Several times in this course, you will be asked to write or record a good explanation of something mathematical.  Sometimes you are explaining a solution, and sometimes you are explaining an algorithm.

When explaining an algorithm, you should keep in mind:

• if you are supposed to choose your own numbers for the explanation, make sure you choose ones that will let you illustrate well the algorithm you are explaining. 

• Thus, if your algorithm involves trading/exchanging/etc., make sure that the numbers you choose will give you a good variety of such steps
• If your task includes modeling the algorithm with manipulatives, make sure that you choose numbers small enough that you will be able to fit all of the manipulatives you need on the screen at once
• As much as possible within the guidelines above, make sure that all of the digits are different--this makes it easier to keep track of what's going on (if there's more than one 4x5 in your problem, there's a confusion about which is which, and indeed, if you are adding, it is ideal to not have sums of 11, since then it is harder to distinguish between the tens and ones place).

• Use correct language:
  
• place value language; know what place value you are working with, and name it correctly.  Name place values wherever it is not undesirable to do so.
• parts of an operation: know which is the divisor, and which is the quotient, know what a product and a factor are, and use those words correctly
• In operations (subtraction and division) make sure that you are reading the problem in the right order: if you are saying divide __ by ___ or subtract ___ from __, make sure you have the numbers in the right order.
• Don't use "borrow" or "carry", instead use: trade, exchange, rename, regroup and record
• Explain the trading steps explicitly.  For example: Now we trade 1 hundred for 10 tens, and that's a fair trade because 100=10 tens.
  
• Go from concrete to abstract
• If you have both a manipulative and a number way of showing the algorithm, do each step first with the manipulative, and then with the numbers, and
• explain each step in the number work as a way of recording what is happening with the manipulatives
• Make the connections step by step--don't wait until you have done the whole problem with the manipulatives and then make the connections: do a step with the manipulatives, and then the same thing with the numbers.  My husband describes one of his friends' masters thesis as dropping a wall on the reader one brick at a time.  I want you to drop an algorithm wall, very carefully, on your listener, one brick at a time.
• Think about what your audience knows:
• If you are showing how a student who is not yet at the derived facts stage would solve a problem with small numbers, you should be counting, not using known facts
• If you are showing how to do a multi-digit algorithm, you may assume that your audience knows the basic facts for that algorithm, but not that they know the procedure for the algorithm you are explaining
• If you are showing the multi-digit algorithm for multiplication, you can assume that your audience knows the multidigit algorithms for addition and subtraction, but not multiplication or division.
• Do not assume that your audience knows what should be recorded where or why.
  
• Feel free to copy me.  That's what the examples are there for.  You don't need to reinvent the wheel, you just need to understand it well enough that you can do it right.

Good explanations are your stock in trade as a teacher.  Even if you have a very student-directed class, you need to be able to make good, concise, complete explanations at the point where you are summarizing, and there will be students for which nothing else works (you will also find students for which this does not work.  If you find something that works all the time, patent it.)

I insist that you practice giving good explanations.  You should practice your explanations 2-3 times before you record them.  You will be glad later, when you are teaching, if you have the words practiced so your explanations come out right the first time. If your explanations are not nearly perfect, I will insist that you redo them before I am willing to grade them.