If children are taught the traditional algorithms too early, before they have made sense of the process, it becomes procedural not conceptual. The goal is understanding! We need to allow children time to develop understandings of the concepts: addition, subtraction, place value and then come up with methods that make sense to them.
Our goals for students should be more than just memorizing procedures. They can do so much more. We want them to develop understanding and strategies that make sense to them. From that point, they can then develop computational fluency.
An example of an efficient strategy for subtraction would be to count up. Counting by ones would be an efficient strategy for adding 58+3, but it is not an efficient strategy for adding 82 + 39. Knowledge of basic facts is directly related to accuracy, as is organization. If a student is trying to organize an algorithm that does not make sense to him, he is likely to make errors. Students need to verify each answer, by asking themselves, “Does that make sense?” or “Is that reasonable?” When students focus only on the digits and not the entire numbers they are more likely to get answers that do not make sense. Flexibility is key! No one method is best for all problems. They need to think about each problem and choose a strategy that works for that problem. Often when students are taught algorithms before they are developmentally ready they use it as the only strategy. Then you see third graders taking the problem 100-98 and crossing out the 1 in the hundred’s place, making the 0 in the ten’s place a 10, crossing it out to make a 9 and finally making the 0 in the ones place a 10. That is not a very efficient strategy for that particular problem. It is much easier to think about 98 and 100 and think about how far apart they are, perhaps using a hundred’s chart as a visual. We do not want children to think that there is only one way, when in fact there are several efficient strategies that student’s could choose from.
These are the 3 categories of strategies for solving equations with large numbers. These are the same for addition, subtraction, multiplication and division. Direct Modeling is the first stage. Often students will go back to direct modeling if they get to a new type of problem or one that is difficult for them. This is a good indication that they are trying to make sense of the problem. For example: A student may have developed a couple of personal strategies that work well for them when adding two, 2-digit numbers when the sum is less than 100, but when he encounters a problem where the sum is greater than 100, he may need to use direct modeling by using base-ten pieces. We will be going over some examples of invented strategies in this power point. The next slide shows examples of the US traditional algorithms for addition and subtraction.
These are examples of how many of us were taught to add and subtract large numbers. Notice how they focus on the digits and not the whole number. The place value is obscured, rather than brought out. While we probably all learned and have used the phrases “carry the one” and “borrow from the eight”, these terms are misleading. First, we aren’t actually dealing with a one and an eight – we’re dealing with a ten and an 80. And second, carrying and borrowing aren’t mathematical terms and don’t accurately describe what is happening.
After reading the rules, Click the mouse and the four problems for the participants to solve will fly in! Please solve at least 2 of the equations. Be sure to record your thinking and be prepared to share your strategies with the group. You may want to give participants transparencies to record their work on, that makes it a little easier for sharing. Walk around the group as they are sharing and observe strategies. When selecting people to share be sure to have a variety of strategies represented.
Have some participants share their strategies for one of the problems on the previous slide. If they did their work on a transparency sheet they can put that on the overhead and explain their thinking to the group. Have several different people share. Be sure that several strategies are represented. While they are doing the sharing just show the Title of the slide. Then switch to the overhead for sharing. After a few people have shared, switch back to the power point. Click the mouse to make the different questions show. Stop after each question and briefly discuss. You can use different strategies for sharing such as: whole group, small group, share with a partner. The following slides show some examples of alternative algorithms for addition. Many of you used the same or similar strategies.
A common way that students will add is from tens to ones. In reading and writing they start on the left, so it is natural for them to want to do the same in math. This slide shows the thinking clearly explained and recorded. We are NOT saying to teach this instead of the traditional algorithm. We are showing ways that many students will try to do themselves. Students are often not able to clearly explain their thinking. If you know what they are trying to explain, you will be able to help them record it efficiently. This can also be done by adding ones then tens and combining, which is the basis of our traditional algorithm. Recording it as above, brings out the place value, rather than obscuring it. It is important to note that it is mathematically incorrect to write this problem this way: 40+30 = 70+14 = 84 because the equal sign means that what is on each side of the = has the same value. In addition to the ways recorded on the slide, it would also be OK to record this problem using either of these ways. You may want to show these to your teachers on an overhead. 40 + 30 = 70 6 + 8 = 14 40 + 30 70 + 14 84 70 + 14 = 84
This is similar to the previous strategy, but instead of breaking both numbers into tens and ones, you keep one number intact and break apart the second number. Both the commutative and distributive properties can be applied here. In order to use this strategy efficiently students must be able to break apart numbers and count on by tens. Again, it is important to note that it is mathematically incorrect to write this problem this way: 46+30 = 76+ 8 = 84 because the equal sign means that what is on each side of the = has the same value. In addition to the ways recorded on the slide, it would also be OK to record this strategy using either of these ways. You may want to show these to your teachers on an overhead. 46 +30 76 46 + 30 76 + 4 80 + 4 84 + 4 80 + 4 84
If a student is trying to explain this strategy, base ten pieces can be used as a visual for others to understand what they are doing. This strategy will not work for all students, and is more commonly used when one of the numbers is close to a multiple of ten.
