We Ask, We Listen, We Learn is one of the early blog posts I wrote. (I posted it on January 5, 2015.) The post includes interviews of several students answering the question: How much is 12.6 x 10?
One of the video clips shows Mallika Scott interviewing Natasha, who explains how she decided on the incorrect answer of 12.60. Natasha explains, “Any number that’s times 10, you can add a zero at the end.”
Even though I wrote the blog more than three years ago, Westley Young recently posted a comment that addresses, in part, the “add a zero” strategy:
Westley (who Tweets as @Cukalu) raised issues that have long concerned me. One issue is my concern about students in the elementary grades noticing a pattern for multiplying whole numbers by 10 and then later erroneously applying the pattern to decimals, as Natasha did.
My second issue is my struggle with what to do about my concern.
In elementary math, students have experiences looking for patterns. This can be good. They learn algorithms. This can also be good. Their language is often imprecise. This is not so good, but they are in the process of learning and learning is messy. Teaching is also messy.
Many thoughts race through my head as I reflect on this video clip of Natasha and on Westley’s comment. What should I do as a teacher? How can I help Natasha and students like her reconsider?
My First Thought
What if I said something like this to Natasha:
If Natasha answered yes, I’d want to hear why she thought that was true. (I think it’s important to ask students to explain their reasoning, even when they’re correct. Maybe even especially when they’re correct.) Then I might say something like: “I wonder how I could multiply a number by 10 and wind up with an answer that’s the same as the number before I multiplied.” Could this help? Where might this have led?
My Second Thought
What if I said something like this to Natasha:
Yikes, then I’d be going down a semantic hole that does more to stimulate eye rolls from students than deep mathematical thinking. Natasha wasn’t adding zero in the sense that she was performing the operation of addition; instead, she was writing/tacking on/appending (or however you would describe it) a zero to a number. For 8 x 10, she’d do this and get 80. For 12.6, she did this and got 12.60.
My Third Thought
What if I said something like this to Natasha:
This is the idea that I’d like students to understand, as Westley pointed out in his comment. In a class setting, maybe I’d write that on the board, have students talk with a partner first, and then ask students to share their ideas with the class about why what I said is mathematically true. I think they’d agree.
Well, suppose students agreed with all of these statements. Would that help them think about multiplying by 10 in a different way? I’m not sure. In my teaching experience, teaching by telling often isn’t effective. My statements may be clear to me, but statements like these are too often mathematical blah-blah to students.
The Role of Contexts
OK, another thought. Mark Chubb (who tweets as @MarkChubb3) wrote a blog titled The Importance of Contexts and Visuals. In the section about using contexts, he wrote:
When I read this, I said to myself, “Yes!”
And then I was reminded of another clip from the same interview with Mallika and Natasha. A few questions after Natasha solved 12.6 x 10, Mallika asked her to solve a word problem: One pen costs $1.39. How much do ten pens cost?
Here Natasha again faces multiplying by 10. She restates what she said when multiplying 12.6 x 10, “Each time when you multiply the number by 10, you just add a zero to the end.” But here, in the context of buying pens, she gets the correct answer of $13.90. Thinking about what Mark wrote, the context of the problem invited Natasha to give an answer that made sense instead of what she did with the naked number problem. Take a look.
Where’s My Thinking Now?
I’m still mulling. As Mark wrote, when multiplying decimal numbers by 10, if we ground students’ experiences in contexts, they have a better chance of applying reasoning to make sense, rather than applying a pattern that they learned for whole number. (It might be interesting to talk with students about why the “tack on a zero” works with whole numbers and not with decimals. But that would be later.)
Also, in response to Westley’s comment about the importance of what’s really happening mathematically when we multiply whole numbers by 10, I think I need to acknowledge students’ recognition of a pattern but continue to help them understand the pattern mathematically. Too many students see the pattern of multiplying by 10 as pure good luck, especially when they don’t have that ease when multiplying by other numbers.
I know that I haven’t addressed another part of Westley’s comment about students “moving the decimal point” as a strategy for multiplying decimal numbers by 10. I’ve heard that explanation also and agree that the decimal doesn’t actually move. Maybe another blog?
I’m still thinking. I welcome your thoughts.
A Postscript
I realize that one example doesn’t prove much. Here are video clips of two other students, Jennifer and Sergio, answering the same two questions that Natasha answered. For 12.6 x 10, Sergio asked to skip it, claiming he couldn’t figure out the answer without paper and pencil, while Jennifer gave the incorrect answer of 120.6. For the contextual problem of buying 10 pens at $1.39 each, Sergio and Jennifer both get the answer correct, each reasoning in a different way.
