Do Numbers Make Sense?

It is my undocumented, unresearched, unvarnished opinion that most people (adults and children) who are math phobic lack a fundamental understanding of place value and number sense; the Hierarchy of Numbers.  In children, it most commonly manifests in one or more areas:

  • math avoidance / uncertainty / fear
  • errors when reading larger numbers
  • inability to identify the place value of a digit
  • uncertainty when comparing numbers (>, <, =)
  • errors in exchanging
  • errors in computations where a 0 place-holder is required
  • overreliance on tricks and shortcuts
  • confusion in estimation and rounding

Oftentimes these manifestations are not recognized as being rooted in a lack of the foundational understanding of the hierarchy, so the remediation chosen by the teacher is specific to the manifested error; we treat the symptom rather than the root cause.

Montessori’s elegant math materials evidence her belief that a firm foundation in numeration is vital.  From Golden Beads to the Decimal Division Board, each material reinforces the hierarchy of numbers while teaching a computational concept.  Yet some children can work successfully with these materials without fully internalizing the physical reality of number sense.

I have come to the conclusion that one reason that more work is not done on the hierarchy is the lack of engaging activities and problem sets to reinforce hierarchical concepts.  The newest release to the Lockhart Learning website is the first step to filling that need: 30 Hierarchy of Numbers: Command Cards and Story Problems.  These cards are arranged in three levels: matching quantity to symbol, hierarchical sequencing, and place value in larger numbers.  Each card is designed to deeply connect the child with the hierarchy through a Montessori material that is likely in your classroom right now.   Each card takes a unique approach to the hierarchy – no 2 cards are exactly alike.  This means that the cards can be used diagnostically.  If a child struggles with one particular question, there are likely one or two specific skills that should be revisited.  Please go to the website’s MATH page and “Get a Sneak Peek” for more details and examples of the types of commands and problems in this card set.

It is my intention to continue to work in coming weeks on additional math materials to address this problem.  I plan to design command cards / problems that can be used each time a new material is introduced to refresh the child’s understanding of the hierarchy in the context of the new material.

Next up on my drawing board is writing relevant story problems.  We have so little time and so much to teach; it seems like we should be taking advantage of every opportunity for cross-curricular applications, to enhance relevance for our children and to get twice the learning from a single activity.  Story problems are ripe for just this application.  Rather than calculating “how many cookies each friend gets”, why not calculate the average amount of rainfall in a given city or the population density of different parts of the world?

I plan to work on these math projects, releasing each to the website as soon as it is developed.  This means that they will trickle out over the next year.  If you are interested in staying informed about new releases, please be sure to sign up for my e-mail newsletter. (Go to the Lockhart Learning home page and click on the invitation in green text.)

Also, if you are planning on attending the AMS conference in Orlando in March 2013, please consider attending my talk, To Infinity and Beyond, which will discuss a wide variety of methods for improving our use of Montessori and teacher-made math materials.  I would love to meet you and hear about your experiences!


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