Mathematics Update
January, 2011
First and Second Grades
Brenda Hartshorn
Dear Families,
I’m still working with Shutterfly in order to be able to add email addresses to the math website I've put together back in December. Unfortunately, I'm unable to add any email addresses to this site, and the folks at Shutterfly have not called me back after several calls to them, and speaking to a live person. Seems to be a technical problem they are unsure how to correct.
Soooo…here's an update for our math lessons this month. I'm sorry I'm able to send you photos of the math “in action,” as your children have been engaged in a variety of measurement, and number sense activities. If I can get Shutterfly site to take new email addresses, I'll be sure to let you know.
Second Grade Mathematics:
Double-digit addition and subtraction word problems have been our focus for much of this month. In order for young mathematicians to understand how to go about solving these problems, they need to understand the action and the "story" behind the numbers. For example, "Suzy collected seashells. She had 47 shells on Monday. On Wednesday she went to the beach and gathered 35 new shells. How many shells does she have now? Questions asked are: Is this a putting together or adding problem? Or is it a taking apart or subtraction problem? How would you set this problem up as an expression with the plus or minus sign? 35-47, 47-35, 47+35 or 35+47? Is there more than one expression that fits this story problem? About what will the answer be? (This requires some estimation work and an understanding of place value and how a two-digit number can be broken up into groups of tens and single ones) About 40+35 or about 50+30.
Many of us adults learned the standard algorithm for adding and subtraction two-digit numbers. We learned the terms "carrying" and "borrowing" for regrouping tens and ones as we solved computation problems that were set up in a vertical manner, as follows.
Now, there is a place for this algorithm, but it's not in first or second grade. Children need to first learn and have a solid understanding of place value and operation, such as addition or subtraction in conjunction with place value. Place value involves knowing that 36 has two digits: a 3 and a 6, but that these digits do not have single digit value. Their value is based on their "place" in the number. The 3 is in the tens place, so it symbolizes 3 tens, which we call 30. The 6 resembles 6 single ones. Most second graders are still learning that the 3 is really called a 30 or 3 tens, and should not be called a 3—otherwise this can lead to misunderstanding 36 as a total of 9, and other misunderstandings once the operations of addition or subtraction begin to happen.
So, the algorithm many of us adults learned to use to get the "right" answer, didn't always make sense to us. We merely memorized a procedure that was intended for us to get the correct sum or difference, but did not help us to develop number sense and an understanding of place value. As a matter of fact, the language used to teach us this algorithm created more misunderstandings for many people. An example goes like this: "You have 47-29, you take the 9 away from the 7, but it's not large enough, so you go over to the 4 and you cross it out and make it a 3. Then you add a little 1 to the 7, and you have 17 so now you can take 9 away from 17, which gives you 8, and you put that down here. Then you go back up and take the 2 away from the 3 that you just made, and you get a 1 and you put it down her next to the 8, so you have 18." This method fails to help children truly understand that the 40 was transposed into a 30+10 next to the 7, so that it was really 47=30+17 so that 30-20 and 17-9 would result in the correct answer.
Now, students will be introduced to standard algorithms for addition and subtraction by fourth grade, but only after it has been determined that a child truly understands place value and why this algorithm works and can be used, but still needs to be used in conjunction with number sense so that the "reasonableness" of an answer can be determined.
When your child is asked to solve a two-digit addition or subtraction problem at school or at home, these are the kinds of paper and pencil thinking we are learning to show:
34+17
30+10=40
4+7=11
40+11=51
OR
34+10=44
44+7 is like 44+6+1
44+6 gets me to the next decade of 50
50+1=51
34-17
30-10=20
20-7+13
13+4=17
OR
34-10=24
24-7 is the same as 24-4-3, so
24-4=20
20-3=17
This may be a bit confusing to us as adults, but your children have been learning to make meaning of double-digit numbers and how they can be pulled apart and put back together in ways that make sense and use what they know about their addition and subtractions facts up to 10 and up to 20.
When your child brings math homework home requiring addition or subtraction of double-digit numbers through word or story problems, please encourage him or her to talk about what is happening to the numbers, setting up the expression on paper, and then showing the steps in solving it, such as breaking numbers down into groups of tens and ones, adding on or subtracting by tens, getting to the next decade, etc.
