Yes, this is a tongue-in-cheek post, but considering how we spend our lives here trying to help folks master their core math facts, I thought this would be a wonderful story to share about the need for mastering your core math facts!
Yes, this is a tongue-in-cheek post, but considering how we spend our lives here trying to help folks master their core math facts, I thought this would be a wonderful story to share about the need for mastering your core math facts!
I’m intrigued by the negative responses to the Common Core standards. The opposition is so passionate that often their arguments explode in a flaming ball of general vitriol that makes it hard to see what specific aspects they’re opposed to. So, in an attempt to engender productive discussion, here here is my attempt to present Common Core concerns simply and clearly.
Inherent in my approach is the assumption that in the grand scheme of things everyone’s actually on the same side–the kids’ side. And our kids will be better served when everyone stops fighting and instead works together productively to find common ground. (pardon the pun and cue the sappy music for group hug)
Please jump in to clarify or add concerns, and I will edit accordingly.
Many folks see the standards as federal control and they hate being constrained and dictated to by The Man.
In many cases, efforts to address the standards have been inept or offensive.
The standards are too high for struggling students or populations, so individual states, districts, and schools would like more flexibility to lower them. Others think the standards are too low.
Clear cut standards could create greater pressure on performance. This pressure could tempt teachers to try “teaching to the test,” even though such methods generate worse test results*. Performance pressure could also prompt additional testing throughout the year, which would detract from instructional time.
The standards were not created in an ideal way, using ANSI guidelines for example. It’s costly to implement new standards. They might track too much information about students.
Every once in a while I encounter a savvy educator who is opposed to memorizing math facts–or at least they appear to be. Just today I saw a fearful article that exclaimed “Memorization Can Inhibit Fluency” and “Memorization . . . can be damaging.” Such educators are doing a wonderful job of championing critical math skills such as number sense, comprehension, and problem-solving, but by attacking the vital skill of automaticity, they’re unwittingly undermining the very processes they intend to champion.
Before I go any further, let me jump to the punchline. Because I know that if you’re one of these educators, you’re already getting ready to give me your very passionate point of view. So . . . if as an educator you have a negativity towards memorization, I would suggest that it’s because you haven’t seen it done well. [It can also be that you're mostly opposed to memorized process instruction, and in your enthusiasm you lump in all memorization. If so, please read on.]
When done brilliantly, core fact memorization is wildly beneficial. So my call to action is this–run a simple experiment with Big Brainz. Run some kids through our program. All the way through our program. Then measure the impact on number sense over the next year. I can guarantee the comprehension and application of those kids will skyrocket because I see it happen every single day with every student who runs through our program.
It permanently accelerates comprehension and problem-solving development by freeing up working memory. It improves number sense! It reduces pressure and catapults confidence. The kids love it. Oh, and it actually delivers automaticity–every time.
Again, until you put us to the test it’s likely you will be defensive and won’t really see anything I’m saying. So seriously, please put us to the test.
Now getting back to how negativity towards memorization undermines overall math success. Imagine a basketball coach telling his team, “To win games you need great defense and shooting–so stop learning how to dribble and pass.” That would make no sense. Great ball-handling only helps everything else you want to do in basketball, and automaticity only helps comprehension and problem-solving.
Sidenote: Just like in math, a basketball player absolutely needs to progress as quickly as possible from static dribbling drills to dynamic dribbling drills. And even static dribbling drills should be motivational and productive. But showering players with negativity towards drilling in general is going to undermine the overall skill, handicapping all their future success because they’ll constantly have to be thinking and looking at the ball, rather than their teammates or the hoop.
NCTM, TEKS, Common Core, and The National Mathematics Advisory Panel, all agree that “curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills,” further clarifying that computational fluency means “automatic (i.e., quick and effortless) recall of facts.” That’s memorization.
Furthermore, that same Final Report specifies that arguing about the relative merits of these different facets of math is “misguided” because “these capabilities are mutually supportive, each facilitating learning of the others.”
Memorization Accelerates Comprehension and Problem-Solving
Students need number sense, comprehension, and application, and a child who enters fourth grade already automatic with core math facts is going to be able to develop these math concepts FASTER than a student who doesn’t. They will be more confident, but they will also have more available working memory.
Our own research confirms that memorization accelerates comprehension and problem-solving. Specifically, giving student populations the gift of automaticity catapults their year-end math scores. Fluency accelerates overall math learning because once they’re fluent, students permanently have more available working memory with which to learn new concepts and applications. If you’re going to chop down trees all day, please take a few minutes to sharpen the ax.