An open number line, as shown in the slide, or base ten pieces are good ways to visually explain this strategy to others. Again, we are NOT saying to teach these strategies. We are providing visuals so that you can help students who are approximating these strategies to explain and record their thinking.
Number Oriented: Invented strategies deal with the whole number, not isolated digits as the traditional algorithm does. Place value is the focus of invented strategies. Whereas, in the traditional algorithm place value is obscured by things such as, “ Carry the one (it is not a one).” Left Handed rather than right handed: Most young children will naturally start at the left of the problem, because they are used to starting at the left for reading and writing. Either way is workable, sometimes starting at the left is more efficient and sometimes starting at the right is more efficient. Flexible rather than rigid: Different strategies work better for different problems. If we are thinking about the numbers, rather than the digits, this is more apparent. Look at the two problems for a moment. Did you use the same strategy on both? Many times students who have been drilled in one procedure will automatically go to that procedure without even thinking about the numbers. On the second problem, 526 + 98, it is quicker to think 526 + 100 and then take away 2, than it would be to do the traditional algorithm. For some problems the traditional algorithm may be the more efficient way. The key is flexibility and allowing students time to develop understandings!
When our focus of math instruction was on teaching the procedures of a standard algorithm, we could feel confident we were doing a good job when we did the things listed on the left. We used manipulative to model the steps and perhaps even had students play games with the base-ten pieces. Then we clearly explain the steps and came up with catchy phrases to help students remember the steps. Our students chanted “we always, always, always add the right side first”. Because students needed lots of practice to remember the steps, we provided and corrected pages and pages of practice – monitoring and reteaching along the way. Because the focus of instruction has shifted away from procedures toward developing concepts, our role and the ways we know we’re doing a good job have to change. We still need to provide manipulatives and models like base-ten pieces and hundreds charts, but instead of modeling the process for students, we guide their thinking. Rather than showing the steps, we provide many opportunities for students to do their own thinking and share their strategies. As students are trying to make sense of adding or subtracting larger numbers, we need to recognize the direction of their thinking and help them complete their approximations of a strategy. One of our most important roles is to help students learn mathematically accurate ways of recording their thinking. And finally, when we see that students are ready, it is our job to gently press them toward using more efficient strategies.
When mathematics becomes about memorizing isolated rules, thinking stops! We want to develop students who think and use what they already know to help them solve problems.
After reading the rules, Click the mouse and the four problems for the participants to solve will fly in! Please solve at least 2 of the equations. Be sure to record your thinking and be prepared to share your strategies with the group. You may want to give participants transparencies to record their work on, that makes it a little easier for sharing. Walk around the group as they are sharing and observe strategies. When selecting people to share be sure to have a variety of strategies represented.
Have some participants share their strategies for one of the problems on the previous slide. If they did their work on a transparency sheet they can put that on the overhead and explain their thinking to the group. Have several different people share. Be sure that several strategies are represented. While they are doing the sharing just show the Title of the slide. Then switch to the overhead for sharing. After a few people have shared, switch back to the power point. Click the mouse to make the different questions show. Stop after each question and briefly discuss. You can use different strategies for sharing such as: whole group, small group, share with a partner. The following slides show some examples of alternative algorithms for subtraction. Many of you used the same or similar strategies.
