Arrays are introduced early in the elementary curriculum. As a bridge from counting to skip counting, to multiplication, and eventually the area model, they quickly become one of the most useful tools in learning elementary mathematics. But how do teachers, parents, and students use arrays to build an understanding of multiplication?
Arrays in math are used to organize a number of items in an arrangement that is easy to count, allows for skip-counting, and bridges counting ideas to multiplication or other patterns. Most arrays in mathematics are in a rectangular formation with straight and regular rows and columns.
Rectangular arrangements allow a bridge to understanding the area model of multiplication. Arrays offer multiple entry points to figuring out the solutions to problems and offer opportunities for children, teens, and adults to explore mathematical patterns at multiple levels. (Opportunities with these characteristics are often called “low floor, high ceiling” activities.) Arrays can be found all around us throughout our days and activities, providing a chance to practice answering “how many” and to explore and extend a variety of mathematical and pattern-making skills both inside and outside of the classroom.
Here are a few ideas about how parents can use arrays to support their children’s mathematical development, as well as encouraging a sense of exploration and playfulness.
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Keep it simple and lighthearted: “How Many?”
The best way to use arrays to extend mathematical thinking at any level is to ask “How Many?” Those two simple words open a world of exploration, playfulness, as well as opportunities to count whatever it is the child sees in the array. Your child may or may not surprise you with what they notice and what they are able to do. Allow the child to answer the question as they see fit and to use whatever strategies they are comfortable with in that moment. Follow up with questions such as “Great! And how many red ones” or “Can you use skip-counting to find the same number,” but only after the child has found success using their own strategies and observations. We want to keep this playful and fun, and we want the child to experience success, encouraging them to want to achieve higher levels of understanding and more and more success in mathematical sense-making. Research from the past few decades has been crystal clear that both children and adults learn more efficiently, reliably, faster, and retain the information longer when learning happens through play than through most any other mechanism.
Build up mathematical ideas strategically
Watch for a progression of mathematical understanding related to arrays. A child may first count one-by-one, perhaps even touching each item as they count. They may even first count haphazardly. Allow the child to first answer “how many” using whatever strategy they wish. Then gently encourage them to see if they can get the same answer using a slightly more “advanced” strategy. With repetition, the connections and understanding grows. The child will learn number patterns before either of you notice, and you may even find that they have mastered a few multiplication facts without the use of other memorization tools.
- Haphazard counting: the child may count the items without any rhyme, reason, or pattern in the counting. This may often lead to errors in counting, as it’s likely the child will miss some items and/or double-count others.
- Patterned counting: the child may follow a pattern, strategy, or order to count each item once. Perhaps across each row. Perhaps down each column. This strategy of counting is much more reliable.
- Addition: the child may add row by row or column by column.
- Skip counting: the child may use skip-counting by either rows or columns. Similarly, the child may use repeated addition strategies.
- Multiplication: the child may begin to see the rows or columns as groups and multiply the number of rows by the number of columns (or columns by rows).
Know that learning is not necessarily linear or consistent
Whenever you are working with your child, it is useful to remember that learning does not always go in one direction. Our brains are fascinating organs, and children’s brains work fast and furious to make and strengthen connections and bring ideas and patterns together. However, while the connections are being formed and strengthened, the most “advanced” techniques may not be the easiest for your child’s brain to access on any given day or in any given situation. This is why it’s important to consistently allow your child to use whatever strategy is available to them at that moment. If your child uses skip counting, or even multiplication, one day, don’t be surprised or discouraged if they default to counting one-by-one the next day. This is normal and your child is still learning. The process of reverting to earlier methods even as you learn more efficient or advanced ones can be extremely useful in building connections and understanding.
Talk about how the arrangement makes things easier or harder to count
Some things are hard to count. Other things are easy. Arrays in mathematics make things easier for most humans to count accurately (much like arrays in computer science make things possible for logic-minded computers to count). When you count groups of objects, talk with your child about what makes it easier or harder to count those objects. Ask what would help if they wanted to skip-count the objects. You may even find that your child eventually starts rearranging the items into an array before counting in order to make it easier.
Find and create arrays everywhere
Arrays are all around us. Look in your kitchen for muffin tins or how you arrange cookies on the tray before baking. Look at ceiling tiles when you’re in an office or school space. Check out the buttons on a calculator or the dots on a die. Create arrays out of paperclips, small balls of modeling clay or silly putty (affiliate links), blocks, etc. (Arrays made out of household objects can be a powerful tool for word problems!) As your child’s understanding advances, leave some blanks in the arrays and watch as your child bridges to subtracting out the missing pieces or reverts to counting one-by-one. And if you’re stuck or looking for more inspiration, follow hashtags like #arraychat or #howmany, or check out the counting book How Many? (affiliate link) by Christopher Danielson.
For more inspiration and more ideas, check out the videos below. And above all, remember that
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