Chapter 8 - Homework Corrections

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:

x + x + 23-x + 23-x +12-y +12-y +y =70

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:

2x + 46 -2x + 2y + 24 - 2y = 70

...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."

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!"

John wrote "Sometimes on the test I really don't know what I'm doing on a particular problem. But when I get the test back and have to do that darned problem all over again, then I'm forced to figure out what I did wrong and how to fix it. And I feel like I have accomplished something. Maybe, if I do this enough, I'll start getting them right the first time around. That would be cool -Wow! No more homework corrections!"

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."

"Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence."

Ibn Khaldun (1332-1406)