I love this mistake. From their work, what can we infer about what the student knows? Where are they getting confused? How would you help? (Also, if you don’t feel as if you have enough information, feel free to jump in with want information you’d like.)

Elementary teachers? You out there?

## 19 replies on ““27 – 11 = 38””

Do we get to talk about teacher mistakes here, too? That’s no number sentence. It’s an algorithm with a missing value.

That said, it does seem the student has interpreted the intended number sentence as 27-__=11 and then made a classic error. But there is also a possibility that the student interpreted the number sentence to be 27-11=__, or as __-11=27.

Either way, I am generally opposed to tasks that ask students to complete holes in algorithms and I favor tasks that ask students to perform calculations in unambiguous ways (when we are assessing their ability to compute, which should not be the sole mode of assessment).

One of the things we focus on when evaluating learner’s work is asking: “What question is the learner answering?”

http://deltascape.blogspot.com/2011/03/whose-job-is-it-anyway.html

In this case, I would guess that the learner has seen an approach for finding the missing value in

___ – 27 = 11 and thinks it can also apply to

27 – ___ = 11.

In other words, the learner did find a value that is a distance of 11 away from 27, just not in the right direction. Now it’s the teacher’s job to build on what the learner can do and further support what the learners is trying to do.

Without seeing more, I think that this student knows that addition is ‘related’ to subtraction, and that the numbers 38, 27, and 11 can be used to write several number sentences. He knows he can use addition to solve subtraction sentences.

As his teacher, I would focus on developing number sense and unpacking the information that is given in the number sentence. I’d go back to the number line to ensure that he has a solid conceptual understanding of the problem. He should be able to visualize these problems.

Check the number sentence you wrote. Is it true that 27-38 leaves you 11? Can you re-write your number sentence so that it is true? Does this match the original sentence?

Look at a number line. What is the starting value? Where you do want to end up? How far apart are those numbers?

I am on the same boat as Christopher here. VERY BAD PROBLEM and it is NOT a number sentence. Although I will say I think the student has been very used to adding numbers like this in this algorithmic style so what the student did was “line up the numbesr” and then start at the ones and work your way down. They performed the algorithm in the way that seems right in their mind.

POSITIVE NOTES:

The student’s work quite clearly shows 27 – 11 = 38, but nobody has commented on that, yet.

I agree with most of what @Rachel wrote, but I would ask the student to check their work by finding 27-11 again. Should the answer be larger or smaller than 27? Then perhaps they will just be able to do the problem. If not, well, 11 didn’t work, try subtracting a different number and see what happens.

While this student is working, my hypothesis is that they do not understand the concept of the unknown value, and when they found an unknown, they just filled it in with the next available number. I want to check this hypothesis more, so see how the student does with _____ – 15 = 31. Also check to see if they can correctly explain: “Does 17 make this number sentence true? Why or why not.”

NEGATIVE NOTES:

Isn’t this a number sentence? Does it matter that the sentence is written vertically instead of horizontally? 27 – ___ = 11? Or is that also not a number sentence? Is the problem that understanding this means interpreting the horizontal bar as equals? I could see that.

I disagree with the assertion that this is an algorithm. Basically for the same reason that I think it is a number sentence. If the operation of subtraction had to be inferred, I would feel differently. Or the problem was multiplication and it was set up with lots of blanks below the line indicating but not saying that the students should use partial products. But it seems ok as is, and more importantly, I don’t think the presentation of the problem is causing the student to get the wrong answer.

I really like the mistake in this question. Many children at this level of math already know about the concept of addition and subtraction, but they might have a hard time on applying the concepts while they are solving the questions by themselves. Especially, the question that has a blank (or the number that the children supposed to figure out) in the middle of the equation is very tricky for them to work on because that question does not ask for the outcome of the equation. Since children are mostly working on the ways of figuring out the outcomes in the equation (such as 5-2=?), it will take sometimes to switch their ways of solving this kind of question. I also think that the children get confused with the question between (unknown value – 15=11) and (27-unknown value =11). It still means that the children are confused when they supposed to apply the concept of addition or subtraction in this kind of question.

Therefore, I will guide the students to solve this kind of question with blocks in the beginning and provide smaller number than 27 just to teach the ways of solving. I would grab 27 blocks and ask the children to remove the numbers of blocks until they get 11. Then, we will discuss about the ways we solve and I will bring the equation format (Standard Algorithm) to re-do the ways of solving. At this moment, as Andrew mentioned in his comment, I will emphasize the importance of the unknown value that supposed to be smaller than 27 since it is a subtraction question.

From their work, what can we infer about what the student knows? Where are they getting confused? How would you help?