I heard one educator lament that pressure can impede working memory so when a student most needs to recall a fact, they can’t and then they’re stuck. This misses the point. If the fact had been actually memorized, it wouldn’t have needed working memory. It would be effortless and automatic, even under duress. When core facts become trusted friends rather than distant memories, they are always available to help diminish stress AND free up precious working memory.
Of course they still need comprehension and application skills as well. They need all three.
I’ve heard some educators express concern that rote memorization is creating robots and therefore impeding the development of number sense. Not true. Phrases like “robots” and “drill and kill” have a nice, scary ring to them, but they imply a false principle. Mastering core skills does not in any way inhibit or undermine higher-order thinking. Quite the opposite, it liberates it. Freeing up working memory absolutely helps students be more successful at conceptual understanding and problem-solving. We’ve run tests where students master multiplication before learning ANY comprehension, and they still do significantly better when they are taught the comprehension.
I’ve heard some educators complain that the pressure of memorization can give students a bad experience. So stop doing it badly and start doing it well! It’s not memorization that creates bad experiences any more than math instruction in general creates bad experiences. It’s bad instruction and bad memorization that create bad experiences. And be very aware that the frustration kids feel when they have to stop and think about every single basic calculation they encounter is always significant. You don’t remove that frustration simply by telling them it’s OK.
Common Core Emphasis
I’ve heard passionate educators exclaim that Common Core raises the priority of comprehension and application while lowering the emphasis on memorization. Absolutely false. Common Core significantly raised the priority of memorization, bumping it an entire grade earlier AND increasing the specificity of recalling facts “from memory.” Now, they ALSO raised and clarified the priority of comprehension and application . . . because all three are necessary and mutually beneficial.
Once again, successful math students “must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills.” “Debates regarding the relative importance of these aspects of mathematical knowledge are misguided” because “these capabilities are mutually supportive, each facilitating learning of the others.” [Final Report, 2008 emphasis added]
Educators with a wonderful and valuable passion for number sense often express that the best way to develop automaticity is to simply use the facts regularly, combined with solid comprehension. I really like the sound of that, and I really wish it were true. But it’s not. It doesn’t work.
We’ve worked with millions of students, and the majority of them will never develop automaticity with their core facts simply through continued usage and exposure. Some students will. And all students will master some. But most students at some point will need to buckle down and focus to become automatic with just their single digit facts.
However, they CAN acquire this precious skill in a matter of hours–at least with Big Brainz they can. EVERY student on earth should be given the precious few hours needed to acquire this precious skill and permanently expand their mathematical working memory.
Wrapping up, let me also point out that it CAN be perfectly accurate to rant and rave against memorization IF you get just a bit more specific. Rant and rave against ineffective memorization–techniques that generate a negative experience and/or don’t actually generate fluency (which is most of them). Blast these traditional methods out of the water.
You’re also welcome to rail against step-by-step process instruction that less effective teachers often fall into.
Don’t Change or Undermine Common Core or TEKS Requirement
But at the same time be very careful not to undermine the critical skill of automaticity. Be true to all the research behind the very specific common core requirement: “by the end of Grade 3, know from memory all products of two one-digit numbers.” Don’t undermine or change this requirement. And don’t think it’s importance is small just because it can be written concisely. The best scientific research in the world found that the #1 “Critical Foundation for Algebra” is whole-number fluency, the automatic and effortless recall of core facts, because it frees up working memory to focus on higher-order concepts and application. Give this precious gift to every single student.
Sharpen the Saw
Your students are going to be chopping away at comprehension and problem-solving 180 hours a year, every year. Give them the few hours needed (and a sufficiently powerful fluency program) to permanently sharpen their saw.
In 2011 Andrea Kuszewski wrote a great article about how we can increase our intelligence: http://blogs.scientificamerican.com/guest-blog/2011/03/07/you-can-increase-your-intelligence-5-ways-to-maximize-your-cognitive-potential/
My metaphor for what Andrea said would be: cross-training. If you’re constantly pushing your body (or mind) in new and challenging ways, it will continue to strengthen and improve. Once it gets used to a certain activity, it stops building new muscle and can actually decline as it gets more efficient.
Big Brainz happens to fulfill most of the applicable steps for expanding your intelligence, and the key tie-in for me, was the fact that increasing your mental capacity in one area, allows you to be more capable in other areas.
The cross-training metaphor kicks in again–getting good at any sport will have a positive effect in other sports. In education that would imply that mastering your multiplication facts with Big Brainz would likely make you a better reader.
