Fly on Math Teacher's Wall - Put the Value in Place Value
A group of math bloggers have gotten together to join forces in ‘squashing’ some math misconceptions. For the inaugural blog hop we are tackling the big idea of Place Value. So, read through my thoughts and then at the bottom is a link to another blogger and their thoughts on Place Value.
Before we start, answer this question (I’ll come back to it later, but get your answer now):
How many tens are in 243?
For far too long textbooks have only focused on the ‘place’ a digit is in when writing lessons for place value. Typical questions sound like this, “In the number 64,235, what place is the 4 in?” Questions like that may be important, but they do not emphasis Place Value. Even questions that ask students to tell the value of a digit do not emphasize Place Value. I suggest that they might actually cause misconceptions surrounding the big idea of developing Place Value as a whole, i.e. questions like “What is the value of the 2 in 64,235?” are leading children to see digits in isolation without developing an understanding of how the digits around it compare. When we break Place Value down into its parts and focus on them in isolation, focusing on the place a digit is in or focusing on the value of the digit, it does not give children a full picture of what Place Value is about. The Number and Operations in Base Ten progressions document, makes the point that the underlying understanding we need to help children comprehend is the relationship between the values of the places; “the value represented by each place is always 10 times the value represented by the place to its immediate right” (p.2). The digits within each place represent 0 through 9 of those units and when we get 10 of those units it makes one of the next highest unit. Van de Walle (2013) says it like this; we need children to be able to understand how 100 can be 10 tens or 100 ones.
Let me share a story about my son. One night he was helping my husband and I count out some cash that we were going to be depositing and he counted out “1 hundred, 2 hundred, 3 hundred, 4 hundred, 5 hundred, 6 hundred, 7 hundred, 8 hundred, 9 hundred, 10 hundred, 11 hundred…” (See TMWYK’s Ten Hundred Doras as another example of children describing numbers in this way.) As he was counting out the physical hundred dollar bills it made complete sense to call it 10 hundred, NOT 1 thousand. But when it is written as 1000, how often do we call it 10 hundred? We don’t. Not on numbers like 1000, but we do on numbers like 1500 (did you say that as 15 hundred or as 1 thousand 5 hundred???). It’s an interesting thing to think about…we ask kids to do all kinds of bundling activities (count the items and when you reach 10 you bundle them up and now it is called a 1 of the next highest unit). But how often do we let them use their informal, yet mathematically solid, counting of those bundles and continue on with their counting? Usually we don’t, we stop them, and have them turn those 10 hundreds into 1 thousand, then they can continue counting. When we stop their connections to cool patterns they are noticing, it creates children, and adults, who, when asked how many tens are in 243 respond with ‘4.’
The answer to “How many tens are in 243?” is not 4. If I had 243 t-shirts and decided I wanted to bundle them into rolls of 10 t-shirts, would I only have 4 rolls?
Cathy Fosnot’s Contexts for Learning Mathematics has a fabulous T-shirt factory unit. In which, students have to help Grandma Eudora organize all the t-shirts she is making (and along the way start to understand place value). If Grandma Eudora had 52 t-shrits laying around unbundled, you have the kids bundle the t-shirts into rolls of 10 to make them more organized and easier for Grandma to count. As they do each bundle of 10, record the status of the t-shirts in a T-chart:
This activity does more than just develop place value; it also helps you facilitate some awesome discussions about composing and decomposing numbers. As you move into larger numbers of t-shirts, you can use the chart to help you lead discussions about the patterns they see and connect them to the ideas of place value.
Okay, so, back to our original question; How many tens are in 243? If Grandma Eudora had 243 t-shirts, when we bundle them up there would be 24 bundles of ten in those 243 t-shirts. i.e. 24 tens in 243.
Let’s try another. How many hundreds in 2000? Hint: it is not 0. I also like to encourage kids to think in terms of money when working on place value. For example, if you have $2000, how many hundreds would you need to make that amount of money? How many tens are in $2000? How many ones? How many tenths (or dimes)?
Doing activities in which students look for patterns and relationships helps them develop a more cohesive understanding of place value that is not developed in the types of tasks most textbooks use for Place Value.
Want to try another one? How many tenths are in 1036.5? Hint: it is not 5.
I will leave you with these questions to help you with your state of disequilibrium (or maybe these will put you into a state of disequilibrium)…
How many thousands are in 1036.5?
How many hundreds are in 1036.5?