From this problem, you can definitely tell that the student sees a relationship between 27, 11, and 38. However, they’re getting confused about the concepts of subtraction and addition and their relationships. Or the student might have made a error from rushing and not realizing that they were adding rather than subtracting, as they wrote. It’s also troubling to see the student’s reasoning that is a number sentence, as it is not.

To help, I would have the student try double checking their answers to see if 27-38 gives you 11 and if 27-11 gives you 38. I feel as though sometimes students don’t see the value of double checking answers or working backwards in what they did. I know as a kid, I didn’t like explaining my work or double checking.

In my opinion, I think the student may understand how to find the answer (and the students know how to do addition and subtraction), but he was confuse by how the question represent. This type of question is tricky because students may not understand what the question is really looking for. While students first learn how to work on addition or subtraction, the math problem are usually easy and simply. For example, 10-1=9/ 1+9=10. But when the math problem was represent in another way like the upper example, students may feel confuse and have not idea how to work on it. They may thought the question is asking them to add up/ subtract the number, and they will come up a wrong answer like the upper example. In order to help the students understand how to work on those questions, I think first teachers need to explain what the question is looking for. Then, they can teach students to use number line and find out the difference between 27 and 11. By explaining to students and drawing the number line, I think students will be able to have a clear picture on why the answer should be 16.

Students may know that how to do the addition in some way, but they may not understand what the minus and plus signs mean. In addition, the direction may cause some confusion even though I don’t know why the sentences are stated in the way. If the sentence was saying like “I have 27 apples and I gave some of them to my friend, and I left 11 apples on my hand. How many apples did I give away?” Or elementary school teacher should teach children the difference between subtraction and addition. If this question is for elementary school, drawing picture is much easier for children to understand. I would like to recommend that teacher should give children an opportunity to understand the concept of addition and subtraction and plus and minus sign as well, then should give this kind of question with easier question sentence.

Does this student have a clear understanding of additions and subtractions? From their work, we know for sure that this student understand the relationship between the three numbers: 11,27, and 38. the work they’ve shown to the right makes us question whether this student knows the difference between addition and subtraction, and whether this student fully understands how and when to use the operations.

To correct this mistake, we should first clarify concepts that this student might be mistaken:

addition= getting bigger (increasing)

subtraction= getting smaller (decreasing)

From this student’s work since 27 and 11 created 38, which is greater than 27 and 11, we can infer that the work that the student did is addition. “27 plus 11 equals 38”

Then we look back at the original question and explain it easier for the student to understand. I would recommend explaining that when a number is subtracted from 27, the leftover (aka the answer) is 11. “If we have 27 candies, and someone takes a certain number of those candies that we have 11 left, how many did that ‘someone’ take?” Making the question more visual might help the student understand the question better and guide them figure out the question.

I would be

I would like to ask what age this student was, and what grade that this subtraction question was asked? I believe there is a strong correspondence between the age and the mistake that was made. If this child is at an age where they were exposed to story problems and not this specific set up, then it would be easy for the child to make that mistake. But if they are in, lets say 5th grade, then there is obviously a piece of knowledge that was missed in the earlier grades.

I think that a teacher should really just sit down with the student and go back to the basics. Go back to using visuals to explain the subtraction problem and allow the student to show the teacher how they got there answer. From there you can correct the misconception.

Also, I would like to point out that the child clearly knows how to add. They might just have a miscommunication or misunderstanding of the concept of subtraction. They will do great with negative numbers 😉

To start off with, the child set up the problem correctly to find the missing number. They know that in order to find the missing number it is like a puzzle and have to subtract 11 from 27 to get the middle number. The child then goes on to set up the problem and instead of subtract they added. since the child made this mistake I am wondering if they understand the symbols (+,-) and when to use each concept.

If I was this child’s teacher I would use a mathematical tool to help (blocks). I would set up 11 blocks and ask the child to count with me how many more blocks are needed to get to 27. I would clarify that this is the way we add to see mow much more. then I would set up 27 block and as the child to take away 11 and count with me how many are left over. the answer would be 16 again and I would make sure the child knows that this is subtracting to get a total. I think this was the childs inital goal but confused themselves when finding out the missing number with adding instead of subtracting.

Seeing student’s work, he/she correctly shows the relationships among 27,11, and 38, so I can infer that he/she understands how to find the answer and the relationships between addition and subtraction. This mistake drives me to think about the situation when children may kind of know about the concepts of addition and subtraction but feel confused when they apply concepts into solving some specific questions.

