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Fluency Without Fear – the Best Ways to Learn Math Facts – by Jo Boaler and others at Stanford

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Here is a wonderful piece by Stanford math professor Jo Boaler.  I offer it not as part of the Common Core debate but because it lives in a world so far beyond that debate.  Whether or not you are a math person, so to speak, you will enjoy this.

Never mind the longer paragraphs.  It’s still easy to read.  Here’s the beginning:

A few years ago a British politician, Stephen Byers, made a harmless error in an interview. The right honorable minister was asked to give the answer to 7 x 8 and he gave the answer of 54, instead of the correct 56. His error prompted widespread ridicule in the national media, accompanied by calls for a stronger emphasis on ‘times table’ memorization in schools. This past September the Conservative education minister for England, a man with no education experience, insisted that all students in England memorize all their times tables up to 12 x 12 by the age of 9. This requirement has now been placed into the UK’s mathematics curriculum and will result, I predict, in rising levels of math anxiety and students turning away from mathematics in record numbers. The US is moving in the opposite direction, as the new Common Core State Standards (CCSS) de-emphasize the rote memorization of math facts. Unfortunately misinterpretations of the meaning of the word ‘fluency’ in the CCSS are commonplace and publishers continue to emphasize rote memorization, encouraging the persistence of damaging classroom practices across the United States. Mathematics facts are important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging. The English minister’s mistake when he was asked 7 x 8 prompted calls for more memorization. This was ironic as his mistake revealed the limitations of memorization without ‘number sense’. People with number sense are those who can use numbers flexibly. When asked to solve 7 x 8 someone with number sense may have memorized 56 but they would also be able to work out that 7 x 7 is 49 and then add 7 to make 56, or they may work out ten 7’s and subtract two 7’s (70-14). They would not have to rely on a distant memory. Math facts, themselves, are a small part of mathematics and they are best learned through the use of numbers in different ways and situations.  Unfortunately many classrooms focus on math facts in unproductive ways, giving students the impression that math facts are the essence of mathematics, and, even worse that the fast recall of math facts is what it means to be a strong mathematics student. Both of these ideas are wrong and it is critical that we remove them from classrooms, as they play a large role in the production of math anxious and disaffected students. It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact. But I learned math facts through using them in different mathematical situations, not by practicing them and being tested on them. I grew up in the progressive era of England, when primary schools focused on the ‘whole child’ and I was not presented with tables of addition, subtraction or multiplication facts to memorize in school. This has never held me back at any time or place in my life, even though I am a mathematics education professor. That is because I have number sense, something that is much more important for students to learn, and that includes learning of math facts along with deep understanding of numbers and the ways they relate to each other.

read the rest at Fluency Without Fear.


  1. We can debate interminably the question of “fluency,” in computation. I think many people have some notion of automaticity when it comes to “simple” or “basic” arithmetic facts. You ask a kid how much 3+9 is or what is 7×4 and expect the answers 12 and 28 to be produced in less than a second without hesitation. The concern is usually that giving more time means the student is using some sort of counting strategy, and then we are warned of dire consequences for the student in question when it comes to learning “higher mathematics” if s/he has to rely on fingers or anything else that boils down to counting.

    But of course, there are developmental issues, stages of thinking that kids can go through if allowed to do so. At some point, we hope to see a child progress from grabbing a bunch of plastic objects, counting 3 into a pile, then 9 into another, then pushing them together and counting them all and arriving at 12, to perhaps counting 3, then “counting on” another 9, starting at 4, to arrive at 12. Later, as the child begins to understand the commutative property of addition and uses that with some common sense, s/he may adjust the second strategy to counting 3 more onto 9, the larger addend, and getting to 12 sooner. But eventually, we want that child to instantly answer that 3 + 9 is 12.

    So is the path to that goal to drill the child in those arithmetic facts as soon as possible? I think that for many Americans, that’s the only way, be it via flash cards, computerized drills, timed worksheets, etc. To such people, I suggest checking out a quality book on elementary math methods (anything by John van de Walle for primary grades will do) and starting to think about whether drill is simply the only way to get to the goal, and if not, whether it’s the best way.

    Unfortunately, most such folks have no patience for the notion that thinking and recall are connected. Hence, they will scorn ideas like using math facts already mastered by a child (e.g., doubles, which many kids learn before school as part of board games, etc.) upon which to build new facts (e.g., doubles plus or minus one), or using understanding of base 10 (e.g., 10 plus any one-digit number gets you into the “teens”; 9 plus any one-digit number gets you one less than 10 plus that number, etc.). And so they demand that teachers spend precious instructional time in classrooms drilling kids. The more drill, the better. The more demanding the task (high-pressure timed sheets of arithmetic are favorite here), the better. Kids who don’t or won’t crack down and learn to spew back those facts automatically are bad or stupid or both. Teachers who won’t crack that particular whip are fuzzy (and probably soft on Communism and Islamists, too).

    There are loads of games that are built around giving kids opportunities to practice math facts. Some textbooks for elementary grades offer lots of kid-friendly games that help kids practice their facts. But teachers often ignore those ancillary books and parents often won’t sit with their kids to play those games when they’re assigned as homework (my son, now nearly 20, used to have the oft-praised, oft-maligned INVESTIGATIONS IN NUMBER, SPACE, AND DATA series from TERC through fourth grade, and he and I played a lot of those games. He breezed through math in high school despite lack of repetitive drill. Shocking, I know).

    I honestly believe that there is more underlying the American romance with drill in math class than concern with education: the “No pain, no gain” mentality is still dominant in our culture, part of the Puritan ethic that pervades far too much of American life. While there’s nothing to prevent wise teachers and parents from adopting/supporting blended approaches to elementary mathematics education, the tendency is for many parents and teachers to rebel against anything that doesn’t seem like it’s toughminded and “character building” (i.e., pain-producing). That’s not simply philosophy at work: it’s psychology bleeding into how we think about children and education. And as long as that viewpoint is pervasive, we’re going to continue to spin our wheels with math and much else, and ensure that many children learn to despise one of the most beautiful, thoughtful areas of human endeavor.

  2. wrightstar says:

    It all began with counting. I am a professional mathematician (US Army Intelligence, Universities of Wisconsin, Texas, California, S. Florida, ATT Bell Labs, etc) and I count occasionally. Counting is logically valid.

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