How many tens are in 1036.5?
How many ones are in 1036.5?
How many tenths are in 1036.5?
How many hundredths are in 1036.5?
Post your thoughts in the comments about how you figured out the number of tenths in 1036.5, and then continue on in your journey to increase your understanding of Place Value by visiting Teaching Math by Hart. She is next in the blog hop.
About The Author
“How many tens are in 243?”
One of the things about math is the need for preciseness (or precision, I don’t mind), not only in definitions but also in language generally. The question above is not a precise question. It has (at least) two interpretations:
1: In the place value representation system, how many tens are indicated? Answer 4
2: As a number, regardless of its representation, how many tens do I need to make this number? Answer 24, and 3 “ones” left over.
Both answers are correct !
One must be more careful with the wording, so it would be more precise to ask “How many groups of ten can I get out of 243?”, or “How many tens do I need to add together to make 243?”.
Regarding place value it seems that very little notice is taken of the actual number words, which are in English almost entirely a match to the numerical place value representation. French is a bit more problematical, say quatre-vingt-deux.
Yes, Howard, you are right. With children I would be way more cautious with how I present the question. But I intentionally ask it as “How many tens are in 243?” with adults to bring out the fact that we over emphasize the place a digit holds.
Thank you for this post! I love that you linked to the progressions document because so much of what is being taught at the younger grade levels is a foundational understanding that will be critical to performing higher level math in later years. I also loved your point about formal vs. informal language when counting in “real life”. When I read 1500 in your post I read it as “fifteen hundred”. In the classroom we spend a lot of time teaching students to say “one thousand five hundred” and, separately, a good deal of time having students use place value to understand that 15 hundreds would be the number 1,500- we need to spend more time allowing our students to flow seamlessly between these different terms for a variety of values.
Thanks for the great post! The Math Spot
Great post. Thanks for the activity ideas for helping students get a better grasp on place value. I think my big take away is that we need to provide more opportunities to play with numbers and place value in a variety of ways, not just go straight for the minimal collection every time.
I think you have written about the biggest misconception adults have about place value. I have had many teachers argue with me that if there is a zero in the tens place than that number has no tens.
Tara
The Math Maniac
How many tens are in 243? I agree with howardat58. Let’s picture the entire amount as individual things, say sticks. It’s all in how they are bundled and how the bundles are grouped. If we bundle by tens for easier counting, we indeed end up with 24 bundles of ten, with 3 strays. If we group ten bundles together for hundreds, we now have 2 groups of hundreds, four bundles of ten, and those same 3 sticks left out of the bundles and groups.
The activities designed to help children understand this process of grouping by noticing and using patterns (T-shirt activity) are great for playing with the notion of place value and internalizing the concept. Much better than the textbook question which leaves it up to individual interpretation. In my years of teaching, I noticed that some children “get” place value almost intuitively, most need this kind of play with the concept, and a few really struggle for awhile even with manipulatives, play and analogies.
I skipped over all the replies because I wanted to see if I figured this out correctly. The number of tenths in 1036.5 is 10,365. There are 10 tenths in every one whole so if there are 1036 whole ones that would make 10360 tenths. We then have 1/2 of one whole so that is 5 more tenths making the total 10365 tenths in the number.
Nice explanation, Sandy.
I would agree with howard58. When working with younger children, who are still in the concrete phase of thinking, the question being asked needs to be more concrete than abstract. I see it all the time with my third graders the questions being asked are not taking into consideration the developmental readiness for such thinking. As teachers we can bridge that gap but wording is critical.
I think the goal of this question is to see if the student understands place value and has the flexibility to manipulate a multiple-digit number in equivalent forms. Sometimes discussing “pure math”, the language can be a hinder due to the interpretations. If I add context to this question, I might be able to get a better assessment data. ” I have $243 in $10 bills and $1 bills. How many of each kind of bills could I have?” Students might not give “24 ten-dollar bills and 3 dollar bills” as the answer, but if you think about it, any correct combination in 10s and 1s will give me the evidence about the student’s understanding of place value.
I also found this article from Achieve the Core very helpful: Think About Place Value in Grade 2. http://achievethecore.org/content/upload/Thinking_About_Place_Value_in_Grade_Two.pdf
Thank you for the article recommendation, it looks great. I agree that putting the question into a context makes much more sense and I have a great lesson I use with the kiddos for that…I’ll have to write a post about it soon!