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Two Digit Addition and Subtraction
Question
I have had questions from multiple teachers and I am curious myself as to why students who touch the model, regardless of how they respond to the questions or the strategy used, score an "N". Can you explain to my why touching the model automatically warrants "Needs a Prerequisite"? I have provided an example below. The student knew the tens and ones that resulted from 33-14 and could put them together as a single number and their strategy was efficient (knows parts to break up tens). Please help me understand how to respond to this question. I appreciate your time.
Answer
from Kathy Richardson
I am so glad you let me know this is an issue for your teachers.
Let me try to explain why touching or moving the model is so important.
First of all, if the child touches/moves the model for one of the problems but not the other, they would get an "I". But if they need to touch it both times, it appears they are at a very basic level of needing the model to get answers. A child who knows parts should not need the model to figure anything out. I am wondering if the children who know a lot about numbers are using the model, not because they need it, but because they want to be sure they are right. Perhaps the teachers need to interrupt the children if they begin to move the model to ask them to see if they can do it without the model. If they know the parts and understand what is going on they shouldn't need the model.
We are trying to move children to the place where they can work with symbolic problems. The models are supposed to help them understand what is going on when they add and subtract but not to be used as counters so they can go through the motions without thinking about what is happening. The concrete models should help children develop more and more efficient strategies as they learn to take numbers apart and as they recognize particular relationships among the numbers. The models need to be aids to thinking, not tools for getting answers or to demonstrate memorized procedures. Children should become less and less dependent on the use of the models for any particular concept, ultimately not needing the model at all. There are 4 stages of using models that children move through.
Could you check back with some of the teachers to see if they think the children really know what is going on but like the security of using the models? Or do they think the children are using the models without thinking much about what they are doing thus staying dependent on the model?
I would love to hear more about what they think is going on with the children.
Question
I am confused between the top three choices we are given to choose from for the strategies they used. The choices are:
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Knows parts to make/add tens
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Counts to make/add tens
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Visualizes written problem
What kind of answer would get the top strategy? I keep thinking that they have to know parts to ten (for instance, in 28 + 16, 8 + 2 more gives me ten and 6 ones are left over), but my fellow teachers are saying that if they know to combine the tens and then combine the ones and then regroup then I could choose the first strategy.
Here is an example of what most of my kids tell me when they explain:
(28+16)
First, I added 20 plus 10 to get 30. Then I added 8 + 6 to get 14. I can't have 14 ones, so I added the ten from 14 to the 30 to get 40 and then I have four ones left over so the answer is 44.
Is this kind of answer worthy for the top strategy choice?
As for the second strategy (counts to make/add tens) I am just not sure what the difference is between that one and the first one.
Answer
from Kathy Richardson
First, I want to give you some information about the thinking behind the assessment that I think will help you interpret what the kids know.
First of all, this assessment is different from most other assessments of 2-digit addition and subtraction because we are looking for the level of understanding of the underlying mathematics rather than the ability to get right answers. We are looking to see if the children think of numbers as tens and ones and use that knowledge to arrive at the answer.
-Knows parts to make/add tens
The child is making and adding tens and knows the combinations used. (So if they added 8 and 6, they would know the answer is 14 without needing to count. Or if they made a ten they would know they needed 2 to make a ten and 4 ones would be left over.)
-Counts to make/add tens
The child is making and adding tens but needs to figure out the combinations used. (For example, they might count on to add 8 + 6 or if they took 2 of the 6 to make a ten, they would need to figure out that 4 ones would be left over)
The assessment does not distinguish between different ways kids use tens and ones to add.
So kids might add the tens and then the ones and then add the 30 and the 14.
They might add ten to 28 to get 38 and then break up the 6 to get 40 + 4.
They might pretend that 28 is 30 and add 16 to get 46 and then take off the 2 they added on to change 28 to 30.
All of these would be the first or second choice depending on whether they knew the "facts" or not. You are right that adding the tens and then the ones is not as sophisticated as breaking up the 6 to make a ten would be. I would make a note of that. Then, I would challenge the kids to see if they could find a way to add the 16 without breaking up the 28. Some won't be there yet so it won't be a requirement. I would just say, "I am wondering if anyone can figure out a way to add the 16 to the 28 without breaking up the 28." I would have them all build the 2 tens and 8 ones and see what they can come up with.
-Visualizes written problem
The child is not thinking about the model in front of them but is thinking about the symbols as though the problem were written down.
What concerns me most about the response you say is typical for your students is that it seems like the kids are thinking about "doing a problem" rather than about combining tens.
Do you think they are visualizing a written problem when they answer or are they just using the language they are used to using in the classroom?
