Daniel Willingham--Science & Education
Hypothesis non fingo
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On fidget spinners & speeded math practice

7/2/2017

 
I was very pleased to collaborate with Daniel Ansari, (@NumCog) a renowned authority on the cognition of mathematics, for this blog. 

Just in case you have been away from this planet for the last few months, ‘Fidget Spinners’ are the latest toy sensation. Some have suggested (without any evidence) that this new gadget is "perfect for children with attention deficit hyperactivity disorder, autism, anxiety." Although there's no evidence for that, kids love them, which has prompted a flurry of interest in possible educational applications (see here), and educators have come up with creative ways of integrating spinners into educational activities (when they are not banning them, see here).
 
One such idea was the subject of a tweet by Dan on June 14th. The idea is simple: students use the spinner as a timer and try to solve as many math fact problems as possible while it is spinning.
Picture
This seems to us a simple, harmless and perhaps even fun thing to do, and most people on Twitter took it that way. Most, but not all.
 
Negative responses fell into two categories. One suggested that timed practice would lead to math anxiety. The other suggested that this kind of practice might legitimize the much maligned ‘drill and kill’ approach to teaching math.
 
If a teacher doesn’t like an activity, that’s obviously reason enough not to use it as far as we’re concerned—we’re not in the business of advocating for particular classroom work. But we can point to the research literature bearing on the two common concerns, and based on this research, we don’t think they have merit.
 
Regarding anxiety: This issue has been raised most prominently by Professor Jo Boaler of Stanford University. For example, she argued in a recent blog that  “…timed tests are a major cause of this debilitating, often lifelong condition [referring to math anxiety].”  
 
First, let’s note that the fidget spinner worksheet offers timed practice, not timed assessment, which Boaler mentions. It seems to us that in a zero-stakes situation like a worksheet, the main agent of anxiety would be social comparison, an issue that teachers have plenty of experience handling.
 
Second, when it comes to timed assessments, the evidence for an anxiety link is still lacking. Boaler cites Ramirez et al (2013) in her blog. This article examined the relationship between working memory and math anxiety and showed, perhaps counterintuitively, that math anxiety impacts students with high working memory more than it does those with relatively lower working memory capacity. The authors argue that because math anxiety affects working memory, through intrusive thoughts and ruminations (“I can’t do this”, “I am terrible at math”), that students who typically use working-memory-demanding strategies are hit the hardest. These data say nothing about speeded math practice--the measure of math achievement used by Ramirez et al was untimed.
 
In a review of Boaler’s book, Mathematical Mindset , Victoria Simms (@DrVicSimms) writes “…she discusses a purported causal connection between drill practice and long-term mathematical anxiety, a claim for which she provides no evidence, beyond a reference to ‘Boaler (2014c)’ (p. 38). After due investigation it appears that this reference is an online article which repeats the same claim, this time referencing ‘Boaler (2014)’, an article which does not appear in the reference list, or on Boaler’s website.”
 
Again, it seems obvious to us that if a teacher feels that this sort of activity would make her students anxious, she won’t use it. But it’s not accurate to claim that research shows that this sort of activity generally makes students anxious.
 
What of the second concern, that students should focus on developing a conceptual understanding of math rather than being able to recall math facts speedily?
 
Arguments for speeded recall of math facts are not arguments against building students' conceptual understanding of mathematics.  As @MrReddyMath  put it:
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But cognitive scientists have long argued that there is an iterative, bidirectional relationship between the development of procedural math skills (such as being able to recall your math facts) and conceptual understanding (such as understanding the inverse relationship between addition and subtraction). That was the conclusion of the final report of the National Mathematics Advisory Panel in 2008.
 
It was also the conclusion of Professor Bethany Rittle-Johnson, a developmental psychologist at Vanderbilt who has extensively researched the relationship between procedures and concepts in math learning.  When asked about the debate regarding memorizing math facts vs. developing conceptual understanding in a 2016 interview she said “Actually, I think it's a silly argument because the evidence is pretty clear that children really need to do both things. Understanding is super-important, but understanding relies on knowing enough that you can understand it. If you have to spend all your time figuring out what two plus three is, then you can't notice relationships between number pairs, [for example].” Practicing math facts should be one of the methods used to help students build solid foundations to scaffold their learning of mathematics.
 
Fine, but why, you might ask, apply the pressure of timing practice? Does speed matter?
 
