1. Deconstruction and Reconstruction of the Multiplication Principle
Elise Lockwood John Caughman
Oregon State University Portland State University
Zackery Reed
Oregon State University
2016 PNWMAA Meeting

2. The Multiplication Principle
“Product” Principle
“Fundamental” Counting Principle
Red
Blue
Green
Pink
Jeep
Red Jeep
Blue Jeep
Green Jeep
Pink Jeep
Tesla
Red Tesla
Blue Tesla
Green Tesla
Pink Tesla VW
Red VW
Blue VW
Green VW
Pink VW
It underlies many of the counting formulas that students encounter

3. The Multiplication Principle
)Discuss these statements for a bit…) This should lead to questions…are these even saying the same thing? Why the need for different kind of language? Are these mathematically equivalent/sufficient, etc.
Is this as varied as it gets?
Pedagogically what’s best?

4. Statements of the MP
We conducted a textbook analysis of statements of the MP
Large # of texts (60+)
Even larger # of statements (70+)
Encountered a *LOT* of variety….
We identified 3 types of statements
Structural statements
Operational statements
Bridge statements

5. The Multiplication Principle
)Discuss these statements for a bit…) This should lead to questions…are these even saying the same thing? Why the need for different kind of language? Are these mathematically equivalent/sufficient, etc.
Is this as varied as it gets?
Pedagogically what’s best?

6. Summary of 3 Statement Types
Some textbooks had both structural and operational statement in their narratives…

7. 3 Features of the MP
We also identified 3 central features common to the language used in many statements of the MP:
Require independence of # of options
Allow for dependence of option sets
Require composite outcomes to be distinct

8. Feature 1: Independence of # of options
How many possible outcomes are there if …. I choose 2 different cards from a standard 52-card deck, where the first is a face-card (J,Q,K) and the second is a heart?
Maybe it is 12 x 13 =156? (# face cards x # hearts)
Or maybe it is not!
If I first choose one of the 3 heart face cards, there are only 12 choices for the second card.
If I first choose one of the 9 non-heart face cards, there are 13 choices for the second card.
So there are 3x12 + 9x13 = 153 possible outcomes.
For a problem like the outfits problem, this statement works perfectly
Problematic for students, perhaps…there’s nothing wrong with this statement,

9. Feature 1: Independence of # of options
)Discuss these statements for a bit…) This should lead to questions…are these even saying the same thing? Why the need for different kind of language? Are these mathematically equivalent/sufficient, etc.
Is this as varied as it gets?
Pedagogically what’s best?

10. Feature 2: Dependence of Option Sets
How many different ways are there to arrange the letters in the word MATH?
The cardinalities are independent, even though the sets themselves may not be independent
Pretty simple should just involve 4!,
We want the MP to be able to solve something like that
If I pick M to be first or A to be first, the sets differ, even though there are the same number of options

11. Feature 2: Dependence of Option Sets
AHM _
AHT _
AHMT
AHTM
HM_ _
MH_ _
MT_ _
AH_ _
AM_ _
HA_ _
TA_ _
TH_ _
TM_ _
AT _ _
MA_ _
HT_ _
Set of Outcomes
AMH _
AMT _
AMHT
AMTH
A_ _ _
H_ _ _
T _ _
M_ _ _
AHMT AMHT ATHM
AHTM AMTH ATMH
ATH _
ATM _
ATHM
ATMH
HAM _
HAT _
HAMT
HATM
HAMT HMAT HTAM
HATM HMTA HTMA
HMA _
HMT _
HMAT
HMTA
_ _ _ _
HTA _
HTM _
HTAM
HTMA
MAHT MHAT MTAH
MATH MHTA MTHA
MAH _
MAT _
MAHT
MATH
MHA _
MHT _
MHAT
MHTA
TAHM THAM TMAH
TAMH THMA TMHA
MTA _
MTH _
MTAH
MTHA
TAH _
TAM _
TAHM
TAMH
THA _
THM _
THAM
THMA
4 x 3 x 2 x 1 = 24
TMA _
TMH _
TMAH
TMHA

12. Feature 2: Dependence of Option Sets
Set of Outcomes
A_ _ _
H_ _ _
T _ _
M_ _ _
{H, M, T}
AHMT AMHT ATHM
AHTM AMTH ATMH
HAMT HMAT HTAM
HATM HMTA HTMA
{A, M, T}
_ _ _ _
MAHT MHAT MTAH
MATH MHTA MTHA
{A, H, T}
TAHM THAM TMAH
TAMH THMA TMHA
{A, H, M}
4 x 3 x 2 x 1 = 24
{A, H, M, T}

13. Feature 2: Dependence of Option Sets
)Discuss these statements for a bit…) This should lead to questions…are these even saying the same thing? Why the need for different kind of language? Are these mathematically equivalent/sufficient, etc.
Is this as varied as it gets?
Pedagogically what’s best?

