Module 3 Place Value
Developing Whole-Number Place-Value Concepts
pg 245 questions 2
2) Describe the three types of physical models for base-ten concepts. What is the significance of the differences between these models?
Each of these models can be used within the elementary classroom, just at different learning stages. Ideally, the groupable model would be the first model that would make most sense to start teaching the idea of base-ten place value to young students. As listed earlier, this model allows for the student to actually see and touch the ten objects in one area. The student can learn by being able to touch the objects and count. The difference in the groupable and the pre-grouped is the ability to be able to physically separate the grouped objects. The pre-grouped are just that, they cannot be separated. The hardest and the model with the least recommendation for early learners is the non-proportional model. This model is hard for the student to understand because it does not increase by exactly 10 times the number from the ones value. The students do not have the ability to see the gradual increase in the size or shape of the object. The student would have to understand that a certain object is representative of the ones value, tens, hundreds, etc.
4)What are some of the ways the hundreds chart can be used to identify and use place-value concepts?
- Use for skip counting by evens, odds, or tens (this can be used by placing number cards in the chart and removing them to have the students practice counting by particular patterns, use of place-value of the ones place
- Lessons on decade number recognition
- Two digit number recognition and introduction to tens place-value
- Introduction of basic number sequencing can discuss numbers by ones place, tens or decade numbers
Power Point:
I was fortunate enough to sit down with my second grader and watch her try to figure out the riddles that were given in the presentation. I have recorded her answers and our conversation as well as a photo of her work. I started by telling her that she could use the counters if needed. At first she declined, but at the riddles became a little more involved or more difficult she relied on the counters to help her see how to write the question.
Her responses and our conversation are in sequence of the riddles
1). Gracie “I have the answer, it’s 43”
How did you get your answer? She looked at her answer and said,
Gracie “Can I write it out?
Me – “Sure.”
She then came up with the answer of 63
2). I thought it was interesting that she divided up all the tens numbers into 12 tens
to add. As you can see she forgot to add the 6 ones and added it at the end to make 406.
3). She said, “I should put the 300 at the top of the 30. Then I add it up.”
4). She had to pause and really think about this question. At first she wanted to add the 22 to the total as 341. She looked at her answer and told me, “I don’t think so.” I asked her why. She asked if she could use the ten-blocks and started counting. She had decided to count out 22 tens and then counted by 10. She then subtracted the 220 from 341 to get 121. She decided that she had 1 hundreds
5). On this one, she struggled to write 13 tens and lined the numbers up wrong and then added. We discussed looking more closely, reread the riddle and she corrected her original problem and answer.
6). She understood the riddle, but when she had to borrow she forgot to borrow from the hundred place as well. This left her with the answer of 185. We talked about looking closely at her math and seeing if there were any mistakes. She quickly caught that she forgot to borrow and came up with the correct answer.
7). She decided she didn’t even know where to start on this problem, and I noticed she was tired by this point. I didn’t force her to finish. I thought it was a great problem to stretch the thinking of the students.
-I think using riddles introduces the student to reading the problem in word form which is an early introduction to solving word problems. This allows for the student to write the numbers down and come up with solutions based on what they have read.
-I also enjoyed reading and hearing about the points made on early learners of numbers. It is important if we are teaching younger aged children that we remember their capabilities and don’t expect them to grasp a concept or activity they aren’t ready to understand. I love the idea that there is an alternative to using base-ten blocks such as a tens frame that gives students a tangible and concrete way to think about counting.
Case Studies:
- Dawn’s case 11 – (1)Why does it make sense to Andrew to have “5 and 10” follow 59? (2) What does he understand? (3) What is he missing?
- Andrew is at the point in number sense where he understand the ones place-value. He still sees the sequential numbers as 59 ones, then the next obvious number would be 50 10 ones. He is at the pre-base-ten understanding. It is defined as ‘unitary because he doesn’t yet understand there are grouping of ten, he only sees the number as units of ones.
- Andrew does understand order of numbers or the composition of numbers and the process of counting on. He understand the count-by-ones approach.
- Andrew does not understand the process of ‘bridging the decade’ or the pattern needed to achieve the numbers following 59. He doesn’t understand the process of moving to another set numbers. Realistically through counting by ones it makes sense to say 59, 510 because he doesn’t understand the next number in sequence or the bridges of ten, 50, 60, 70, etc.
(2) In Danielle’s case 15, the children came up with many ways to write “one hundred ninety-five.” What sense do you see in each one?
– 1095 – understands basic unitary form of counting. There isn’t an understanding of base ten.
