Common student errors

What are error patterns?

Anytime children learn to perform an algorithm without fully understanding it, they will forget how it works, and some children will develop particular errors that they will make consistently. If you see an error that is made only once, then it's probably a careless error--one that comes from not paying attention rather than from a significant misunderstanding.  If you see an error that is made several times, it's likely a result of a misunderstanding.  Interviews with children about problems are an even better way of learning about the thinking that is producing errors, but we can often figure out thinking that could lead to a pattern of errors by looking at examples of student work. 

The error patterns I'll be sharing here are ones that are fairly common for children to make from the standard algorithms for addition and subtraction.  Any algorithm that children learn without fully understanding it will lead to error patterns, but the error patterns that you find with the standard algorithms have been studied in some detail, so those are the ones we'll be looking at. If you would like to read further about error patterns in arithmetic, I recommend the book Error Patterns in Computation by Robert B. Ashlock.

Why do children make errors, and how should a teacher react to errors??

Children make errors that show a lack of understanding because they don't know the content thoroughly. Often (usually) people will think that they know things more thoroughly than they do.  We tend to believe that we know something when we recognize it--when we look at a problem and its solution and it looks familiar, we think we know that problem and solution.  Sometimes we do know what we need to about that problem, but often if we try to solve the problem without any outside help, we find that we didn't know the content as well as we had thought.  We (people) tend to confuse familiarity with mastery.

Mastery of content means several things: we know how to solve it, do it, or explain it independently. We have practiced it until we can do it smoothly.  We have thought about it so that we understand not just what to do, but why we are performing the steps we are. Mastery means we have seen the concept in several different contexts, and we can recognize it  in different places where it appears. 

As teachers...

• We should expect that children will need repeated practice and experience with important ideas over time--their knowledge won't become thorough and permanent after only a few weeks or months.
• When children make mistakes that indicate a lack of understanding of why the addition or subtraction process works the way it does, we should make sure that we include a review of why the correct steps make sense as well as a review of what the correct steps are. 
• When we revisit important content such as addition and subtraction, we need to think about how to explain why we do what we do, which may mean breaking out manipulatives again, or finding meaningful contexts to clarify the meaning of the problem. 

Examples of error patterns: Addition

This example of student work shows an understanding of ones, tens and hundreds, but not of how those are related. In this example, the ones, tens and hundreds have been added separately, but no trading/exchanging has taken place. This work does not show an understanding that when you have 10 or more in a place value, then you have to trade for the next larger place value.  This is one of the most typical student errors.  Not trading and regrouping is a common misunderstanding, and it shows a fragility in the students understanding of place value. Notice that in this example you can't tell in what order the student is performing the steps--it would look the same left to right or right to left.

This example of student work shows that the student has made mistakes in remembering the steps of the standard algorithm.  It seems likely that the student doesn't understand the steps that they are misremembering.  This student is working from right to left (you can tell this because the number work in the right columns is affecting the sums to the left).  This child is "carrying" the ones place of each sum over rather than the ones place of the sum. I use the word carrying here to indicate that the child isn't thinking through the logic that 10 ones is equal to a ten, but instead is recalling a set of steps whereby one of the digits is moved to the next column.

Subtraction

This is  possibly the most common of the subtraction errors. This child has subtracted the numbers in each column without consideration for whether they are in the minuend (top) or subtrahend (bottom).  The different roles of the minuend and subtrahend is somewhat fragile knowledge for some children, and in the context of multi-digit addition, which is more abstract and less familiar, some students will reverse the order of subtraction.

Another common place where children run into trouble is with subtraction where a middle digit is 0.  The double-trade (hundreds for tens, and then tens for ones) is something that students may do incorrectly, even if they can subtract correctly in other situations.  There are several different ways to incorrectly "borrow" in order to subtract--the examples to the left are two different mistakes a student might make. In both cases, the mistake is a sign that students have learned a set of steps to subtract without understanding those steps in a way that would let them apply those steps to this type of problem.

