New comment in one of my Tabitha posts, from Steve Prosser (who, not coincidentally, has an app he’d like to sell you),

It is important starting around entry to first grade, IMHO, for children to: a) have memorized the patterns of basic arithmetic equations, and to b) understand how more complex problems can break down into simpler step-by-step arithmetic. Doing this in a way that promotes right-minded learning (pattern recognition) is vital. I’ve done my best attempting this for my daughter with my app – mathflashapp – and her performance through the third grade suggest this is the right track.

I have lots to say here and struggled with whether to say it in the comments or on the main part of the blog. So forgive me if this gets too detailed.

The post in question was really about the relationship between research findings and a child’s development. I wasn’t expressing a belief or an opinion in that post, nor a philosophy. I was using the example to make the research come alive.

This points to a more general principle here on the blog. I am interested in examples (whether my own or others’) when they either:

(1) Present a puzzling case that needs explaining, or

(2) Illustrate research findings,

and when they

(3) Are offered with enough detail that others may propose alternate interpretations or hypotheses.

In Boolean algebra, I strive for (puzzling OR illustrative) AND detailed.

Claims of the sort, “It worked for me and I’m OK” or “I did this with my class; they seem to be unharmed” are not particularly helpful to the cause of each of us learning something.

I have no beef with fact memorization, nor with apps that help students to memorize these facts. But we have good research evidence that far more time is spent on low level rule recitation, practice and review in American mathematics classrooms than in other countries with more successful mathematics education programs.

One of the agendas of this blog (there are many) is to explore what other possibilities are within our grasp at a variety of levels. So I’m never going to devote much space to techniques for memorizing arithmetic facts. There’s no new ground for me to cover there. Yes, an app makes it more efficient and gives instantaneous feedback. It’s a marginally better training device than flashcards would be. I have no problem with that.

But I don’t think we learn much from it as a field.

On the other hand, I know for sure that few people outside of the hard-core elementary math education circles know anything substantive about CGI. We have a lot to learn from that project. Examples can help bring the findings to life, and can help people understand the importance of these findings.

Selling a group of math teachers on the proposition that it would be nice if more students knew their addition and multiplication facts? That is not a particularly difficult challenge.

Helping them (and me) to understand the thinking that’s going on in kids’ minds as they learn new stuff? That’s a life’s work.

See also my post on setting norms.

Provocative post, Chris. If I’ve not already mentioned the work of the late John van DeWalle here (I’ve no doubt you’re aware of it), let me do so now. I think he had some of the best notions of how to address the concerns of those in and out of K-5 classrooms about getting those facts into kids heads. And he didn’t seem to be big on rote memorization, from what I’ve seen. I urge interested readers to look through any of his K-8 methods books to see the sorts of possibilities he advocates.

For my part, I agree that there’s much more to be gained in trying to understand how kids think about basic arithmetic, the sorts of errors that they are prone to, the buggy algorithms they develop, and what sorts of learning experiences are effective in giving students a strong foundation for doing and thinking intelligently about both basic arithmetic in particular and mathematics in general. We have at least a century of evidence that suggests strongly that if we want to kill interest in and appreciation for mathematics in the vast majority of our citizens, it would be hard to improve on what’s been done in most US K-5 classrooms.

Making mathematics into a bunch ‘o facts and procedures to be memorized with or without understanding (mostly the latter) has long been mastered by American pedagogues. I seriously doubt there is any untapped ground when it comes to turning the beauty and power of mathematics into sheer tedium and drudgery.

The path towards sanity is not to drop “facts,” and despite two decades or so of rhetoric to the contrary, few mathematics educators advocate for anything of the kind. Rather, most have been looking at sane and effective methods for integrating basic number sense, mental math, estimation, problem solving methods, and mathematical habits of mind into curricula and teaching methods that don’t ignore the advantages of having facility with basic arithmetic facts and operations.

I could go on at length but will resist the temptation to expound upon the entire history of the Math Wars over the last two decades. Instead, I’ll close by suggesting that the last thing the world needs is another drill-and-practice math ap. If that’s what was required to “spark a love of mathematics,” we’d be home by now. That we’re not, not even close, is the best refutation of such nonsense.

“Buggy algorithms”. Don’t get me started, MPG, on the relative bugginess of various algorithms (especially the “standard” ones).

Tonight’s bedtime conversation with my 7-year-old daughter:

G: Dad, did you know that in some countries girls aren’t allowed to go to school? They won’t even get to know what 9 plus 9 is.

Me: Do

youknow 9 plus 9?G: No, but I know 10 plus 10 is 20. We do that one a lot.

Me: Okay, so what’s 9 plus 9?

G: 18.

Me: How’d you get that?

G: I counted two down. They make JUST THE GIRLS stay home and do chores ALL DAY.

I’m confident that my daughter will become more fluent with her single-digit addition facts. In the meantime, she’s learning strategies and developing number sense. I’ll take that over flashcards, even the ones with retina display, anyday.

Compare that with a conversation from earlier in the year:

My daughter comes home from school and tells me she hates math. She completed 13 questions on “The Mad Minute” and compares herself to her best friend who completed all 30.

My 4 year old daughter watched the worst DVD today that was supposed to teach math. . I rented it, thinking it couldn’t hurt. Not sure if I should say the name of it here but it basically flashed math facts across the screen with a lame storyline. There was no time given for her to think or process what she was “learning.” Even if it was supposed to help with math facts, the order of the facts presented didn’t make sense to me. . 2+2=4 flew across the screen followed by 4-3=1 or something like that. Even if it is just a drill exercise, why not at least include fact families to show that you are undoing what you just did? No more silly DVDs for us.

We count everything. . if we’re doing an art activity, we count how many googly eyes we need for our animals, etc. I think that is more meaningful. Having said that, we count a LOT (poor kid 🙂 so eventually she’ll get her math facts down.

Two quick place value anecdotes: The other day she was counting something and she said “forty-eight, forty-nine, forty-ten. . ” reminded me of some comments made here by you or somebody else about the language of math – she knew what “fifty” meant though she used a different (and perhaps more meaningful) name for it. Also we were doing an activity and she had to read the number 19, so I tried to get her to say it aloud. I said to her, “It’s one – _____,” expecting her to say “nine” and just recognizing the one and nine individually. You probably can fill in the blank. . she said “one ten” and of course she was right. . 19 has one ten in it. It just amazed me that she could recognize that at age 4. I need to do more research on early number sense. It’s amazing watching her learn.