Is it Common Core or is it Good Number Sense? 

Although the criticism of Common Core standards has died down in recent years, it’s still all too common for some to question math lessons or homework problems.  I hope that the information found in this blog entry will make the goals of math learning in Smithfield (and in Rhode Island and the rest of the country!) more clear.

While visiting elementary classrooms this past week, I observed 5th-grade students using base-10 blocks to determine the answer to a division problem as a precurser to learning the standard algorithm. In this activity, the students figure out how many 100-block squares, 10-block strips, and single squares can be distributed evenly.  For example, if you are dividing 768 by 8, how many 100-block squares, 10-block strips, and single squares would each of the 8 groups have? Someone with good number sense can easily tell you that you won’t have any 100-block squares in each of the 8 groupings. You’d need a total of 800 to have one of these squares in each group. The next step is to determine how many 10-block strips you’d need.  You could simply take some of these strips and pass them out to the 8 groups. You’d stop at 9 per group before running out of strips that could be distributed evenly. Some could figure this out even quicker, knowing that 9 x 8 = 72 and if each of the 9 are strips have 10 blocks, this would be 720. There’d be 48 blocks left to distribute so if you know your 8-table, you know that you’d need 6 single blocks per group.

Base-10 Manipulative

Using the standard algorithm, one actually applies exactly the same thinking, except that for many students, going through the algorithm is a robotic exercise with very little thought going into how the algorithm works.  For this reason, the algorithm is taught after exercises with manipulatives and conceptual learning. Parents, though, might question why we don’t simply go directly to the short cuts.  

As adults, there are times when we use standard math algorithms (short cuts) and times when we apply good math sense.  If you were adding ½ cup sugar to ¾ cup sugar, you’d know you’d have a cup and a quarter, not because you found the least common denominator, converted both numbers to equivalent fractions with the least common denominator, then converted to a mixed number and fraction, like this:   1/2     +     3/4     =     2/4   +    3/4    =    5/4    = 1  ¼ 

Instead, you would know because you would “deconstruct” (in your head) ¾ as being ½ + ¼ then you would recognize (grouping!) that ½  + ½ = 1, so the end result would be 1 and ¼. Of course we do want to teach our students the standard algorithm of finding the least common denominator, but we DON’T want them to only know that way! 

Second graders do not work with fractions but, instead, work with whole numbers and grouping 10’s.    For 9 + 4 …. We know that 9 + 1 = 10 and 4 is just made up of 3 + 1 so we’d have one ten and another 3 so the answer would be 13:  9 + 4 = 9 + 1 + 3 = 10 + 3. I know parents know this but if you are doing it automatically, realize that your child isn’t, so we’re working on teaching them that. 

Here’s another real life example:  When adding and subtracting money, we group by dollars (100 cents).  If I added 85 cents to 30 cents, I know in my head that another 15 cents would make the 85 cents a dollar and I also know that 30 cents is 15 cents plus 15 cents, so I’d group the 85 cents with one of the 15 cents to make a dollar, leaving 15 cents.  So, the answer would be a dollar and 15 cents. 

The way this second grade standard is written is: Fluently add and subtract within 20 using mental strategies (counting on, making ten (e.g. 8+6=8+2+4=10+4=14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 =9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6+7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Now, you might say, “who in their right mind would subtract 4 from 13 using 3 steps like the example just provided?????”  Of course, one wouldn’t do this in practice, but as we teach these young students the strategy we want them to do it out the long way.  Once they learn it, it can be done in their head. Through this work, we are building mental math skills and making great mathematicians.

Leave a Reply

Your email address will not be published. Required fields are marked *