It is very subtle, but I think it would be easier to tell they were thinking about groups of tens if they said:
First, I added 2 tens and 1 more ten and I got 3 tens. Then I added the 8 and the 6 and I got 14. Then I took the ten out of the 14 and put it with the other tens and that made 4 tens and I had 4 left so that makes 4 tens and 4 left over. That makes 44.
Remember, the assessment is intended to give us information so we can provide kids with the instruction they need to move forward. It is not about judging you or the children. As you give the assessments, try to think about "What do they need from me?"
I hope this helped. I really love getting questions from teachers. Please feel free to send me any other questions that come up.
Question
I am using AMC Anywhere. I completed Part 2 of Assessment 9, Two-Digit Addition and Subtraction with a student, and they received an "N". Do I go back to do another assessment or go back to Part 1 of the assessment if Part 2 was given first? Ex: I assessed using Part 2 of Two-Digit Addition and Subtraction. Do I go back to Assessment 8, Grouping Tens, or Part 1 of Two-Digit Addition and Subtraction?
Answer
from Kathy Richardson
If a student gets an "N" on any assessment, it means they need a prerequisite because they show no understanding of the concepts in that assessment. In the case of Two-Digit, it is appropriate to go back to Grouping Tens to see if there is any understanding of tens and ones.
Question
I have a question about AMC Assessment 9, Two Digit Addition and Subtraction. I am struggling to find the connection between the student responses from the interview and the recommended stations. In the other eight AMC assessments it is easy for me to see where a child's instructional level is an I or a P what stations to use based on the Linking Assessment and Instruction section of the AMC Concept book. Those connects are not so easy for me to make in AMC Assessment 9. For example, on Part One of Student Interview, the heading in the Summarizing Instructional Needs is Adding Up Tens, With Model, and Relationships/Procedures. When I look on pages 62-65 of the book, trying to find stations to connect, it lists Forming and Counting Groups (Instruct and Practice), Two-Digit Addition and Subtraction (Practice), Comparative Subtraction (Practice), and Combining and Separating 1 Ten and Some More (Practice). I do not see experiences that are recommended from the Assessment 9 book that are instructing and/or practicing the critical learning phases that are being asked in the interview. HELP!
Answer
from Kathy Richardson
I hope the following information will help make things clearer.
At this level, children need to apply what they have learned about parts of numbers, number relationships, and numbers as tens and ones to solve problems. It is really integrating and using what they have learned thus far.
I think the Goals in this case are as important as the Critical Learning Phases when planning instruction.
Goals
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To use what is known about single digit numbers to add and subtract two-digit numbers
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To use what is known about numbers as tens and ones to add and subtract two-digit number
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To describe how they solve problems
Critical Learning Phases
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Tells how many to make the next ten
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Combines numbers by reorganizing into tens and leftovers when necessary
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Breaks apart tens when necessary and reorganizes them into tens and ones
Children solve problems in the following ways:
I • Counts all or on. Children are thinking of numbers as a collection of ones instead of as groups of tens and leftover ones.
f they count all- they need more work with learning about tens and ones. Go back to Grouping Games and Organizing into Tens and Ones. They need to practice organizing into tens and ones until they can make tens and know the total number of tens without counting.
P • Uses tens and ones when solving problems but they need to figure out the combinations needed to make tens and leftovers and do not easily keep in mind the number of tens they have formed or have left.
If they have to figure out the number needed to make a ten or need to count by tens, or need the model, they need to practice adding and subtracting. (p. 64-65) If they really struggle, they can practice using 1 ten and some more as listed on p. 65).
They need to practice adding and subtracting using models until they can do the problems without models. And can apply what they know to problems presented symbolically.
A• Adds and Subtracts using tens and ones without needing to figure out the combinations or totals and without needing a model.
AMC Two-Digit Addition/Subtraction assesses the following levels:
1. Making Tens: Going Back( + 20 and + 12) , See p. 61 - 62
I• Needs instruction
Counts all or on
Need experiences learning to form and count groups
P• Needs practice
Figures out tens or counts by tens (See p. 63)
Needs experiences organizing into tens and ones
A• Ready to apply
Adds with ease
Move on - and provide instruction according to what they did on the rest of the assessment
2. Making/ Breaking Tens Using Models
I• Needs instruction
Counts all or on
Needs experiences organizing into tens and ones (see p. 63)
P• Needs practice
Figures out tens or counts by tens (See p. 64)
Needs experiences adding and subtracting 2 digit numbers
A• Ready to apply
Adds with ease
Move on - if need models, continue practice to point don't need
models (See p. 64).
Also go on to Comparative Subtraction (See p. 65).
Making/Breaking Tens - Presented Symbolically
Children need to apply what they know about adding and subtracting tens when using models to what they do with symbols.