Sergio: 12.6 x 10 | Sergio: Buying Pens problem |
Jennifer: 12.6 x 10 | Jennifer: Buying Pens problem |
Natasha agreed that 12.60 is the same as 12.6
I watched the video. She explained that any number, you can add a zero at the end and it won’t change the number.
Human beings can’t multiply decimals. We are not calculators.
I get students to work with whole numbers. So 20 x 60 is 2 times 6, then count up the zeros and put them in a list at the end of the 12 to make 1200.
That way , when they learn about powers, they know that a power is just a list of numbers to multiply together. 10^6 is 10x10x10x10x10x10 They have already been doing that so now understand powers.
In working with Natasha in your original video, I think I would suggest to her that we look at “columns” and the value of each digit in 12.6 and 12.60. This would hopefully enable her to see that both numbers are one ten, two units/ones and 6 tenths. If her understanding of decimals was not clear a drawing or even manipulatives would make it clear that the number is not changed by adding the 0. The same process could be followed looking at 8 and 80 or 16 and 160 which might enable her to see the change in place value that happens when multiplying by ten. It is interesting that she is able to complete the contextual,money calculation correctly. I am not sure that children always understand the connection between decimals and money. Maybe this is a problem peculiar to my country where money and decimals are taught as separate “sections”. That highlights the issue of teaching maths in topics and the fact that this prevents children from making the necessary connections but if Natasha was a child in my class, I would consider that as a reason for her being able to do the one calculation correctly but not the other.
Thanks for your comments and suggestions. I think that the context of money is one of the few ways I know to help students bring meaning to decimal numbers, at least to hundredths. I’d love to have others to draw on as well.
These examples hammer home to me the importance of context and then connecting the context to the symbols in valid language… and getting the students to practice using that language. Hearing the student say 6 x 10 was 60 … but translating that to 6/10… the young man *gets* the “ten times larger* in context… but didn’t get it w/0…
Would estimating help in this situation? If you got her to see that 12.6 is close to 13 and then multiply by 10 she would know the actual answer is close to 130…?
Hmmm, interesting thought. Estimating is always a good idea, I think. Thanks.
How can the relationship between repeated addition and multiplication assist understanding in these situations? Combining with an estimation strategy could be beneficial, too.
I think it helps with whole numbers, but adding 12.6 ten times seems so clunky. But maybe. Estimation makes sense. BTW, the most common wrong answers students give to 12.60 is 120.6, which is close but not accurate. So estimation wouldn’t help there. My, this gets more complicated. Thanks for the comment.
I always tell my students to move the decimal point in questions like this.
But adding zeros to the ends of numbers is what people understand.
Four times an orange is four oranges.
Twelve times one hundred is twelve hundred.
Seventeen times a million is seventeen million.
People want to stick stuff on the end. That’s what their intuition says.
If I tell young children to multiply twelve by one thousand by first of all moving the 1 to the ten thousand place, they will be baffled and wonder if maths is supposed to be this hard.
Your comment reminded me that my goal is to find ways for students to notice patterns, see relationships, and make generalizations for themselves. I find that this supports sense making more than telling.
Thank you for your thoughts on this. I’ve been asked to do a number talk tomorrow that looks at 4 x 5, 4 x 50, 4 x 500, 4 x 5000. I am wondering if I should ask them what patterns they notice and what would happen if you did 4 x 0.5 – what would you predict.
This seems like a really fine idea. Is this with students? Teachers? It might also be helpful to ask them to think about 4 x 1/2, which they might be able to reason is 1/2 + 1/2 + 1/2 + 1/2, or 2. This might be a way for them to verify that 4 x 0.5 has to be 2. And then maybe 4 x 0.05? Please let me know how it goes. And thanks for posting.
I’ve been working i=on the same idea with a student I tutor. I like the idea of estimating as well as money. I’ve also modelled the multiplication with base 10 materials, to show why the digit is in a new place value. 5 x 10 is 5 groups of 10.
I am vehemently against tricks such a “adding a zero” (because, as eye-rolled above, anything plus zero still results in the same number) or “moving a decimal” (again, as stated above decimals don’t move…they live between whole numbers and fractions). Both of these are dangerous as they do not encourage conceptual understanding.
Having students explore patterns using place value charts is always helpful, but can sometimes still result in overgeneralization as students think about “moving digits” without the understanding of why.