Examples are as follows: 46+55
Breaking a number into tens and ones: 40+6 or 10+10+10+10+6 or
20+20+6 or 30+10+6
And 50+5 Or 10+10+10+10+10+5 or
20+30+5 or 40+10+5
So 40+50=90 and 6+5 = 5+5+1 = 11
90+11=101
Adding or subtracting by tens (Also known as Skip counting by tens or Jump Ten):
46+55
46, 56, 66, 76, 86, 96 96 +5 is 100 and 1 more
or 101
Getting to next decade: 46+55 46+4 more gets to 50, so then 50+51=101
If you're interested in reading more about teaching or not teaching algorithms to young children, I'll include below a summary of a research article that has been a catalyst for further research, which has resulted in similar findings. It is titled, "To Teach or Not to Teach Algorithms."
First Grade Mathematics:
Much of this month's earlier work was learning about linear measurement, using both standard and non-standard units. Cubes, tiles, paperclips and inches were used to measure lengths of several objects and tape lines set up around the classroom. Careful and accurate measuring, recording these measurements, along with comparing different lengths measured were all part of our work together.
Most recently we've been working with several activities and games for learning skip counting patterns for twos, (both even and odd numbers: 2, 4, 6, 8, etc and 1, 3, 5, 7, etc.), counting on by 5's past 100, and counting on by 10's past 100. Counting backwards by ones has also been a big part of our daily counting practice.
In addition to counting forward and backward counting patterns, we've been spending a lot of time with games and activities that are helping us learn all the addition and subtraction combinations up to 10. By the end of first grade, students are expected to know all combinations of two addends to equal 10, such as 0+10, 1+9, 2+8, 3+7, 4+6 and 5+5. They also are expected to know other combinations for all numbers less than 10, such as 0+7, 1+6, 2+5, 3+4 for 7. Subtraction problems expected to be learned and understood without the use of fingers or "counting on" or "counting back," include 2-1, 4-2, 4-3, 10-5, 10-2, etc.
As always, if you have any questions or comments, please contact me at bhart@madriver.com or call me at 496-3742, ext. 122.
"To Teach or Not to Teach Algorithms"
A summary of research originally presented by Constance Kamii and re-explored by several individuals since first published.
Students were asked to solve the following problem, which was presented horizontally, 6+35+185. All of the students were told to answer the question without paper and pencil. The results were as follows:
Three classes of second graders were looked at.
Class 1: All students had been taught the old-fashioned carrying algorithm for addition—12% got the addition problem correct.
Class 2: Students had not been taught the algorithm in school, but some had seen it at home---26% got the question right.
Class 3: All students had not been exposed to the algorithm at all---45% of the students in this group answered the question correctly.
Three different classed of third graders were also asked to solve this math problem without the use of paper or pencil.
Classes 1 and 2: All students in both classes had been taught the standard algorithm or carrying or regrouping—32% of one class and 20% of the other class got the correct answer.
Class 3: None of the children in this class had been exposed to the algorithm at school or at home—50% of the students got the correct answer.
Four different grade four classes were asked to answer the same addition question without using pencil or paper. All of the students in these four classrooms had been taught the standard algorithm for double-digit addition problems. In each of these classes, 30% or fewer students gave the correct answer.
There was also quite a lot of concern about the incorrect answers that were given by students in the classes that had been taught algorithms. Especially, in the grade two class. Many of the students who had learned algorithms were very far from the correct answer, suggesting that the students were simply focusing on using the algorithm and did not really understand place value.
It's important to note again that 30% of the fourth graders were able to answer the problem correctly, and all had been taught algorithms. On the other hand, 45% of the students in grade two that had never been exposed to algorithms were able to answer the question correctly.
Conclusions reached by the original researchers were:
" Algorithms are harmful to children's development of numerical reasoning for two reasons: (a) they "unteach" place value and discourage children from developing number sense, and (b) they force children to give up on their own thinking."
Now, "harmful," is a pretty strong word, and I'm not sure I can agree with that 100%, but I do think that teaching children algorithms as "short cuts" to get the right answer can lead to confusion about the "wholeness" of a double digit number, impede estimation skills from developing, and often stops children from "thinking mathematically" about the reasonableness of an answer, or finding more than one way to prove or check his/her math work.