-10095- understanding of the basic unitary form of counting. There isn’t an understading of base ten.
-195- the only student who has start to the understanding of place-value and the correct way to write numbers beyond the ones place.
– 1395 and 1295 – understands understands basic unitary form of counting. There isn’t an understanding of base ten. It is possible here the students misunderstood the number the teacher had said considereing they added in the numbers of 2 and 3.
In Muriel’s case 14, the children talk about different kinds of zeros. Explain what they mean by this.
I think that Beth is starting to understand base-ten and place-value. I think she is well beyond the unitary form of counting. She is able to break different numbers down and explain the place value or value of the different numbers. She is really thinking about zero and it’s value when she breaks 30 down by saying that 30 without the 0 would be 3. The same holds true when she explains that the 1 in 15 is a 10 not a 1 – justifiably an understanding of base-ten place-value.
Yessica brings up the fact that she notices that on a calculator the numbers for ones value are written as 07 – she says that when written this way, 07 means simply, 7. She also describes that the 0 in 70 represents 7 tens. Overall, I think the students are being challenged to think beyond basic pre-place-value of numbers and are starting to understand the place-value of base-ten. They are exploring and are able to answer many of their own questions by continuous monologue with the teacher facilitating and group discussion.
Donna’s case 12. Use cubes or counters to do the bean-counting activity Donna describes. What mathematics is highlighted as you do this work?
I think of being able to collect data and grouping the data. The students are using unitary counting by ones
The students can use base-ten counting when grouping by 10s and counting on when counting past the evenly divided counters.
The students are practicing cardinality.
The students are practicing part-part-whole relationships.
Depending on how the students are counting the whole parts, they could be practicing skip counting, counting by 5’s or by 10’s.
The only draw back could be the cost. I know I asked a few teachers if they use these in their classrooms and they didn’t. They did say they would like the link and are interested in learning more. I would think if you are truly interested in getting these in your classroom and your school will not fund them because they use the base-ten blocks, you could research some grants to write for the funding of the blocks.
Place – Value Videos
Why might Cena Be able to write two-digit numerals correctly without being able to explain the meaning of the digits?
1)Cena can write the number correctly but she doesn’t understand the meaning of the digits because she is still in the pre-place-value idea of understanding. She doesn’t yet have a concept of the tens place-value, but does understand the ones place value. Therefore every digit to her can be explained in the ones place-value only. She understands the unitary form of counting. Every tile has a number but only a ones value.
What reasons might there be for the differences in Cena’s performance during the whole class lesson as compared to the individual interview?
2)Assuming the videos of Cena were made at the same time of the academic year (one wasn’t made at the end of the academic year and the last one made at the beginning of the academic year), Cena understanding may be different because she was working with groupings of the same objects rather than non-porprotional objects and she was able to see the placement of the same objects grouped together on the board and circled. She was able to divide that there were 4 groups the same and 9 ones that were not able to be grouped.
How would you describe Jonathan’s understanding of place value as compared to Cena’s understanding?
3)Jonathan understands that the stars on the board can be grouped and counted by 10’s but he has to stop and think how many 10’s are needed to divide the number 24 into 2 tens and 4 ones. This shows he is starting to understand the concept of base-10 place value where as Cena is still using unitary form of counting.
The tens frame counting concept gives students a direct concrete example of the counting process. It does NOT require students have a conceptual understanding of visual or auditory processing. The tens frame allows the student to actually touch and feel each object and see the placement on an outlined area. The student can see when a complete frame is full and count on to another frame.
The ten frame allows for the student to have an actual visualization of how the number is made and how the number can be taken apart or deconstructed. Students can use ten frames to explain the process of addition and subtraction (adding and taking away). They also start the early concept of algebraic function of being able to find the missing addend. They allow the student to visually see the number in which they are looking.
Questions:
- Are Digi blocks something you use where you work? I have never seen them until now and the teachers I tell about them seem very interested, but haven’t heard of them before.
- I know you specifically work with children with special needs. You have mentioned using the ten-frames with them in a previous post. Have you used the base-ten counters with them as well? If so, which ones do you and they prefer?
- What do you think will be the hardest part of introducing base-ten to young students?
- Dr. Higgins showed us a 0-99 chart, and in our readings the chart was a 1-100. Have you ever seen the 0-99 chart? I can honestly say I haven’t. Do you think both would be beneficial to use in the classroom or would it be better to just display one or the other?
- If you teach older students do you think you can use the base-ten manipulatives or do you think you might have resistance since it is something so widely used in the lower grades?