The circuit that you built is designed for proximity detection. It doesn't really make sense for what you're trying to do. So, to clarify, you're not supposed to use a 555 timer or a microcontroller? If that's the case, then the Arduino is definitely out of the question. What do you mean you'll use "counters" for counting?
Using a circuit based on reflection is going to run you into a lot of problems. What are you reflecting off of? Did you plan to reflect on something on the other side of a doorway or the person walking through? If you're reflecting on the person walking through, then you deal with the possibility of the person wearing something that will absorb all the IR light.
I'd recommend you put your IR transmitter on one end and your receiver on the other, and watch for the break. maybe you can have a capacitor discharge when the beam breaks giving you a solid point of contact...
Additionally, I think you'll find most people here are adverse to doing school assignments. It's not that we don't mind helping, it's usually that we get people who want us to do all the work: "Hey I have to solve this problem, how do I do it?" It's a bit offensive to us that someone does so little and wants success. If you pass this class, what good is it to you if you didn't learn anything? On the other hand, if you are having trouble solving a specific problem such as "Why didn't this work?" or "I'm looking for ideas on how to...." people will be much more inclined to help you out. The suggestions I laid out above are ideas, but you still have to do the design work to make a functional circuit.
If you just want a solution, you'll likely get little help here.