Working the place value in a deep way is so important and having students reason can really help. So the ideas about using whole numbers or estimating are two I wholeheartedly support.
Interestingly…I asked my son both questions, and he used the same combination strategy for both, but could make more sense of the non-contextualized question. “12.6 x 10…well 12 x 10 is 120, so it’s 126.”
Hi, for one of my homework assignments my teacher had us read about this. I found it to be really helpful as a future teacher. I want to teach elementary school and this is very helpful. Also, my sister is 5 and struggles with math. However, I know as she gets older I can use your examples to better help her. Thank you!
Hello,
I am in a college math class that shows how we teach children in elementary school. As part of our class homework during thanksgiving we were to come on here and read your blog. I am so fascinated by it and feel there is so many aspects I can take with me when I become a teacher. Children get so confused and I believe the examples you gave can really help them be successful. I also, have a 5 year old sister who struggles with math and I have a feeling when she gets a little older fractions and decimals will be hard for her. I will definitely will be using your questions you asked to help her understand. Thank you for the great post!
Thanks for your comment. And good luck with your teaching career. It’s a profession of amazing honor.
I agree with your take on this concept. I as a student of mathematics sometimes struggle with the understand of place values. I can see how making sure a student understands the reasoning behind the problem is important. Adding a zero does sometimes work for example in addition, however, this concept does not always work for every area of math, and this is where small understanding problems can lead to bigger problems in the future.
And for place value with younger students, see here:
Hello, I am in a college math class where we are supposed to learn about the concepts of teaching elementary students and I find this very interesting, especially since it is a glimpse of what we would have to see going into this profession. What was fascinating was that most of the students get the same answer but explained it in very different ways. Then when it comes to fractions and more difficult subjects the students tend to give up. Which reminds me of my sister and how I try to help her but she responds with the way I am teaching her is incorrect and there is only one way to solve for that. Also, I agree with what you had stated about how most students do not understand by just listening because the point does not get across. Thank you.
Thanks for your comment, Jacqueline. And welcome to the teaching profession. For me, it’s been an incredible honor and worth all the hard work.
Your blog was very interesting to read because I for one have simply memorized what to do with certain math problems without truly understanding the reason to why we do algorithms and why it works. The questions that you have here would be very beneficial for students to hear and asked to answer because they will actually have to think it through. Being forced to actually explain why is a good learning practice because it allows them to conceptually understand what they are doing, making math easier and more understandible.
My early math learning was also mostly memorizing, and I think I’ve gotten better from teaching younger students. It’s all about thinking and explaining.
Hello my name is Eileen, I am currently taking a Math for Elementary teachers class in college and one of our assignments was to respond to your post. Aside from this class I work with intervention students and I see they struggle with making making sense of this concept. After reading and watching the videos, I very much agree with all the concepts you talk about in your blog. I would love to see what kind of engaging activities teachers could use to helps students make sense of this concept.
Thanks for your comment. If you’re interested in seeing how younger students reason, I posted some other video here:
I believe it means that children learn a certain algorithm and stick with it, without even understanding the meaning of the mathematical math problem, and trying to show that the students are right to a certain extent. Also, show how children apply certain algorithms to problems even if they don’t apply.
All true.
Very interesting blog, I feel even as adults we have that pattern stuck in our brains of adding a 0 at the end of every number. How can we undo this or set up students not to follow that “just adding zero” trick? The example of adding context to problem such as the “pen problem” is a great idea so we aren’t just “telling” students statements that they may not comprehend, but they will be able to relate and discover a different trick.
I think the notion in math class ought to be “do only what makes sense to you” and we need to push for them making sense. I don’t think that tacking on a zero is the problem as much as not knowing why it works or not having another way to justify the answer. It’s complicated. And contexts can help, though I confess that I can’t think of a context other than money that serves.
Something I learned from a colleague that has helped me with students: Do only what makes sense to you. Then I ask them to explain how it makes sense.
Good morning Mrs. Marilyn, I am a freshman at Santa Ana Community college and I have a math class and got to see your blog. I agree with your statement “I think it’s important to ask students to explain their reasoning, even when they’re correct. Maybe even especially when they’re correct.” I believe that this exercise would help students even when they don’t get the same answer because they wouldn’t be so hard on themselves as if they weren’t to use this exercise some students would assume that math isn’t for them. Thank you.
You’re welcome!
