Today I had the pleasure of spending the day at the Northeastern Intermediate Unit at a math training for elementary teachers. The two Math Curriculum Specialists at NEIU, Karla and Leeta, do a great job and today was no exception. There was lots of great discussion on how to structure lessons in ways that make math more rigorous and build conceptual understanding for students.
At one point in the training we were asked to classify different mathematical problems as requiring either "low-level thinking" or "high-level thinking." At the conclusion of this we discussed why we chose the categories for each problem. This conversation concerned me.
Many times other participants reasoned that a problem required high-level thinking for reasons such as these:
- "None of my students could do that"
- "That problem is really hard"
- "There was more than one step"
To explain why they thought problems were low-level thinking problems, these explanations were given:
- "My second graders can do that, so it can't require high level thinking"
- "Everybody in my class could do this"
- "That's something that's taught in the earlier grades"
What worries me about the above comments is that they imply that any multi-step problem makes students think at a higher level, that younger students shouldn't be required to think at higher levels, and that only our best students can reason mathematically. These assumptions are simply false.
False assumption #1 - All multi step problems require higher thinking
It's dangerous to confuse the number of steps in a problem with how rigorous it is. Higher-level thinking is analyzing, synthesizing, and evaluating. A bunch of steps that require nothing more than computation still allow a child to know nothing more than memorized procedures. Little mathematical understanding is required.
We shouldn't confuse how difficult a problem is with how rigorous it is. Students can struggle with problems for a variety of reasons: lack of vocabulary knowledge, reading problems, etc. If you asked me to find the square root of (45x - 3/π), I would have a great deal of difficulty. That doesn't mean that it requires analysis, synthesis, or evaluation.
False assumption #2 - High-level thinking should only be done in the upper grades
If we don't teach our students to think when they are in the lower grades, what makes us think that they will know how to do it when they get older? One of the hardest things for me as a teacher is to get my students used to having to analyze and evaluate their thinking when I ask them "why?" Students should be required to think at all times.
Too often we only give first grade students questions such as, 4+3=?
We should spend more time asking, "How many different ways can you make the number seven?"
The latter question is just as grade-level appropriate, and it allows students to think instead of just manipulating numbers to find the right answer. It allows for learning, and not just memorizing.
Don't think I'm saying that knowing math facts is not important. It is. But understanding math is important, also, and often ignored.
False assumption #3 - Only the smart kids are capable of mathematical reasoning
Again, I think this misunderstanding comes from the confusion between difficulty and rigor. I would think that even struggling 5th grade students would be able to come up with several answers to the second question above with little difficulty. That doesn't change the fact that it requires a different level of thinking than basic recall and/or calculation problems.
Many elementary teachers struggle with math. Many would even admit that they are not great at math. We'll ignore the fact for right now that we'd find an elementary teacher who claims to not be able to pass an 8th grade reading test to be completely incompetent, but an elementary teacher who claims to not be able to do 8th grade math completely normal. I'm sure that will be the subject of a future blog post.
Because of their own background, the way they were taught, and/or their perception that math is about numbers and getting correct answers (it's not), the belief is out there that our "smartest" kids are the only ones who are capable of being good at math.
Math is not about numbers. Being able to multiply numerators and denominators may get you the correct answer to a fraction multiplication problem, but it does not show that you understand what multiplying fractions really is, or what problems you face in your life that may require you to use this skill. Being able to Divide, Multiply, Subtract, and Bring Down may get you the correct answer to a division problem, but it won't show that you know that division is putting items into equal groups.
These memorized procedures aren't math, but in the United States we rarely require more than that from our students. This was shown pretty clearly in the 2007 TIMMS study in which math and science teaching and learning were examined in countries around the globe. The kids who aren't good at memorizing these procedures that we often teach out of context and without relevance are not bad at math.
All students are capable of higher-level thinking. Some are capable of doing this higher-level thinking with more difficult problems, but all students should be required to think.
The alternative is that we develop a generation of students who can't.
Photo Credit - flickr.com: Sidereal