A few weeks ago, I wrote an article for TheAtlantic.com describing some of the problems with how math is currently being taught. Specifically, some math programs strive to teach students to think like «little mathematicians» before giving them the analytic tools they need to actually solve problems.
Some of us had hoped the situation would improve this school year, as 45 states and the District Columbia adopted the new Common Core Standards. But here are two discouraging emails I received recently. The first was from a parent:
They implemented Common Core this year in our school system in Tennessee. I have a third grader who loved math and got A’s in math until this year, where he struggles to get a C. He struggles with «explaining» how he got his answer after using «mental math.» In fact, I had no idea how to explain it! It’s math 2+2=4. I can’t explain it, it just is.
The second email came from a teacher in another state:
I am teaching the traditional algorithm this year to my third graders, but was told next year with Common Core I will not be allowed to. They should use mental math, and other strategies, to add. Crazy! I am so outraged that I have decided my child is NOT going to public schools until Common Core falls flat.
So just what are the Common Core Standards for math? They are a set of guidelines written for both math and English language arts under the auspices of National Governors Association and the Council of Chief State School Officers. Where they are adopted, the Common Core standards will replace state standards in these subject areas, establishing more common ground for schools nationwide.
To read newspaper coverage of the new standards, you’d think they were raising the bar for math proficiency, not lowering it. «More is expected of the students,» one article declares. «While they still have to memorize or have fluency in key math functions and do the math with speed and accuracy, they will have to demonstrate a deeper understanding of key concepts before moving on.»
But what does this mean in practice? Another recent article explains, «This curriculum puts an emphasis on critical thinking, rather than memorization, and collaborative learning.» In other words, instead of simply teaching multiplication tables, schools are adopting «an ‘inquiry method’ of learning, in which children are supposed to discover the knowledge for themselves.» An educator quoted in the article admits that this approach could be frustrating for students: «Yes. Solving a problem is not easy. Learning is not easy.»
With 100 pages of explicit instruction about what should be taught and when, one would expect the Common Core Standards to make problem-solving easier. Instead, one father quoted in the aforementioned article complains, «For the first time, I have three children who are struggling in math.» Why?
Let’s look first at the 97 pages of what are called «Content Standards.» Many of these standards require that students to be able to explain why a particular procedure works. It’s not enough for a student to be able to divide one fraction by another. He or she must also «use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9, because 3/4 of 8/9 is 2/3.»
It’s an odd pedagogical agenda, based on a belief that conceptual understanding must come before practical skills can be mastered. As this thinking goes, students must be able to explain the «why» of a procedure. Otherwise, solving a math problem becomes a «mere calculation» and the student is viewed as not having true understanding.
This approach not only complicates the simplest of math problems; it also leads to delays. Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them «understand» the conceptual underpinnings.
This brings us now to the final three pages of the 100-page document, called «Standards for Mathematical Practice.» While this discussion is short, the points it includes are often the focus of webinars and seminars on the new Common Core methods:
- Make sense of problem solving and persevere in solving them
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated reasoning
These guidelines seem reasonable enough. But on closer inspection, these things are essentially habits of mind that ought to develop naturally as a student learns to do actual math. For example, there’s nothing wrong with the first point: «Make sense of problem solving and persevering in solving them.» But these standards are being interpreted to mean that students «make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution.»
This is a rather high expectation for students in K- 6. True habits of mind develop with time and maturity. An algebra student, for instance, can take a theoretical scenario such as «John is 2 times as old as Jill will be in 3 years» and express it in mathematical symbols. In lower grades, this kind of connection between numbers and ideas is very hard to make. The Common Core standards seem to presume that even very young students can, and should, learn to make sophisticated leaps in reasoning, like little children dressing in their parents’ clothes.
As the Common Core makes its way into real-life classrooms, I hope teachers are able to adjust its guidelines as they fit. I hope, for instance, that teachers will still be allowed to introduce the standard method for addition and subtraction in second grade rather than waiting until fourth. I also hope that teachers who favor direct instruction over an inquiry-based approach will be given this freedom.
Unfortunately, the emails and newspaper articles I’ve been seeing may herald a new era where more and more students are given a flimsy make-believe version of mathematics, without the ability to solve actual math problems. After all, where the Common Core goes, textbook publishers are probably not too far behind.