![conviction, both orally and in written form (see Just to Cross the Street strategy.]
figs. 3–5). The children even began to make up Julian. I just knew the answer. [Fluency]
their own strategies and explain them with enthusi- Hannah. I started on 9 and counted 8 more.
asm and pride. They named their strategies [Counting Up]
“Jenna’s Strategy” or “Jack’s Favorite Strategy.” Mark. I took 8 + 8 and got 16. Then I added 1
Every day, the knowledge base that we were build- more to make 17. [Doubles Plus One]
ing became stronger. It was as if I had given my
students a license to think. To my students, equa- The more strategies we learned, the longer our
tions were not just equations anymore; they were Mental Math time took. Every minute was worth it.
numbers that they could manipulate in any way My students seemed to be picturing one another’s
that made sense to them. The following dialogue strategies mentally. This combination of an intense
occurred during a Mental Math session. I have study of strategies and a daily opportunity for prac-
noted the strategies that the students used. tice added up to success. Even now, well after our
study of the strategies has ended (although we still
Me. Sarah had 9 fish. Her mother gave her 8 review and practice each day during Mental Math),
more fish. How many fish did Sarah now have? my students use their solid base of number sense to
Evan. Seventeen, because 9 + 9 is 18. One less embrace every new mathematics challenge that
would be 17. [Doubles Minus One] comes along.
Jenna. I know 9 + 9 is 18 and 8 + 8 is 16. . . . In
the middle is 17. [This strategy later became
known as Jenna’s Favorite Strategy or The Dou- Another Discovery
bles Sandwich.] Many positive things came out of our journey, such
Jack. I took 2 from the 9 and gave it to the 8, as the number sense and mental mathematics flu-
which made 10. Then I took the 7 left over and put ency I have already mentioned. I made yet another
it with the 10, which gave me 17. [Make Ten] discovery, however: insight into my students and
Peter. Okay, I took 100 plus 100, which gave me their abilities. I learned that I could not judge a stu-
200. I took 200 – 400, which gave me negative 200. dent by others’ impressions of them. Some students
Then I took negative 200 + 209, which gave me 9. who seemed “lower ability” to their first-grade teach-
Then I added 1, which made 10. That’s the Make ers actually were my best thinkers. They were able to
Ten strategy. Then I added the 7 that were left, help others understand concepts. Conversely, some
which gave me 17. [Peter loved to work his way students who were named top mathematics students
away from an answer and come back to it. We in the previous year had difficulty. These few stu-
named his method the Walk All around the World dents were very good at performing algorithms or
Figure 2
Subtraction strategies that the students used
Subtraction Strategies Our Interpretations and Descriptions
Counting Back Beginning with the minuend, count back the number you are subtract-
ing; we would see 9 – 3 and think, “9 . . . 8, 7, 6” for an answer of 6.
Counting Up Beginning with the number you are subtracting, count up to the other
number; we would see 12 – 9 and think, “9 . . . 10, 11, 12.” Our answer
would be 3 because we counted three numbers.
Doubles We would see 14 – 7 and think 7 + 7 = 14.
Think Addition We learned to think of related addition problems when confronted with
subtraction facts; we would see 7 – 5 and think 5 + 2 = 7.
Fact Families Similar to Think Addition above, we would think of the fact family to
recall the “missing number.” For a problem such as 8 – 5, we would
recall 5 + 3 = 8, 3 + 5 = 8, 8 – 5 = 3, 8 – 3 = 5.
Subtracting from Ten In equations with 10 as a minuend, we would mentally picture 10
(10 fingers, 10 frames, and so on) to learn what remained when some
were taken away.
