* Bansho

What is Bansho?

a. A type of sushi

b. A form of martial arts

c. A math teaching strategy

It’s a clever math strategy!  Bansho was recently featured in Teaching Children Mathematics, a NCTM publication.  Originally developed in Japan, this powerful visual strategy has been used successfully in Thailand and for this article, in Texas.  Daphyne Miller is the featured teacher.

How does it work?  Bansho organizes the math learning process visually (such as across a board or wall), encouraging student-generated ideas and discussion.  The board space is divided into sections that correspond to 3 phases in a lesson: activating prior knowledge, exploring a problem, and discussing/ extending the problem.  Students connect their ideas to others’ work throughout the process.  Teachers must anticipate student responses, provide hands on materials, and monitor student work and interactions.

What does it look like?  Depending on the problem being solved, a pictorial representation of the problem is on the left side, along with keywords and related vocabulary.  The center section features student work, organized in columns to show a progression from concrete to abstract reasoning.  On the far right, student work and teacher input extend the learning.  Check out Thinking of Teaching blog for cool images.


How could this be adapted to support special needs students? 

  1. If you look at most Bansho illustrations, you’ll see lots of handwritten work.  Using digital tools to capture student work would help those kids who struggle to spell or even draw.  Smart boards could be an easy adaptation.
  2. Provide visual cues for students to communicate during all three phases of the lessons.  These could be as basic as index cards printed with cues: “Look at your partner.  Ask her to tell you her number sentence.  Ask her to write that sentence for you.”
  3. Pair students carefully.  Use a buddy system but don’t wear out the “helpers!”
  4. Encourage students to use their special interests when extending the math problem to varied topics.  This will likely make it easier for them to share their ideas with the group.  Allow video recordings for those kiddos who are reluctant to share in a large group.
  5. Be creative in reducing the visual clutter of a Bansho display.  This could include digital instead of paper worksheets, using a smart board for the entire display, or placing the three sections or phases onto separate boards.

* Mathematics discourse

Mathematics discoursequotation marks 2

That’s another name for providing students adequate time to discuss and reason in math instruction.  The current issue of Teaching Children Mathematics describes Tracy Shannon’s delightful classroom where math instruction is engaging, hands on, and allows students time to talk and solve interesting problems.  The authors, Gina Gresham and Tracy Shannon, provide a useful framework for other teachers who want their students to improve achievement and motivation.  Is it reasonable to think this can be accomplished, given all the time and testing pressures faced by classroom teachers?  YES!

Here is Shannon’s sample schedule of a 75 minute block of math instructional time:

  • 15 minutes: whole-group, teacher-led instruction
  • 10-12 minutes: Station 1- Teacher-led activity with small groups
  • 10-12 minutes: Station 2- Hands-on game activity with small groups
  • 10-12 minutes: Station 3- Computer activity with small groups
  • 15-20 minutes: Whole-group wrap-up lesson and exit slip

It’s obvious that this approach requires careful planning and access to digital resources, in addition to developing a community of students who support and encourage one another.  The authors also discuss the challenges of mathematics discourse, including how much struggle to allow students as they solve problems.  Teaching with mathematics discourse requires a skillful teacher and most likely, a solid team of professionals.

* Backing up, moving forward in math

Backing Up and Moving Forward” is an insightful article in this month’s Teaching Children Mathematics.  The premise of the article is that carefully worded math problems can assist teachers in first assessing student performance and then determining the next steps: do students move forward or back up?  The authors, Barlow et al., share their experiences with 5th graders’ understanding of dividing fractions and how teachers determined what those next steps should be.

In the process of designing their lessons, the authors share some features of effective word problems.  In this experiment, the word problem was fashioned around “Chef Frederick” as he made dessert.  Do students have familiarity with this kind of cooking experience?  Does the problem support varied response styles, such as visual, manipulative, or written?  measuring-cups

The authors also established a 4 point scale of student performance, from exceeding expectations to lacking fundamental understanding.  From my perspective as a special educator, I can predict that many of my kiddos would fall in the lower end of the proficiency scale.  Asking students to demonstrate their understanding visually is an effective indicator of performance.  Both teachers and students can better see the reasoning process and where it might have broken down.  In one example, a student did not know how to represent a simple fraction; obviously, dividing fractions was introduced before that student had sufficient prior learning.

I’d love to see a follow-up article on how classroom teachers could address major gaps in mathematical understanding.   It takes time to replace misconceptions.  It takes willingness on both the student’s and teacher’s part to tackle the process.  What happens to the struggling learners while the remaining kids move forward?  It is encouraging that these educators recognize the need to fill in those gaps, instead of simply pushing forward.  If this assessment and repair process occurred routinely at all grade levels, perhaps we wouldn’t see as many kids who are partially memorizing algorithms and procedures without true understanding.


