When this email promoting a webinar appeared in
my inbox, I was intrigued … and terrified. 
What do you mean “reconsider using worksheets”?  These are the life blood of math classes.  How else are students supposed to practice
basic skills?

Intrigued, I signed up for the webinar.  Terrified, I listened carefully.

The webinar “Why We Should Reconsider Using Worksheets (And
What We Should Be Doing Instead)” was offered by Robert Kaplinsky, a math
educator from Southern California.  I
follow him on Twitter (@robertkaplinsky) so I had a good sense of his approach to mathematics
education.

In his webinar he presented his concerns with the use of
worksheets:

Full disclosure – I have used worksheets as busy work.  I’ll admit it, it’s an easy sub plan.  But I also have experienced his other three
points.  When students are given a
worksheet, they go into “git ‘er done” mode often without deeply understanding
the concepts behind the problems.

While there can be a role for traditional worksheets in some
situations, Kaplinsky promotes as an alternative “open middle problems.”  Here’s an example:

Notice that there is not just one solution.  There are, in fact, many.  This type of problem encourages students to
uncover the mathematical concepts behind the problem.  These are best done in groups where students
can talk through the problems.  Then,
sharing among the groups reveals the variety of solutions and leads the students
to a deeper understanding of the concept.

Here are the benefits he sees to these types of problems:

No longer terrified, but fully intrigued, I decided to try this
with my Algebra 2 Trig classes.  On
September 25 my Period 5 sophomores would be taking the NWEA MAP test with Sara
Anderson’s sophomores while I would have a block period with our combined 34
juniors!  Instead of giving them an 80-minute
study hall (which I’m sure would not have resulted in much studying), I decided
to test out the open middle problems.

I projected these problems in front of the class while groups
armed with whiteboards went at it:

While these problems do not reflect Algebra 2
Trig content, I thought that something familiar would be a good way to introduce
this type of problem.

What I found most interesting was that the students very
quickly focused in on the concepts behind the problems.  For the inequality, they knew that the first
box had to be a negative number.  Instead
of having them practice row after row of “multiplying or dividing an inequality
by a negative number” worksheet problems, this open middle problem brought to
the surface and had them apply what they had learned in middle school: multiplication
or division of an inequality by a negative number reverses the direction of the
inequality sign.

I also enjoyed watching them talk about math.  So often students just want to write out the
steps without explaining their thinking.

I had found Kaplinsky’s “benefits” to be true.

Here’s a link to his website with tons of open
middle problems.

The following week I had planned to teach piecewise
functions – a challenging topic for many students as it requires them to graph
only a “piece” of a function.  And if it’s
a linear function where the domain does not included the y-intercept, well, you can just forget about that!

I did my usual flipped “video the night before” and “exercises
in class the next day” on Tuesday.  But on the block period, I gave them my own version of open middle piecewise functions problems:

What I saw was students referring back to their work from
the previous day, explaining the concepts to each other, asking questions, and
being creative.  Multiple concepts beyond piecewise functions were
reinforced: slope, different types of functions, what a function is, how to
restrict the domain, among others. 
Success!

While I will still continue to use worksheets as an optional
review for skills-based problems, I am looking for other opportunities to
integrate open middle problems into the curriculum.

I am also wondering if this sort of problem could be used in
disciplines other than math.  Any ideas?