Question: What are array cards? How are they used in Investigations? How are they similar to or different from flashcards? What benefits or advantages do they offer compared to using flashcards?
Answer: In Investigations, arrays and Array Cards help introduce children to the concept of multiplication. They provide an image of what multiplication means, of what it looks like. They help students to practice and to develop fluency with the facts, specifically, and with multiplication and division, more generally.
The Array Cards used in Investigations are rectangles, divided into squares, used to model multiplication. An Array Card for 4 times 5 is a rectangular arrangement of 4 rows of 5 (or 5 rows of 4) squares. On one side of the card the multiplication facts are listed ("4 x 5" and "5 x 4"), and each individual square can be seen. On the other side, only the total (20) is shown. (See Figure 1.)
(Page 41, Things That Come in Groups.)
Looking at the marked side, children can count the total number of squares (20), or count by one dimension (5, 10, 15... or 4, 8, 12...). When faced with the side of the card showing only the total, students must reason about what they know about that total, and the factor pairs (or dimensions) that make a rectangle that size. For instance, a 4 by 5 rectangle looks quite different than a 2 by 10, although both equal 20. So Array Cards provide students with an area model of multiplication, as well as a model that shows multiplication as equal groups.
How are Array Cards used in Investigations?
Much of the early work with Array Cards involves students in playing multiplication and division games. As the Teacher Note on Array Games states, "Using the arrays in this way gives students a variety of opportunities to use a concrete model for multiplication and division, and in the process of playing the games, they become more familiar with the 'multiplication pairs.'" (Page 28, Arrays and Shares.)
Game 1: Multiplication Pairs. Players lay out the Array Cards with some facing dimensions up, some facing totals up. Player 1 chooses and points to an Array Card. If the dimensions are showing the player gives the total; if the total is showing the player names the dimensions. If the answer is correct, Player 1 takes this Array Card. Play continues until all Array Cards have been picked up.
Game 2: Count and Compare. This game is basically the equivalent of the familiar card game, War. Deal out all the Array Cards, totals face down. Players compare their top Array Card (still total side down) to see which is larger (has the most squares), without counting by 1's. The player with the larger array takes the cards.
Game 3: Small Array/Big Array. This "game encourages students to build a larger array from two or three smaller arrays... creating a concrete model for how a multiplication situation such as 7 X 9 can be pulled apart into smaller, more manageable components, such as 3 X 9 and 4 X 9." (Page 28, Arrays and Shares.)
For more on the role Arrays play as students develop fluency with multiplication and division, see the Ask an Author, Teaching Multiplication and Division.
How do Array Cards compare to Flashcards?
In fact, array cards are quite similar to flashcards, but they provide certain advantages:
Array Cards give students a visual image of multiplication.
Flashcards typically show only a problem to be solved, such as "6 X 7 =". Array cards connect the numbers in the problem to an image -- 6 rows of 7, or 7 rows of 6. The totals are also connected to the size and shape of different rectangles. Seeing what the facts look like, and how they are related to other facts, can help children learn and use them fluently.
Array Cards provide an entry or starting point for all students.
With flashcards, if students hit a fact they do not know or cannot remember, they are stuck. With array cards students can count the squares, or "skip count" by rows or columns ("5, 10, 15" for a 3 X 5 for example). As students come across these facts they don't yet know, they are practicing important math (counting, skip counting) and internalizing strategies that will help them remember the fact fluently.
It should be said that Investigations does not expect nor want children to count all the squares as their main strategy for learning their multiplication facts. The teacher's role is pushing kids to the next level of understanding. For example, teachers should help children who are counting by ones to see that they can count groups of squares if they count the rows or columns. Students who always count each row can be encouraged to see that that 7 X 5 has a 5 X 5 in it, so they could start their skip counting at 25. (See Figure 2.)
It should also be said that array cards are not useful only for students who are struggling. The idea that an array can represent multiplication and division problems is a powerful model for much harder problems. For example, consider a 3rd grade activity that asks students to figure out how many rectangles could be made with 120 squares. (They have been doing a computer activity that uses a 12 by 10 rectangle.) Students who have a solid understanding of the array model use strategies that are quite sophisticated and impressive. "I know from the computer that 12 by 10 works. If I double one side and halve the other I have a rectangle that's still 120 squares. So I doubled 10 and got 20, and half of 12 is 6. So a 6 by 20 rectangle works." Some students might need to cut an actual 12 by 10 rectangle in half and reassemble it to convince themselves of this. The student in the example above can reason like this mentally because of the power of the array as a model. (See Figure 3 below.)
