Many parents and teachers have complained to me that their children can’t subtract. While most of us managed to get through grade school generally getting subtraction, the whole concept seems to be beyond our children’s abilities.
In the past I’ve blamed Math Investigations and it’s approach to teaching subtraction. But those complaints have continued this year, even with our new instructional resource. In fact, the complaints have gotten louder, with many parents upset that Connects moves through material quickly and doesn’t give kids the time to understand and learn procedures.
This pace, which is leading to frustration and a profound lack of understanding, is intentional. Not because of Connects, but the PWCS Math Department.
When the Math Wars erupted in Prince William County one issue stood out in the fray – the issue of the Standard Algorithms, which representatives of the PWCS Math Department and the creators of Math Investigations persistently and incorrectly call the US or American Algorithms. The Standard Algorithms for Addition and Subtraction are the procedures we all remember from grade school where you stack the numbers by place and then add or subtract by place, carrying or borrowing from other places as needed.
The issue over the Standard Algorithms wasn’t just whether they’re taught to not, but whether they should be taught. The creators of Math Investigations and the PWCS Math Department are of the opinion that the Standard Algorithms are dangerous; that they’re confusing, obscure the meaning of the numbers, and require students to do little more than blindly apply rules they don’t understand. Those aren’t my words, they’re their words (TERC.edu, The Algorithm Issue).
“But our ultimate goal is computational fluency and there are other efficient, accurate algorithms that students understand better and that can be notated in a way that doesn’t obscure the meaning of the numbers. Using these methods, students more easily achieve computational fluency rather than blindly applying poorly understood rules.”
I contend that the Standard Algorithms, if properly taught, are not confusing and require a thorough understanding of place value to properly execute. How else can you carry or borrow, without understanding that 15 is 5 ones and 1 ten, so the 5 ones go in the ones place and the 1 ten gets carried to the tens place?
I’m not alone in holding that opinion as it’s shared by the man who led the committee responsible for writing the Common Core State Standards for Mathematics, with which the VA SOLs are aligned. Initially the CCSS required the Standard Algorithms for Addition and Subtraction in 2nd and 3rd grade; they were pushed back to 4th grade in the final draft. The CCSS’s writers felt that pushing mastery of the Standard Algorithms for Addition and Subtraction to 4th grade wasn’t a significant change as achieving fluency with them and mastery of them by 4th grade would require extensive instruction in them prior to 4th grade. Again, those are their words, not mine, as Dr William McCallum noted in his comments to an article on the CCSS published in The Atlantic, A New Kind of Problem. Dr McCallum is a Math professor at the University of AZ and led the team that wrote the CCSS for Math.
“I also don’t understand why the author believes that the standard algorithm is not an algorithm based on place value and the properties of operation. What exactly is it based on if not those?“
“The intent is to allow the standard algorithm in earlier grades, but not require it until Grade 4. Programs that choose not to mention the standard algorithm at all until Grade 4 will have a responsibility to show that their approach works; that is, that it supports the development of fluency with the standard algorithm in Grade 4.”
So why can’t Johnny subtract and why does the new program, supposedly based on Connects, rush through things without giving kids time to actually learn them?
When the PWC School Board adopted Math Investigations as our Math curriculum, the instructional approach Investigations is based on became the instructional approach used in our classrooms. Because Investigations doesn’t teach the Standard Algorithms, our children have not been taught the Standard Algorithms. Only teachers who bucked the PWCS lesson plans and used alternate materials taught the Standard Algorithms to fluency or mastery.
Math Connects, on the other hand, teaches the Standard Algorithms for Addition and Subtraction, starting in 2nd grade. By 3rd, 4th, and 5th grades, Connects students have already been taught the Standard Algorithms for Addition and Subtraction, so the lessons on adding and subtracting in those units focus on expanding the procedures to larger numbers – NOT learning how the standard algorithms work or why. How they work and why is taught extensively in 2nd grade and reinforced briefly in subsequent grades.
You might think that the lesson pacing for this year would be adapted to include more lessons on how the standard algorithms work and why. After all, there’s no reason why we couldn’t pull the materials introducing the Standard Algorithms for Addition and Subtraction from 2nd grade and expand and use them in 3rd, 4th, and 5th grades. Since our children by and large have not been taught these processes, it would make sense to provide them as much background and understanding as possible.
You might think that, but you’d be wrong.
In 4th grade our children get 3 1/2 days of lessons on the Standard Algorithm for Addition; 3 of those days use Investigations as the instructional resource. Those lessons are – studying the US Algorithm for Addition (MI), Using starter problems to help solve addition problems (MI), Standard Algorithm for Addition (Connects 1/2 day), Comparing and contrasting two different methods of adding (MI 1/2 day), and Solve addition problems two different ways (MI – 1/2 day). Note that the emphasis here isn’t on understanding and becoming fluent with the Standard Algorithm for Addition, which most of our children have not been exposed to. There are no lessons building the Standard Algorithms from stacking and adding without carrying to stacking and adding with carrying. Instead, our kids jump right into a lesson expanding the Standard Algorithm for Addition to the thousands, with carrying, even though they’ve never done more then “study” it, and that “study” has been limited to noting how it’s confusing and obscures place value.
Have any of you noticed that your children constantly try to add from left to right, starting with the largest number first, rather than from right to left, starting with the 1’s first? That’s because that’s how Investigations teaches addition.
It’s even worse with subtraction, which is one of the areas many parents and teachers felt was deficient in Math Investigations. Remember, prior to this year most of our children have not seen and do not understand the Standard Algorithm for Subtraction. Our 4th grade students this year get dumped right into 2 1/2 days of subtracting with the Standard Algorithm, with no lessons building it or explaining it.
Have any of you noticed that your children constantly try add up from the bottom number to the top number when subtracting? That’s because that’s how Investigations teaches subtraction – you add up from the bottom number to the top number (or from the smaller number to the larger number). That’s why situations where they have to borrow is so confusing to them.
There was no reason for this. The school division knew transition would be challenging, especially where the Standard Algorithms for Addition and Subtraction were involved. There’s is no reason why the materials from 2nd grade that introduce, explain, and build the standard algorithms could not have been inserted into the lesson pacing for 3rd, 4th, and 5th grades and expanded to meet the SOL expectations for those grades. But that wasn’t done, and the net effect is that our children are confused and frustrated.