A Mathematical Approach To Housetraining

A Formula For Success

Being an Engineer myself, I have found that any time I can apply a mathematical equation to one of life's problems, I am then able to understand the variables involved and better able to find the solution. So, what about a formula for training puppies? 

House training is a learned behavior. It is up to the pet owner to teach it. Dogs learn by repetition. Every learned behavior can be taught through a process of repetitive action followed by reward. So, let’s look at a mathematical approach to achieving the learned behavior of house breaking.  

In mathematical terms we can assign the variable “N” to the value that represents the magical number of successful repetitions you have to get to before the behavior is learned. That number is different for every dog and every behavior. Additionally, achieving that number is more complicated than simply executing that behavior “N” number of times.  

For house training, we want to teach the dog to eliminate outside of the house in a specific area we will designate as the “success” area. If it were a simple mathematical formula, then when puppy eliminated in the success area “N” times, he would be trained. If that were the case, we could just leave puppy outside, in the success area, until he eliminated “N” times and from then on he would be ready to live in the house. But, that type of training does not work as there is no reward involved in the puppy’s mind. He just goes when he needs to. He does it where he is.  

Somehow you need to teach puppy that going in certain places please you and in other places displease you. Further, because you can not reason with puppy, or explain to puppy anything, you must be careful not to show your displeasure for past indiscretions. In other words, you can not punish your puppy for something it did not do the instant before the punishment. If you do not heed this advice you will be forever trying to housebreak your puppy as you will soon see. So, let’s take a look at a mathematical formula to describe the process for training this behavior.  

We will let “N” represent the number we need to achieve before the behavior is learned. For housebreaking our hypothetical puppy we will arbitrarily assign N a value of 300.  

For our equation to have any meaning, we need to pick units which we can understand, for this behavior we will pick units of weeks. This means all parts of our equation need to have variables that can be expressed in terms of weeks or factors thereof.  

Now, for our puppy to learn this behavior he has to have the ability to experience reward for proper behavior and to do this he has to have a choice. His choice for house training behavior will be to choose between going in the house and holding it when he is in the house. We will call this variable element “C” for choice. It should be obvious that for “C” to have any value other than zero, the puppy needs to be in the house. In fact, “C” is a function of the average number of times a day the puppy will eliminate times the number of days per week the puppy is in the house so that he can make a choice. In other words, “C” represents the number of choice opportunities per week.  

Next we have to develop a term for the part mistakes in the house play in the learning process. Given the puppy has choices, obviously good choices, or successes will help training and bad choices, failures will set training back. We will call mistakes in the house, or accidents wrongful elimination. Only un-witnessed wrongful elimination will be considered a setback and we will call that variable “W”.  

Obviously un-witnessed wrongful elimination allows the puppy to believe that what it has done is okay and so we will need to apply a factor to it which represents the degree of setback to the learning process that occurs. We will call this the “un-witnessed elimination factor”, or “P”, and “P” will be able to take on one of two values. The reason for this is that there are two ways a pet owner may handle a wrongful elimination or mistake. The first is when the owner simply cleans the mess up and disinfects the area. In this case we assign P = 1.5. The second is when the owner makes the accident worse by inappropriately punishing the puppy for something the owner did not witness. For this P = 3.0. Yes, an un-witnessed accident is a setback to the learning process, but an inappropriately punished un-witnessed accident is twice as big a setback.  

What about when the owner witnesses the accident? Well, that is called a success. Any time a wrongful elimination is caught in progress, the owner has the opportunity to turn it into a positive learning experience. The owner can tell the puppy NO! in a tone that tells the puppy this is not okay and then immediately take the puppy outside. This action will count well towards the learning process.  

Finally, we will assign to our formula a Forget Factor, “F”. Obviously, the more frequent exposure the puppy has in any given week to the “choice” zone, the more opportunity he has to learn, as is reflected in the larger value for “C”. Conversely, the more days the puppy does not access the “choice” zone, the more opportunity he has to “forget” what he has learned. Since it is virtually impossible to housebreak a puppy that does not spend at least 1 day a week in the house, I will allow the forget factor to take on values of .1 to .6 for the remaining 6 days maximum of non-house time which the puppy may experience.  

In Summary:  

    N = Number of Learned Behavior Experiences Required For Behavior To Be Learned  

    C = Number of Choices In A Week or Learning Opportunities  

    W = Number of Wrongful Eliminations In A Week or Negative Learning Opportunities  

    P = Un-witnessed Eliminations Factor, value = 1.5 for correct handling or 3.0 for incorrect handling 

    F = Forget Factor, value equal 0 for 7 days in the house, .1 for 6 days, .2 for 5 days, .3 for 4 days, .4 for 3 days, .5 for 2 days and .6 for 1 day. This is a multiplication factor representing the opportunity the puppy has to forget what he has learned so far.  

