Closing the Teach For America Blogging Gap
Jan 20 2014

Long Division in Pictures

Perhaps your students’ mathematical thinking rivals the discussion in this Abbott and Costello skit.  I always enjoy that one, and it’s funny in part because it speaks to our befuddlement with math algorithms.  I don’t remember learning long division so I can’t comment on how well I understood it at the time.  However, if I did understand the meaning behind the mechanics, it was lost over the years such that this Montessori lesson seemed exciting and new.  I’m guessing that textbooks include some good graphics and descriptions to conceptually illustrate the algorithm (since I’ve not taught such young grades in a traditional setting I’m not sure what form these take), but I also wanted to share this series of pictures from my training.

A note that in Montessori lessons units, tens, and hundreds are represented using green, blue, and red respectively.  Before this lesson children learn place value (including the colors) and work with division more concretely (think Cuisenaire rods — I still remember that day-glo orange).  They also do a “first passage” with the board and beads in which they do all the mechanical distributing and exchanging but record only the quotient.    I’ve included the complete recording so you can see how the materials lead into each step of the algorithm.   Let me know what you think!

One day all children in this nation will have the opportunity to attain an excellent education.

Have a great week!

3 Responses

  1. Meghank

    I don’t know about this, but I had a great time teaching fourth graders to do basic division 9/2, 15/5, 16/3, etc, using toothpicks. I taught lower grades, but I volunteered for a Saturday “test prep” session for fourth grade. When they started to pull out their calculators, the other volunteer teacher and I drew a line, and brought out the manipulatives. They loved it! Turns out, teachers are no longer using manipulatives to teach math after second grade, even though fourth graders still need and love it. The reason is simple: standardized testing starts in 3rd grade, and calculators are allowed on the tests. It’s very sad that teachers are allowing kids as young as 8 to use calculators instead of really understanding math. Even if the “Common Core” changes this, which I doubt it will, teachers like you (middle school) will still be dealing with the fallout from this (kids unable to do basic arithmetic without a calculator) for many years to come.

    Also, our principal walked in on our tutoring session and said we didn’t have time to use manipulatives, to pull out the calculators so they could really get ready for the big test.

    • Educator

      I agree that kids of all ages are drawn to manipulatives and that these help the students to understand concepts and estimate reasonable solutions — even if they go on to use calculators on a test or in daily life.

      That said I definitely understand the pressure to conform to a culture of kill-and-drill teaching focused on passing tests. It’s too bad that testing has so much influence on teaching, and that schools in low-income communities have the most to lose and so are at most risk for falling into these short-term traps.

      The Common Core standards themselves won’t change teaching, as math practice standards* are not new, but I do read an increased emphasis on reasoning. To the extent that test-makers incorporate such skills and mindsets into assessments, possibly through performance exams and open-ended assessment items (which have their own set of challenges from economic, logistical, and equity standpoints) math teaching will have to change.

      I don’t foresee overnight change, however, as It is very hard at the upper grades to encourage a focus on mathematical practice if “tricks” and skills have been emphasized in previous grades. Not only may students lack a tool kit for discovery-based learning, they also have preconceptions about math, math class, and what “good” math teachers do. Teachers also need training, support, practice, and time to adapt to new teaching and assessment methods.

      New York state is preparing its education stakeholders for a drop in scores as schools and teachers adapt to the new methods of assessment, already used in some states such as Kentucky where the initial drop-off in scores was quite dramatic and year-to-year growth incremental at best.

      * 1. Make sense of problems and persevere in solving them.
      2. Reason abstractly and quantitatively.
      3. Construct viable arguments and critique the reasoning of others.
      4. Model with mathematics.
      5. Use appropriate tools strategically.
      6. Attend to precision.
      7. Look for and make use of structure.
      8. Look for and express regularity in repeated reasoning.

    • sixth-grade teachers

      I have been reading about methods for teaching and understanding whole-number division since the concept of division is a foundation to fraction operations which are taught in my grade. You and the other teacher found a way of teaching (with toothpicks) that was engaging and valuable to the students.

      Something I found interesting in my reading is the distinction between distributive division and group division (also called measurement or quotative division) and experts argue that both types should be used when framing division:

      - Distributive division: also called paritive or sharing. Number of groups is known, number in each group is unknown e.g. 4 friends share 12 apples equally. How many does each receive? (partition or share the apples) More commonly encountered.

      - Group division: also called measurement, quotative, or chunking and based on repeated subtraction. Number in each group is known, number of groups is unknown e.g. Marie is cutting a 20-foot ribbon into 2-foot lengths. How many lengths of ribbon will she have? or John has 20 graham crackers and is preparing plates of 4 crackers each. How many plates can he prepare? Less commonly encountered but useful foundation for fraction division.

      Also students should be able to interpret remainders meaningfully: sometimes the quotient should be rounded up in case of a remainder (22 students on a trip, 5 students per van, how many vans are needed?), sometimes rounded down ($22 pizza budget, $5/pizza, how many pizzas before tax / tip if not possible to subdivide), and sometimes written as a fraction or decimal ($22 spent on 5 identical picture frames, amount spent on each frame). Not only useful from practical life standpoint but again useful in fraction operations.

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Metro Atlanta
Middle School

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