* 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.

 

* 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-

* 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.

* The spaces in between: fractions and decimals

The latest edition of Teaching Children Mathematics (TCM) is solely focused on teaching fractions and decimals.  And it’s a winner!  We don’t need a journal to tell us that many kids are struggling in this area of math.  Looking back over my own teaching history, it’s interesting to trace the evolution of teaching “between the wholes,” as TCM refers to it.  It’s a miracle that any of my students understood decimals and fractions back in the day.  I really didn’t.  It was considered a big leap when teaching supply companies created an abundance of pie- and strip-shaped manipulatives and games.  But these materials served to reinforce misunderstandings by focusing solely on area models of fractions.  Students memorized shapes and attached a meaningless value to them.  They rarely understood the relationship of fractional parts to real life.

One of the most powerful differences in this improved instruction is student exploration of part-whole relationships. My favorite article focuses on “French Fry Tasks,” where students are asked to “share” paper fries using estimation, with increasingly larger groups of people.  Some constraints are in place: no rulers and no folding the paper.  This strategy has a number of advantages, but a primary one is understanding that repeated sharing results in smaller sizes.  Should we tell kids that the denominator increases as the size shrinks or let them discover this through careful experimentation?  Duh.  TCM also includes two articles which provide excellent tools for assessing student understanding of fractions and decimals, along with strategies for improving their performance.  french-fries-155679_640

The strategies outlined in TCM are not unknown in the current teaching landscape.  So why aren’t they an integral part of math instruction?   Not all teachers DO know how to teach fractions and decimals effectively.  But I think the greatest deterrent is the clock.  It takes time for students to develop these understandings.  It takes a lot of student interaction and exploration.  It takes classroom management skills to keep these tasks meaningful.  It takes reasonably-sized groups of kids.  And it means slowing down the mighty rush to “get through” the curriculum before tests are administered.  Without administrative leadership, this lenghtier process of learning will not occur.  And for that administrative support to exist, it takes a broader (political) climate that rejects our current dash through the curriculum.  Let’s hear it for the french fries!