So, here is what our formula may look like:  

    Length of Time to Housetrain Puppy (in weeks) = (1+F){N/(C-PW)}  

For instance, let us say for this particular puppy to “learn” that he cannot go in the house “N” must equal 300.  Suppose too, that the puppy is only allowed to be in the house on weekends, Saturday and Sunday. Suppose on the average he eliminates 6 times every day. So on the weekend (2 days), C = 6x2 = 12. This puppy has 12 learning opportunities per week.  

Also, since on Monday through Friday puppy is not allowed a choice, (that’s 5 days of opportunity to forget), let’s assign F the value of .5. Let us also suppose our puppy has 4 un-witnessed wrongful eliminations on an average weekend, this assigns W a value of 4. Let us also say that the owner inappropriately punishes the puppy for an un-witnessed wrongful elimination, so P = 3.0.  

Then, using these values let’s compute the result: 

    Length of Time to House train Puppy = (1+.5){300/(12-3.0(4))}   

     = (1.5){300/(12-12)} = (1.5){300/0} = ERROR! 

Oops! As you can see, we will be trying to divide by zero. That simply won’t work. The answer is infinite, we will never get this puppy house trained if he only gets to come in the house on weekends and we don’t see one third of his eliminations and then punish him for any time we didn’t see him have an accident. We need to do better than this.  

What if we don’t punish the puppy for un-witnessed accidents but everything else remains the same? In this case, P = 1.5. 

    Length of Time to House train Puppy = (1+.5){300/(12-1.5(4))}  

    = (1.5){300/(12-6)} = (1.5){300/6} = (1.5){50} = 75  

Well, 75 weeks is almost a year and a half of weekend only training. But, that is better than never. You can see very quickly the effect inappropriate punishment has on a puppy’s behavior learning capability.  

Now, what if this same puppy had no un-witnessed wrongful eliminations on the weekend?  

    Length of Time to House train Puppy = (1+.5){300/(12-1.5(0))}  

    = (1.5){300/(12-0)} = (1.5){300/12} = (1.5){25} = 37.5  

Wow, only 37.5 weeks to a house trained puppy just by making sure to watch puppy’s every move. This is a very optimistic number, as it requires absolutely no accidents during the times puppy has a choice. This would require a very diligent owner.  

Now let’s say this puppy is in the house every day, but the owner works a 9 to 5 job. Let’s say the owner free feeds and waters the puppy, so the puppy has no predictable elimination schedule. While owner is at work, puppy has 3 un-witnessed wrongful eliminations every day. Given this puppy also eliminates 6 times daily, C = 6x7 = 42 and W = 3x7 = 21. Since puppy is in the house every day, the Forget Factor, F = 0. Let’s also say this owner does not inappropriately punish her puppy when she gets home from work for his accidents, so P = 1.5.  

    Length of Time to House train Puppy = (1+.5){300/(42-1.5(21))}  

    = (1.5){300/(42-31.5)} = (1.5){300/10.5} = (1.5){28.57} = 42.86  

Well, that’s not too bad, but what if this owner put the puppy on a schedule or could come home at lunch time and prevent just one mistake daily? Now W = 14 instead of 21.  

    Length of Time to House train Puppy = (1+.5){300/(42-1.5(14))}  

    = (1.5){300/(42-21)} = (1.5){300/21} = (1.5){14.29} = 21.43  

Wow, by preventing just one more accident she cut the time in half! Already you can see the benefit of putting puppy on a feeding and watering schedule so you can better predict his elimination times and thereby prevent un-witnessed wrongful eliminations. Also, you can see the benefit of supervising your puppy as closely as possible, as opposed to giving your untrained puppy free run of the entire house.  

Of course, there is no exact science here, no mathematical equation with guaranteed answers, but the facts do support the trend in the results presented here. In short, the more opportunity for your puppy to be in the house, and the better you are a preventing mistakes by frequently and consistently taking him to the appropriate spot, praising him when he goes, the sooner he will be able to learn what you are trying to teach him. Additionally, every accident undoes some of the learning process. Every time you skip a day in the training process by leaving your puppy outside, undoes some of the learning as well.  

Now you have a formula for success,  but it's up to you to fill in the variables .... Good Luck!
 

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