As far as I concerned, I think the child may confused at the concept of the unknown value in that he/she sets up the solution correctly and vertically, but when he/she tries to figure out the middle number, he/she just skip it and go directly to the other available number. I also wonder if he/she understands the unknown value in the vertical bars, rather than horizontal ones. So, I’d like to write the solution horizontally, such as 27-___=11, and ask him/her to solve it. I also think it’s vital to ask student to explain why they way he solve it, which tests if he understand the concept of unknown value. To reinforce child’s understanding, I will translate this operation to something in our daily life to help student understand what it means to. For example, i will say “If I got 27 gummy bears in my pocket”, writing down the 27 on the board. And keeping saying that, “then my little brother comes and grabs some”, writing down the blank box vertically and the minus sign. “Finally i got 11 gummy bears left, could you help me to figure out how many does my brother takes away? ” writing down the bar, and 11.

Moreover, to critically learn about child’s problem is important to adjust my teaching method as well!!

I am also very curious which grade the child who answered this was in. The student clearly understood that to find the answer, he or she needs to work with the numbers that the worksheet provided. I think that the student understands what 1+7 and 2+1 is, but does not understand that subtraction is a different than addition. If 11 was actually a negative 11, then the answer as 38 would be correct.

Yet, I would say that the student attacked this question at a unique angle, but not one that found him the correct answer. When I saw this, I immediately thought “what is this simplest way to rearrange this question?” I wrote on a scratch piece of paper, 27 – _ = 11. All this is is taking the question and writing it horizontally instead of vertically. Then as a teacher, I would suggest the “counting down” method to my younger students. I would instruct them to count from 27 to 11 and see how many fingers were lifted as they counted down. I found that 16 fingers were lifted, which means that the answer is 16. To challenge the students, I would ask “what other ways can you rearrange this question.” They would hopefully say, 11 + _ = 27, which is also making the question horizontal instead of vertical, but would challenge them to know that you start from the bottom and change the subtraction sign to an addition sign. Then I would instruct them to “count up,” and see how many fingers to count from 11 to 27. The answer would able be 16 in this situation, which shows to my classroom that there are different strategies to solving these questions.

I think the child has a little concept to get the answer for this question but he/she does not really understand the concept. He/she knows it will get answer when 27 subtract by 11. I think he/she just did a careless mistake on the adding instead of subtracting.

I can say that if the child can understand more on this concept and do more practices on it. he/she will do well on the mathematical problems. This child really has a great logic to think about these types of problems that is a little tricky to him/her.

I would suggest him/her to think about what he/she did. Why 27 subtract by 11 that you get 38, why not other answers? I will try to lead him/her to do it correctly and explain the concepts and rules in mathematics to them. For example, you should follow the rules of the signs. Otherwise you will think it does not make sense. Also, he/she can try to use the answer to prove that correct or not. It can make sure to get the correct answer. So when he/she put 38 into the box, the bottom one should be -11 but not 11.

This student made a really interesting mistake that is worth pointing out. I would guess this is from an early elementary level, considering it’s still adding and subtracting. The thing that makes math confusing sometimes is the way that a question is phrased and asked. For example, here it’s asking you to find the number that you add to 27 that would get you 38. I would assume if you phrased it differently, like by saying what is 38-27, this child would be able to get the correct answer with no problem. Because of the way this problem is phrased, it might’ve made the student more confused. I think this might be the introduction to subtraction, which is probably why this question is asked this way so students can practice doing addition and subtraction simultaneously.

I think the student made a really simple mistake by mistaking subtraction for addition. In doing so, this show that this kid is in the early stages of learning adding and subtracting. When looking at this problem you see where the kid made the mistake. I think when asking student questions like this the wording should be very different until they really understand the concept. Then you can go into different ways to ask the questions to see if they understand. For example this question is asking to fill in the missing spot in the number line. The way it was written is not even a number line. A number line would be like 27- __ =11. I do see that the kid in this problem made a mistake and it is something that should be looked at and corrected right away so it’s not something they carry with them and also so that they understand more clearer. However, in my eyes the bigger problem is how students are being asked questions and the way questions are being step up is the bigger problem.

I think it is some common mistake children will make that they mixed up the function of “-“and “+”. They simply thought the numbers in one equation can change their positions freely but still keep the equation works. The student get confused about concepts of addition and fraction. We can help him understand by making him realize the mistakes. We can ask him to compare “38” and “27” first to see which one is bigger then he will know that 27-38 couldn’t be 11. To teach him use the answer to prove the equation is important. Then we can reinterpret the question to be there are 27 blocks in total, how many left if we take way 11 blocks. Transferring this question into daily life will be helpful that the student won’t say 38 anymore because he know that blocks are getting fewer not more. We need to help children build clear understanding of addition and subtraction by looking at real life cases rather than working on equations only.