I believe that students in most classes do what we call "homework". Some of them do this in study halls during school hours, and frequently students do their homework at home. But in my experience, most students come to school with their homework, find out if they have done it correctly, and that is the end of it.

In my writing-intensive geometry class I expected the students to correct their homework. And by correcting it, I meant that they were to do far more than just crossing out their wrong answer, but also to write the correct answer, explain why their answer was wrong, and why they made that mistake. They were also expected to write a complete explanation of the correct way to do the problem, as you will see in the remainder of this page.

One question in a homework assignment was as
follows: "*A circle is inscribed in a quadrilateral. The lengths of
the sides of the quadrilateral are as follows: AB = 23, CD = 12. ABCD
is circumscribed around the circle. Find the perimeter of
quadrilateral ABCD.*" Karen had gotten the problem wrong on the
homework when she first tried it, because she had not thought of
using two variables, and was overwhelmed with the problem of trying
to label the diagram and work with just one variable. In class the next day, as we were discussing the homework, I
suggested that those students who had trouble should try the problem
again, using 2 variables. In her HW corrections, Karen drew the
following diagram, and used algebra to find the answer:

Here is her written description of her successful
method in solving the problem: "*First, I
drew the diagram above and labeled all the segments. This time, I
used two variables, x and y, which really helped. Here is my first
equation: *

*I was worried because there were two variables,
and also because the equation was so long! So, when I'd tried it
earlier, in my homework, I guess I just got overwhelmed and gave up
too soon. But when I had a second chance (and some hints!) I realized
I shouldn't worry about the long equation - I just need simplify it
and see what happens, so I crossed my fingers and hoped for the best.
It was very cool because when you see that long equation you think
'oh no!'; I simplified it a bit and this is what I got:*

*...which is very cool, because all the x- and
y-terms cancel out and you are left with the perimeter = 70! It was
so amazing - just when I was about to give up, it all came out great!
Next time, I hope I can stay calm and get it right on the first
try!"*

The students were asked to do their homework using a regular pencil, and then make corrections in red pencil so that they can quickly spot areas in which they are having trouble. These corrections prove extremely useful when studying for tests.

I believe that all of this helps students to learn, and to cultivate habits of mind that will serve them well in later years. The students themselves, though they grew weary of writing homework and test corrections (see chapter 3), nevertheless saw their value, as you will see in their comments below:

"*I chose this piece to put in my portfolio
because I feel that homework is something that shows the amount of
effort you put into the class on your own time. I am proud to say
that I always do my homework, and if I did not understand a problem,
I make sure that it is explained in class by a classmate, or by the
teacher. I always show my work, so that I can study my homework
papers before a test, especially the parts where I made a mistake, so
I don't make that mistake again!*" Serena

Danny had a great learning plan, which helped him
to be an A student: "*I think homework is a great factor in how
well you do in class. Homework also reflects how much work you do as
a student. I feel that I put a lot of effort into my homework and
notes. In addition, I have a code for my homework: If I feel the
homework problem is challenging and easy to forget for the test, I
circle it and so when I study for the test, I start with the circled
problems; I know exactly which problems to study. If I circle and
star the problem, this means either I still don't understand it or
that I have a question.*"

Elise wrote a long essay on this topic; when
reading it I was very impressed with her effort and her tenacity. She
did very well in my class, so I knew that her effort and tenacity
paid off! She wrote: "*I do all of my homework in a notebook, and
my test corrections too. This ways I always have my previous work to
look back on, especially when studying for a test. I always take time
to show all of my work in each problem, so if I find out that my
answer is incorrect I can go back through my work and find out where
I went wrong so I don't make that mistake again. I write down all the
steps in the solution when I do each problem too, because that way I
can see exactly where I went wrong when I do my
corrections*."

Janine added some nice drawings, to add color to her comments about her homework, which had actually needed very few corrections!