Teaching Children Mathematics / March 2004 365](https://image.slidesharecdn.com/mathstrategies-120507121001-phpapp02/85/Math-Strategies-4-320.jpg)
Math Strategies
- 1. Learning Strategies for Addition and Subtraction Facts: The Road to Fluency and the License to Think eaching the basic facts seemed like the log- attention should focus on strategies for computing T ical thing to do. Wouldn’t a study of the basic facts make mathematics computation much easier for my students in the future? How could I with whole numbers so that students develop flexibil- ity and computational fluency. Students will generate a range of interesting and useful strategies for solving help my students memorize and internalize this computational problems, which should be shared and seemingly rote information? How could I get rid of discussed” (NCTM 2000, p. 35). I believed that I was finger counting and move on to mental computation? definitely on the right track by focusing on strategies. As I embarked on my first year of teaching second My next step was to research effective strategies for grade following many years of teaching first grade, learning basic facts. Facts That Last: A Balanced these questions rolled through my head. Approach to Mathematics by Larry Leutzinger I had spent the summer poring through the cur- (1999a, 1999b) was an invaluable resource. After riculum for grade 2. After familiarizing myself with reading these wonderful books, I set a goal for my the mathematics concepts that I would be teaching, I students to memorize the basic facts and “move on,” decided to begin the school year with an intense study but I was in for quite a surprise. I was completely of strategies for learning and remembering the basic unaware of the impact that this experience would addition and subtraction facts. I looked to Principles have on my students and their number sense. and Standards for School Mathematics (NCTM 2000) for guidance. I was even more excited about my choice of topics after reading the following quote: The Journey Begins “As children in prekindergarten through grade 2 The first week of school, I began the unit with great develop an understanding of whole numbers and the excitement. I chose the “doubles” for a starting point. operations of addition and subtraction, instructional We worked with this concept for a few days; I wanted to be sure that everyone understood it. The By Lisa Buchholz first day, we used manipulatives to create equations Lisa Buchholz teaches first grade at Abraham Lincoln School in Glen Ellyn, Illi- showing doubles. The next day, we illustrated and nois. She is interested in mathematics journaling and giving her students the wrote about everyday situations in which we can see opportunity to share their thinking. doubles, such as “5 fingers plus 5 fingers equals 10 fingers” or “2 arms plus 2 legs equals 4 limbs.” On the third day, the children wrote their own definitions for “doubles” in their mathematics journals. They 362 Teaching Children Mathematics / March 2004 Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
- 2. also created equations that showed doubles. My stu- share our mathematics excitement and learning, I dents of average to lower-average ability had equa- devised homework for the Doubles Plus One strat- tions such as 3 + 3 = 6 and 7 + 7 = 14. My higher- egy. The assignment asked the children to write an ability students had equations such as 175 + 175 = explanation of the strategy and to create problems 350 and 324 + 324 = 648. These students particularly that could be categorized as Doubles Plus One. The enjoyed the open-ended structure of the mathematics homework also included a parent information sheet, journal assignment, and their enthusiasm ran high. which explained the strategy and gave specific exam- We had begun to really learn and apply doubles ples of its use. The children were excited to share this instead of merely recognizing them. If I had been homework and show off their new “math brains,” a paying closer attention, I would have seen the begin- term they had invented. ning of our number-sense explosion. During our Doubles Plus One exploration, I After making sure that the students had a full recalled seeing the same strategy in the district- grasp of the doubles, we started working with com- adopted textbook, which I was not using. The book binations related to “Doubles Plus One.” Using devoted only one page to the concept of Doubles manipulatives, illustrations, and journal writing, we Plus One. Devoting only one page to this strategy began to train our brains to see “5 + 6” but think is like expecting someone to successfully ride a “5 + 5 + 1.” (I say “we” because I was learning as two-wheeler after one experience. The sharpest, well. I was taught not to think about equations but to most capable child might attempt it once and memorize their answers. This “journey” was an understand all its nuances and complexities, but the awakening for me too.) Soon my students began to majority of children need time and experience to compute Doubles Plus One equations mentally, with learn and apply the strategy. The page in the book very little effort. Even my lowest-ability students would have presented the strategy, but would the were solving the equations with ease. students internalize and apply it or would they only In an attempt to integrate home and school and to recognize it? In Elementary and Middle School Teaching Children Mathematics / March 2004 363