Figure 3 The entire grid shows a 12 by 10 array. Imagine cutting the grid at the darkened line. Now move the resulting bottom half and place it to the right of the top half, creating a 6 by 20 array.
Array Cards allow students to work on other, important, areas of mathematics at the same time as they are practicing their facts.
- Commutativity. Array Cards give students a visual, practical way to explore and make sense of the commutative property -- the idea that 6 X 7 = 7 X 6. Students say things like, "All you have to do is turn the card -- if I turn this 6 by 7, now it's a 7 by 6. It still has 42 squares."
- Distributivity. Array Cards give students a visual, practical way to explore and make sense of the distributive property -- the idea that 7 X 5 = (5 X 5) + (2 X 5). "I know 5 X 5 is 25. 7 X 5 is two more rows of 5. 25 and 10 is 35." (See Figure 2.)
- Terms. Array Cards enable students to explore, see, and understand in a new way terms such as square numbers. "Hey, the card for 6 X 6 is a square! That's why 36 is called a square number!" Other terms that work with arrays can illuminate in a meaningful way for students: prime (numbers that have only 1 rectangle) and composite (numbers that have many rectangles) numbers, factor, factor pair, multiple, and dimensions.
Array Cards allow for a focus on meaning and understanding.
The focus of work with flash cards is often purely speed and memorization. There is little meaning to what children are doing; they spit out an answer that goes with the number sentence. The problem with this type of learning is
- Students who cannot remember a fact may have no sense that there is a way to figure it out. You either know it or you don't.
Students who "know" their facts often are unable to use that knowledge meaningfully in other situations. As an example of this, consider the following example, from an article by Thomas C. O'Brien:
"In the 1980s my colleague Shirley Casey and I published the findings of a simple research study dealing with grades 4, 5, and 6 in which math was being taught as present-day critics would wish. Pupils in these grades were asked, 'What is 6 x 3?' Virtually 100% of the answers were correct. But when we asked the children, 'Give me a real-life situation or a story problem for 6 x 3 = 18,' 74% of fourth-graders and 84% of fifth-graders could not do so! Furthermore, more than half of the erroneous stories involved straight addition: 'On Monday I bought 6 doughnuts. On Tuesday I bought 3 doughnuts. So 6 x 3 is 18."
If You Want to to Use Flashcards (or Worksheets)...
Some teachers and parents really want to use particular kinds of materials, like flashcards or worksheets. What follows are some "ideas about how to use them in ways that meaningfully complement the [school's] curriculum and what it is trying to accomplish in mathematics. For example... try the following:
- If you use workbook sheets, suggest that children only solve 'the problems which have even answers, or the ones where you have to borrow when you subtract [or]... the two problems which have the greatest and least answer on the page [or] the problems where there are exactly two digits in the answer' (Raphel, 2000b, p. 7). This helps children search for patterns among problems, think about every problem on the page, practice estimating, and build number sense. Ask children how they figured out which problems they needed to solve, and how they solved the ones they did.
- If you use flashcards, encourage children to tell you how they knew, how they figured it out, or how they remember (or might learn to remember) each fact. You might ask, 'Did you use something you knew?' or 'How would knowing 3 X 3 help you with 3 X 4?'
- You might sort through the flashcards to think about which facts your child already knows, and which facts he or she still needs to work on. You can both see mastery develop as you weed out more and more cards. A homework assignment that supports this work with facts is described in Annette Raphel's book: How many different answers are there in the times tables from 1 X 1 to 9 X 9? As she notes, children are often relieved to know there are only 36; it makes the task feel within reach (Raphel, 2000b, p. 24).
- Children might do activities with flashcards, such as drawing 10 cards randomly and putting them in order from least to most. Or, they pick a fact that they don't yet know and thinking of one or more different ways to figure it out." (Murray, 2002, p. 101-102)
For example, here are some ways to figure out and remember 8 X 6.
- 8 times 6 is the same as (4 X 6) + (4 X 6). That's 24 plus 24, or 48.
- 8 times 6 is the same as (5 X 6) plus (3 X 6). 30 plus 18 is 48.
- 8 times 6 is the same as 6 X 8. I know 5 X 8 = 40. Add on one more group of 8 and it's 48.
Megan Murray, TERC
With thanks to Katie Bloomfield and Cornelia Tierney.
Raphel, Annette. (2000). Math Homework That Counts: Grades 4-6. Sausalito, CA: Math Solutions Publications.
Murray, Megan. (2002). Schools and Families: Creating a Math Partnership. Glenview, IL: Pearson.
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.