What we have actually found is that mastering multiplication has a disproportionately positive impact on everything a student does because it significantly boosts the students’ confidence and work ethic. Sure, by simply mastering core multiplication facts, the student has removed the most prolific stumbling block in math. But they’ve also removed a giant billboard from their life that was constantly telling them “You’re stupid,” or “You’re no good at math.”
By conquering the ubiquitous challenge of fact fluency, they’ve chopped that sign down and burned it, replacing it with a sign that constantly tells them, “Wow, you NAILED that really hard thing. You’re wicked smart! With a little elbow grease, you can do anything!!”
Transforming students from discouraged to flourishing–THAT is why we do what we do here at Big Brainz.
I just stumbled across an article incorrectly stating that mastering multiplication tables is LESS important in the Common Core.
Here’s what Common Core really says:
The more you care about developing higher-order mathematics, the more important fluency becomes–here’s the comment I left:
It’s completely incorrect to state or imply that Common Core does not stress or require basic fact memorization. In truth, Common Core actually elevates the priority of automatic recall–now requiring automatic recall by the end of grade 3. Automaticity is obviously not the end goal of mathematics, but it IS a critical step.
This is a critical clarification because massive research shows that the average student was already failing at fluency in Idaho (and everywhere else in the country). Anything making it an even lower priority will only exacerbate the problem of trying to teach algebra to 7th-graders who have to stop every 5 seconds to calculate 6×8.
Understanding and strategy are partners with fluency–not enemies. Implying otherwise is like saying that you’re going to win football games by tossing out the offensive line so that the quarterback has more room to maneuver. They may not be sexy, but they win games.
Automaticity doesn’t impede or detract from strategy and understanding–on the contrary it helps it enormously. The real problem with fluency is that most teachers are still using primitive tools, spending 15 to 30 hours, only to average 59% fluency by 5th-grade. That means they’re spending too many precious hours without even getting to the finish line!
This drives us crazy because as experts in fact fluency, our average 3rd-grader masters fluency in 4.6 hours and tests above 95%. Don’t toss fluency, but do toss stone age tools like flash cards, flash games, and worksheets. I’d rather not even mention our product’s name because I don’t want this to be self-serving. I’d rather focus on the key point that total fluency is critical, and it’s quite speedy and guaranteed if you just stop and Google a truly powerful solution.
Lack of Assessment and Reporting.
Fact fluency is not directly measured in end-of-year testing. Schools tease at assessment with worksheets and games, but they never get in and find out which facts students can recall automatically and which ones they can’t.
Principals and teachers aren’t eager to promote failing students, but because the depth of failure is not clearly reported in black and white, the problem is easily swept under the rug.
If schools published a letter grade for the math fluency of each class, we would see an instant and dramatic improvement. Few educators would be willing to tolerate an average grade of “F” for their classes, but with fact fluency they do tolerate across-the-board failure because they never really know that’s how bad their students are doing.
Math fact fluency is easily the most un-diagnosed cause of overall math failure later in life, and could be quickly remedied if it were simply assessed and reported.
If you read Part 1, you know that mastering multiplication is hard. But it’s not impossible, right? I mean any decent educator with a deck of flash cards can whip just about any student in to shape eventually, right?
But teachers rarely get to teach 1-on-1 and when they switch to teaching large groups, their instruction has to generalize. They turn to worksheets, random multiplication games, or flash cards. However, all of these methods lack the benefits that intelligent instruction can provide.
Specifically, worksheets, flash cards, and most games aren’t able to prioritize fact presentation based on how well the student knows each individual fact. And second, they aren’t able to be strategic in how they move knowledge form short-term to long-term memory.
Random drill and kill–a surefire recipe for failure.
Because it’s hard. It’s difficult. It requires mental effort and consistency.
Even Timez Attack requires a significant amount of effort. It’s fun and rewarding, but it’s active engagement. And folks today favor passive engagement like watching TV, playing casual games, or clicking “Like” on Facebook.
I guess this is kind of a short post, but in the end, effort really is the only real reason a child doesn’t master their multiplication tables. I’ll start giving more practical solutions in the next segment.
Multiplication worksheets are the most common way that schools measure math fact fluency, yet aggregating response times from so many different facts rarely tells you which ones a student is actually fluent with and which ones they are still calculating.
