Fluency

Transparent head silhouette with interlocking gold and silver gears inside representing brain function

Connecting Numeracy and CT: The Structure was Always There – Part 3
Transparent head silhouette with interlocking gold and silver gears inside representing brain function

In the last two posts (Part 1 and Part 2), we looked at what fluency actually is and the three structures that sit underneath it. In this post, I want to make a connection that might surprise you.

Those same three structures — place value, properties of operations, and equivalence — are also at the heart of computational thinking.

I know. Stay with me.

What is Computational Thinking, Actually

Computational thinking isn’t about coding. It’s about a particular way of approaching problems — one that computer scientists and mathematicians have always shared. It has four moves:

  • Decomposition — breaking a complex problem into smaller, manageable parts
  • Pattern recognition — noticing what stays the same across different contexts
  • Abstraction — stripping away surface detail to see the underlying structure
  • Logical sequencing — ordering steps where each one depends on the one before

Sound familiar? It should. When a student solves 27 × 4 by breaking it into

20×420\times 4

and

7×47 \times 4

they are decomposing. When they notice that the same move works with 27 × ¼, they are recognizing a pattern across abstractions. When they express that as a(b + c) = ab + ac, they have abstracted. These aren’t separate skills layered on top of mathematics. They are mathematics — the structural kind.

Where Numeracy Fits

Numeracy is what happens when a student can take those structures into the world and use them confidently. Not just in a math class, on a familiar problem type, with a procedure they’ve practiced. But in a grocery store, a spreadsheet, a situation they’ve never seen before.

That flexibility — the ability to deploy structure in unfamiliar contexts — is what separates a student who has memorized math from a student who understands it.

So fluency, numeracy, and computational thinking aren’t three different goals pulling in different directions. They’re three expressions of the same underlying capacity: the ability to reason from structure.

What this Means for Teaching

If the structures are the through-line, then the question worth asking about any lesson, any task, any resource is: does this make the structure visible?

Not just — is it a good activity? Not just — will students enjoy it? But: will students walk away having experienced something about how numbers behave that they can carry forward? Will they be able to use that understanding the next time they encounter something unfamiliar?

That’s the goal. Not fluency for its own sake. Not computational thinking as a trendy add-on. But students who see the structure — in numbers, in patterns, in problems they’ve never met before — and know what to do with it.

And when those students finish a challenging task and they tell you they want to do more — because somewhere along the way, they started asking “What if…?”, run with it. They’re posing rich questions and ideas to explore because they’re curious. Many of their ideas will likely involve some mathematical modeling. Another huge win.

So, the end of the lesson is not, necessarily, the end of a lesson. But it is the beginning of a mathematician.

Connecting Numeracy and CT: The Structure was Always There – Part 1

When students compute, are they executing steps or reasoning from relationships? Most of us were taught the former. But the latter is where mathematical power lives.

Feel the Structure

We all want our students to become fluent with facts. There are many ways we can try to get them there. Some focus on strategy development and building mathematical relationships. Others focus on speed and memorization. One way builds confidence and connections. The other creates anxiety.

To build fluency, we need to understand what it is.

If fluency is represented by this walnut.

When students are able to apply strategies to multiple contexts to solve problems, they are fluent. They are also working their way toward numeracy: having the confidence to use basic maths at work and in everyday life.

Let’s look at some ways to solve 27 x 4:

What’s the underlying structure of each? They all break the factors into friendlier numbers and multiply, then put them back together. Decomposing and Composing. All with the underlying structure of the Distributive Property. One uses the Place Value structure as well (top left). Another also uses Equivalence as a structure (bottom left). But all use the structure of distributive property. Naming the property isn’t something students in grades 3 -5 need to focus on. They just need to understand what’s happing and apply it.

Sometimes teachers go with timed tests rather than focusing on strategies because they think that the strategies are replaced by fluency. It’s not really the strategies, though. It’s the underlying structure of these strategies. And fluency doesn’t replace the structures, it compresses them.

In my next post, I’ll discuss the three big structures that encompass all of K-5 mathematics.

Blogarithm Posts

Last year I had the honor of being asked to write four posts for NCTM’s Math Teaching in the Middle School Blog: Blogarithm (one of the coolest math blog names out there).  They were posted every two weeks from November through the end of December (which just shows that I can post more frequently if someone is reminding me every other week that my next post is due (thanks Clayton).

Pythagorean Decanomial

The four posts are a reflection of a lesson I taught with a 6th grade teacher, in September of last year, who was worried (and rightfully so) that her students didn’t know their multiplication facts.  After a long conference, we decided to teach a lesson together.  I modeled some pedagogical ideas and she supported students by asking questions (certain restrictions may have applied).

Links to the four posts are below.

  1. Building Multiplication Fluency in Middle School
  2. Building Multiplication Fluency in Middle School Part 2
  3. Building Multiplication Fluency in Middle School Part 3
  4. Building Multiplication Fluency in Middle School Part 4

While you’re at the Blogarithm site check out some other guest bloggers’ posts.  Cathy Yenca has some great posts on Formative Feedback, Vertical Value Part 1 and Part 2, and 3-Act Tasks

Math Students are Bleeding Out!

