Category Archives: Thoughts on Learning

Why School Districts’ Adoption of “Common Core Curricula” is not Common Core

This year, our school district adopted a new math curriculum.  According to several of the teachers who served on the adoption committee, the chosen materials were head and shoulders above and beyond what other curricula were providing to meet the mathematical demands of the new Common Core.

When I received the materials, I looked forward to a program that could satisfy the Common Core’s demands that kids problem solve, persevere, seek short cuts, and critique mathematical ideas.  But during the trainings, and in several communications from our principal and data team leaders, we received the expectation that we teach the program “with fidelity.”  This included an expectation that we teach the same material on the same day as the other grade level teachers in our school.  Ironically, I was even asked to skip an entire module so that I could begin the next unit in tandem with my fellow fourth grade teachers.

Now set aside your concerns that skipping modules is the antithesis of “fidelity”.  Set aside your doubts about the district, who in recent years has provided training on formative assessment and how that guides differentiated instruction.

The point that I am trying to make is that the way our district adopted this “Common Core” curriculum was decidedly not aligned with the values and skills the common core is designed to teach.  First of all, the Common Core asks students to “use tools and make strategies.”  The district is implicitly asking teachers not to create any new tools for student measurement and not to use other teacher strategies they may have picked up in their years or decades of teaching service.

Secondly, students are asked to “look for shortcuts”.  Certainly, there have been several lessons where I think to myself, “Why are you teaching this if these students, who are clearly bored, already get this?”  The answer is because the district wants standardized adoption and the principal has insisted that we teach in synch with our colleagues, regardless of the needs and abilities of our students.

Finally, the Common Core asks kids to “make sense”, to “argue”, and to “critique reasons.”  In this vein, when my colleagues and I have tried to make sense of these demands, we scratch our heads.  When we critique these policies, when we argue our points, we are rebuffed.  In other words, math is a dialogue, not a dictate.

As parents, I hope that you understand that the Common Core asks our students to be able to do some pretty sophisticated and amazing things.  All I am asking from our district is that teachers be allowed the latitude, flexibility, creativity, and autonomy to teach in ways that students need.

Math Anxiety and Fluency

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Math anxiety is real and it affects about half of the U.S. population. Brain imaging has shown us that stress gets in the way of working memory, which is typically required to do mathematics. In fact, math anxiety seems to affect people with a high amount of working memory most! One of the main sources of math anxiety is the stress produced by timed tests (Boaler, Education Week, 7/3/2012).

I used to have my students take timed multiplication and division tests to practice fact (computational) fluency, but I don’t do this anymore. Time is too heavily factored into the measure of fluency in that equation. Rather, I have students learn and practice strategies for working with numbers so that they can be fluent.

Here’s a tip for developing the habit of seeing patterns in numbers: spend more time looking for patterns in math homework and throughout school mathematics. School math is designed to illustrate patterns! Though this is sometimes a weakness in that a framework is often given and problems are explicitly trying to illustrate relationships, this will help students (who attend to these patterns) develop strategic thinking.

Take for example two problems from last night’s homework.

  1. Kaden divided 96 by 12 and then multiplied by 6. Write an expression to show the problem. Then, solve the problem and write an equation.

When most students first approach a problem like this, they begin to decode or solve it step by step. First they do 96 divided by 12 getting… what is that anyway? And then they multiply by 6.

When I look at that problem, the first thing I notice is that 96 is being divided and multiplied. These are inverse operations – meaning that they undo each other. If the problem was 96 divided by 12 times 12, I hope we would just leave it at 96 because we all know this doing and undoing relationship. In this case, the situation is only slightly less humorous (mathematically speaking). If we are going to divide 96 by 12 and then multiply by 6, then we are undoing the dividing by 6! This leaves a much simpler expression: 96 / 2 (96 divided by 2). The equation I would write would read 96 / 12 * 6 = 96 /2 = 48. That’s number fluency.

  1. 81 – (9 * 7)

Here too, most of my students given this for homework followed the “proper steps” and dutifully multiplied 9 * 7 and then subtracted that from 81. This misses a beautiful relationship embedded in the problem. If you notice that 81 is also a multiple of 9, you can rewrite the problem in the following way.

81 – (9 * 7) = (9 * 9) – (9 * 7)

We are subtracting 7 groups of 9 from 9 groups of 9. The difference is 9 * 2.

When most people think about what mathematics is, they think about memorization and procedures. I think this is what it felt like coming through the system for a lot of people in my generation. Today, in many classrooms around the world, math is taught around understanding ideas, discovering relationships, exploring patterns, and developing strategic thinking and problem solving skills.

The difference between memorizing a bunch of rules and procedures and coming to understand mathematical ideas and principles is profound. Memorization requires no real reflection or ownership. There is often very little connection between the learner and the material. As a result, the learning is shallow and has very little application outside of school for that student. Teaching number fluency with conceptual understanding on the other hand requires more engagement with the material. It challenges students to come to terms with a deeply interconnected and vibrant discipline. I believe it also opens the door to a real relationship with mathematics that can help students avoid math anxiety.

Mathematics is sometimes called the science of pattern and order. This is exactly what draws people to math – let’s keep that perception of the subject alive when we are actually doing math work.

Math Smarts

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“Teaching is mostly listening, learning is mostly telling.”
-Deborah Meier

As we dive in to the new school year, those of us in the Olympia School district will be meeting our new math curriculum, Bridges, for the first time.  Many parents will be trying to decode the words “Common Core” in the context of this new way of teaching.  The most difficult part for many will be allowing their children to struggle with (and through) the math.  This will require us to redefine the job of teachers.  This will also force us to redefine what “math smarts” are.  Even though this may be challenging, the upside is that more kids will have a stronger understanding of math concepts and more will identify themselves as mathematicians.

