All those long multiplication tables. Timed tests and 鈥渕ad minutes鈥� of worksheet problem-solving. Fluency drills.
Somehow, getting kids to know their basic math facts continues to be at the heart of some of the loudest disagreements in mathematics education.
Let鈥檚 put this old misconception to bed: The cognitive science about math learning indicates that, yes, students do need to develop fluency with their multiplication tables and single-digit addition鈥攕ometimes called 鈥渘umber combinations鈥濃�攁nd be able to recall them automatically. The main reason why? Having these facts at their fingertips frees up working memory for students to attend to problem-solving, applying procedures to more difficult problems, and other tasks.
The real problem is that research doesn鈥檛 point to a clear recipe for how to help students develop their math facts.
Adding to the challenge, some teachers labor under the idea that explicit work on fact fluency isn鈥檛 fun for kids, is stressful, or ignores deeper conceptual math. In part that is a product of bad practice鈥攍ike hours on computer programs or poorly crafted timed exercises鈥攂ut some of it, some teachers note, is also the product of adult baggage.
鈥淢ultiplication facts seem boring to us because we know them,鈥� noted Dylan Kane, a 7th grade math teacher in the Lake County district in Leadville, Colo. 鈥淏ut successfully learning new things is motivating for students. Because it鈥檚 old and boring for us doesn鈥檛 mean it鈥檚 old and boring for them.鈥�
For this story, 澳门跑狗论坛 examined dozens of studies, spoke to researchers on cognition, and interviewed practicing teachers. We asked them how they approach fact fluency鈥攁nd how to balance this piece of the math puzzle against all the other things that should happen in the elementary math classroom.
In general, they said, math-fact-fluency work can and should be a purposeful part of the math classroom. But it doesn鈥檛 have to take forever鈥攐r be a drag: Keep the practice relatively short but consistent, keep it well-sequenced, and don鈥檛 try to do too much at once, they adivsed.
1. Focus on student mastery of a few facts at a time鈥攁nd space practice out
Whether students are building their fact fluency with flashcards, worksheets, dice games, or other common tools, there鈥檚 a tendency to try to do too much at once, cautions Nicole McNeil, a professor of psychology and the director of lab on cognition, learning, and development at the University of Notre Dame.
鈥淲hen you use all the number combinations randomly, that is too much for a kid to remember. You have to structure it in the way we know works for retrieval practice鈥攖hree facts, maybe four, maybe two,鈥� she said. 鈥淵ou can鈥檛 become fluent with all of them at once.鈥�
One teaching approach here is called Teachers present a new math-fact flashcard alongside one that a student already knows. As the teacher quizzes the students, the teacher gradually adds more known facts, increasing the time between the known facts and the new one the student is learning.
Many online games purport to help with fact fluency, but as usual, the devil is in the details, and teachers need to know precisely how these programs work, said Kane, the Colorado teacher. He likes an online computer program he uses with his middle school students who still struggle with math facts. It has a limited number of flashcards, and if a student gets a math fact wrong, the computer adds it back into the pile for the next day鈥檚 work. (The program only allows kids to go through the cards once a day.) It also starts with smaller numbers, gradually adding more difficult ones as kids鈥� fluency develops.
In effect, the program handles the spaced repetition element that seems to help kids remember their math facts. 鈥淭hat increases the probability that getting the fact right yesterday will help them retrieve it successfully today,鈥� he said.
In all, Kane said, teachers also need to be aware that fact fluency takes time to develop and isn鈥檛 a one-and-done.
鈥淚 think in too many places, this is seen as 鈥榦ne unit,鈥� and there isn鈥檛 enough regular, steady, low-dose practice throughout the year,鈥� he said. 鈥淚t needs to be a long-term project where students are making gradual, incremental, but steady progress that they can see.鈥�
2. Teach strategies鈥攂ut know that they have limitations
Many students will ultimately intuit strategies that help them learn their addition facts. But explicit teaching of these backup strategies can be helpful鈥攍ike 鈥渃ounting on,鈥� in which students count up from one of the numbers to be added (i.e., 6+5 is counted out as 鈥�7, 8, 9, 10, 11"). Decomposition, in which students learn that they can break larger numbers down to smaller ones to make it easier to add, can also help kids who struggle.
This is probably because it helps students to unite conceptual knowledge about how whole numbers work with the facts themselves.
In addition to regular retrieval practice, 鈥淚 do think it is also helpful to give kids some of these backup strategies,鈥� said Notre Dame鈥檚 McNeil. 鈥淎ctivating existing knowledge is part of what helps it stick when you鈥檙e trying to use retrieval practice, because you can anchor it to some of your existing long-term memory.鈥�
Multiplication can be trickier because fewer strategies seem to transfer across different number sets. The strategies also take longer than in addition, and it鈥檚 not as clear whether teaching strategies explicitly helps students reach automatic recall.
Doubling is one of the strategies that Brian Bushart, a 4th grade teacher in the West Irondequoit Central school district in New York state, teaches explicitly. As early as 1st grade, students learn doubles like 2+2, 3+3, 4+4, and so on鈥攅ssentially, the two鈥檚 times table. They can build on that knowledge when moving into multiplication.