Good afternoon,
I’m currently taking a math class that is specifically for people who would like to become elementary teachers. Right now we are working on decimals and I see that the mistakes the children make in the video are easily made. When they are multiplying whole numbers it is easy for them just to add a zero to the answer but on the other hand when using decimals it can be tricky of where to put the decimal or adding a zero to make it easier. As written in your blog, children look for patterns on solving certain math problems. This has its pros and cons. Pros because as they do their homework or take tests they apply their knowledge to all the problems. Con because the more they solve the problems without getting asked how they did it is not going to help them because they are solving the problem in their head and not out loud. When someone asks for help or clarification on the problem they won’t know how to respond. Therefore I agree with your blog on asking the children to explain their problem solving skills and would love to hear more!
I’ve dedicated most of my teaching to listening to students explain. For more videos, check here:
Hi Fatima,
After reading your comment, I totally agree! We are taught these shortcuts while learning a new concept (aka, _ x 10 means add a zero at the end!) that are meant to help us get through math easier, but in reality this could cause future confusion once newer math topics are taught.
It’s so important to ask students not if they got the answer, but rather HOW they got their answer and WHY they believe their answer is correct. No two brains think alike, so by asking these questions we (as future educators) can view this problem in a new perspective!
Hi Mrs. Burns,
My name is Amber Vang and part of my homework assignment was to reply to your blog. I just wanted to say as a future teacher that your key takeaways were really helpful. As I read your blog as well as watched the videos I got an insight as to how you viewed certain topics which I thought was very interesting. I loved how you gave examples on how you would have asked Natasha how 12.60 was equal to 12.6 and adding the part where you asked to explain why she thought the way she did. I believe that this allows multiple ways on how to really make a student use their knowledge on an equation without seeking for help on the first try. I will definitely be using this tactic in my future classroom. Thank you.
Thanks for writing and I’m pleased that the videos gave you insights.
Hello, my name is Thy Do. I was not born in America so I have never attended Elementary School here. I do not know how American children learn to spell and calculate. I would like to be a mathematics teacher in elementary school. Through this blog, I can figure something out how they learn. This trick is one of the best examples to teach students when they study deeply understand how to multiply in decimal number. We should not teach them this trick at the first or when they just know about decimal numbers. In Vietnam, we support students use a piece of paper and pencil to try every single problem. If they feel more confident, the teacher will teach them those tricks. A trick is very dangerous for children because maybe they solve it correctly but they will not know why they have that result.
Thanks for writing, and good luck with becoming a teacher.
Hello my name is Brenda, I am currently taking a Math for Elementary teachers class in college and one of our assignments was to respond to your post. I though your blog was very interesting. I do confess I was one of those student who were tough to “just add a 0” when multiplying by 10. Ass I seen your blog I would also like to learn and also share that when multiplying by zero to not only take that easy step to get the problem done. I look forward to seeing more of your blogs.
I appreciate your comment. Thanks.
Hi Brenda, I can definitely agree with you, I would also make math much more harder than it was especially when it was something simple like adding the zero during multiplication. I think that her blog was very interesting with great information that we can also include in our future classrooms!
Reading this blog really made me think! Before reading it, I automatically agreed in my head that the “multiply by 10, add a 0” statement was true. After reading further into the blog, I realized how wrong I have been. I agree with what you have to say about teaching the concept of place values and how we are supposed to explain to students when it is okay to just add 0 and when it is not okay, (aka decimals).
Ah, hearing that my blog made you think made my day. Thanks for your comment.
I strongly agree with your way of thinking when it comes down to adding a zero but adding a zero will not alway work and when it doesn’t, it can lead to misunderstandings and problems to future students. Here you showed a perfect example of a students making a mistake on the problem due to using an algorithm. In this case it comes down to knowing where your place values are and knowing decimal places.
Thanks for your feedback.
Hi my name is Elizabeth Antunez, and I am a student at Santa Ana College. I am currently taking Math for Elementary teachers there, and I found very interesting how in your blog you mention students finding patterns to mathematics, this particular section caught my attention because my math professor was touching this subject during class, and it’s amazing how there are many different methods in completing math by different patterns. I really enjoyed reading your blog! I know that blogs like this will help me later on to know what students and other teachers are feeling towards a particular area.
I’m always amazed at the variety of ways students think about the same problem and explain their reasoning. And I think that having students explain should be part of every math class. Thanks for your comment.
Hi Elizabeth, I totally agree with you when you said that it is amazing how there are different methods in completing math by different patterns because I personally thought that there was only certain ways to solve a problem but in reality there are many different methods out there to help you complete a math problem.
