September 27th, 2011
Our math quick tips continue on this sunny Tuesday with another number sense routine by Jessica Shumway, exclusive to the Stenhouse blog. Her new book, Number Sense Routines, is still available for full preview on the Stenhouse website!
Using Algebra and Arithmetic Routines to Improve Number Sense
What goes in the blank?
Pose this problem to your students. Many of them will write 14 in the blank. Some will add 7+7+8 and put 22 in the blank. Others say that the equation is impossible. Some might answer 6.
During a Cognitively Guided Instruction training, my math coach Debbie Gates challenged me to present this problem to some fifth grade students. I was surprised that many of them wrote 14 or 22 in the blank. Some of them wrote 7+7= 14+8=22. Through their elementary school years many of these fifth grade students had developed misconceptions about the equal sign and what equality means.
A couple of years later I was part of a study group that read Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School by Carpenter, Franke, and Levi. It really got me thinking deeply about students’ misconceptions about the equal sign as well as the critical importance of encouraging students to think relationally. I saw these factors as critical to my students’ number sense development, so of course, we made it a Number Sense Routine in our classroom.
At the beginning of the school year, I simply started with a series of True/False Statements that Carpenter el al. suggested in their book (see page 16 of Thinking Mathematically):
I wrote these equations on the board (one at a time), and the students discussed whether each statement is true or false and, of course, explained how they know. This was the beginning of our conversation about equality and what the equal sign means. Many students believed that the second and third equations were false! I recorded students’ ideas about equality during the course of our discussion:
We did similar series of equations like these over the next two weeks. Other examples include:
True or False?
True or False?
True or False?
As we worked through these daily routines, we continued our discussions about equality and the meaning of the equal sign, but students also began to dive into important ideas like the commutative property of addition and multiplication, how to compose and decompose amounts, and relationships among each side of the equal sign (relational thinking). This was exciting!
At this point, I began using equations like these:
My students agreed that the first equation is false. Some solved for both sides and said that 20 does not equal 19. One student said, “I knew right away that it was false because there is a three on both sides, but 16 is one less than 17.” This student was thinking relationally—this is a critical component to algebraic reasoning! Additionally, the student was using her number sense and looking at relationships among the numbers. She actually did not need to solve for both sides, rather understanding the relationships on each side of the equation helped her figure out that the statement is false. Her explanation of her reasoning helped other students look at the other two problems in the same manner.
Later, after changing the routine from True/False Statements to Open Number Sentences, I had students come up with their own Open Number Sentences that I could use for future Number Sense Routines:
Luis’s Open Number Sentence encourages his peers to use relational thinking rather than solve for both sides. Students who use relational thinking will likely use a compensation strategy, which is a strong mental math strategy.
These routines helped my students negotiate meaning around the equal sign, dispel misconceptions about the equal sign, learn to think relationally, and use important arithmetic and mental math strategies. Through our various conversations each day for several weeks, I watched my students’ number sense flourish. I found these algebra routines to be extremely effective! The students immediately get sucked into the discussions and lose themselves (and find algebraic thinking!) in debate and negotiations about the mathematics. Additionally, a focus on the symbolic (the equations) was a nice addition to our visually focused dot cards and ten-frames routines. At many points in our discussions, I found it helpful to students to link the dot cards with our True/False Statements. This routine is extremely rich and can be adapted and enriched in a number of ways!