Many students will want to start on by subtracting the tens. Splitting the number being subtracted into tens and ones makes this much easier. A good visual for this is a hundred’s chart. Just as with addition, it is important to note that it is mathematically incorrect to write this problem this way: 73 – 40 = 33 – 3 = 30 – 3 = 27 because the equal sign means that what is on each side of the = has the same value. In addition to the ways recorded on the slide, it would also be OK to record this problem using either of these ways. You may want to show these to your teachers on an overhead. 73 -40 73 – 40 33 – 3 30 – 3 27 33 - 3 30 - 3 27
This strategy involves rounding to the nearest ten. An open number line or base ten pieces are good visuals to show this strategy
Adding some to the initial number so that the ones are the same makes for an easier subtraction problem. The key to remember is that if you add in some extra at the beginning you must take them away at the end. This can be explained as loaning ones to make an easier problem, but must be paid back before you are finished. A hundred’s chart is a good visual for this strategy.
Not all subtraction problems are take away. Many represent the difference. Often counting up is an efficient strategy for subtraction. An open number line is a great way to record your thinking when counting up. At the end of the problem, you must add up all the jumps you made to get the answer.
Another example of counting up involves just adding tens to get close. Then subtracting to get back to the target number. Again, an open number line is a great visual for this.
This strategy uses negative numbers. However, most young children will not call them that. They will say things like: 3 – 6 = 3 less, or 3 – 6 = 3 in the hole, or 3 – 6 = minus 3, or 3 – 6 = 3 not enough. Most young students who do this will need help with how to record it efficiently.
Go back and try using some of the strategies on the subtraction handout to solve one or more of these problems. I encourage you to try the strategies that are not entirely clear to you. By working through some problems you will develop a greater understanding of the strategy. Then you will be better prepared to assist students who may be trying to use a similar strategy.
Be sure to present subtraction in a variety of forms. Not exclusively take away. When problems are written horizontally students will often stop and read the problem and not focus solely on the digits. If students are using the traditional algorithm, be sure to have them explain why it works and verify their answer. If they really understand the process they should be able to explain it and find another way to check their work.
Place value is an integral part of invented algorithms. When the strategies are build on understanding, students are less likely to come up with answers that do not make sense. Often when students are taught the traditional algorithm without full understanding of the concepts they make errors. When students are thinking about the numbers, they are less likely to come up with answers that do not make sense.
One challenge is the transition from direct modeling (base ten pieces, etc…) to invented strategies. These ideas are ways that can help with that issue. Hundreds’ charts and open number lines are good visuals to use when making the transition.
Developing new understandings requires time and multiple opportunities. Do not try to rush kids through this stage.
We are not saying that the traditional algorithms are bad. The problems occur when they are introduced too early, before students have developed adequate number concepts and place value concepts to fully understand the algorithm. Then they become isolated processes that stop students from thinking.
Remember our goals are for students are conceptual understanding and computational fluency. Meaning is the most important thing!
Transcript
1.
Alternative Algorithms for Addition and Subtraction If we don’t teach them the standard way, how will they learn to compute?
Children’s first methods are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestors did. By trying to bypass the constructive process, we prevent them from making sense of arithmetic.
Fluency demands more of students than memorizing a single procedure does. Fluency rests on a well-build mathematical foundation that involves:
Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily.
Accuracy depends on careful recording, knowledge of basic number combinations and other important number relationships, and verifying results.
Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to choose an appropriate strategy for a specific problem.
5.
Stages for Adding and Subtracting Large Numbers
Direct Modeling: The use of manipulatives or drawings along with counting to represent the meaning of the problem.
Invented Strategies: Any strategy other than the traditional algorithm and does not involve direct modeling or counting by ones. These are also called personal or flexible strategies or alternative algorithms.
U.S. Traditional Algorithms: The traditional algorithms for addition and subtraction require an understanding of regrouping, exchanging 10 in one place value position for 1 in the position to the left - or the reverse, exchanging 1 for 10 in the position to the right.
6.
What do we mean by U.S. Traditional Algorithms?
Now that we have discussed some alternative methods for solving subtraction equations, let’s return to the problems we solved earlier. Go back and try to solve one or more of the problems using some of the ways on the subtraction handout. Try using a strategy that is different from what you used earlier.
Use story problems frequently. Example: Presents and Parcels picture problems from Grade 2 Bridges
Multiple opportunities
Not every task must be a story problem. When students are engaged in figuring out a new strategy, bare problems are fine. Examples: Base-ten bank, work place games such as Handfuls of treasure and Scoop 100 from Grade 2 Bridges.
29.
Suggestions for Using/Teaching Traditional Algorithms
Delay! Delay! Delay!
Spend most of your time on invented strategies. The understanding students gain from working with invented strategies will make it much easier for them to understand the traditional algorithm.
If you teach them, begin with models only, then models with the written record, and lastly the written numerals only.
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