Hello Mrs. Burns,
In a class setting, talking in partners would really help students understand the reasoning behind the concepts. That part of the article stood out to me because in the math class I’m taking we’re learning about the importance of students working together during class. I’ve learned that when students work together they are more likely to make connections with what they’re learning and are also more likely to remember the lesson. Thanks for your article.
Yes, I think that talking is essential to the learning process, so I find as many ways as possible for students to talk. Have your heard about Think, Pair, Share? First students think on their own, then talk with a partner, then share with the class. Powerful.
Good afternoon,
Hello, my name is Odalis and I’m currently taking a class for Math Elementary teachers. After reading your blog I found it interesting when Natisha would explain how she added a zero when multiplying by 10. I believe that it is important to view students reasoning of how they solve the problem just how you mentioned because there are various ways to solving a problem. Also, it is important to give different examples just how I watched different videos on the examples that students do when they multiply by 10 just add zero at the end. Thank you for this great post!
Glad you found it helpful.
Hi Odalis ,
I found this blog intresting and informative. I found it intresting how natasha applied a zero when multiplying 12.6 x 10= 12.60 which is the incorrect answer. But when she multiplied $1.39 times ten pens she got the answer correct $13.90. It’s intresting how students apply a zero when they don’t see the problem visually. But when it was visually they processed the problem in a correct way rather than just adding zero.
I believe after reading this blog I believe an endless amount of people will benefit. The reason I say an endless amount is because as a future math teacher I believe this will help me to get them to understand in a more thorough manner. Prior to reading your blog I didn’t really stress the importance of understanding why an answer is what it is I always believed as long as you got the correct answer all is good. Thank you for enlightening me.
Ahhhh, understanding (aka making sense) is what it’s all about for me in the classroom. Good luck!
Hi!
I can personally relate with you!
I have struggled with math all of life and when i have gotten a chance to work in groups or even in partners I seem to feel more confident and with more knowledge regarding the topic. I believe that we all learn from one another, and that team collaboration is perfect. After all, we get to share our thoughts and most importantly learn good skills that will eventually be beneficial when working alone. For instance, I know i like to do my multiplication through the lattice method, but my classmates do the standard algorithm. For instance, if i didn’t know the standard algorithm they would be nice to teach me the method and vice-versa. Thank you for sharing!
I strongly agree with your reasoning behind sharing with a partner. I have personally always struggled with math, but sharing with peers has always helped me learn from my mistakes and grow my mathematical skills. I believe teachers should implement more pair sharing in their lessons, to help students learn different techniques/ methods while solving a math problem. Thank you for sharing these incredible points!
The benefit of a partner is a good tip to remember when you start teaching. It’s hard to do a good job alone.
I have, as-well, found that sharing with a partner helps me a lot. Although I can try to figure the answer on my own, its always great to be able and talk to your partner to double check your answer. Also this method even allows you to find out different ways your peers solved the problem, and can even change the way you break down a similar problem next time.
This blogpost gives me a further explanation on why reasoning can be a better solution in math problems rather than algorithms. Mark Chubb’s approach about starting with a context with reasoning before introducing an algorithm is something we do in a mathematical concepts class I am currently taking taking. I agree that this process helps build an understanding of number sense and avoids following patterns that do not apply to a context.
Contexts are so important, a way to bring meaning to abstractions. At least to help.
I agree with you Nathalie! Mark Chubb’s approach abut context with reasoning before introducing an algorithm is very helpful for students in math classes.
After reading this blog I have learned a lot from it and how to incorporate it in my future classroom. Also a lot of people will benefit from this because the children will get the chance to fully understand why and answer to a problem is the answer, and how to get the answer. I wish my teachers in the past had read this post because they would have explained to me why the answer to certain problems was what it is and explained to me in depth. Thank you for allowing me to apply this knowledge in my future classroom.
Thanks for your comment. I remember when I first started teaching math when I was still smarting from the many math classes I took in college when I struggled to keep up. I decided to do it differentlyin my own class. I’m still learning how.
Yes, understanding “why” is what it’s all about. Good luck in your future classroom.
Hello Ms. Burns,
Even though this is an increasing concern for teachers, we’re forgetting that the students actually did the calculations and have keen observation skills to even notice that you can add a zero to a whole number when multiplying it by ten. I think that teachers should still commend students on their efforts and observation skills.
The reason some students fail to see why 12.6 x 10 is 126, and not 12.60 is because they still don’t understand the basic concept of place values and number sense. If students understood those two things, they would see that there is a difference in the decimal when multiplying by ten. One of the most basic rules we learn in math is that anything times one is itself. So why would anything times ten be itself? That doesn’t make sense. Once students realize this, they wouldn’t say 12.6 x 10 is 12.60 anymore.