Some of the assignments we did in my class are not part of the traditional curriculum, and were not in our textbook. One of these was one I called "Diagonals of a Polygon" In this assignment, the students were asked to consider polygons, such as triangle, quadrilateral, pentagon etc., and discover how many diagonals each polygon had. For example, a square has two diagonals - one from the upper left corner to the bottom right, and the other from the upper right corner to the bottom left. The students explored triangles (which have no diagonals), quadrilaterals, which have two diagonals, pentagons, which have 5 diagonals (which form a star), etc. I will leave it up to you to explore this idea and discover for yourself how many diagonals other polygons have, and if there is a formula for finding that. If you like, you can learn a lot more about this topic at the following web page: http://www.mathopenref.com/polygondiagonal.html. This website gave me the following formula for finding the number of diagonals of any polygon:

Formula for the number of diagonals

"The number of diagonals from a single vertex is three less than the number of vertices or sides, or (n-3).

There are N vertices, which gives us n(n-3) diagonals

But each diagonal has two ends, so this would
count each one twice. So as a final step we divide by 2, for the
final formula: where **n **is the number of sides (or
vertices)

And then the formula is:** n(n-3) divided by 2
**

Another way of writing that is:

In her homework corrections on this assignment,
Julie wrote: "*This worksheet was fun because it makes you feel
like you invented the formula all be yourself! I think this was a
great assignment because it was really hands-on, and also it was cool
that it wasn't even in the textbook! I liked figuring it out because
nobody helped me and I figured almost all of it out myself. The
mistake I made was a really small one, and writing the correction was
easy. For some reason, I when I multiplied n times n I wrote 2n!
Silly me; of course n times n is n squared! Even though I didn't get
it 100 per cent correct, I feel like I learned a lot
anyway."*

The students constructed and explored a variety of polygons and wrote explanations and essays about what they discovered. In exploring a pentagon, Jared wrote the following comments:

*"**This project was fun because it made me
feel like I 'invented' or figured out a formula all by myself. I
thought this was a very good project because it was more of a
'hands-on' thing, not just 'reading it in a book and taking their
word for it' thing. I always think you learn more this way because
you can see from your own experience the 'how' and the 'why' of it.
And if you forget the formula, you can figure it out again. I really
like the challenge of finding patterns in numbers and stuff. It's
actually like it's yourself who is the teacher, which is
cool."*

In his homework corrections, Jared wrote "*I
chose to put this piece because I feel that homework is something
that shows the amount of effort you put into a class on your own
time. I am proud of my corrections because I worked really hard on
them*."

On "Parent Night", when the students' parents came
to my classroom, I showed them many of the writing assignments that
their sons and daughters had completed. The parents were amazed at
the amount of work the students had done, and were fascinated by the
student papers that I shared with them. I asked the parents to write
a comment on how they felt about this class, and the
"writing-intensive" approach. One parent wrote: "*I enjoyed reading
Nicole's reflections, and seeing some of the assignments that she
completed. I was especially interested (and impressed) with the
Homework Corrections that the students are required to write. I don't
think we did that when I was in school, but what a great idea! It is
so important for the students to figure out what they are doing
wrong, so that they can do it right next time.*"

Another parent wrote "*I think it's wonderful
that the students review and correct their work, and even explain
where they went wrong. It is so much better than just bemoaning their
mistakes and then moving on without getting to the heart of the
problem*!"

And even the students agreed; even though they
thought the corrections were a "pain", they began to see that the
work they did achieved results. Kelli wrote "*At first I was very
unwilling to do the HW corrections; what a pain! But after completing
a few, I found that it really helped. I realized that although I
might get some other problem wrong in the future, I sure wouldn't get
that one wrong again*!"

Perhaps the comment that pleased me the most was
Karen's, when she wrote: "*In doing my corrections this year, I
think that I really got to the root of my mistakes, which helped me
learn. This is something that I really liked about this class. In
past math classes, I never had this valuable experience, and I think
that it has made a really big difference.*"

Ibn Khaldun (1332-1406)

**Go To Homepage** **Go To Introduction**
**1) Constructions** **2) Clock Problem** **3) Test Corrections** **4) ASN Explain** **5) Thoughts About Slope** **6) What is Proof?**

**7) Similar Triangles** **8) Homework Corrections** **9) Quads Midpoints** **10) Quads Congruence** **11) Polygons**

**12) Polygons Into Circles** **13) Area and Perimeter** **14) Writing About Grading** **15) Locus** **16) Extra Credit Projects**

**17) Homework Reflections** **18) Students' Overall Reflections** **19) Parents' Evaluate Method** **20) In Conclusion**