Developing Whole-Number Place-Value Concepts

So, as children develop place value understanding, there are different phases that they must go through and successfully accomplish/understand before they can move on to more complex understanding.  These are the phases:

  • Pre-Place-Value Understanding – This is when children can count to a certain number as they count by one’s but may not necessarily understand what the numbers stand for.  For example:  a child may be able to count 23 jelly beans but doesn’t really understand that there are twenty and three jelly beans, or two sets of ten and three jelly beans.  This is also known as Pre-Base-Ten Understanding as which required unitary counting.
  • Base-Ten Understanding – At this level, students understand groups of 10 and can count these as a single object.  Students will be able to count by tens and then continue counting in what is “left over” as the ones.  With this understanding, it is important to give students lots of opportunity to count in different ways and express these numbers with numerals and words to make that connection.  This is also when place-value understanding is specified.  It is important to have students explore how to group objects for counting in this phase.  So that they “know” that there are groups of ten because they created them, such as banded straws or cups of counters, or rods of cubes.
  • Equivalent Understanding – At this stage of understanding, children see units of ten and hundred and do not need to singularly count each object to know that they can count with these objects as groups, but they must have had adequate time learning with groupable models first.  For example, 1 hundreds-flat , 3 tens-rods, and 2 ones-cubes is equal to 132.  This is also the stage at which children learn to count and work with numbers that are represented with nonproportional models.

Learning and Modeling – It is important that teachers provide students with adequate work and experience with each stage before they move them forward.  The suggestions of counting cubes, links, crayons, shoes, students, toothpicks, buttons, beans, plastic chips, craft sticks, beans, washers, etc. was great.  I wouldn’t have thought of counting anything and everything really.  Also, linking number word forms with numerals is an important factor which needs to be included and continued as learning progresses.  Once students have mastered equivalent understanding, the activity 11.2 in which students count various objects and record them as a number word and as tens and ones is powerful.  Additionally, with that same worksheet, understanding of ten-ness is reinforced with counting “ungrouped” objects and recording the numbers, counting out a given number of objects and filling in tens-frames to represent, and “looping” objects in groups to create a certain number.

Benchmark Numbers – Benchmark numbers are key number of recognition such as multiples of 10, 100, and later on special numbers such as multiples of 25.  This will assist students as they learn to work computations with numbers.  Using a hundreds chart or a number line can assist in this understanding.  For example, when students are multiplying 45 x 4 they can also see that as 45 + 45 + 45 + 45.  It would be easy for students to figure out this by using benchmark numbers.  So, students could say I am multiplying 50 x 4 or adding 50 together 4 times and then I am taking away those 5’s that were missing so taking away 20.  50 x 2 = 100 x 2 = 200 – 20 = 180.


Questions for Katie (Meaning of Operations and Basic Facts)

questionmarksSolving Word Problems and Math Matters

1.  I had trouble with the addition/subtraction word problems and sometimes categorized the problem incorrectly.  So, I’m thinking I’m working out an addition problem when it is really a subtraction problem.  I think it had more to do with the way I would choose to solve the problem.  Did you have this issue with any of the word problem categories?

2.  I really liked the use of Bar Diagrams.  I felt like it gave a visual (almost tangible if you will) meaning to Part-Part-Whole problems.  I really saw the relationship building that is explicitly shown with this method.  It gives students more meaning than just addition or subtraction and shows relationships between the operations and how they work together.  What do you think about using this method?

I think that using a variety of these teaching strategies for helping students learn to understand and solve word problems would be my preferred approach?  Katie, what do you think, would you prefer to use one method over all others or a combination?

Storyline Online

3.  What book did you use from the Storyline Online?  I used Library Lion.  It is a cute book that I had read before (I realized once I was half way through) but I didn’t think it particularly lent itself to “math.”  I think there are other books that are so much more of a tie to math?  What do you think Katie, have you used other good math books?


4.  After everything we have read, what do you think about Timed Facts Tests for math?

Case Studies

5.  After reading the Case Studies, what do you think about giving students the choice of which manipulatives they can use to help problem solve?  Would this cause more problems with keeping students focused on the problem solving versus walking around the room investigating available manipulatives?



Final Articles – How can we help our students learn/know/recall their basic facts without resorting to memorization?

We should use the Chinese philosophy and us 10 as a bridge because of its importance in the base-ten numeration system.  Also, we could introduce the basic facts in units, such as 6+ unit, the 7+ unit, and so on.  These units are categorized by the know entity (addends) instead of the unknown entity (sums).  Lastly, the article reveals that we should teach different strategies that are not in textbooks but that can help children see the patterns among addition and subtraction facts (still using 10 as a bridge).