Reteaching and correcting:

The best way to start reteaching is to try to understand exactly what the child is thinking, and the best way to do that is to ask them. Suppose, for instance, you are working with a child who is making that first subtraction error. Start the discussion with a similar problem, and ask them to solve it for you, and watch their process. When I'm doing this, I usually follow these steps:

• watch the child solve the problem
• ask the child questions about the error points--what did you do here? how did you do it? what does this part of the problem mean? does your process make sense to you?
• bring out manipulatives or something else to very concretely model the numbers and the problem (if I know what tools the child has used to do such problems concretely in the past, I'll use the tools or manipulatives that are familiar to the child. If I don't know, I'll use base 10 blocks if they are available, and something similar to the Montessori stamp game if base 10 blocks are not available)
• coach the child to solve the problem using the manipulatives
• coach the child to write down the process, emphasizing the step where they made the error
• watch while the child solves 1-2 problems correctly following the same process
• assign the child to solve several (usually 3) more problems independently using the same process. Emphasize that for just these few problems, you want them to do all of the process again (manipulatives as well as numbers)
• if the child continues to be successful, let them choose to stop using the manipulatives, but remind them to write down the key steps. Follow up to make sure the are still remembering after a few days.

There are a lot of branch points when you are discussing things, so there are a lot of ways the discussion might go. I'm going to give you an example of just one way it might go:

Teacher: I want to see what you're thinking when you subtract. Can you show me how you would solve 43 - 15?
Child shows:

Teacher: "When you got 2 here, how did you get it?"
Child: 5 minus 3 is 2
Teacher: The 5 is down by the minus sign, so really we should be doing 3 minus 5. What does it mean to do minus 5?
Child: Take away 5?
Teacher takes out manipulatives and makes 4 tens and 3: So if this is 43, and here are the 3 ones, and these are 4 tens, how can we take 5 ones away?
Child: Take away these (3 ones) and take 2 away from this (ten).
Teacher: Good idea--what can we do so we can take 2 away from the ten?
Child: Trade it in for some ones?
Teacher: right--how many ones do we trade it for?
Child: 10?
Teacher: yes, we always have to trade a 10 for 10 ones. Show me how you could make the trade.
The child trades a ten for 10 ones.
Teacher: OK, so now how many 10s are there?
Child: 3
Teacher: and how many 1's?
Child: 13
Teacher: good, let's write this down on the numbers. Cross out the 4 in 43 and change it to show 3 tens
[Child changes tens place]
Teacher: now cross out this 3 (pointing to the 3 in the ones place) and change it to show 13.
[Child changes ones place]
Teacher: So does that show how many 10s we have? and how many ones we have
Child: yes
Teacher: OK, now you can take away the 5 ones.
[Child takes away the 5 ones]
Teacher: good. How many ones are left?
Child: 8
Teacher: You can write that down in the answer.
[Child writes 8]
Teacher: How much more do we need to take away?
Child: ten
Teacher: Yes, so take away 10 and write down how many are left
[Child completes problem]

Teacher: That looks good. Show me how you would do the same thing for the problem: 52 - 16. Start by making 52 with the manipulatives.
[Child does the problem, and the teacher prompts at the trading step for the child to make the trade first with the manipulatives, and then write it down before subtracting]

Teacher: Good work. I want you to do 3 more problems like that for me, just that same way. I'll come check in with you in a little while

[child does 3 more problems correctly]

Teacher: This looks good. Tomorrow when you do more problems like this you can keep using the manipulatives if you want to, or you can just do it on paper, but imagine in your mind what the trade looks like and make sure you write down the trading step every time, even if you're not doing it with the manipulatives.

[Teacher checks in with the child for 2 more days to make sure the error stays corrected]

Often a teacher won't have this much one-on-one time for error correction, so this may need to be done with groups of children rather than individuals. This means the teacher can't get one-on-one answers at each point, but coaching through doing a problem concretely as well as numerically can still be useful.

Sometimes the class has been solving problems in other ways that the children understand well and do not make mistakes with, and the teacher can use those to go back to as reference points rather than using physical manipulatives.