How do I know if my child is ready to do arithmetic using an algorithm (paper and pencil procedures)?Understanding how our place value system works is one of the most important concepts for young students to acquire and it should not be assumed that just because a child can count to big numbers or accurately compute using a paper and pencil procedure that they truly understand this concept. Both Kathy Richardson and John Van de Walle spend a great deal of time addressing this issue in their books because it is so critical. If a student does not understand our place value system on an abstract level they will be forced to resort to one more than counting strategies because they do not have the understanding to use the short cuts of counting by tens and multiples of tens that are built into our place value system. They are stuck in additive thinking and cannot progress to multiplicative thinking, which is a severe handicap to their computational fluency. Deciding when a student is ready to move to purely paper and pencil procedures with understanding is one of the most difficult decisions teachers and parents face. If we rush students we risk teaching them to just follow a sequence of steps with no real understanding of how it works but on the other hand we do not want to hold a child back just because they are in a lower grade if they are developmentally ready. That would not be fair to them. Different students will develop this understanding at a truly abstract level at different times, but most students will develop it sometime late in first grade or during second grade. That means that younger students doing this type of work will need to either use counters, models such as the hundreds chart or number line, or count by ones mentally. Once a student has this understanding, they will begin to make jumps of numbers greater than one and we want to foster the idea of making jumps that utilize ten or multiples of ten because they are useful and build on this idea of place value (tens and ones) and promote computational fluency. The challenge then is to find a way to know when a child is ready to do this type of work with true understanding. We are beginning to use Kathy Richardson’s Assessing Math Concepts to help us determine this, but it is not an exact science. Children are often able to disguise their lack of understanding of place value by following directions, using tens and ones models (such as linker cubes or base ten blocks) in prescribed ways, and using the language of place value. The diagnostic task presented here is designed to help you look more closely at children’s understanding of place value. It is not suggested as a definitive test but as a means of obtaining information for the thoughtful teacher or parent. The tasks are designed for one-on-one settings (interviews). They should not be used as instructional activities. This task is referred to as the Digit Correspondence Task and has been used widely in the study of place-value development. Place some blocks on the table (in this case I am using 36). Ask the child to count the blocks, and then tell the child to write the number that tells how many there are. Circle the 6 in 36 and ask, “Does this part of your 36 have anything to do with how many blocks there are?” Then circle the 3 and repeat the question exactly. Do not give clues. Based on responses to the Digit Correspondence Task, Ross (1989) has identified five distinct levels of understanding of place value. · Single numeral – the child writes 36 but views it as a single numeral. The individual digits 3 and 6 have no meaning by themselves. · Position names – the child identifies correctly the tens and ones positions but still makes no connections between the individual digits and the blocks. · Face value – The child matches 6 blocks with the 6 and 3 blocks with the 3. · Transition to place value – The 6 is matched with 6 blocks and the 3 with the remaining 30 blocks but not as 3 groups of 10. · Full understanding – the 3 is correlated with 3 groups of ten blocks and the 6 with 6 single blocks. Full understanding indicates this child is ready to begin work symbolically with understanding. Manipulative and model support may still be beneficial in many situations however. Students who consistently count by ones most likely have not yet developed base-ten grouping concepts. That does not mean they should not to continue to solve problems involving two-digit numbers. In fact, it may be the combination of place value activities and solving problems in concert that finally helps that student to come to gain this understanding. The NCTM Standards authors also suggest a blending of place value work and computation. “It is not necessary to wait for students to fully develop place-value understandings before giving them opportunities to solve problems with two- and three-digit numbers. When such problems arise in interesting contexts, students can often invent ways to solve them that incorporate and deepen their understanding of place value, especially when students have the opportunities to discuss and explain their invented strategies.” (p. 82, NCTM) If you have students who persist in counting by ones, doing activities with the ten frames or carefully worded story problems can often encourage students to make use of the place value concept of tens and ones. Show Your Thinking- Not Just another Procedure It has been the need to see a student’s thinking that has caused teachers to instruct students to show their work. This has usually meant to show the standard algorithm as evidence that the student could compute correctly. However, there is more than one way for students to show their thinking. Ways to show thinking: (Progression from Concrete to Symbolic)