Let me explain.  There’s a math epidemic (remember Ebola 2014+).  Students are bleeding out from the gashes of their misconceptions of mathematics.  The lack of teaching conceptual understanding along with sacrificed opportunities to make mathematical connections is the double edged sword.  This is an epidemic, and some teachers, school systems and educational leaders are treating it like it’s a tiny scratch, instead of the pervasive threat to mathematical achievement that it is.

Here’s a familiar scenario:  A school’s test scores come back after the spring testing season (or mid-terms).  The scores show little growth from the previous year in the area of mathematics, and any change is not in a positive direction.  The knee-jerk reaction to the valid question, “What can we do to fix this?”  is to look for programs and technology that will fix the problem.  These are the same individuals who, way back in August, looked us all in the eye and, with the greatest of sincerity, reminded us that the single most important factor determining student success is the quality of the teacher.  Not new programs.  Instead of growing the quality of teachers, we get programs that:

  • push speed over comprehension (imagine if we taught reading this way).
  • define fluency based on digits rather than efficiency, flexibility, and accuracy.
  • use technology to separate us from our students when we know that what we really need is to spend more time listening to them and creating an interactive classroom with technology as a support for this human interaction
  • are essentially Band-Aids

I often hear the phrase “back to the basics” in times like these.   I’ve heard parents, administrators, and even a few teachers say this.  I think everyone would agree that “back to the basics” should mean that students become computationally fluent.   The idea of going back to this implies that we were doing something right before.  And we all know that’s not true.  After all we have generations of adults who are not computationally fluent and/or have extreme math anxiety.  And how did that happen?

Answer 1:  Timed tests.  My sixth grade teacher called them speed tests.  We did them every day, right after lunch.  (I was never in the top 10).

Answer 2:  Algorithms memorized by students with no understanding, presented by teachers with little understanding other than from a teachers’ edition.

Answer 3:  Little or no real problem solving.  Naked computation all around.  No wonder students were turned off by mathematics!

Answer 4:  No interaction.  Math is a social activity.  If you talk to any engineer, designer, architect, mathematician, statistician, etc.  They aren’t doing their work in an office silently sitting in rows.  They’re constantly talking to one another about the mathematics they’re using.  The idea that all of us are smarter than one of us makes so much sense in the real world and it should make sense in the classroom as well.

If we went back to teaching math like we did 20-30 years ago (I think that’s what some of these folks were implying when they said “back to the basics.”  We’d still be in the same boat.  Anyone ever watch How old is the shepherd?.  That was popularized over 20 years ago and the results haven’t changed.  Going back is not an option.  Building fluency is.

So what do we need to do in math classrooms?  I have a few ideas to stop the bleeding and these are certainly not original to me.

keep calm

  1. Apply pressure to the wound. Give up on the ineffective treatment, not the patient.  Apply pressure to stop the bleeding.  Focus on tasks and activities that build number sense.  Number Talks, Math Talks, Estimation 180, Visual Patterns any or all of these can be put in place at any level.  And the best part is, students can easily be trained to begin to apply the pressure themselves.  They have the power to stop the bleeding!
  2. Close the wound. This can only happen with stitches.  And it takes time to get the hang of it.  The wound has to be closed with the thread of understanding.  We can’t understand for them, so the wound has to be closed with the help of the students.  The students create this thread as we stitch and we can’t do it without them.  How do they create this thread of understanding?  We have to stop telling so much and instead “be less helpful.”  If we tell students too much, the thread breaks.
  3. Treat any symptoms that may show up after the initial treatments above:

Symptoms

 Name

Treatment

Students may begin to rely on rote procedures with no foundational understanding

Sometimes unintentionally caused by parents & other adults trying to help.

 Misconceptionitis Identify the misconception(s) and re-build understandings using the CRA (Concrete Representational Abstract) model

Students are finding unreasonable solutions to tasks & problems and they often seem unaware; clueless

 Unreasonableness

This is often attributed to students just not thinking enough.  Treatment should include a DAILY diet rich in estimation – prescribe www.estimation180.com

Students count (often on fingers when computing or rely on a calculator for the simplest of calculations and even then, they can get incorrect answers.

 Influencia

This is often diagnosed along with unreasonableness (see above).  Its roots lie in naked computation and memorization of facts rather than allowing students to build strategies and practice those strategies until they become fluent.  First, counting strategies are the lowest level strategies.  Students need to build more efficient strategies by exercising with  investigations of number relationships through number talks, math talks, and strategy building.  Stop giving speed tests.
Students have strategies for computation, but are not applying them in problem solving situations No Solvia

Students need a heavy dose of problem solving every day.  This must involve students engaging in the Big 8 Standards for Mathematical Practice.  Problem solving tasks every day.  Hydrate often with student reasoning.  Adopt the classroom mantra: “The answer isn’t good enough.”

Begin new concepts with a problem before any formal instruction on the topic.  See what students can do before assuming what they can’t do.

I’m a teacher and I know many of you reading this are the choir that need no preaching to.  If you’re interested in saving the patient, stopping the bleeding, and raising math achievement, click on some of the links in this post.  There’s so much to learn from those smarter than me.  Also check out #MTBoS on Twitter.  Lots of math goodness from the best out there.

Click here and here to learn more about strategy development.  Great stuff from www.nzmaths.co.nz!