It does. When working a complex problem you not only want to pull simple math facts from memory, you want to do so quickly, so that the other work can proceed apace. Indeed, adults with stronger higher-level math achievement retrieve math facts faster (Hecht, 1999).
 
And speed matters not just in using math facts but in learning them. Methe et al (2012) conducted a meta-analysis of interventions for basic math in single-case research and reported “we found interventions involving practice under speeded conditions and a carefully controlled instructional sequence produced the strongest effects,” echoing results from Powell et al (2009) who reported that timed practice (vs. untimed) was crucial to an intervention for struggling 3rd-graders to learn math facts, and Fuchs et al. (2013) reporting similar results for 1st graders..  
 
It is clear, as is the case with any learning, that such speeded practice of math facts must be adaptive and appropriate for the level of the learning, and should be scaled gradually. And like everything else in a classroom, it will ideally be engaging. That’s challenging when you’re trying to develop automaticity, because it implies a certain amount of repetition. That’s why we liked the fidget spinner idea; it’s a little twist on a familiar task. It won’t be to every teacher’s taste, but we can say that there is no evidence it will prompt the problems that some feared.
Michael Pershan link
7/3/2017 10:34:08 am

I really like your "both matter" perspective in this piece. I teach math both to elementary and secondary students, and I think fact mastery really does matter, and in a way is harder to achieve.

In the context of my own teaching, I think a lot about the best way to help students come to "just know" arithmetic facts. I was really intrigued by this part of your post:

"And speed matters not just in using math facts but in learning them."

Why? Why should speed help with learning? Does it always? When does it, when doesn't it?

Teachers are often skeptical about research, and one reason is that we know that teaching contexts can vary a great deal. For example, many of the intervention literature involves one-on-one tutoring rather than whole-class instruction. Many of the speed activities involve flashcards with immediate feedback rather than timed worksheets. And many involve students with learning disabilities. All of these variables interact in confusing ways. How can we know that the research you're citing is relevant to the spinner activity?

In light of all these contextual variations, I think it's important for scientific communicators to provide EXPLANATIONS (or models, or reasons, etc.) rather than results. The results may or may not apply, given our particular contexts. But the fundamental thing that research can provide teachers is UNDERSTANDING, which we can then apply across contexts.

So: why should timed practice help learning?

In one study I read (about fluency software) I learned that students with learning disability did not improve their addition fluency through untimed practice. Why? Because during untimed practice, the students simply DERIVED the facts rather than trying to RECALL them. In other words, you'd see a lot of kids in front of screen counting out 3 + 9 with their fingers instead of trying to recall them from memory. The kids were already pretty good at using this strategy, and the untimed practice allowed them to keep doing what they were good at.

I see this in my own students too. It's not so much that timed practice is helpful for learning directly, as much as it creates a context in which kids practice the things you'd like them to practice.

A solution is timed practice with immediate fact instruction. (You got 3 + 9 wrong? OK, 3 + 9 = 12. Try again.)

But would this solution also work with the worksheet/spinner activity?

Maybe, maybe not. It depends on the students, as you mention at the end of the piece. But it's not just that it needs to "scale appropriately." There's a rationale behind the scaling. The practice won't be effective unless kids are attempting to recall the facts directly -- that's my understanding.

The worst case scenario is that teachers give kids a full worksheet of problems, and kids can't directly recall ANY of them. Instead, kids work on using strategies to derive the facts. The teacher says to solve as many as you can, but the students can only correctly answer that many questions using direct recall -- with strategies, there's not enough time. Time pressure (along with the long list of problems) generates anxiety, which makes it harder still to answer problems correctly. None of this produces fact fluency.

Based on talking to colleagues and other math educators, this worst case scenario is in fact prevalent in US classrooms. These "Mad Minute" activities could be used appropriately, but they are instead often given to novices who are not prepared to draw on their mostly memorized facts for the activity. And, I think, this probably does generate feelings of helplessness and anxiety.

I want to emphasize that I don't know this to be generally true -- it's something I suspect, based on experience. My sense is that research has not yet studied these issues at the level at which I'm speculating, but if there is good work here I'd be eager to know about it.

My point is that without the rationale for WHY speed could help learning, we have no way of applying the principle to our teaching. Speed is not an instructional condition worth promoting, as it can mean so many things in so many different contexts. It's only when we start sharing an explanation for why/how/when research supports timed practice that we get something applicable to the classroom.

Sorry for the long comment! Thanks for the thought-provoking and important piece.

John Golden link
7/3/2017 11:27:38 am

I feel like this post is both a little naive and lacking context.