In the second article (The Road to Fluency and the License to Think), the author/teacher started out the year using a whole new method to teach basic math facts.  She followed a schedule of teaching addition strategies:  doubles, doubles plus one, doubles plus two, doubles minus one, doubles minus two, combinations of ten, counting up, add one to nine, make ten, adding ten, and commutative property.  Following this schedule, she would follow a certain “routine” as she taught each new strategy:  introduce and explore using manipulatives, create illustrations, use math journals, and assign homework including a parent information page.  Finally, she would complete each day with a Mental Math session in which students would share with the class their “strategy” for solving the problem.  This helped give other students strategies that they had not before considered and gave a whole “thinking math” feeling to the classroom environment.  Students would even name their particular method for figuring out problems – which made things more interesting for them and created more of an investment on their part.

In the final article, many different “types” of multiplication problems were discussed:  repeated addition, scalar, rate, cartesian produce, and area.  And the importance of helping children understand that multiplication has a variety of meanings and is not just a sequence of isolated facts.  This article that the the most effective sequence of instruction for multiplication facts is:

  • introduce multiplication with real problems involving repeated addition
  • use hands-on materials that truly represent the problem (ie markers in the ex)
  • then, substitute blocks for problem items as a next step
  • then, substitute student drawings for tangible manipulatives
  • then, advance to students using tally marks or other representations instead of drawing true items
  • give students many thinking strategies to simplify problems such as 8 times a number can be 5 times a number added to 3 times that number or 4 times a number added to 4 times a number
  • give many opportunities for practice with games, flash cards, dice, calculator, computer-based practice, concentration cards, etc.
  • finally, reteach those who really struggle by beginning showing them how much thy DO KNOW …. using a known and unknown multiplication fact table so that they can feel proud and confident in their current abilities and not be discourage about what they need to improve upon

Cube Nets

Cube Nets

That was fun!  I did ok with this activity …. in 5 minutes I got 23 out of 24 correct.  All nets must have 6 “pieces” or sides to fold up to make a cube and the sides cannot all be in a line, at least 2 must be arranged on another line graph.  Nets with squares all in a row will not work because it will not provide for a top and/or bottom.  Also, nets with cubes that are in a cube will not work because it will not be able to fold up to cover all sides.  Without folding is there a quick way to tell if a net will work?  Well, I couldn’t fold the figures that we worked with on-line, so yes.  But, I’m not sure how.  I suppose one must just need to picture the squares and fold them in your head.

Katie, how did you do with this?  I know that Sunee was so good with the 3-dimensional figures at our table last week.  I wonder how she did with this too.

Making Meaning for Multiplication and Division

Case 8 – Bunnies and Eggs

The bunny problems were incredible …. I couldn’t believe that kindergartners could figure that out so easily.  Using creativity with their manipulatives such as keys and blocks was very resourceful.  All the children, Jason, Rashad, Carlita, Kenya, and Flora all put together some sort of three groups of four – some did count them all up, but ultilmately, they were all using multiplication strategies.  Jason used blocks, Rashad and Carlita used keys, Kenya used Unifix cubes, and Flora used rocks – all using a different manipulative to help them solve the problem.  Junior was making similar groupings but had trouble counting each component of the group and instead kept counting groups.  He did not see the big picture that each group represented a basket of smaller items.

Case 9 – Easy Multiplication

In this case, 2nd graders were given an incomplete rectangle to work with but they did very well with it.  They saw very quickly that there were 5 groups of 3.  But, interesting that Christopher wrote that as 3+3+3+3+3 instead of counting by 5’s.  Then Patrick identified a different way of seeing the groups as 3 times 4 plus 3.  I noticed that Nisha would announce the different thinking processes to the class – this gives students other ways to understand and learn.  Just like when we ask several different students to come up to the board to write how they solved a problem – we can see different thinking strategies and are thus exposed to new ideas.  This information sharing was also done when she asked them to share their strategies in small groups.  Further, visual displays were used and recorded to help understanding and strategy sharing even more.  Then, that was when the students started connecting their ideas to other ideas that they had heard – WOW!  It was noted that the students used lots of repeated addition to accomplish their multiplication.

Case 10 – Candy Canes in Packages

Janine’s group struggled to make send of buying candy canes for 609 students at the price of $0.29 for every package of 6 .  One of the girls, Letitia, struggled to keep up with adding groups instead of single numbers.  Janine kept seeing her understanding but then it would fly away out of her head and she would be confused again.  This was similar to Junior’s problem in that they couldn’t keep the idea of “groups” in their head and could only “see” single items instead of things together.