- 3. Figure 1 Addition strategies that the students used Addition Strategies Our Interpretations and Descriptions Doubles Adding two of the same number together, such as 5 + 5 or 7 + 7 Doubles Plus One Finding “hidden” doubles in expressions where one addend is one more than the other, such as 5 + 6 (thinking 5 + 5 + 1) Doubles Plus Two Finding “hidden” doubles in expressions where one addend is two more than the other, such as 5 + 7 (thinking 5 + 5 + 2) Doubles Minus One Locating doubles in expressions where one addend is one more than the other, such as 5 + 6 (but thinking 6 + 6 – 1 versus 5 + 5 + 1) Doubles Minus Two Locating doubles in expressions where one addend is two more than the other, such as 5 + 7 (but thinking 7 + 7 – 2) Combinations of Ten We learned to recognize expressions equaling 10 such as 6 + 4 and 7 + 3 for use in other strategies; we would picture our ten fingers. Counting Up This strategy was used only when adding 1 or 2 to a given number; we would see 9 + 2 and think, “9 . . . 10, 11.” Add One to Nine Used when adding 9 to any number. This was how we “primed” ourselves for the Make Ten strategy; we would see 6 + 9 and think 6 + 10 – 1. Make Ten Turning more difficult expressions into expressions equaling 10 and then adding the “leftovers”; we would see 7 + 4 and think 7 + 3 + 1. Adding Ten Adding 10 to any number increases the digit in the tens place by one: 5 + 10 = 15, 12 + 10 = 22. Commutative Property Any given addends have the same sum regardless of their order: 8 + 7 = 7 + 8. Mathematics: Teaching Developmentally, Van de answer mentally. I called on one student to answer Walle (2001) advises not to expect students to have the problem and explain how he or she solved it. an understanding of an introduced strategy after Then I asked if anyone had a different way of solv- just one activity or experience. He states that stu- ing the problem. I called on volunteers until we had dents should use a strategy for several days so they heard five or six different ways to solve the prob- can internalize it. I wholeheartedly agree. lem. If the children had trouble understanding someone’s “strategy,” I wrote the student’s expla- nation on the board step by step. A chorus of “Oh, Falling into the Routine I get it now” or “That makes sense” usually fol- After we studied Doubles Plus One, my lesson lowed this illustration. We named this time of day plans for the strategies seemed to fall into a consis- “Mental Math.” This was a daily chance for the tent pattern. My “mini unit” for each strategy fol- children to apply their strategies. Over time, I fea- lowed this sequence of events: tured two-digit numbers in the story problems. The children simply carried over their knowledge of the • Introduce and explore the strategy using strategies to the new problem. manipulatives. Many of the strategies we studied are featured • Create illustrations of the strategy. in Leutzinger (1999a, 1999b) and Van de Walle • Use mathematics journals: Write a definition of (2001). We tailored some of these strategies to the strategy and create problems that match its meet our needs and generated some strategies our- criterion. selves. With the combination of the “strategy” • Assign homework including a Parent Informa- lessons (and related activities) and Mental Math tion Page. sessions, the children demonstrated an amazing command of the world of numbers. They actively Additionally, each day I read a story problem used the strategies I had taught them (see figs. 1 aloud and asked students to come up with an and 2) and explained them with confidence and 364 Teaching Children Mathematics / March 2004
- 4. conviction, both orally and in written form (see Just to Cross the Street strategy.] figs. 3–5). The children even began to make up Julian. I just knew the answer. [Fluency] their own strategies and explain them with enthusi- Hannah. I started on 9 and counted 8 more. asm and pride. They named their strategies [Counting Up] “Jenna’s Strategy” or “Jack’s Favorite Strategy.” Mark. I took 8 + 8 and got 16. Then I added 1 Every day, the knowledge base that we were build- more to make 17. [Doubles Plus One] ing became stronger. It was as if I had given my students a license to think. To my students, equa- The more strategies we learned, the longer our tions were not just equations anymore; they were Mental Math time took. Every minute was worth it. numbers that they could manipulate in any way My students seemed to be picturing one another’s that made sense to them. The following dialogue strategies mentally. This combination of an intense occurred during a Mental Math session. I have study of strategies and a daily opportunity for prac- noted the strategies that the students used. tice added up to success. Even now, well after our study of the strategies has ended (although we still Me. Sarah had 9 fish. Her mother gave her 8 review and practice each day during Mental Math), more fish. How many fish did Sarah now have? my students use their solid base of number sense to Evan. Seventeen, because 9 + 9 is 18. One less embrace every new mathematics challenge that would be 17. [Doubles Minus One] comes along. Jenna. I know 9 + 9 is 18 and 8 + 8 is 16. . . . In the middle is 17. [This strategy later became known as Jenna’s Favorite Strategy or The Dou- Another Discovery bles Sandwich.] Many positive things came out of our journey, such Jack. I took 2 from the 9 and gave it to the 8, as the number sense and mental mathematics flu- which made 10. Then I took the 7 left over and put ency I have already mentioned. I made yet another it with the 10, which gave me 17. [Make Ten] discovery, however: insight into my students and Peter. Okay, I took 100 plus 100, which gave me their abilities. I learned that I could not judge a stu- 200. I took 200 – 400, which gave me negative 200. dent by others’ impressions of them. Some students Then I took negative 200 + 209, which gave me 9. who seemed “lower ability” to their first-grade teach- Then I added 1, which made 10. That’s the Make ers actually were my best thinkers. They were able to Ten strategy. Then I added the 7 that were left, help others understand concepts. Conversely, some which gave me 17. [Peter loved to work his way students who were named top mathematics students away from an answer and come back to it. We in the previous year had difficulty. These few stu- named his method the Walk All around the World dents were very good at performing algorithms or Figure 2 Subtraction strategies that the students used Subtraction Strategies Our Interpretations and Descriptions Counting Back Beginning with the minuend, count back the number you are subtract- ing; we would see 9 – 3 and think, “9 . . . 