Lesson #2 - Augsburg College

An Introduction
Hi and Welcome!
Hi! We spent one year teaching Challenge Math to fifth graders in Northfield, Minnesota, and we created
this curriculum to provide you with fun lessons that you can use when you teach Challenge Math. The lessons
are organized by topic, and we recommend that you spend two to four weeks on a topic before moving on to the
next one. We’ve provided you with more lessons than it would be possible to teach in a year, so feel free to pick
out your favorites.
We were all math majors at Carleton College, so we think that math is pretty cool. Math, though, is not just
fun at the college level; it’s fun at the elementary and middle school levels as well. Math teaches students how
to use their imaginations to solve problems, and thus encourages them to think creatively. Math also teaches students
how to organize and explain their thoughts, which is a skill that they can apply to all areas of their lives.
It’s true that math can be hard and sometimes frustrating for students, but as long as the lesson remains fun, with
your guidance your students will work past their frustrations and achieve success!
Where are Fifth Graders Developmentally?
Fifth graders are in a period of transition. From approximately the ages of 6-12, children develop the ability
to think about concrete events, which they can physically see and manipulate. By the end of this developmental
stage, children have the ability to observe more than one quality of an object at a time and can classify objects
according to multiple characteristics. They should also understand the concept of conservation by the end of this
stage. A classic example of conservation is that the volume of a short, fat glass can be the same as the volume
of a tall thin glass, even though one of the glasses may look bigger. Fifth graders should be able to tell you how
the taller glass compensates for the shorter one. They should have also developed a sense of identity (the idea
that if nothing has been added to or subtracted from an initial quantity, the initial amount will remain the same).
Finally, 5th graders should have developed a sense of reversibility; they can explain how one effect can reverse
or negate another, such as addition negating subtraction or multiplication negating division.
These breakthroughs are very important because they are necessary for the development of mathematical
reasoning. Many adults never progress beyond this stage of development because the next stage can require
a bit of difficult intellectual digging. Challenge Math provides the perfect opportunity to help students begin
this next stage, which is commonly referred to as the formal thinking stage. In this stage, students develop the
mental processes necessary for thinking about abstract events, which are not visible or tangible. Thinking abstractly
can prove to be much more difficult than thinking about concrete objects or events. It requires a mental
cataloguing of ideas that must be practiced and encouraged. The ability to think abstractly is important, if not
necessary, for success in all areas of education. Developing this ability can begin with opening one’s mind to
the possibility that there are multiple ways to view a problem and approach it. Challenge Math is a great setting
for developing these new ways of thinking because it provides opportunities to discover solutions through
multiple avenues.
Where are Fifth Graders Academically?
Within any group of students, a range of mathematical ability is inevitable. In a class of thirty students, it is
likely that there are three or more broad categories of ability, such as under-achieving, over-achieving, and one
ore more categories between these two. But what constitutes standard achievement?
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