Digits versus Numbers

Rory was playing the Osmo Numbers Game (he was pretty spoiled by Santa this year!), and he was getting really frustrated because the game wanted him to make the number 12, and he kept doing 1 and 2 (instead of 10 and 2) and not getting it right. This led me to wonder how I could help him understand the difference between digits and their place in numbers.

osmo1

Oliver learning numbers with Osmo.

Rory is in kindergarten and has no understanding of place value yet, although he should start learning about numbers greater than 10, in school soon. To help him realize that 12 is different from 1 and 2, he needs help conceptualizing the idea that one 10 is different from one 1.

“They must be able to conceptualize place value; the understanding that the same numeral represents different amounts depending on which position it is in.” Charlesworth, 2012.

osmo2

Rory trying to make 14 with the digits 1 and a 4; then making 14 by adding 10 and some more.

Knowing the difference between digits and numbers is a developmental milestone and comes with a strong understanding of place value. I often have primary teachers ask, if I have any ideas on how to make this difference more apparent, and I do!

There are 3 main strategies you can use to develop this understanding:

1) Concrete representation of numbers

Use manipulatives to build the numbers. Showing the difference between 1 and 2 and 10 and 2 using a rekenrek, counters or unifix cubes clearly demonstrates that the digit in the ones position is different from the digit in the tens position. (If this doesn’t make sense, watch the video below!) Grouping or bundling things into groups of 10, or using base 10 blocks, is the best way for students to visualize the difference. Make sure you give lots of time for them to practice counting objects and grouping them into bundles of 10.

“A set of ten should figure prominently in the discussion of the teen numbers” Van de Walle, 2014.

2) Patterns

Write the numbers vertically and ask the students to notice any patterns they see. You will be surprised at your student’s or child’s ability to see that numbers repeat from 0-9, or that all the teen numbers have a one in front of it. Bring in the 100 chart! Now you’ve got a starting place for inquiry…why do they all have a one in front? Your goal is to have students come away with the understanding that digits mean different things when they are in different places.

3) Addition

In order to build number sense, you want students to think flexibly about numbers. In other words, can they decompose a number and can they do it in more than one way? When introducing numbers greater than 10, you want your child to decompose them into 10 and some more. This is another reason I love the rekenrek, for its amazing ability to show a number as 10 and some more.

“Mapping the teens number names to a ten and one structure is an important idea.” Van de Walle, 2014.

Counters are great too because they give students practice creating that group of 10: count out 13 ones, but if you group your tens…you have one ten and 3 more. Representing this as an addition sentence is another great connection for your kids to make and is the beginning of learning signs and symbols to represent math problems.

Here is a video of Rory and I learning about numbers greater than 10. Watching it back, I would have done a few things differently (like have my husband look after Oliver!), but I still think it gives you an idea of the 3 strategies in action (concrete manipulatives, patterns and addition)!

Do you have other suggestions of how to build the understanding that digits are different from numbers? I’d love to hear them! Please share them in the comments section below.

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The Importance of Place Value in Primary

The importance of place value

I always start the school year with number sense. Why? Because it gives me a good indication of each student’s comfort with numbers. Number sense is the one strand that is always interwoven into other strands in mathematics. It is also the most telling predictor of a child’s success in math. If they have a good number sense – they can work flexibly and easily with numbers in most situations. If they don’t, I will have an idea of how to construct the rest of my lessons so that those students finish the year feeling more confident with numbers.

So what concept do most teachers start with? Place value. Place value is one of the hardest yet most important skills for primary students to master.

“Place value is the understanding that the same numeral represents different amounts depending on which position it is in.” (Charlesworth, 2012)

“The most critical period in this development occurs in grades pre-K to 2” (Van de Walle, 2014).

The understanding of place value follows this progression:

Level 1:        Initial concept of 10 (see 10 as ten units).

Level 2:        Intermediate concept of 10 (see 10 as a unit of ten ones but rely on
physical or mental models to do so)

Level 3:        Facile concept of 10 (easily work with units of 10 without the use of
                      models)

Place value development carries on through the elementary years by increasing the place value to build bigger numbers. But it is more than just knowing how to read big numbers; it is crucial to the learning of trading rules that underlie whole number operations. As a result, it is the most important building block for number sense.

Sadly, all too many children have a rote understanding of place value without truly understanding what different place values actually mean. They are not able to go back and forth between equivalent representations of the same number (i.e. 31 could be three groups of 10 and one 1or it could be 2 groups of 10 and 11 ones).

Introducing place value

Rory has just started kindergarten (my baby!!!!) and he has no understanding of place value (and nor should he!). He does have a good understanding of conservation of number though, meaning he understands that different arrangements of counters still mean the same number. To introduce him to place value, I did the following:

1)   I asked him which was larger 63 or 36 to see if he has any understanding of what he’s counting or place value. He didn’t. Interviewing children is a great way to gauge their ability and find out what they’re thinking and why.

2)   I gave him a bunch of unifix cubes (a great model to start with!) and asked him to count as high as he could. This is a great test for cardinality and one-to-one correspondence. Both he and I were surprised that he made it to 39 without any assistance!

3)   I asked him to now group the cubes into groups of 10 which he did easily. He quickly noticed if a group of 10 was incomplete (the benefit of a proportional model!).

4)   We finished by noticing that the number of groups of 10 corresponds to the place value in the number. This went way over his head, but I felt better for saying it!

If this were a classroom lesson, I would complete it with a comparison of one ten tower to one cube and engage the students in a discussion of what they notice.

Do’s and Don’ts for Teaching Place Value

So how do you teach something this important? Here are some do’s and don’ts

DO:

  • DO use proportional manipulatives (the hundreds should be bigger than the tens which should be bigger than the ones)
  • DO use many different manipulatives so the child doesn’t think PV can only be represented with one type (i.e. base 10 blocks, unifix cubes, paperclips, counters)
  • DO integrate PV teaching with estimation and computation tasks
  • DO integrate PV teaching with measurement tasks (and use different units to measure!)
  • DO work on PV skills throughout the school year (not just in September!)
  • DO question and assess students constantly to be sure they are really understanding the concepts and not just answering in a rote manner
  • DO engage students in composing and decomposing #’s in a wide variety of ways
  • DO allow children to invent their own computation strategies

DON’T:

  • DON’T give worksheets which encourage a rote understanding not a concrete understanding (i.e 6 tens and 4 ones is 64 in all.)
  • DON’T teach PV abstractly, tie it to a manipulative
  • DON’T rush into operations involving regrouping tasks i.e. borrowing and carrying (mental math rarely involves regrouping as a strategy) and you run the risk of students not conceptualizing place value
  • DON’T start computation with numbers that don’t require regrouping (i.e. 23 +45); otherwise students will think that 56 + 35 = 811
  • DON’T rely on one type of manipulative (i.e. base 10 blocks) or children will memorize that a rod is ten and a small cube is one without actually understanding how many ones make a rod
  • DON’T teach PV in isolation, make it rich by integrating it with other subjects (see the link below for suggestions)

A true understanding of place value doesn’t usually happen until the concrete stage of development (7-8 years old), but it doesn’t mean you shouldn’t provide lots of opportunities for its development. What can you do? COUNT EVERYTHING!!!!

Want more suggestions? Click here!

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