The teacher can raise a discussion in class about “adding a zero to the end of the number” to a number when multiplying by ten. Asking students why this method works (place values), and if there are situations that it doesn’t work (when working with decimals). If anything, we can show students repeated-addition for multiplication so they can review what it means to be multiplying by another number.
It’s important to approach math from multiple perspectives, so that students can think about ideas in their full complexity. It’s always important to talk about the “why.” Thanks for your comment.
Hi, As well as other few classmates, I personally also felt like there was no more clear reason as to why an answer is the answer of that particular problem. I figured if I got that answer correct and my teacher said it was correct, it was all good. However, after reading your blog I was able to understand things from a different perspective. And that’s what it’s all about. You sharing this information with us is allowing us to share it with others and so on. I’m also a very strong believer in “teamwork”. I think that by teachers and professors allowing students to work in groups it allows them to learn from each other in a socially manner. Kids are open to different ways to develop different solving tecginiques that they can surely benefit from. Thank you so much!! 🙂
For me, the teacher was always the answer book. Then I found out that life has no answer book, so I’d best start thinking.
Hi, I really enjoyed reading your blog I found it informative. This gave me an idea of how a child processes different math problems. I could apply this to my future teaching career. I found it interesting at the beginning Natasha got the answer wrong when multiplying 12.6 x 10 she got 12.60 she just added the zero. But compared to the word problems she got it correctly.
The context pushed Natasha to make sense. Naked numbers often don’t.
Hi Karla,
I agree with you that sometimes our teachers have told us the answer was correct without asking for a process. I think it is very important for everyone to understand why we get the answer we get and also how we get it. But it was very intresting how a visual also makes a difference for example the pen problem they got it correct.
I very much enjoyed reading this blog. I myself use many of the patterns Learned in elementary school without really thinking about the why behind them anymore. Giving visual context is a great idea as many children are visual learners. Building on what Mark Chubb wrote, when we allow students to use context they are familiar with it makes it easier to learn these new concepts. It also helps to use your students interest to help them learn. Some like cars and you can use that in your lesson. As old school as it sounds, I remember using block to represent ones, tens, and hundreds. This, or a visual equivalent on a computer, could vastly help the students understand these patterns better. In my current math class my professor visually draws out the value places in columns and labels them 10^2 10^3 and so on. Which made learning a much more complex concept, bases of 5, way easier for me. I am happily taking all of these strategies for when I have my own students.
Hi. I found it interesting how the students each got a different answer when it came to solving 12.6 x 10 yet the same answer for 1.39 x 10. It’s interesting how making it a word problem could actually get students to give the correct answer. It’s really interesting to see the difference in thinking between each child. I really like this article and it’s true that students aren’t taught this concept the right way. I agree that having the students explain how they got their answer is good and having them figure it out on their own in a way by questioning them. The place values I think would especially make them think more about the math they are doing because they weren’t taking it into consideration for the first problem.
Hello, I am a college student who is taking a math class to become an elementary teacher and I am also a tutor who tutors fourth and fifth graders. I really enjoyed and agreed with what you talked about, especially since I used to struggled with understanding how you multiply by 10 with decimals and any topics requiring decimals related. As a tutor I would like to know what is one way I can explain to a student how this concept in a different or simpler way.
Hello, I am a college student who is taking a college course to become an elementary teacher and I am as well a tutor for 4th and 5th graders. I really enjoyed and agree what you talked about, especially since I used to struggled a lot when it comes to decimals especially multiply and the place values where decimals goes. As a tutor I would like know how to explain to a student a different or simpler way to explain the misconception with multiplying by 10 with decimal and why some student answer just add zero.
Ah, it’s so hard to know how to help a student I’ve never met. That said, what’s important is for the student to do the explaining about how to multiply by 10. Offer a suggestion and ask the student to explain in his or her own words. When I work on how to explain something, I have to make sense of it for myself, and that’s when the learning happens.
Hello, I am a college student who is taking a math class to become an elementary teacher, as well as a tutor for elementary students. I really enjoyed reading what you were talking about and watching the example videos, especially since before I used to struggled a lot with multiplying decimals and where the place value goes when multiplying decimals, like by 10 and other 2 digit numbers. As a tutor and a future teacher, what is one way I can explain to a student to help them understand more about multiplying by 10 with decimal and why some students know why they add a zero when multiply decimals by 10.