If we allow students to show their thinking in a variety of ways they will be able to show their thinking in ways that make sense to them. In other words, we need to let the students show their thinking in ways that make sense to them or else we run the risk of just teaching a new procedure. Even though many of the student strategies at first seem slow and cumbersome, the increase in students’ understanding that math should make sense and they can figure it out on their own saves time in future grades in terms of remediation and re-teaching. Give the time in the early grades for students to develop the key place value understandings and to become fluent with number relations (basic facts) and they will be more successful with symbolic computation in the upper grades. Why not just teach the standard algorithm right away? If you teach the standard algorithm (or alternative algorithms for that matter) right away, students will often times only use that one method forgoing all others even when others are easier and faster. For instance, the equation that solves the problem, “There were 75 students on the playground. All 48 of the second graders left to go in. How many students are still on the playground?” could easily be solved using mental math by counting first to 50 from 48 and then jumping from 50 to 75, (which many students can do mentally because they are such friendly numbers) allowing the student to arrive at the solution of 27. Contrast that with the standard algorithm. First you line up the columns. Then you trade 1 ten from the 70 and make the 70 into 60 and the 5 into 15. Finally you subtract the ones (15 – 8 = 7) and then the tens (6 – 4 = 2 or 60 – 40 = 20) arriving at the exact same place of a solution of 27. This is not to say we do not ever want to develop these algorithms, they can be helpful, but if we teach the algorithms first it is very unlikely these students will ever develop the flexibility using number sense that we are looking for. Instead of first looking at the numbers and thinking what is the best way to do this arithmetic, they will often blindly follow the procedure of the algorithm. We see this in the way many of our older students and adults compute with whole numbers. Even when given a relatively simple equation to figure out mentally such as 102 – 97 many students will automatically write out the equation in columns and begin the standard algorithm or procedure. Teaching procedures before number sense is developed often limits their number sense and mental math flexibility in the long run. Please see article “What do you mean by Basics Plus?” for a more detailed explanation of this topic. Why have Flexible methods for computation? “As students move from third grade to fifth grade, they should consolidate and practice a small number of computational algorithms for addition, subtraction, multiplication, and division that they understand and can use routinely…Having access to more than one method for each operation allows students to choose an approach that works and best fits the numbers of a particular problem.” (NCTM)
Benefits of allowing flexible methods:
Below I have listed three strategies for working on this concept that would be appropriate for students in Kindergarten through third grade.
1) Counting Activities Have students count collections of objects and keep track of how many. As the collections grow in size they can begin to use ten frames or cups to group the objects into tens and extras. This is an activity that can begin in pre-school and is appropriate for some students even in third grade.
2) Context Based Activities Students are given a context-based situation similar to this one from Fosnot and Dolk. In this situation, the story is that the students are going to be working in a candy factory. Their job is to sort the candy into the boxes (ten sections) and extras. Each child is given a set of “candy” that they then count and sort. They report the number of candies and the number of boxes (tens) and extras (ones). Older students should be encouraged to look for patterns between the number of candies and the number of boxes and extras. A project like this may take a series of days to complete or may be revisited off and on throughout the year to see individual progress. Fosnot and Dolk have written a series of books that have many ideas for these projects based on a math concept (big idea).
3) Utilize Change Unknown Problems from CGI Problem Structures Pose problems that use the change unknown problem structure to promote the action of counting up Example: Joe had 34 baseball cards. His uncle gave him some more. Now he has 78 cards. How many did his uncle give him?
The open number line is a great way to foster jumps of 10 (and multiples of ten) with this type of problem.
34 78
A typical progression is: Counting on by ones 34 + (1 + 1 … + 1) = 78 Counting on by tens and ones 34 + (10 + 10 + 10 + 10 + 4) = 78 Counting on by multiples of tens and ones 34 + (40 + 4) = 78, so his uncle gave him 4 tens and 4 or 44 cards. I would not expect to see jumps of ten like this until spring of first grade or second grade for most students, but there are kids younger than that capable of this type of thinking when given the opportunity. Choose the numbers carefully to foster this type of thinking I would suggest progressing in this order, but you may choose to mix the problems based on observations of your child or students. 1. Problems with both addends multiples of ten. Example: In above problem use the numbers 30 to 70 2. Problems with the change only a change of tens. Example: In above problem use the numbers 34 to 74 3. Problems with a change of tens and ones. Example: In above problem use the numbers 34 to 78 4. Problems with a change of tens and ones over a decade. Example: In above problem use the numbers 34 to 82 The added bonus is that as students solve problems like this they are not only developing their problem solving and communication skills, but they are also developing their sense of place value and becoming more automatic with their number combinations (basic facts) without having to resort to activities such as drilling with Holey cards or timed tests. It is a serious error to work for mastery of non-regrouping problems before tackling regrouping. To keep these problems separate has been the documented source of many error patterns. “Bad Habits” that children learn must then be unlearned. Better to just select numbers carefully and have students figure out the distance using tools (manipulatives) and models (number lines, hundreds chart) to help them figure out the difference in ways that make sense to them. The long-term goal is computational fluency which values efficiency, but do not push students to prematurely abandon manipulative approaches.
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