||Jul 13, 2003 by Teacher Education
|| This lesson is one in a series of mathematics investigations sponsored by
Annenberg/CPB as part of the Private Universe Project in Mathematics. It
involves interactive learning using virtual manipulatives as well as an archived
discussion board for teachers. The underlying assumption is that knowledge and
competence develop most effectively in situations where students work on
challenging problems, discussing various strategies, argue about conflicting
ideas, and regularly present justifications for their solutions to each other
and to the entire class. It also includes a videotape/streaming video of
children attempting the same problem.
|| The goal of this project and this problem in particular is to participate in and
observe students participating in a hands-on mathematical problem solving
activity. The goal is also to allow teachers (in-service or pre-service) to
reflect on the nature of mathematical learning.
Elementary-aged children come up with a variety of mathematically sound proofs
for the number of combinations in the Towers problem--and then generalize their
solution to apply to all towers. Results can be used by the children to
hypothesize about towers three-high or five-high.
|Target Student Population:
|| This site would be appropriate for both pre-service and in-service teachers. It
could be used as part of a professional development program in mathematics
It could be easily used by elementary-aged students.
|Prerequisite Knowledge or Skills:
|| Much of the prerequisite knowledge is available to participants through a
bibliography which accompanies the task. Online articles, and references to
those which are not available online are linked to the problem solving activity.
Cooperative groups skills are needed for students solving problems this way.
|Type of Material:
|| Interactive activity and discussion board; simulation.
|| The material could be used by an mathematics methods class or a content area
course in combinatorics. The material at this site is appropriate for
professional development of elementary mathematics teachers. Everything needed
to try out the activity with elementary chidlren is provided along with links to
publications showing the research base.
|| Either a shockwave plug-in or the ability to download is required for a
stand-alone Mac program. Links to download and install the Shockwave plug-in
are available for both Macs and PC. In order to view the videos that accompany
activity, participants can use streaming video available from Annenberg or
|| The strength of this site is the breadth of the activity and the accompanying
support materials. It is also indicative of the other materials available at
the overarching website for Annenberg/CPB.
The site provides the opportunity for participants to solve a mathematical
problem, reflect on it, and read theoretical analyses of the problem and the
underlying information on student learning. The interactive discussion would be
a plus, but it appears to be available only in the archive.
Math problems requiring justification through interaction between kids should
help them develop awareness when they have wrong answers. The ability to
explain to each other the reason for their answers should be more valued than
simply getting the right answer. In this activity, there is more than one way
to distinguish patterns that lead to a right answer, thus introducing the
excitement of creative thinking among the children and aiding students in
looking at different approaches to a problem. The tower block activity should
help kids move beyond the goal of getting work done toward deeper metacognition
that is required for real understanding of math and science.
|| Because the material has been available for sometime, there does not appear to
be the opportunity to participate in the online discussion that connects to the
activity. However, the activity stands alone as a learning opportunity.
According to Alex Griswold, one of the authors, "selection of the original
of 25 students in this study was random" and they made "a conscious effort to
make their group representative of the entire student body." But "today they
are typical young adults in all respects,
except that they love math more that
most students and feel comfortable doing it." Unfortunately, the reflections
"of the same students, now seniors in high school, reflecting back on the
experience," is not available on the web. More evidence may be required for
elementary teachers who are faced with parents who expect math to be drill and
Potential Effectiveness as a Teaching Tool
|| Everything necessary for a strong learning opportunity is available in one
place--the video materials (either by streaming or by purchasing the cassettes),
the virtual manipulatives (on the web, or available as a download for Macs),
the problem itself, and the opportunity to read archived discussion by teachers.
In a transcript of the video, one student declares "We're not scared of being
wrong. They liked us being wrong, like when we were wrong they asked us why we
thought of doing that. And you like always had like logical reasons behind why
you did it, so it like turns into something." Too few of our college students
come with these important attitudes about learning. The potential effectiveness
of teaching tools that give children such attitudes toward learning are
|| The technology may put off some viewers.
The Blocks program as a stand-alone application that can be downloaded for
Macintosh is superior to the shockwave version because the towers can be printed
for analysis and discussion. The fact that Shockwave version cannot be printed
means that children have no concrete evidence to use when they try to convince
others that they have found all possible towers four cubes high.
Ease of Use for Both Students and Faculty
|| Each of the components of the website is useful on its own merits. Together
they provide a well-rounded package for investigating children's learning in
mathematics and developing one's own approach to problem solving.
The Tower Problem provides an engaging activity that teachers can repeat with
their own students at appropriate grade levels. The resources provide one
important step for math teaching,
with the work not done in isolation or as an
end to itself, but as part of a continuum toward getting students aware of the
logic behind the math they learn.
|| The technology involved may be daunting to some newcomers to web objects. Those
with dialup connections will not be able to use the streaming video technology
for viewing videos. The streaming video schedule may not work for some
participants, requiring that they purchase the videotapes.
A warning should be added to clarify that the Mac version for download can be
used to print the towers whereas the Shockwave version cannot be printed. The
Mac version is for OS 9 but it works well in the Classic Environment from OS 10.
|Other Issues and Comments:
|| This is only one problem in a comprehensive site that provides learning tools
for teachers and use in higher education.
It is not clear how to convince parents and teachers that children who are
reflective and confident about their math skills are needed. But the Block
Towers exercise is certainly a step in the right direction, with a link to
longitudinal research that reveals important outcomes from this approach to
|Comments from Author:
||Thank you for considering "The Blocks Problem" from Private Universe
Project in Mathematics. We feel honored to have our resource included
in your site. I'm very happy with the review as well, and would only
like to clarify that the online discussion ("channel-talk") is indeed
still active; that is, it is available for use should someone come
with a discussion topic. Our teacher professional development
workshops, of which Private Universe Project in Math is an example,
have "rolling enrollment." That is, they continue to be broadcast,
and teachers can continue to register to participate,
for years after
the initial broadcast. There hasn't been discussion on the list in
the past year, but it is still available as a forum; we have other
series which have "sprung to life" again years later when a new group
of interested teachers signs on. Since the apparent inactivity of the
discussion list was a concern for the reviewer, I wanted to clarify