* Anonymous scholars

This month’s Teaching Children Mathematics features an excellent article entitled “Learning From the Unknown Student.”   What’s that about?   The idea is to expose students to effective strategies and prompt analysis of others’ mathematical reasoning by using “anonymous scholar” work.  The unknown student provides an avenue for sharing an alternate problem- solving approach without leading students to believe it’s the Teacher Way of doing math.  I typically employ a variation of this strategy in writing and social skills, but it is equally effective in math.

Let’s say I want a student to recognize a common error in her writing, perhaps an abundance of incomplete sentences.  But this kiddo does not see those mistakes and is already hyper-sensitive about correction.  That’s when I introduce “a student from last year” whose writing is replete with the same errors.  Now my student becomes a helpful editor and delights in using effective strategies to catch those errors, such as reading the sample out loud and using a rubric or checklist.  I have found that students are much more relaxed about revisions and editing when they have sliced-and-diced someone else’s work.  Where do I find these student samples?  Some are actually students from last year.  Others are copied from Google images or a search such as “writing samples, grade 2.”

For those students who struggle to add a specific feature to their writing, such as an effective opening sentence, I will use commercially-prepared mentor texts (Empowering Writers is a good choice) and graphic organizers with built-in prompts (usually created by me).  There’s no point in replicating my students’ dismal classroom experiences, where other kids seem to write effortlessly.  Those scholars are not anonymous.

In social skills instruction, I tell anecdotes or write social stories about anonymous scholars who struggle to make friends or follow directions.  I have also referred to “a student at another school, but I can’t tell you his name.”  It’s amazing how my students immediately verbalize highly effective strategies for dealing with these issues.  For the younger set, we watch puppets literally wrestle with familiar social and academic glitches.  Sometimes I wonder what kind of teacher I am, since Rocky the raccoon and Sandy the pup never learn to take turns, listen to others, or manage their frustration!

In the examples I’ve shared, there is a downside to using anonymous scholars.  A student with very low self-esteem may attempt to build his confidence on the back of that pitiful kid who can’t add 1 + 1.  However, I think that is more easily managed than erasing 50% of the answers on a math page, then telling the kids they are improving.  Who would believe that?  silhouette

* School-wide bulletin boards

A recent “Coaches’ Corner” feature of Teaching Children Mathematics encourages the use of bulletin boards as teaching tools.  Robyn Silbey shares a cool example for grades K-5 by using a single prompt (What is equal to 10?) divided into columns for all classes to respond.  This tool seems appropriate for expanding algebraic instruction and thinking to all elementary grades, so here’s my proposed bulletin board.  Teachers could rotate through small group displays so that all kids get “on the wall” in one year.

bulleting board image.jpg


* The Math Diet

In a recent edition of Teaching Children Mathematics, authors Kateri Thunder and Alisha N. Demchak make a strong case for a “Math Diet” to grow healthy mathematical reasoning in kids.  Just as researchers have identified five foundational components of healthy reading reading instruction, the Math Diet is meant to provide students a balanced framework for navigating math through elementary school.

Here are the five components of the Math Diet as described by Thunder and Demchak:

  1. Counting
  2. Subitizing
  3. Conceptual understanding
  4. Strategic Competence
  5. Procedural fluency

I hear someone mumbling that EVERYONE knows counting is a given, and it’s certainly not esoteric.  BUT, there is a lot more to counting than meets the eye.  Many kids are able to process the more abstract features of counting (understanding order irrelevance, for instance), while struggling students never seem to get past using their fingers.  The key to a healthy Math Diet is to start early and start right.

In my experience, it’s hard to “feed” kids this Math Diet once they have acquired a taste for junk food math skills.  And as I’ve noted before, elementary teachers themselves may not have been exposed to healthy Math Diets.  That’s one reason I joined The National Council of Teachers of Mathematics:  my personal Math Diet was sketchy.  maths-

* Multiplying fractions with visuals

In a previous post, I expressed my concern about the use of some models and processes for representing multiplication of decimals and fractions. A recent issue of Teaching Children Mathematics provides a better description of why my students and I were both confused by the representations being used by some classroom teachers.  As we plowed through classwork and homework, kids were increasingly anxious as I tried to inject some meaning into the process.  “My teacher says to color these boxes!” the kids would exclaim in desperation.  “We aren’t doing it that way!”  When asked what the boxes meant, the kids had no idea.

