Wednesday, February 3, 2016

Subtraction Strategies in grades 1 and 2, and how they impact learning in the upper elementary grades.

Subtraction is the first nearly universal struggle math learners stumble upon in the early elementary years, before division, fractions, and eventually algebra.  It doesn't work like addition, it is not commutative (i.e. the turn-around rule does not work in subtraction), the answer is called the "difference" (confusing word in math), and we use about ten different words to represent it (minus, take away, subtract, minuend/subtrahend, how many less, decreased by, how many left over, how many fewer,  how many more, what is the difference, etc!).

By the time students reach third grade, they have been introduced to a variety of different subtraction strategies (see grade 2, unit 3 for a list), but as the authors of the Everyday Math units tell us, we should not expect every student to be able to master every subtraction strategy.  Students are introduced to subtraction in kindergarten, and practice only basic subtraction in first grade.  By the end of second grade, they should have tried all the strategies introduced, and they likely will have latched onto a couple.  In third grade, we really want them to be considering strategies to use beyond counting on their fingers.

Third grade involves a lot of subtraction practice in units 2 and 3, where your students should be "building their understanding of place value to develop methods for subtracting 2- and 3-digit numbers." (<-- from the Mathematical Background: Content section in the grade 3 TLG)  Here, expand-and-trade takes front and center as a major algorithm for students to practice and use.  This is an important piece of their understanding for future computation and problem solving, and really tests their understanding of place value.

It is not intended to be an algorithm they latch onto for life; expand-and-trade is like training wheels for the more traditional vertical subtraction most adults are familiar with.  Parents may see their child struggle with expand-and-trade, and not have the faintest idea how to help them.  "That's not how I learned it," they may say to themselves, or express to you in a frustrated note attached to a homework assignment.  Unfortunately, the temptation is to just go ahead and teach the traditional vertical subtraction method instead, because it makes more sense to the adult.  But do what you can to prevent this.  Expand-and-trade is designed to slow down the steps and deconstruct the numbers so students can see precisely when they are subtracting ones, tens and hundreds.  When they eventually transition to more efficient methods, they are well-versed in what each step of subtraction represents, and are better equipped to catch their own errors when they happen and know when their answers are in the ballpark or not.  In short, they become more confident with their subtraction.

It is the confidence that we are aiming at, because that confidence carries over to the rest of their math learning.  If they can add and subtract with confidence, then learning how to multiply and divide will not produce as much anxiety.  When they can multiply and divide with confidence, fractions are less confusing and intimidating.

Subtracting numbers is one of the most important things your students will learn how to do while in first, second and third grade.  The more opportunities your students have to work on subtraction in the early years, the better they will do in the later years.  And your fourth, fifth and sixth grade teaching colleagues will really notice a difference.

Here is a link to the VLC with some resources related to subtraction:

Subtraction Resources on the VLC

Lastly, don't forget to access the Grade Level Resources option toward the bottom of the grey menu when you log onto ConnectED.  There are lots of tutorial videos and visual aides and other resources in this section that can help you provide subtraction practice opportunities for your students.  Contact me if you need help accessing these tools.

Tuesday, February 2, 2016

What's Happening in Classrooms: Fantastic Turn-and -Talks

Recently I have visited a lot of math lessons where turn-and-talks are frequent and intentional, and I want us to celebrate and encourage that practice.

Teaching math with intentional student discourse involves taking risks, but they are risks worth taking..  These are some of these risks and ways to overcome them:

It might be disruptive and might not work.  When you ask your students to talk to each other during a lesson, you run the risk of creating a noisy, disruptive environment.   For this reason, we need to have students practice a specific protocol for turn-and-talks.  Make sure they are using indoor voices.  It is also helpful to have explicit instructions for what to discuss, and make sure they only last a minute or so.  I encourage everyone to have students practice math turn-and-talks, because math is something students don't always feel comfortable talking about.  You could say, for example:

OK, friends, let's practice a turn-and-talk.  Here's what I would like you to discuss.  13 plus 12.  What is the answer, and what are two different ways to show or prove your answer is correct.  Ready? OK, turn and talk.

You might wish to add extra protocols, like making sure each partner has a chance to talk.

OK, one, two, three, eyes on me.  (Wait for all to respond stop talking).. Now I would like you to turn-and-talk again, just to make sure both partners had a chance to speak.  Turn and talk again now please.

Practicing and re-teaching protocols for math talk is really important during the middle of the year to maintain the most productive and comfortable learning environment.

I used to wonder if too many turn-and-talks might mean too many transitions and disruptions to learning.  Could Turn-and-talks could get old?  This is actually not something to worry about.  You can have students turn and talk repeatedly in one mini-lesson, three or four times in just a few minutes.  As long as you encourage them to use the protocols, if there are things to talk about, give them the opportunity to do so.  The more they do this, the more comfortable they will become communicating their thinking with one another, with you, to their class, and on paper.  The benefits of turn-and-talks are especially great in math classes and every class should be utilizing them (in my humble opinion..).  It is a good idea to insert a little space or discussion between turn-and-talks, but the actually risk of doing them too often is almost zero.

Not enough turn-and-talks is the greatest risk of all... When your students are not communicating with each other about their thinking in math on a regular basis, they are not getting opportunities to reflect on their work, to reflect on their own thinking, and to hear the thinking of their peers.  Critiquing and analyzing the mathematical thinking of others is an important mathematical practice, and turn-and-talks are a critical way to make that happen.

It's important to diversify students' conversational experiences in math.  Turn-and-talks, small group work, class discussions, peer conferences, partner work-- They all provide excellent opportunities for students to reflect on their own thinking and learn from their peers.  But Turn-and-talks are special, in that they offer students a chance to communicate directly with a partner about an idea that is right there in the moment.  They only take a few seconds, and they keep your students engaged in your lesson.  If you would like support from a colleague in using turn-and-talks in math lessons, chances are good you have someone right next door or across the hall who can help you... Or you can ask your friendly math strategist to come model turn-and-talks in math.