Teachers wouldn't use them if students were anxious - that's just not true. a) we see anxious students. Maybe there's not a research study that's figured out how to assess this, stupefyingly, but we know it from deep and repeated anecdotal, qualitative evidence. Even among my students at a mildly competitive entrance 4 year university, many non-STEM majors have these stories. b) if teachers thought they were supposed to give them and/or thought it would be good for the students anyway, they'd give them.

The lack of context is that this is argued as if we have a majority of classrooms where these timed assessments are not given. You're arguing, I think, against people who are trying to change the status quo of timed assessments. They are in no danger, sadly to me, of going away. Their proponents need no encouraging, though you do so here.

Just as a teacher, I think about assessing what you want to know. You find out how many in a minute or fidget. But what you want to know, are which facts are being recalled. Which could help you and the students to know what to work on next. Why not give out the sheet and ask the students to circle the ones they know by memory? Why not give out the sheet and ask students to identify 5 they want to know by memory but don't? Why not give out the sheets like a word search, "find all the computations that sum to 8?" All of these would be more useful for learning.

I was in a middle school last fall where I couldn't complete the sheet in the time given. (I have my multiplication facts.) What are the students to make of their experience?

Finally, your justification seems based on the fact that what's important is the conceptual understanding, and you're defending speed because it supports that. Then lets put our combined efforts into communicating that it is understanding that matters. Not speed for its own sake.

educationrealist link
7/3/2017 03:29:40 pm

I wholeheartedly concur with Michael Pershan's post, and most of John Golden's post, but I wanted to add another reason why teachers doubt this sort of research without being the "oh my god, the anxiety will hurt the children" sort:

A few years ago, I taught a math class designed purely to help kids pass a high school graduation test, and since the test was on middle school math, I had all the time in the world to work with the kids on conceptual understanding vs. math facts, speed vs accuracy, and so on.

There were only 18 kids in the class, so I had time to go one on one, small group, and so on.

I instantly noticed that, while most of the kids knew most of their math facts, I had two distinct sets of outliers:

a) knew math facts cold, every one, but very shaky on abstraction. That is, could say 12x11=132, but 12x + 5=29 was nearly insurmountable.

b) could not memorize math facts. More specifically, did not and could not come up with 2x4 after hours of practice. Couldn't even memorize it verbally. That is, after 90 minutes of intense focus, writing down, visualizing, saying it, could not say 2x2=4, 2x3=6, 2x4=8. They could go 2,4,6,8 but not the other.

In each small group, there were subcategories. So in the first group, there was one boy who was the fastest fact recaller in the group, but could never, no matter what I did, solve the second equation. He also had a very low IQ (sub 85). But most of these kids could solve the same problem if they turned it into a question: "I multiply a numer by 12 and add 5, get 29". "TWO!" they'd say, faster than I could write it out. This has, consistently, proved the most consistent success story I've had. A number of students since have survived algebra because I taught them to restate the problem in the way their brains could handle.

The second group was, for the most part, fatally challenged in math, weak in every way. But one girl was the best algebra student in the class, provided you gave her a calculator to do math facts. She could graph lines, solve for points, and even factor simple quadratics.

Now, the rest of the kids did get better at their math facts. Not a lot better, but some marginal improvement. For the most part, though, they knew what they knew. What might have helped is instruction earlier in their math careers to get them past the 10 or 12 math facts they stumbled on (9+7, 6x9, etc.)

So how can researchers help teachers? First, it could acknowledge the existence of the groups I mention above, which I've seen repeatedly through my career. They could research best practices with these groups, as well as diagnosis. They could acknowledge the realities of IQ, as well as the bizarre ways it can play out (eg, some low IQ kids will be great with math facts, others not so much).

If elementary teachers could sort out the kids who will not ever be able to get their math facts, it would help them be tougher and more focused on the kids who can.

At the same time, all teachers know kids who are weak on math facts but excellent on other aspects of math, which again makes us skeptical when researchers argue that math facts are an essential, uncompromisingly necessary avenue to success.

Renhard Heggert II
7/20/2017 05:49:29 am

I`m flabbergasted. I have seen pictures of these things and read about it but I still don`t understand what they do. It`s a ball bearing with a piece of plastic, it doesn`t do anything. Whoever invented this should work for NASA, you have to be a genuis to make something so pointless and get people to actually buy it. I can`t belive people spend money on it, there`s just too many dumb people on this planet.


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