14. Feature 3: Distinct composite outcomes
How many 3-letter ‘words’ can be made using the letters a, b, c, d, e, f,
If the word must contain e, and no repetition of letters is allowed?
3 x 5 x 4
If the word must contain e, and repetition of letters is allowed? e
___ ___ ___ e e

15. Feature 3: Distinct composite outcomes
How many 3-letter ‘words’ can be made using the letters a, b, c, d, e, f,
If the word must contain e, and no repetition of letters is allowed?
If the word must contain e, and repetition of letters is allowed?
3 x 6 x 6
Notice that there ARE 3*6*6 ways to complete the process I just described. e
___ ___ ___ e e

16. Feature 3: Distinct composite outcomes
A purely operational statement does not work on that last problem
3 x 6 x 6 overcounts
Consider two ways of completing the process
The “eae” password is counted too many times
By an operational statement of the MP, there are 3 x 6 x 6 = 108 ways of completing the process.
This is true, but this is not equal to the number of distinguishable desirable outcomes. e a e e a e
___ ___ ___
___ ___ ___

17. Feature 3: Distinct composite outcomes
A purely operational statement does not work on that last problem
By an operational statement of the MP, there are 3 x 6 x 6 = 108 ways of completing the process.
This is true, but this is not equal to the number of distinguishable desirable outcomes.
Simply multiplying by applying an operational statement on the problem would yield an incorrect answer

18. Feature 3: Distinct composite outcomes
Bridge Statements address these issues
It’s worth pointint out that no statemetn of the MP would work to solve tha tproblem…
There are subtle aspects
Not all statements are created equal
Pause here and point out that these are some subtle issues, and it’s hard for students to interpret a statement like this.

19. Summary of 3 Features
Some textbooks had both structural and operational statement in their narratives…

20. Students’ Understanding of the MP
We had 2 calculus students reinvent MP in an 8-session teaching experiment
Steffe & Thompson, 2000; Gravemeier, 1999
They solved some initial counting problems
If you had to write a rule for when you are going to use multiplication to solve counting problems, what would you write?
We wanted to know how students might come to understand the MP
I’m going to talk through Initial, intermediate, and final statements of the MP
Particular tasks designed to elicit key ideas

21. Initial MP Statement
Use multiplication in counting problems when… there is a certain statement shown to exist and what follows has to be true as well.

22. Coin, Die, Deck Task
How many ways are there to flip a coin, roll a die, and select a card from a standard deck?
C: And off of those six possible options there will be 52 options for what cards you can pull from a deck.

23. Coin, Die, Deck Task
How many ways are there to flip a coin, roll a die, and select a card from a standard deck?
C: So let's, we've definitely come to the conclusion that if their groups are equal we multiply. 
P: If we're combining equal groups we're multiplying. 

24. Statement 2
For each possible pathway to an outcome there is an equal number of options leading to that path but without repeating the same pathway more than once.

25. Push for More General Language
We asked if they could articulate language that was more general than a “pathway”

26. Statement 3
If for every selection towards a specific outcome there is no difference in the number of options, regardless of previous selections, then you multiply the number of all the options in each selection together to get the total number of possible outcomes.
This still could potentially succumb to overcounting, so we introduced the 3-letter sequence problem with an e and repetition allowed, and they fixed it by adding the word “unique”
They did define selections. Options, outcomes

27. The 3-letter ‘Words’ Problem
We introduced a problem involving overcounting

28. Final MP Statement
So that brought us through session 7
If for every selection towards a specific outcome, if there is no difference in the number of options, regardless of the previous selections, then you multiply the number of all the options in each selection together to get the total number of possible unique outcomes.

29. Evaluating Textbook Statements
P: Yeah I feel like, I think this is, holds really strong intuitively for things that are completely separate, like, uh there are no heads and tails on a die, if that makes sense.
But when you have it like where it is overlapping, like I feel like this does work but it's a little hard, you have to do a little more intuitive thinking into it of like the idea that you could overlap people and it doesn't matter, just as long as your cardinality stays the same.
First impressive that they could interpret it, and

30. Conclusions and Implications
The MP is a nuanced idea with subtle and important mathematical issues
If students do not grapple these subtleties, they may apply the MP without understanding potential issues that may arise
Although these statements can be hard to interpret, through engaging with particular tasks our students became attuned to key mathematical issues

31. Conclusions and Implications
Those who teach discrete mathematics could
Take time to explicitly discuss the MP, its importance, and the key ideas
Develop tasks that highlight mathematical aspects of the principle to which students should attend
Get more information on this research from Elise Lockwood’s website! ….
Don’t gloss over it

32. combinatorialthinking.com caughman@pdx.edu
Thank You!!
combinatorialthinking.com

33. The structural statement of Ryser