鈥淵ou can ask students, what if you double again? And you make the connection that if you鈥檙e doubling again, you鈥檙e multiplying by four,鈥� he said. 鈥淵ou鈥檙e showing students that you don鈥檛 need to have a bunch of random skills, because doubling ends up working for a chunk of your multiplication facts.鈥�
Once students learn 3x3 is 9, then they can also make the connection that double that, 6x3, is the same as 2x9 and makes 18, and so on.
鈥淭hese aren鈥檛 just cute strategies, they鈥檙e actually the basis of multiplicative thinking, which is a huge goal we want for students, because they come to us in 3rd and 4th grade with additive thinking,鈥� Bushart said.
3. Keep broader goals about math in mind
Some of the historical drama over math facts grows out of the slipperiness of the term 鈥渇luency,鈥� which in the literature sometimes refers just to math facts and, in other cases, includes learning the standard algorithms for problem-solving (like regrouping or 鈥渃arrying鈥� in multidigit addition). For many teachers, like Jill Milton, an elementary math specialist with the Duxbury school district in Duxbury, Mass., fluency also means an overall kind of flexibility with numbers.
鈥淚f I have a student who is solving 9+5, I want them to have the flexibility to say, 鈥榃ell, I can easily do some compensation and look at that as 10+4,鈥� 鈥� she said. 鈥淚t gives students this step-in to say, 鈥業 can manipulate these numbers and have the control over how I answer the problem.鈥欌��
The goal of automaticity, procedural knowledge, and conceptual knowledge of math are often set up in math education literature as opposing interests, but research suggests that they actually develop together.
鈥淲e want these facts encoded in long-term memory, but we want them as networks of connected ideas, flexible representations that we can draw on in different ways,鈥� Kane, the Colorado teacher, said.
One way to do that is to include number lines and other representations as part of work on math facts, said Daniel Ansari, a professor of psychology and the Canada research chair in developmental cognitive neuroscience at the University of Western Ontario in London, Ontario.
鈥淭he most useful way I like to think of it is from concrete representational to abstract,鈥� he said. 鈥淔rom manipulatives, then you start to do things around number lines and, at the same time, also develop thinking about solving problems in multiple ways and comparing problem-solving strategies. Just practicing question-and-answer alone is not going to do it.鈥�
And when working on fact fluency through traditional activities aimed at fact retrieval, teachers can help shore up students鈥� knowledge of undermining mathematical concepts. Take flashcards, for instance. McNeil says that she always includes the equal sign on the cards she uses for tutoring, so they read 2+5 = _ or _ = 2+5.
Her research indicates that many students struggle with the concept of mathematical equivalence and the importance of the equal sign, which is linked to later success in algebraic thinking.
Similarly, when Bushart does flashcards with students, he lists two facts for each flashcard: 6x7 and 7x6 are really the same math fact, not two unrelated ones. That reinforces the commutative property of mathematics鈥攖hat number order does not matter for addition and multiplication. (It does for subtraction and division.)
Milton, meanwhile, begins many of her lessons with a number talk. Number talks are a common practice in which a teacher puts a question up on the board and has students narrate out loud which strategy they used to solve it, observing and commenting on the different approaches.
There is little empirical research on number talks, and some researchers like McNeil doubt they helps students store math facts in memory. But they likely do help improve students鈥� making sense of numbers, exploring different procedures for solving problems, flexibly selecting among strategies, and math reasoning more generally, she said.
4. Structure timed exercises carefully
For many educators, the idea of timed testing immediately conjures up images of sweaty palms from their own school days. These exercises are included in several popular curriculum series, and many educators remain deeply skeptical about their merits.
Sometimes known as 鈥渕ad minutes,鈥� the exercise requires students to solve a specified number of basic problems in that amount of time. Many teachers鈥攁nd some scholars鈥攃ontend such activities fuel anxiety among kids, despite a lack of conclusive research.
Milton is among them: She finds timed testing 鈥渓azy.鈥� Plus, she said, the exercises don鈥檛 produce much actionable information to determine which strategies students could bolster: 鈥淚f it鈥檚 not one that鈥檚 focused on giving me results, it鈥檚 not useful.鈥�
But there is a benefit to timed exercises, other educators counter: The drills require students to retrieve math facts from memory rather than falling back on the strategies.
鈥淲e have to practice what we want to become fluent with. There is no amount of practice with backup strategy in multiplication that is going to get a student to produce a product in three seconds or less,鈥� said McNeil.
The proponents say the way to gain a benefit out of timed exercises without all the bad vibes is to reduce the pressure. It should focus on fewer facts at a time, and students should try to beat their own time in solving a select sample of facts鈥攏ot get compared with their peers on a poster board hanging on the wall for all to see.
鈥淭he way I introduce it is to ask them to solve as many as they can, then I will sort of go over the answers again orally before the second chance, and then we time it again, and they love beating their time,鈥� said McNeil, who also tutors in a local school district. 鈥淭hey鈥檙e getting so much experience of meeting a goal, and feeling good when they do it, they ask for it at each session.鈥�
Bushart, similarly, tailors students鈥� timed exercise to their own progress. He also says it helps to explain to students why he鈥檚 requiring them to do the timed drills.
鈥淲hen they do timed tasks, I tell them: 鈥楾his is how your brain works. I know you know how to skip count. I know you know how to double. But when you want to remember, you have to practice remembering it, and I have to put you in a situation where you don鈥檛 have a lot of time to do that extra stuff,鈥欌�� he said.
