Learning Multiplication Tables by Fourth Grade
In this article in Teaching Children Mathematics, Gina Kling (Western Michigan University) and Jennifer Bay-Williams (University of Louisville) suggest a strategy for meeting the challenging Common Core standard of knowing from memory all single-digit multiplication facts by the end of third grade. Mastering multiplication facts has been a challenge for generations of math learners. “That was the day I decided I was bad at math,” is a common refrain among adults thinking back to their elementary school days. The methods used – timed tests, tense competitions, and public displays of who mastered multiplication tables and who hadn’t – may be responsible. One teacher remembered, “We learned a song for every fact. I can find any fact quickly, but I still need to sing the song first.”
Kling and Bay-Williams address three essential questions on the road to multiplication mastery:
• What is fluency? It’s been defined as “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.” Note that speed is not on the list. Also, there’s an important distinction between from memory and memorization: really mastering multiplication facts means that students “just know” that 2 x 6 = 12 without having to memorize it, and are so fluent at applying strategies that they do so automatically, without hesitation.
• What approaches successfully build fluency? Kling and Bay-Williams say that conventional methods of teaching the tables don’t build long-term mastery and fluency because they skip the second step in this developmental ladder:
- Phase 1: Modeling and/or counting to find the answer (e.g., finding 6 x 4 by drawing 6 groups of 4 dots and skip-counting the dots);
- Phase 2: Deriving answers using reasoning strategies based on known facts (e.g., solving 6 x 4 by thinking 5 x 4 = 20 and adding one more group of 4);
- Phase 3: Mastery – efficient production of answers (e.g., knowing 6 x 4 = 24).
Traditional approaches (flash cards, drill, timed tests) skip Phase 2. Without that phase, students don’t retain the facts they memorize, and even if they remember them, they can’t apply them fluently because they haven’t developed a feel for the numbers. “Research tells us that students must deliberately progress through these phases,” say Kling and Bay-Williams, “with explicit development of reasoning strategies, which helps students master the facts and gives them a way to regenerate a fact if they have forgotten it. Students make more rapid gains in fact mastery when emphasis is placed on strategic thinking.” Here’s an effective instructional sequence:
- Foundational facts – By the end of second grade, students should know: 2s, 5s, and 10s; addition doubles; 0s and 1s, and multiplication squares (2 x 2, 3 x 3, etc.) – by using story problems, arrays, skip counting, patterns on a hundreds chart, and a multiplication table.
- Derived facts – Building on the foundational facts (which they should know cold), students work on quickly figuring out “nearby” facts by adding or subtracting a group (I don’t know 9 x 6, so I think “10 x 6 = 60” and subtract one group of 6 to get 54); halving and doubling (I don’t know 6 x 8, so I think “3 x 8 = 24” and double that to get 48); using a square product (I don’t know 7 x 6, so I use 6 x 6 = 36 and add one more 6 to get 42); and decomposing a factor (I don’t know 7 x 6, so I break the 7 into 2 and 5, because I know 2 x 6 and 5 x 6, then I add 12 and 30 to get 42).
Underlying all these strategies are the commutative, associative, and distributive properties of multiplication. (Common Core standards don’t ask students to be able to name these properties, only to be to apply them intuitively to make facts easier to solve.)
• What does meaningful practice look like? “There is no doubt that practicing multiplication facts is essential for mastering them (Phase 3),” say Kling and Bay-Williams. But drilling isolated facts doesn’t work. “To maximize precious class time spent practicing facts, embedding that practice in worthwhile mathematical activities is important.” Meaningful practice uses the facts in rich, engaging activities that promote problem solving, reasoning, and communicating mathematical thinking. Games can also deepen mastery of multiplication facts without the anxiety of timed drills and competitions. Here are three games:
- Strive to Derive – 2-4 students have array cards (3s, 4s, 6s, and 9s), uncooked spaghetti or thin sticks, and two teacher-labeled dice, one with 3, 3, 6, 6, 9, 9 and the other with 0, 1, 4, 6, 7, and 8. For example: Lisa rolls a 6 and a 7. She pulls the 6 x 7 array card and places spaghetti to show 5 x 7 and 1 x 7, then says, “Six times seven is five times seven, 35, and one more seven, 42.” She gets a point, then the next player goes.
- Cover It – Two players spread matching array cards so they’re all visible. The first player pulls an array from the middle and gives it to the other player, who must find two arrays that exactly cover the one received. If player 2 succeeds, he or she keeps the three array cards. If player 2 can’t do it, player 1 has a chance and can also win the cards. Players switch roles and continue, saying or writing the combinations they find to cover the original array.
- Multiplication Tetris – The teacher rolls two dice (any kind), and each student decides where and in what orientation to fit that rectangle on the grid paper (for example, 4 x 6) and write the multiplication fact. The teacher continues to roll and students mark out the called rectangle somewhere on the grid. When a student can’t fit a rectangle, he or she is out of the game, and the last students in the game are the winners.