Here are a few tips for maximizing your effectiveness:
1. Assess each individual fact. You can use a stopwatch or let a program like Timez Attack do it for you automatically, but at some point it’s critical to measure which facts they can recall automatically and which ones they have to stop and calculate. If you don’t measure it accurately, you won’t fix it.
2. For your reference, fluent students generally recall AND input a 2-digit response in 1.9 seconds with a standard deviation of .6 seconds. Verbal responses would typically need just 1 second. A fluent student is automatic, but they are also confident. It’s empowering and motivational to KNOW the answer, so don’t shortchange students by settling for fumbled answers.
3. Report the results and do it as a letter grade. Few educators or parents will ignore a student that is clearly failing, yet the majority of students today currently have an “F” in fluency, yet they are promoted without a second thought because the shortcoming is only vaguely discovered and reported–if it’s discovered or reported at all.
4. If you have to use worksheets for assessment for some reason, at least do it in smaller chunks or lock down the average time. 2 seconds per fact will always be a fairly certain demonstration of fluency and 2.5 to 3 seconds per fact for 10 to 12 facts at a time will do a decent job. But if you gave an average student three minutes to answer 60 facts, they would answer the easy half in 1.5 to 2 seconds each, but then take 3 to 8 seconds to calculate others, giving only the illusion of fluency.
5. Don’t settle. Fluency is a core skill, yet the average student today plateaus in the 5th-grade with just 59% mastery–essentially the easy facts. Would you promote a student that had only learned 59% of the alphabet? Timez Attack delivers 95% fluency in a matter of hours, so there’s no need to handicap students for the rest of their math careers.
6. Finally, don’t assume that student fluency will magically improve “somehow” after they leave your class. Adopt the attitude, “If I send a student out the door with poor fluency, I’m likely ensuring they will struggle with math for the rest of their life.”
I’ll wrap up with some general advice. Learning multiplication is hard, so block out the time for it. But it’s also completely doable, so don’t let failure be an option. Timez Attack can knock it out easily for you in a matter of hours. No worries if you prefer a different way but whichever path you choose, make sure your students cross the finish line.
I was in a serious discussion with the head of one of the two largest educational publishers about the incredible possibilities that high-end educational software could offer students. I pressed him to explain why everyone in educational software kept cranking out primitive games instead. His seasoned response? Kids are already so bored in school that by comparison, just a little bit of entertainment is enough.
In other words, if you’re starving, a crust of bread is a feast. Why waste money on steak? Or more generously, if a baloney sandwich will do, why prepare a feast?
So when is good good enough? Let’s use multiplication tables as an example. Are Flash cards good enough? They’re cheap. If you invest the time, they work. Why do anything else? So maybe you’re more motivated than that and you make a little flash game that teaches multiplication. Maybe you go the extra mile and have it track their work and generate reports? While you’re at it, why not Google someone’s thesis from the 80’s so you can call it “Research-Based” and charge a lot of money for it. That’s basically as far as anyone gets. So why in the world would anyone choose to invest millions and millions of dollars over years and years, when viable alternatives are so cheap and easy?
Our answer is that cheap, primitive solutions aren’t good enough because “good enough” can be easily quantified and our students are failing. Only about 1/3 of students in the US are considered Proficient in math. Period. That really ought to immediately end any speculation over whether our current efforts are good enough.
Once every student is achieving needed levels of mastery in the needed time period . . . THEN we can start arguing about whether they’re having too much fun along the way. Until then, can we as an industry please pull our head out of the sand and stop flooding the world with primitive educational software that simply doesn’t do the job?
Now, I understand the complaints about that philosophy myself. “Hey, the software we already have DOES the job. It’s just insanely hard to get schools to realize that and use it.” Yes, schools are unbelievably fragmented, politically divisive, cash-strapped, and justifiably cynical about educational software. You would truly have to be stupendous in ways as-yet-unimagined to cut through all that friction.
So do it. Create programs that aren’t just well-intentioned, evolutionary improvements. Break all the rules. Invest a billion dollars in Algebra and change the world forever. Billion sounds like a big number, but over 10 years that amortizes down to just $25 per student. And that’s just in the US. Spread that across the globe and it’s practically free. Do you think districts would pay a few bucks to yank all of their students up to Proficient in Algebra? Uhh . . . duh! You’d make a fortune and you’d be saving the world at the same time.
So to summarize then, can we please all settle in on the fact that whatever you’re doing, however you’re doing it, we need to massively improve our education technology. Children are still getting left behind. Only a handful are getting ahead. We have the technology to fix it and we have the capital. We just don’t demand it. We’re settling for a crust of bread when we could have a feast.