In an excellent article by Webel, Krupa, and McManus, the authors describe the confusion that results from using visual representations without context.  They emphasize that the visual should clearly depict the differing sizes of the two fractions.  The authors express concern that visuals may be reduced to a series of steps which are no more helpful than blindly following an algorithm.

As I’ve taught students in 4th and 5th grade, all three of these pitfalls have been evident.  Here are some graphics to demonstrate the authors’ perspective (and my students’ anxiety).

Let’s say we want to solve this problem (borrowed from the article):  I have 1/4 of a gallon of orange juice in my fridge.  Then I drank 1/4 of it.  What fraction of the gallon of OJ is left in the fridge?

Here are three possible visuals for this problem.  Multiplying fractions

A and B are very similar and bring back bad memories!  Use a grid to color across and down.  The double-shaded area represents the answer.  Ta-da!  Only it doesn’t answer the question!  In the OJ problem, we need to know what’s LEFT.  C is a superior representation of the problem but a teacher must guide kids into how we arrive at 16 squares in the grid.  That discussion follows logically when you focus on the problem, which has 2 distinct units.

For fragile learners, A and B have great appeal.  Kids can memorize the coloring process without difficulty.  Unfortunately, they are not even close to understanding what multiplication represents.  The “talking it through” process can be very frustrating because they know they’re BEHIND and want to speed along to catch up.  And as we know, THIS ISN’T THE WAY MY TEACHER DOES IT!

* Teaching math teachers

November’s “Teaching Children Mathematics” has an article I’ve been reflecting on for quite some time now.  The author, Kelly McCormick, teaches preservice elementary teachers (PSTs) and describes her unit on fractions.  She makes use of actual fourth grade classroom videos and also has the PSTs solve similar math problems.  Both the teacher and student lessons feature two important components of effective instruction: authentic problem solving and the opportunity to deepen conceptual understanding of fractions.  The PSTs, divided into three groups, use different methods of modeling their solutions.  One of the three groups went off the rails, but a group discussion revealed their misunderstanding (that fractions must be derived from the same whole in order to compare them).  fraction

As a special ed teacher, I appreciate the emphasis on taking adequate time, using hands-on activities and authentic problems, to support student learning.  My concern is that by fourth grade, kids with a math disability, attention problems, language issues, and social weaknesses, are just not comparable to these PSTs or a classroom of typical learners.  The kids I teach are going to struggle to keep up with the work of those who can draw models and talk through their reasoning.  My math-disabled students already have far too many misconceptions about fractions.  And their class is not going to take the time needed for them to catch up.  The class will speed on by, leaving some kiddos in the dust.  I’ve seen that happen repeatedly.

Based on her PST feedback, this class was a turning point for many of these preservice teachers.  One of them even commented that she understood fractions for the first time.  All of that is terrific.  Now how do we shape the reality of school so that our fragile learners understand fractions for the first time?  More thoughts tomorrow.

* Graphics and algorithms, part 2

I made three points in yesterday’s post on this topic:

  1. Students should be allowed access to manipulatives far longer than is typically expected.
  2. Students should not simply memorize algorithms, but should understand the mathematical reasoning behind them.
  3. Visual representations of math processes can be helpful, but can often be more confusing than a standard algorithm.  I have concerns that kids struggle to understand the area model of multiplying decimals and suggested a number line.

Today’s post deals with a terrific process and graphic for teaching division.  It is featured in this month’s “Teaching Children Mathematics,” published by the National Council of Teachers of Mathematics.  Called “division quilts,” it represents a measurement model of division as opposed to “fair shares.”  The fair share model is like dealing cards into equal hands.  The measurement model begins with the dividend, created on graph paper in a rectangular representation.  This is one way to draw 35 ÷ 4.

Division quilts_0001

Next, color groups of 4, using a different color for each adjoining set.  A good question to ask kids is why they should use different colors.Division quilts_0002

By counting the groups of 4, it’s easy to see that there are 8 equal groups.  Finally, number the left-over spaces, which is the remainder.Division quilts_0003

There is a strong relationship between this graphic and the long division algorithm.  For kids struggling with math, that relationship must be explicitly taught.  We want kids to use algorithms, but this process helps explain not only how we got a quotient of 8, but what a remainder means.  Depending upon the context of the problem, kids must decide what to do with that remainder.  I love to use field trip math problems with remainders because the imagery is so vivid.  “If we had parent drivers and placed 35 students in groups of 4 per car, how many cars would we need?”  At first, most kids will say 8.  Then we imagine those last three kids running behind the cars to the museum.  After they exhaust the possible ways for those kids to get smushed on the freeway, most students never forget that remainders are important!