Question: Do students practice computation in Investigations and learn their facts?
Answer: Practice in this program often looks very different than what many adults remember from mathematics classes in their educational past, and therefore lots of folks think it's not there at all. There is much practice that happens within the curriculum, in math class each day. And many of these activities are also used as homework.
1) Explicit work on number (number sense, our number system) and computation is embedded throughout each grade level of Investigations -- in units that specifically address number and computation, as well as those that investigate geometry, measurement, data, and the math of change. Students have many opportunities to develop, compare, use, and practice strategies for solving computation problems. They continually engage in activities that ask them to recall and use important number combinations (also known as "facts") and relationships in meaningful ways.
Particular activities are built into the curriculum to help children learn and practice their facts. For example, students who are working on addition combinations to ten in first grade are playing a lot of related games -- such as Turnover 10 (like a memory game) and Tens Go Fish (you're looking for pairs that make 10 rather than matches) -- that focus on pairs that make ten. When it comes to multiplication facts, third and fourth graders do a lot of work with activities like arranging chairs and with Array Cards. Array Cards are similar to flashcards but these cards are more visual. An Array Card for 4 times 5 is a rectangular arrangement of 4 by 5 (or 5 by 4) squares. On one side the multiplication facts are listed; on the other, the total is listed. You can count the total number of squares (20), you can see the dimensions (4 rows of 5 or 5 rows of 4), and you can see how it looks compared to other facts that equal 20 (it's much more square-looking than a 2x10 for example). It's seeing what the facts look like, and how they are related to other facts, that really helps children learn and use them fluently.
There are also activities which focus explicitly on thinking about which facts are hard for you to remember, and on coming up with ways to help you remember them more easily. Class discussions share those strategies as well. For example, I always had trouble with 8+3 and 9+2 for some reason. It helped me to think about those in terms of their relationship to 10. If I jump to 10 and then think about how many are left to add, they become quite easy (8+2+1 and 9+1+1). Similary, if 6 x 7 is difficult for you, you might find it helpful to remember that you know 3 times 7 (and then double it). Or, if I know my 5's, then I know 5 times 7, so 6 times 7 is just one more 7.
2) There are Routines (K-2) and Ten-Minute Math (3-5) activities provided throughout the curriculum. These activities are an integral part of the Investigations curriculum. They support the development of basic skills, offer regular and consistent computation practice, and provide additional opportunities for teachers to assess students' developing sense of number and computation. In second grade, an example is Today's Number. Each day students find many ways to make Today's Number, which might be the number of days they have been in school or the number of the date. Finding many ways to make 20 offers students a lot of practice with important number facts that equal 20, and in a classroom where this is a regular routine, it also offers a lot of practice with important ideas about addition and subtraction and even multiplication. Consider this example from a second grade teacher:
"Today was the 19th day of school and here are some ways to get to 19 we had today."
10 X 1 + 9
100 - 81
100+500+500+500-1600+19
9+9+1
17+2
"We do this every morning and the kids love it and you can see it challenges children who are at a variety of different levels. We get some pretty lively discussions going as children explain and defend their ways. I can be talking to another teacher in the hall, and the kids just go ahead without me and never notice I am gone. I have learned so much about how they think by just doing this warm up activity every day. One of the things I want you to notice about this activity is the way children are working on their 'basic facts' as they do it. Knowing their combinations is an important goal I have, but we're coming at this from a different perspective. It used to be that children would learn the 2's, then the 3's, and so on. Now they are learning their facts as they use them to solve problems. For example, I'm pretty sure this way to make 19 [9 + 9 + 1] came from our recent work with doubles, and with doubles plus and minus one."
There are similar kinds of activities built into the 3-5 grade units, called Ten-Minute Math, and many include suggestions for homework options. Many teachers we work with address concerns about computational fluency by utilizing these daily routines (eg. Today's Number) and various Ten-Minute Math activities, particularly the ones that promote number sense, estimation and mental computation. (At grades 3-5 an example would be Broken Calculator, where students think about ways to make a number with particular restraints, for example, the "+" is broken.) These activities are vital to the development of students' work with number. When teachers find ways to integrate them on a daily basis they see the impact that they have on students' work with number and computation.
3) Mathematical Games. Investigations includes many games as an integral part of the elementary mathematics curriculum. These games, which explore mathematical content as varied as whole number computation, fractions, sorting, probability, and geometry, are played repeatedly, allowing children to develop efficiency and fluency over time. Watching and talking with students as they play gives teachers a chance to see how their students are thinking, what strategies they are using, how those strategies are developing over time, and what ideas or areas need more support, work, and practice.
These games are designed to address particular content, so a first grader who's working on combinations to 10 might be playing Tens Go Fish and other 10-related games, while a second grader working on combinations to 20 and is therefore playing Close to 20. Close to 100 is an activity first done in the Grade 3 unit Combining and Comparing. For a full description of this game, including instructions and variations, see Close to 100. Also, to see the practice building and connecting over the years, consider that in second grade students learn and play Close to 20; in fourth grade they tackle Close to 1000, and in fifth grade they learn a new version called Close to 0 that involves fractions.
A new resource -- Math Packs -- are sets of math games designed for use in a wide range of settings: at home, at parent evenings, as a library resource, at after school programs or clubs, and in class with a substitute teacher. Each Math Pack contains everything needed to play three or four related games, including directions, score sheets, and a durable set of materials. Math Packs also contain:
- a list of the key math concepts that children practice each time they play
- a range of variations, including ways to adjust the games to provide more or less challenge
- tips for playing each game, so that all players get the most out of learning and playing together!
Math Pack 1
Counting Games: How Many in All?
These games (Collect 15, Collect 25¢, and Ten Turns) engage players in counting and keeping track of their collections of coins and colorful chips. Players need to make sure they always know exactly how many they have; the winner is the player with the most at the end of the game! It's easy to adjust these games so that children can work with numbers up to 20, up to 50, or even larger numbers.
Math Skills: counting; adding; comparing numbers; coin values and equivalencies
Ages: 5-8
ISBN: 076-900-1521
Math Pack 2
Number Games from 10 to 20: More, Less, or Equal?
In these games (Double Compare, Tens Go Fish, Total of Ten, and Close to 20), players combine and compare numbers up to 20. Players need to work accurately; the winners are those who can use their number cards to make the most combinations of 10, the largest sum, and the sum closest to 20!
Math Skills: adding; subtracting; addition and subtraction "facts"
Ages: 5-8
ISBN: 076-900-153X
Math Pack 3
Data Games: Use the Clues!
These games (Guess My Button, Guess My Rule, and Guess My Number), involve players in using clues to discover a secret object, rule, or number. The games are easy to adjust for children of different ages. You can play just about anywhere: at home, in the park, on a bus!
Math Skills: sorting and classifying; forming and testing hypotheses; logical thinking; features of numbers such as even, odd, multiple, and factor.
Ages: 5-12
ISBN: 076-900-1548
Math Pack 4
Computational Games: How Close Can You Get?
In these games (Close to 100/Close to 1000, Close to 0, and The Estimation Game), players are challenged to use their number cards to create math problems with an answer closest to a target number such as 100, 0, .1, or 1000. Players also try to beat the clock as they estimate solutions to difficult problems!
Math Skills: adding, subtracting, multiplying, and dividing with whole numbers and decimals; estimating; place value
Ages: 8-12
ISBN: 076-900-1556
Developed at TERC, Math Packs are available through Pearson.
For more on Investigations' work with addition and multiplication facts, see the articles Addition Pairs Up to 10 + 10 and Multiplication Pairs Up to 12 x 12.
Megan Murray, TERC
June 2000
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.