Case 11 – How Do Kids Think about Division?

Vanessa is considering division in the traditional way in the first problem.  But, in the second problem when she is asked to figure out how many packs of seltzer water to buy if they come in packs of 6 and they need 36 total, she is using the logical inverse of division which is multiplication and really addition to figure how many times she will need to add 6 together to get 36.  It really show me that she has a great understanding of the math concepts and is not worried about using the “correct” operation.

Next, Cory is using his understanding of working with more familiar numbers first and is trying to use 2 first, then use 2 again.  That show me really incredible understanding of what he is doing.  So, he made a mistake, but ultimately, he is trying to divide 36 by 2, then divide 18 by 2, and he comes up with 9 – WOW!  I don’t think I would worry that Matthew is figuring his division problem using multiplication.  If he can see the connection and work successfully, it will only be a matter of time until he can clearly see how division is separate (but still connected to multiplication).

For Further Consideration …..

I created 5 addition/subtraction problems for Olivia (1st grade).  I provided counting bears, Unifix blocks, traditional math counters, markers, crayons, graph paper, and plain paper.  Olivia chose to use only markers and not any of the manipulatives.  With each problem, I asked her what she was doing and why.  She was very thoughtful in her work and took her time.  The only problem that she could not solve correctly was the last comparison problem (and the toughest one):  Olivia has 1 sister.  Lily has 2 sisters.  Lily has how many more sisters than Olivia?  Olivia “figured” 2 with her work.  But, then when I read the question part again “Lily has how many more sisters than you?”  She immediately said 1.  I was not surprised as I picked a friend of her’s and the problem information was true to life.  So, she knew in her mind that she has 1 more sister than her friend.  But I was surprised that knowing this, she missed the problem.  Did she was read the problem wrong or have a misunderstanding with what should be solved?  Something just didn’t work out right in her thinking.  I wish I knew what she was thinking and where the confusion was.  We talked about it and tried to find out what happened, but we couldn’t come up with anything.  Thanks for the help Olivia!

Helping Children Master the Basic Facts


During the Power Point, I worked on many different strategies to help me learn to teach basic math skills.

Addition Strategies:  Using COMMUTATIVE PROPERTY such as 5 + 3 = 3 + 5, called Turnaround Facts, can be helpful to help children learn.  Also, counting on is a beginning step for learning addition facts.  Do not discourage children from counting on their fingers as this is a good beginning strategy and they will stop once they are ready.  Counting on and counting on fingers is useful but it is time-consuming.  So, we want to give children other strategies to help them.  Such as:

  • Adding Zero (especially using manipulatives, so they don’t get confused)
  • One More Than
  • Two More Than
  • Adding Doubles
  • Near Doubles
  • Combinations to Ten (using tens frame is helpful for this)
  • Adding to Ten and Beyond

Subtraction Strategies:  Again, there several ways to help children master subtraction besides just counting back, which again is very time-consuming.

  • Think Addition
  • Down over 10
  • Take from the 10 (ex: 17 – 9 …. 10 – 9 = 1 + 7 = 8) This was a great addition to my subtraction repertoire that I will definitely remember.

Multiplication Strategies:  Using books is especially helpful for laying the foundation for learning multiplication.  I loved this idea and will definitely use the books suggested:  Each Orange Had 8 Slices, Anno’s Mysterious Multiplying Jar, What Comes in 2’s, 3’s, & 4’s, and Amanda Bean’s Amazing Dream.  Then once you have read about things that come in groups, have the class make a list of things that come in groups.  Remind them to use usual groups and not crazy things 🙂  Then, make a class book.  I have done this in kindergarten and it didn’t seem that powerful.  But, I can definitely see how having student create a class book in an older grade (2nd) would be wonderful!  Love this!  Other strategies to use:

  • Circles and Stars Game
  • How Long?  How Many? (with the rods)  I have never seen this before and LOVED the game – how much fun to be learning multiplication without even realizing it!
  • Doubles

Division Strategies:  I used multiplication/division skills when I divided the list of fractions.  I didn’t even think about it.

Finally, what about Time Tests????  They will benefit fewer than they will impede from what I heard because of things like:

  • they will improve skills children already know
  • they will not promote reasoned approaches because students don’t have enough time to think through strategies
  • they will produce few long-lasting results
  • they will reward few but punish many
  • they can give some students a strong dislike for math and a faulty idea of learning math