The job of teachers is shifting.  No longer are we the “Holders of Knowledge” whose purpose is bestowing computation fluency upon our pupils through direct instruction.  Actually, good teaching requires just the opposite: teachers are no longer the center of the classroom.  Good teachers coach their classes to work collaboratively, provide group worthy tasks that help them access key mathematical concepts, and then retreat to observe and assess, intervening only when necessary.

The idea of being “math smart” is now outdated.  First of all, as Carol Dweck points out, it leads to a fixed mindset.  If people say that Johnny has math smarts, it leads him to believe that it is an innate power that he has always had.  You either have it, or you don’t.  Most people think they don’t.  Also, many people mistakenly place far too much emphasis on getting correct answers in math, especially in computation.  This definition must now change to show that math is a verb.  It is something that we do.  The list  of math skills must now include listening, communicating clearly, helping others, asking good questions, improving reading skills, flexible thinking, struggling, persisting, and persevering.

In our new era of “Common Core” and Bridges, the most important thing that parents can do is to avoid telling kids how to do their math.  Rather, model the curiosity and inquiry that we are asking of our kids.  Please, resist the urge to show your kids the shortcuts.  No more “this is the way I learned to do it when I was a kid.”  Instead, take the time to just listen to your kid explain their thinking.  Ask them “why?”  Who knows, maybe you’ll learn something, too.

Sources:
Carol Dweck, Mindset: The New Psychology of Success.
Jo Boaler, How to Teach Maths.
Featherstone et al., Smarter Together.

Blue Brain

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Brain research now confirms that deep conceptual learning requires making mistakes, and that the more positive we feel about mistakes, the more we can learn.  That’s because brain freeze, or what my sixth graders have started calling “blue brain” is real.  Many of my students have such an aversion to making mistakes that instead of digging into new ideas, they become paralyzed, hoping not to make a miss-step instead of trying out new ways of thinking.  I can see why. For sixth graders new to middle school the stakes are high: new friends, lockers (and combination locks), crowded, confusing hallways, five or six teachers, lunchroom rules, free time at lunch but no recess, PE and band lockers, binders, dividers and pencil pouches.  It’s a lot to keep track of and they don’t want to miss out on anything or be the focus of negative attention.  Some of them try so hard to get it all right that they have a hard time doing what they are at school for to begin with:  learning.

This year, about one third of my students tell themselves they are bad a math if they make a mistake, and more than half feel negatively about math mistakes.  If my students are going to grow to the greatest potential they need to break out of this mindset. They need to change their minds about mistakes and learn to see them as learning opportunities, even though the pressure they feel to succeed works to convince them of the opposite.

Luckily, in spite of all their efforts not to make a mistake, they are making mistakes anyway. And they don’t feel happy about it yet.  Rather than forcing them to feel happy, in our class they have started to call the uh-oh feeling you get when you make a mistake ‘blue brain.’  They got the idea of blue brain from a short BBC clip that shows the stop in brain electrical activity following a mistake as blue, and then shows the powerful mistake-fixing activity that follows as red.  The end of the video contains just fifteen seconds on the difference between positive and negative reactions to mistakes, but that was enough to support students to come up with the following list of things they could say to themselves to recover from blue brain more quickly.

  • Try harder next time.
  • Gosh I made a mistake, I’d better fix it.
  • It’s okay everybody makes mistakes.
  • I am learning.
  • Try again.
  • Keep going!
  • I’m just getting smarter.
  • Don’t feel bad, just fix it and move on.
  • That is okay!
  • Keep trying.  You know you can do it.
  • Mistakes make me smarter.
  • I will get it next time.
  • No one is perfect.
  • I’m just learning a whole new way!
  • That’s ok. It’s part of learning.
  • Now you can learn from your mistake.
  • So what?  Just fix it.
  • It’s not the end of the world.
  • You can’t learn without making mistakes.
  • Oh well, I will get another chance.
  • You can bounce back.
  • Let’s find a solution.
  • Don’t worry.
  • We can always try again.

As we enter the second full week of school, most of them aren’t yet happy to have the opportunity to fix mistakes, but at least they are beginning to have some tools and language to help themselves get back into the learning action as quickly as possible.

Algebraic Thinking

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One of the most exciting things for 5th grade students is learning about algebra. They have been anticipating this subject for some time. What’s interesting is that they have been working with abstract mathematics forever, but it just isn’t often called algebra.

When we ask: 5 x 5 = ?, this could be written 5 * 5 = X sometime early on in elementary school. This introduction to variables can be a huge stumbling block for many students. It’s really important to get comfortable with variables in mathematics early on. We can make this problem slightly more difficult by asking it another way.

5 * X = 25 or finally 5X = 25 – now we’re doing simple algebra.

This week we have been looking at systems of equations and solving for all of the variables. These are wonderful problems that help students get familiar with variables in equations.

Here is a sampling of some of the problems we have worked on. Feel free to create some on your own. As long as you have a unique equation for each variable, there will be one and only one answer.

A + B + C = 36

A + A + B = 26

B + B + B = 18

The next problem we tried was

2A + B + C = 48

A + B + C = 40

2B + A = 18

At least one class tried this one

2/5A + B = 17

A + B = 20

And most challengingly some attempted the following

2A + B + C = 30

A + 2B + C = 35

A + B + 2C = 37