Good Evening Ms Burns,
I am taking a math class for future math teachers right now so this topic and blog has been very beneficial to read. I have seen first hand with students in a classroom and with my own nine year old sister just how easy it can be to get confused with multiplication methods and understanding place values. I loved your blog and agree with you when you referred to the importance of talking out their answers and using real life examples such as with money to help better understand such as in your videos. I would love to learn what other methods or ways could be used to help students better understand these principles.
Hello, as I read this I really appreciated how you asked Natasha to explain why she solved the problem the way she did. As someone who is planning to be a teacher, I find it important to understand the patterns students see and how they solve and interpret the problems. I do believe it is important to guide students’ patterns in a direction so that they could understand it. You’re blog was very helpful with seeing how students perceive patterns!
Hello Ms. Burns,
This blogpost really got me thinking about different idea I’ve never thought about. I want to hopefully become an elementary school teacher, and I am so glad I read your post. You made me look at math and solving problems in a whole new direction. I think asking students how they got their correct answer instead of just saying good job is so important. Students may have the algorithm down, but maybe don’t understand why it completely works. If they don’t understand why it works, they might try to use the same algorithm on a whole different problem. I am so glad I read your post, and if you have any advice for me as a future teacher I would love to hear it.
Thank you,
Madison Smith
My advice — keep learning and find colleagues to work with. It’s hard to do alone. I hope my other blogs help, too. Good luck.
An important topic I like that you mentioned in your blog is sharing with a partner. I also find sharing with partner one of the best components to use. Although I can try solving the problem alone, its always great to have a way to double check your answers. As well as realizing a slight mistake you did. This method also allows you to interact with peers and discover different ways to solve the problem. You can use new strategies for the next similar problem.
Hello,
I would just like to say that I really appreciate your blog. There us a lot that I would like to incorporate in my own classroom as a future educator. One thing that really stuck out was how children will look for patterns in order to solve a problem. I understand how this is wrong because it starts giving them the mindset that math is about having a “right” or “wrong” answer. When in reality there are more than one way to get the answer.
Hi, I really enjoyed reading your blog I found it informative. This gave me an idea of how a child process problems. I could apply this to my future teaching career. I found it intresting at the begining Natasha got the answer wrong when multiplying 12.6 x 10 she got 12.60 she added the zero at the end. But when a word problem was given “1.39 x 10 pens” she got the right answer. She processed the problem differently than the first time.
Hello,
I would just like to say that I really appreciate your blog. There us a lot that I would like to incorporate in my own classroom as a future educator. One thing that really stuck out was how children will look for patterns in order to solve a problem. I understand how this is wrong because it starts giving them the mindset that math is about having a “right” or “wrong” answer. When in reality there are more than one way to get the answer.
Hello Ms. Burns!
I’m currently taking a math course about Elementary Math and how to teach it, and I found this blog post so interesting! As we grow into different math courses through middle/high school and college, the ‘_ x 10 means add a zero at the end’ rule’ is engraved into our brains. But we never truly realize..at what point did we learn how this rule applies to decimals?
Teaching children is rewarding yet so interesting, because their young brains are so curious and they will often ask numerous questions to find out more (and possibly challenge their teacher!) about the topic. Helping a student learn a concept is one side, but hearing the student’s explanation on how they got to their solution is a whole different view many adults overlook! We are used to finding the fastest way to the solution that we do not take the time to consider different processing steps to reach the solution!
I am a strong believer in not looking for the answer, but looking for the various ways children will look for the answer! No two brains think alike, and future educators like myself must learn to understand these students will contain endless answers to our “why’s” and “how’s”, even if their explanation is longer than their classmate’s.
Thank you so much for sharing!
Hello Mrs. Burns,
My name is Lilly and I’m currently taking a class for Math Elementary teachers. After reading your blog, I strongly agree with your way of thinking adding a zero when you multiply by ten. I loved how you gave an example to the student, because that is also helping me understand the more how struggle in adding zero and moving a decimal. I also agree with you that most of the student always looks for patterns on solving certain math problems, because they are just doing by the ways they think its more easier and use the patterns that they learn in class. However, thank you for your blog and I will be using this tactic in my future.
Hello Ms. Burns,
After reading your blog and found very interesting ideas. Your blog post was very interesting to read and I myself had different thoughts on developing Math skills. I myself use my Math knowledge sometimes to solve other Math problems because there is many different ways to solve a problem. Everyone learns and teach differently. Your blog should be read by other Math teachers to explain their is different methods or ideas in Math.