8, 7, 6” for an answer of 6. Counting Up Beginning with the number you are subtracting, count up to the other number; we would see 12 – 9 and think, “9 . . . 10, 11, 12.” Our answer would be 3 because we counted three numbers. Doubles We would see 14 – 7 and think 7 + 7 = 14. Think Addition We learned to think of related addition problems when confronted with subtraction facts; we would see 7 – 5 and think 5 + 2 = 7. Fact Families Similar to Think Addition above, we would think of the fact family to recall the “missing number.” For a problem such as 8 – 5, we would recall 5 + 3 = 8, 3 + 5 = 8, 8 – 5 = 3, 8 – 3 = 5. Subtracting from Ten In equations with 10 as a minuend, we would mentally picture 10 (10 fingers, 10 frames, and so on) to learn what remained when some were taken away. Teaching Children Mathematics / March 2004 365
- 5. were already proficient with the addition and sub- Figure 3 traction facts. They could not, however, answer the In her mathematics journal, Tessa demonstrates “how” and “why” questions that I posed daily as part how a Doubles equation is found in a Doubles Plus of their mathematics journaling, such as “How did One equation. you get your answer?” and “Why did this work?” Simply learning what to do is a much easier task than is learning why to do it (Burns 1992). As the year progressed and we explored new con- cepts such as renaming, these students’ shaky foun- dation became more apparent. When solving prob- lems involving regrouping, they could not explain why they crossed out a number and renamed it. They did not really understand mathematics; they under- stood algorithms that they had been taught and had quick recall of the basic facts, showing that they were good at memorizing information. I had uncovered a gap in the mathematics education of these students, a gap that desperately needed to be filled with a better understanding of numbers. A teacher who uses only the mathematics textbook might have a different idea of his or her top mathematics students than would a teacher who probes deeper and expects students to think about their strategies. One student, who I will refer to as Steve, fell par- ticularly hard into this mathematics textbook “void.” On our “Meet the Teacher” day before school began, Steve’s mother informed me that Steve had done Figure 4 third-grade mathematics the previous year. She also told me that he could “borrow” and “carry” to five or Mary Caroline shows that a Doubles equation is found in a Doubles Plus Two equation. six digits. Two weeks into school, she saw Steve struggling and learned that my “lower ability” math- ematics students were helping him with his assign- ments. Steve kept asking, “Can we just do borrowing and carrying? When will we get to borrowing and carrying?” His mother admitted that she wondered what kind of “weird math” I was teaching that had her son so confused. She began to blame his confu- sion on me and my methods until she saw the hole in her son’s mathematics education filled with concepts and strategies. At the end of the year, she stopped in my classroom and said, “Steve has come a long way in understanding numbers. I think he’s ready to han- dle third-grade math now.” I firmly believe that if Steve had been taught with a mathematics textbook and a “dabble at the surface” approach, he really would have difficulty later when his ability to per- form algorithms was no longer enough. Final Thoughts This experience was every teacher’s dream. Not only did I grow in my own understanding of numbers but I now have a fresh enthusiasm for teaching mathe- 366 Teaching Children Mathematics / March 2004
- 6. Figure 5 Sarah shows how the Make Ten strategy works. She begins by creating a problem with two addends equaling more than ten when combined. Then she subtracts from one number to make the other number equal ten. matics. I have transformed from a “page a day” mathematics teacher to a facilitator of mathematics and its concepts. This adventure into number sense took us about two months. In today’s crowded curriculum, that is a considerable amount of time to invest in just one con- cept. My students emerged from this study, however, as amazing thinkers ready to take on any challenge that comes their way. The experience helped my stu- dents become faster and more accurate with mathe- matics flash cards and timed assessments, but the real gain was in number sense. References Burns, Marilyn. About Teaching Mathematics. Sausalito, Calif.: Math Solutions Publications, 1992. Leutzinger, Larry. Facts That Last (Addition): A Balanced Approach to Memorization. Chicago: Creative Publica- tions, 1999a. ———. Facts That Last (Subtraction): A Balanced Approach to Memorization. Chicago: Creative Publica- tions, 1999b. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. New York: Addison Wesley Longman, 2001. Teaching Children Mathematics / March 2004 367