How Big is “Big”?

In the last few weeks I have asked several groups of teachers to indicate where 1 billion would go on this number line:

It has been really interesting to me that many have placed the 1 billion mark in a variety of areas and have had a variety of reasons why.  Many have attempted to use their understanding of place value digits  (there are 12 zeros in 1 trillion and only 9 zeros in 1 billion, so 1 billion should be 3/4 of the way toward a trillion) or their knowledge of prefixes to help (million, billion, trillion… so it must be 2/3 the way along the line). Others thought about how many billions are in a trillion asking themselves, “Is their one-hundred or one-thousand billions in a trillion?” Using this strategy, everyone picked a spot toward the left, but some much closer to zero than others.

Others did something interesting though. They started placing other numbers on the number line to help them make sense of the question. Often placing 500 billion in the middle, then 250 billion at the 1/4 mark and so on until they realized just how close to 0 a billion is when we are considering 1 trillion.

What’s the point?

Really big numbers, and really small numbers (decimal numbers), are difficult to conceptualize!  They are hard to imagine their size!  Think about this:

How long is 1 million seconds?  Without doing ANY calculations would you guess the answer is several minutes, hours, days, weeks, months, years, decades, centuries…?  Can you even imagine a million seconds without calculating anything?

How about 1 billion seconds?  Or 1 trillion seconds?

I bet you’ve started trying to calculate right!  That’s because these numbers are so abstract for us that we can’t imagine them.

Because of this little experiment, I am left wondering three things:

  1. What numbers can/can’t the students in our classrooms conceptualize?
  2. What practices do we do that gets kids to think about digits more than magnitude?
  3. What practices could/should we be including that helps our students make these connections?

What numbers can/can’t the students in our classrooms conceptualize?

Before we start working with operations of any given size, I think we need to spend time making sure our students can visualize and estimate the size of the number.  Working with numbers we can’t imagine doesn’t seem productive for our young students!  In our rush to move our kids into more “complicated” mathematics, we often move too quickly through numbers to include numbers that are too abstract for our students!  We think that if a student can accurately carry out a procedure that they understand the numbers they are working with. However, I’m sure we have all seen many students who produce answers that are completely unreasonable without them noticing. Is this carelessness, or is it a lack of understanding of the magnitude of the numbers involved?  Or possibly that our students aren’t visualizing the size of and relationship between the numbers???


What practices do we do that gets kids to think about digits more than magnitude?

The other day, Jamie Garner shared her frustration on Twitter:


Think about the question from the textbook for a second. Students trying to think about 342 pencils (not sure why they would want that many) should be considering a strategy that makes sense. For example, if you had 342 pencils how many boxes of 10 would that be?  Thinking this way, student should answer 34 or 34.2, or maybe 35 boxes (if you wanted to purchase enough boxes).  However, the teacher’s edition tells us that none of these are the right answer. Take a look:


If our students attempt to make sense of the problem, they will be completely wrong!  In fact, many students will likely answer 4 because they’ve been trained not to think at all about the mathematics, and instead focus their attention on what they think the text wants them to do.

This is one of MANY cases where elementary mathematics focuses on digits over understanding magnitude or relative size.  Here are a few others:


These, along with pretty much any standard algorithm (see Christopher Danielson’s post: Standard Algorithms Unteach Place Value) tell our kids to stop thinking about what makes sense, and instead focus on steps that help kids get an answer without understanding.


What practices could/should we be including that helps our students make these connections?

If we want our students to understand numbers, and their relative size… if we want to help our students develop a conceptual understanding of operations… if we want our students make sense of the math they are learning… then we need to:

  • Use contexts that make sense to our students (not pseudo-contexts like the pencil question above).
  • Provide plenty of experiences where students are making sense of numbers visually.  When we allow our students to access their Spatial Reasoning we are allowing them to see the relationship between numbers and help them make connections between concepts.
  • Provide plenty of experiences estimating with numbers

Below are 2 activities taken from Van de Walle’s Student Centered Mathematics.  Think about how you could adapt these to work with numbers your students are starting to explore (really big or really small numbers).

vdw-number-line1vdw-number-line2


A few questions for you to reflect on:

  • How might you see how well your students understand the numbers that are really large or really small?
  • How are you helping your students develop reasonableness when working with numbers?
  • What visuals are you using in your class that help your students visualize the numbers you are working with?
  • What practices do you use regularly that help with any of the 3 above?

P.S. Here are the answers to the seconds problem I posted earlier:

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Focus on Relational Understanding

Forty years ago, Richard Skemp wrote one of the most important articles, in my opinion, about mathematics, and the teaching and learning of mathematics called Relational Understanding and Instrumental Understanding.  If you haven’t already read the article, I think you need to add this to your summer reading (It’s linked above).

Skemp quite nicely illustrates the fact that many of us have completely different, even contradictory definitions, of the term “understanding”.   Here are the 2 opposing definitions of the word “understanding”:

“Instrumental understanding” can be thought of as knowing the rules and procedures without understanding why those rules or procedures work. Students who have been taught instrumentally can perform calculations, apply procedures… but do not necessarily understand the mathematics behind the rules or procedures.

Instrumental2.png

“Relational understanding”, on the other hand, can be thought of as understanding how and why the rules and procedures work.  Students who are taught relationally are more likely to remember the procedures because they have truly understood why they work, they are more likely to retain their understanding longer, more likely to connect new learning with previous learning, and they are less likely to make careless mistakes.

