem. You have to do a new one.
Griffin (seven years old): But I don’t know what else to do.
Me: Any other number that you haven’t already done is fine.
Me: What about doing 2?
G: I already did that, it’s 8.
Me: No, no. What about doing 2 as the thing you find 1/4 of?
G: Dad, there is no 1/4 of 2. It’d be in the negatives!
I have to believe that some extended time thinking about sharing situations would be a much better use of Griffin’s homework time than expressing the results of this sharing abstractly as of a discrete quantity.
This is a problem with one curriculum, in one order, for all children. My daughter (also 7), who is homeschooled/”unschooled,” has been interested in fractions lately. She figured out how to do things like 1/2 of 5 and 1/4 of 23 a little while ago, and multiply fractions (she usually thinks of it as a fraction of a fraction, which is good I suppose), and add fractions with fairly different denominators (I think the other night she figured out 1/3 + 3/5). I’ve also showed her little puzzles like “3*__ + 4 = 20. What goes in the blank?” (I explained to her that we usually use a letter instead of a blank, but she doesn’t care for that idea (and why would she?)). Anyway, my point is that different kids take an interest in mathematics in different ways, and/or at different times. I’m sure Griffin knows a bunch of things, and has a bunch of interests, that my daughter doesn’t yet. I wish it were easier for institutions to accommodate those kinds of differences.
Yeah, this is part of the problem, Roy. But it’s a more specific critique, too. Your daughter is atypical; her grasp of fractions is pretty far ahead of the curve.
Griffin’s situation is more typical. So yes, I struggle as a curriculum guy who prepares teachers with issues of standardization and differentiation. But I also know from research and from classroom experience that if we’re targeting the mathematical development of most second-graders, we’re unwise to force “fraction of whole number” tasks into their homework. Hitting them with a lot of this before they’re ready leads to all sorts of predictable misconceptions.
So I’m with you, Roy. But as a college teacher, you know the struggles of differentiation for even a much smaller range of abilities and interests than a second grade teacher has to work with, right?
My critique is aimed at the curriculum writers who have chosen bad tasks for most second-graders. The larger system by which all kids need to study the same stuff at the same time? Whole different kettle of fish.
I don’t know Christopher… it seems to me like the homework assignment was exactly what he was ready for. It was YOU, and not the assignment, afterall, that introduced the idea of 1/4 of 2. It appears to me that this particular assignment does for both of your children (Christopher and Roy) precisely what they are ready for: practice the skill of taking a fourth of quantities that are evenly divisible by four and then open up the possibilities for further exploration by posing open ended questions. For Christoper’s son, this might mean reinforcing the skills he just practiced and for Roy’s daughter, this might mean expanding to other whole numbers or even multiplication of fractions.
Ideally, these two children would come together in class and help bolster the understanding of one another: when Christopher’s son tells Roy’s daughter that she did the whole thing wrong, she is presented with the challenge of communicating her thoughts in a manner that is clear to someone with less understanding (a challenge that is hard to come by in the homeschool environment).
Honestly, I say the sooner the better on fractions as concepts, because this is a big area where the meaning is lost in the algorithm on many.
For me, this is a reason I am skeptical of homework in general. Griffin is lucky to have someone at home who can help him deepen his understanding but I wonder about ALL kids. I’m just not sure I see the benefit of sending students away on this type of task.
I can’t see the point of having fractions in a 2nd grade math curriculum. What books are they using (if any)? There are so many more important conceptual issues for 2nd graders to tackle without this push to get them into what some asstard or other has determined is “raising the bar” (mostly because it’s pushing something down to a younger age group. Of course, since the vast majority of such folks believe that the measure of mathematical knowledge is how quickly kids are doing algebra and, ultimately, calculus, the pressure to get arithmetic, integers, and rational numbers “out of the way” increases all the time.
An alternative, which is to give kids more time to really get their teeth into the mathematics of whole number arithmetic, as well as to learn how to problem solve with the mathematics they know and are learning (and could do just by being enticed into thinking about by posing good problems for them), is almost always dismissed as “fuzzy.” What’s fuzzy about it, of course, is that it can’t be so readily assessed on mindless multiple choice Scantron-scored tests of computation.
I think I agree with Christopher and Emily, if that’s possible. There were parts of the homework that were perfect for Griffin and parts that, left unattended, could fester into misconceptions.
It sounds like he was successful in noticing some patterns in finding fourths of whole numbers divisible by four, but was struggling to grasp why some numbers “worked” and “didn’t work” and therefore to generate more that worked. Is that because it’s hard to generate multiples of four (which could be a sign that he’s not ready to think hard about fourths because the whole groups-of-size-four thing isn’t tangible yet)?
The comment that 1/4 of 2 would be negative was more eyebrow-raising because it made me wonder what model is Griffin using to do the 1/4 of 4, 1/4 of 8, etc. Is he thinking about dividing into four equal groups? Is he visualizing it with discrete objects? Or is he using math facts like 4*2 = 8 therefore 1/4 of 8 = 2? It seems pretty clear that he would be perfectly capable, with some thought and discussion, of figuring out a way to fairly share 2 cookies among his mom, dad, sister, and himself. And to say why it’s easier to share 8 Cocoa Puffs or 20 Cocoa Puffs but not 2 Cocoa Puffs with four people.
What makes me agree with Emily is that as a diagnostic task, this one seems to work well… it generated lots of new questions and paths for me to keep thinking about Griffin’s understanding of fractions.
What makes me agree with Christopher is that there was no support to make use of a task Griffin probably could do (share two cookies among 4 friends) and help him get good at thinking about how that task was the same and different from sharing Cocoa Puffs among 4 friends and finally mathematize a little bit what is the same (sharing into 4 equal groups) and what is different (whether you can have fractional parts in the answer). Instead, it seems to have left Griffin feeling like this whole fraction thing is totally weird and about numbers and how they mysteriously behave, instead of hooking that number behavior onto delicious and non-mysterious things.
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