Are you as smart as a 3rd grader? As was mentioned in the previous post, many of us did not do well in high school algebra class. One reason I suggested in that post was because we were taught rules and shortcuts rather than reasoning. Further, algebra was abstract for us, 2D at best. Today’s children attending up-to-date schools begin learning equalities and number sense through the use of math manipulatives.
One of the most important manipulative is the balance shown below.
Students attach blue rectangles to both sides so that the yellow stick ‘balances.’ Even kindergarten students can solve an equation like 3x = 9. Note that earlier, the students explored ways to balance 9 using combinations of two blues on the other side. Today they begin by placing one blue rectangle on 9. Then the teacher tells them to pick up 3 more blues and put all three on the other side on the SAME peg. Of course, they aren’t given 3x = 9 as the problem to solve, but they solve it nonetheless.
The state of Ohio has adopted the common core state standards for teaching as have 45 of 50 states. These standards challenge administrators and teachers to focus on core conceptual understandings and procedures rather than on rote learning. Fortunately because of retirements, newer public school teachers are moving away from rote paper and pencil mathematics and have been trained in both the new standards as well as manipulative mathematics.
Here is a problem for us to solve: Mike has some blocks then finds 5 more. Now he has 12. Write an algebraic expression for this problem.
Did you remember your Algebra 1? Which of these did you write?
a. x + 5 = 12
b. x + 12 = 5
c. x – 5 = 12
Children in the 3rd grade in Ohio had this question on their ‘proficiency’ test and are expected to solve it.
If these 3rd graders had been using the balance for the past 4 years, then they could visually imagine the 12 on one side and one blue at 5 and another [unknown] blue on that same side. Was it that easy for us oldies?
How about another challenge. 18 + y < 22 y = ___
Here it is in 3rd grade representation:
Then there is this problem: x + 5 = y. Make an x, y table of 3 values of each unknown so that the statement is true.
The Ohio 3rd grade problem:
Beyond the physical balance, the student can use the virtual balance on the computer to solve equality problems. One such program allows students to choose increasingly more complex problems for personal enrichment such as the example below.
I’ll let you find x. To use this virtual program, the student drags four x’s to the left and one x onto the right. Then he/she places one 1 atop the 4 left x’s and seven 1’s atop the right x.
Do you remember your Algebra 1 teacher telling you to ‘cross out equal amounts on the left and right side of the equal sign?’ In fact, at this stop in the virtual problem, the student puts back equal cubes and blocks on the right and left. The next frame shows the result.
At this point the more mathematically astute student may jump to the answer by noting a 1:2 relationship of x to 1’s. If not the student removes one x and notes the imbalance; it re-balances only when two 1’s are removed.
Enough algebra for today. If nothing else, at least some old neurons were fired up while reading this post.