Relational2.png

Think of the two types of understanding like this:

 

Shared by David Wees

Students who are taught instrumentally come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill.  Each new skill requires a new set of procedures.  However, those who are taught relationally make connections between and within concepts and skills.  Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.

 

While it might seem obvious that relational understanding is best, it requires us to understand the mathematics in ways that we were never taught in order for us to provide the best experiences for our students. It also means that we need to start with our students’ current understandings instead of starting with the rules and procedures.

Skemp articulates how much of an issue this really is in our educational system when he explains the different types of mismatches that can occur between how students are taught, and how students learn.  Take a look:

Instrumental vs Relational

Notice the top right quadrant for a second.  If a child wants to learn instrumentally (they only want to know the steps/rules to solve today’s problem) and the teacher instead offers tasks/problems that asks the child to think or reason mathematically, the student will likely be frustrated for the short term.  You might see students that lack perseverance, or are eager for assistance because they are not used to thinking for themselves.  However, as their learning progresses, they will come to make sense of their mathematics and their initial frustration will fade.

On the other hand, if a teacher teaches instrumentally but a child wants to learn relationally (they want/need to understand why procedures work) a more serious mismatch will exist.  Students who want to make sense of the concepts they are learning, but are not given the time and conditions to experience mathematics in this way will come to believe that they are not good at mathematics.  These students soon disassociate with mathematics and will stop taking math classes as soon as they can.  These students view themselves as “not a math person” because their experiences have not helped them make sense of the mathematics they were learning.

While the first mismatch might seem frustrating for us as teachers, the frustration is short lived. On the other hand, the second mismatch has life-long consequences!


I’ve been thinking about the various initiatives/ professional development opportunities… that I have been part of, or have been available online or through print that might help us think about how to move from an instrumental understanding to a relational understanding of mathematics.  Here are a few I want to share with you:

Phil Daro’s Against Answer Getting video highlights a few instrumental practices that might be common in some schools.  Below is the “Butterfly” method of adding fractions he shares as an “answer getting” strategy.  While following these simple steps might help our students get the answer to this question, Phil points out that these students will be unable to solve an addition problem with 3 fractions.  These students “understand” how to get the answer, but in no way understand how the answer relates to addends.

Daro - Butterfly.jpg

On the other hand, teachers who teach relationally provide their students with contexts, models (i.e, number lines, arrays…), manipulatives (i.e., cuisenaire rods, pattern blocks…) and visuals to help their students develop a relational understanding.  If you are interested in learning more about helping your students develop a relational understanding of fractions, take a look at a few resources that will help:


Tina Cardone and a group of math teachers across Twitter (part of the #MTBoS) created a document called Nix The Tricks that points out several instrumental “tricks” that do not lead to relational understanding.  For example, “turtle multiplication” is an instrumental strategy that will not help our students understand the mathematics that is happening.  Students can draw a collar and place an egg below, but in no way will this help with future concepts!Turtle mult.jpg

Teachers focused on relational understanding again use representations that allow their students to visualize what is happening.  Connections between representations, strategies and the big ideas behind multiplication are developed over time.

Take a look at some wonderful resources that promote a relational understanding of multiplication:

Each of the above are developmental in nature, they focus on representations and connections.


So how do we make these shifts?  Here are a few of my thoughts:

  1. Notice instrumental teaching practices.
  2. Learn more about how to move from instrumental to relational teaching.
  3. Align assessment practices to expect relational understanding.

Goal 1 – Notice instrumental teaching practices.

Many of these are easy to spot.  Here is a small sample from Pinterest:

 

The rules/procedures shared here ask students to DO without understanding.  The issue is that there are actually countless instrumental practices out there, so my goal is actually much harder than it seems.  Think about something you teach that involves rules or procedures.  How can you help your students develop a relational understanding of this concept?


Goal 2 – Learn more about how to move from instrumental to relational teaching.

I don’t think this is something we can do on our own.  We need the help of professional resources (Marian Small, Van de Walle, Fosnot, and countless others have helped produce resources that are classroom ready, yet help us to see mathematics in ways that we probably didn’t experience as students), mathematics coaches, and the insights of teachers across the world (there is a wonderful community on Twitter waiting to share and learn with you).

I strongly encourage you to look at chapter 1 of Van de Walle’s Teaching Student Centered Mathematics where it will give a clearer view of relational understanding and how to teach so our students can learn relationally.

However we are learning, we need to be able to make new connections, see the concepts in different ways in order for us to know how to provide relational teaching for our students.


Goal 3 – Align assessment practices to expect relational understanding.

This is something I hope to continue to blog about.  If we want our students to have a relational understanding, we need to be clear about what we expect our students to be able to do and understand.  Looking at developmental landscapes, continuums and trajectories will help here.  Below is Cathy Fosnot’s landscape of learning for multiplication and division.  While this might look complicated, there are many different representations, strategies and big ideas that our students need to experience to gain a relational understanding.

Fosnot landscape2
Investigating Multiplication and Division Grades 3-5

Asking questions or problems that expect relational understanding is key as well.  Take a look at one of Marian Small’s slideshows below.  Toward the end of each she shares the difference between questions that focus on knowledge and questions that focus on understanding.


I hope whatever your professional learning looks like this year (at school, on Twitter, professional reading…) there is a focus on helping build your relational understanding of the concepts you teach, and a better understanding of how to build a relational understanding for your students.  This will continue to be my priority this year!