Do students need to memorize basic facts to learn them?
It is important that students are able to perform simple calculations using addition, subtraction, multiplication, and division. It is also important that students understand these operations and know when to use each of them. Memorization focuses on being able to determine the result of a calculation, but on its own, it does not help students recognize the appropriate time to perform them, nor does it provide them with the conceptual understanding to move beyond calculation in their math education. Students need to practice using these facts to solve authentic problems, and need to be able to explain their calculations to become competent and confident users of mathematics in their daily lives.
Recall of facts is an important part of mathematics, but how the facts are made available for recall is more important. Facts such as single digit multiplication and division can be learned by straight memorization, but this has little meaning for students. If these facts are learned through developing common strategies, guided practice and play, then the facts have more personal meaning and are more likely to be recalled when needed. More importantly, if recall does not happen, the hope is that the students will then call on those common strategies for determining the answer. This approach fosters versatility in the learner.
We agree recall of facts is important – but memorization is one of the least effective ways of getting there, and also raises the question, “what do they have?” What happens when one has memorized something, but then forgets? If recall is developed through understanding and strategies, this is no longer an ending-point.
Why does my child have to learn more than one way? This seems confusing and overwhelming.
Multiple strategies is often misunderstood to mean learning many ways to ‘do a question’ so you ‘get the right answer’. But it’s really more about developing understanding and sense-making, which may come about through different means for different learners. Rather than learn by a rote set of different procedures, it’s about flexibility and adapting one’s approach to what makes sense for a particular question or problem. For example, if someone applies a column-algorithm approach to answer (200 – 199) it probably signifies they do not have a strong number sense.
From a learner’s viewpoint, knowledge is something that we all build for ourselves based on making connections with prior knowledge or understanding. By learning a concept through multiple strategies, we are establishing a network of connections for an individual idea. If a single connection breaks down or is forgotten, then we are able to draw on the other connections already established. By teaching/learning in more than one way, we are enabling students to create these multiple connections for their network of understanding, and nurturing a flexible and creative student. Ultimately, if we want to develop creative innovative problem solvers, we need to support students in developing multiple strategies in their learning.
The goal is much more than getting a correct answer. Using several strategies allows for deeper connections and analysis. Students should not be rushed through different approaches without developing proficiency and an understanding of the pros and cons. It is very important that using several strategies does not become a ‘set of procedures to learn’. Multiple strategies should promote and solidify thinking; it should not feel like unnecessary work.
How might I support my child at home to master the basic number facts?
For proficiency to be developed, procedural fluency must be accompanied by understanding, use of strategies, reasoning, and a productive (or say happy?) disposition. For this reason, timed drills have been shown to be ineffective for most learners. Rather, practice should be oriented towards applying strategies and thinking on questions for which recall is not there yet. Flash cards can be used effectively if not relying on remembering (ie. not just flipping the card over), but games can be much more effective as they engage the student while they develop their proficiency.
Practice is important to help build fluency with basic facts. Meaningless drills with no context, and timed activities should be avoided. Mathematics in a school setting should be about open and flexible thinking (more than one way to come to an answer), time, collaboration, creative thought, and play. Mathematics at home should not deviate too far from this design. Games with dice or playing cards can be played to build fluency with basic facts. More advanced problem solving games like CalcuDoKu, KenKen, and Kakuro are just a few examples of engaging challenging tasks that help to develop fluency with basic facts. Even low-level activities should be followed with asking “why does this work?” as a means for deepening understanding and giving meaning to the activity.
The first thing to keep in mind is that while accurate recall and application of facts is very important, speed is less so. I would give students the opportunity to explain how they know their facts. For example, “explain how you know that 7×6 is 42” or “how do you know that 3×5 isn’t 18?” Turning the practice into a conversation engages children more than flash cards or worksheets. It forces them to think about their answers, and it places less emphasis on the answer itself than how they arrived at the answer. I would also ask them questions outside of “homework time” and get them to think about how math facts apply to their daily lives. The more they use their facts authentically, the better they will remember them, the faster they will recall them, and the more meaningful they will become.
As students develop strategies for their basic facts, they need opportunities to check their understanding. Practice can occur in a variety of ways. Problems that apply basic operations and games that utilize knowledge are very useful for reinforcing learning. Flash cards and drill sheets can play a role if used appropriately; they are helpful when used for self assessment and communicating understanding but when used in a manner that encourages only memorization, they are not. Anyone who has ever seen a student struggle on a math test with the aid of a calculator knows there is much more to math than having basic facts readily available. Also, it should be noted that timed drills generally cause more harm than good for a lot of students.
Timed drills should be avoided. Not only can they cause math anxiety, they send the message that “faster equals smarter,” an unproductive believe for learning mathematics at any level, not just K-7. There is no advantage to recalling that 4 times 7 is 28 in 0.25 seconds rather than 2.5 seconds, except as it effects a child’s score on the “mad minute.” Instead, students should practice the basic facts in such a way that it does not turn them off of learning mathematics. This can be done in a variety of ways. For examples, games that emphasize strategic game play over speed or that practice a strategy for a certain set of facts (say, doubles, doubles plus one, near doubles) rather than all the facts at once. Drills and flashcards may have a place: to practice a strategy as this understanding is emerging for students, to have students & teachers identify facts and strategies that require further practice/support.