From time to time, we have students complete math portfolio pieces to show evidence of students' understanding of a concept or skill. Recently, students completed an addition strategy piece.
The portfolio piece had two problems, 298+574, and a word problem, Mrs. Shall has 321 shells in her seashell collection. Miss Russell has 524 shells in her seashell collection. How many seashells do Mrs. Shall and Miss Russell have if they combine their collections?
Students used two different strategies to solve. Research shows that when students make a computational error and use the same strategy the second time they solve, they commonly make the same mistake. In addition, students with a tool box of strategies are better able to approach each problem and use the strategy that is most efficient for the given problem.
In some situations, the most efficient strategy is the traditional algorithm, but in others, to use more mental math, it may be compensation or left to right addition.
This is a student's sample from our Addition Portfolio Piece. You'll notice that the student used two different strategies to solve 298+574, decomposing by place value and left to right addition.
Decomposing By Place Value
(200+500) + (90+70) + (8+4)
700 + 160 + 12 = 872
Decomposing by place value is a strategy used by many mathematicians for mental math. The strategy keeps the place value of the numbers, and gives students the opportunity to solve for partial sums by place value. The strategy, in this situation, avoided the traditional regrouping between place values. You can see the student's understanding of correct algebraic notation, too.
Left to Right Addition
Like decomposing, left to right is a strategy that lends itself to mental math, and keeps the place value of numbers in perspective. You'll notice the student's partial sums are recorded by place value, too. Again, the student has avoided the traditional strategy of regrouping that the traditional algorithm demands in this equation.
Compensation / Creating an Equivalent Problem
In the second problem, the student also shows his command of at third strategy, compensation. He solves 321+524 by creating an equivalent problem, 321+524 = 330+ 515.
321 (+9) 330
+ 524 (-9) 515
When compensating, students create an equivalent problem using landmark numbers. Students can easily solve 330+515 without pencil and paper.
Many of the students on this portfolio piece used compensating for the first problem, too, 298+574. Compensating to create the equivalent problem 300+572 makes solving the problem so much easier. This type of flexibility in thinking is exactly what adults with good number sense do on a daily basis.
298 (+2) 300
+ 574 (-2) 572
We don't avoid the traditional algorithm, but we do insist that students correctly explain it when they solve. In the second problem many students used the traditional algorithm. This strategy was very efficient because it did not require any regrouping.
A few students used the traditional algorithm for the first problem which did require regrouping.
When using this strategy students should be able to explain, "Eight plus four equals 12, I regroup 10 ones and create another group of ten. One group of ten, plus 9 groups of ten, plus seven groups of ten equals 18 groups of ten. I make one group of 100 out of 10 groups of 10. One group of 100, plus 2 groups of 100, plus 5 groups of 100, equals 8 groups of 100. My sum is 872."
Students reach this level of abstract math understanding by first exploring other strategies. One of the earliest strategies students explore is the open number line.
Open Number Line
The open number line is a concrete strategy that third grade students commonly revert to, particularly when they get stuck, or have conflicting sums in two different strategies. When using the open number line, we encourage students to start with the largest addend and then add on. We also encourage them to make the fewest jumps possible. One way of jumping on the open number line is provided in this example.
Regardless of a students strategy, we are working toward efficiency, flexibility, and good number sense. We know that exposing them to many strategies will assist them in reaching this goal.