As a college student, it’s embarrassing to admit, but I still find fractions and decimals confusing. Although, after reading this blog, it actually helped me understand decimals better. Up until this day, I thought the concept of just adding a zero would get you the answer without actually thinking about understanding the equation and mentally solving the problem, other than just applying a pattern. I believe this is a great lesson to be taught to students at a young age before they repeatedly make the mistake to use this method of patterns instead of actually taking the time to understand the problem.
Please do NOT be embarrassed. You have the opportunity to be a better teacher because you struggle to make sense. I hit my math wall at calculus, later than you, but still it was tough. But I worked through and early my degree in math. I don’t want my students to feel deficient in math, ever.
Hi there. I just wanted to say thank you for providing videos in your post. They really helped me understand and listen to the children in a clearer way. As a future teacher I definitely want to keep in mind that children do get the right a swer but sometimes in different ways. I also wanted to thank you for creating this post, I think it is very important for educators to share their outcomes throughout their paths because that’s how we can see our mistakes and help each other out to figure out a better way to present the problem to the children in class.
HI, I am so glad to have found this post today. I was trying to find something along these lines to help some fifth grade teachers that I am working with. I want to share “Do the Math Multiplication C” with them, however, in reviewing the curriculum that we have, I discovered that one of the strategies is “Tack on a Zero” and I was surprised to see such a thing in your books. I think that I will copy this series of blog posts and links to the videos to share with them and tell them to think about this during those lessons instead. Any additionals suggestions would be great (we do have an older copyright of the book, do the newer books still refer to the “tack on a zero” strategy?
Thank you for all that you do to promote student thinking.
No, I haven’t removed the “tack on a zero” strategy from Do The Math. It’s a pattern that students notice and begin to use when multiplying whole numbers by 10, as we all do as adults. I don’t think that tacking on a zero is the problem as much as not knowing why it works or not having another way to figure out and/or justify the answer. I think the premise in math class ought to be “do only what makes sense to you” and we need to push students to make sense of whatever they do in math. When we begin to think about decimals, it’s important for students to start with what they know about whole numbers and then explore which properties and strategies hold true with decimals and which don’t. Here contexts can help, though I confess that I can’t think of a context other than money that serves as well with decimals. It’s complicated. It’s important not to teach a trick but to keep the focus on understanding. I’m not sure this makes sense, so please feel free to send me a message to probe more.
The money context adds another important feature other than intuition-accessible context: it adds a limit to the decimal places. Both 1.39 and 13.90 need to have two digits after the decimal because this is a money problem. Natasha tacked on a zero and also moved the decimal correctly (by instinct?) and so she “got the right answer.” Whether she has learned anything from the experience is anyone’s guess.
Love the Mark Chubb quote and I would suggest that Natasha can learn a lot from the discrepancies between her thinking process on 12.6 x 10 and 1.39 x ’10 pens’.
Adding to my comment:
a strict limit of 2 decimal places makes money problems amenable to Natasha’s memorized rules, which makes sense if you think of whole numbers as a strict limit of 0 decimal places.
Thanks for both of your comments. I continue to think about Natasha. More recently, a student told me that you can’t write $.085 because we don’t have half pennies. Made my heart sing.
In my work as a math coach I frequently face this misunderstanding (memorized rule: just add a zero). I’ve recently been looking for materials to recommend to teachers working with struggling learners and took a look at Marilyn Burns’ Do the Math Now! I was dismayed to see the “Tack on a Zero” rule. Why is it there?!
That’s a good question. Rules and algorithms are an important part of mathematics, but do not replace understanding. I think in Do The Math that we helped students investigate the patterns of multiplying by 10 as a way of understanding what is happening. Honestly, the “tacking on a zero” becomes evident fairly quickly. That said, I’ve found that while it seems obvious to students when multiplying one-digit numbers by 10 that the answer is the number with a zero tacked on, they falter when I ask if it works for a problem like 17 x 10.That’s when we have to return to making sense, not merely following a rule. Teaching is complicated, and I communicate to students that part of learning math is being able to explain why a rule makes sense. I hope this helps.
I find using a place value chart (with PV chips) to see the pattern of x 10 helps. If students have had practice with this pattern with whole numbers, it works for decimals in the same manner. Once they are familiar with the PV chart and how to rename units, I might use the associative property to make the problem 10 x 12.6 and then rename 12.6 as 126 tenths. We then have 10 x 126 tenths or 10 groups of 126 tenths which would be 1260 tenths or 126.0.
Thanks for the suggestion. It seems that every year I try something different, depending on the students, of course. I love adding new ideas to my teaching toolkit.