Reasoning and Proof
Explanations and development in students
Overview
Proof is the essence of mathematics - it is what makes mathematics valid. The making of conjectures about the use of mathematics and the justification of conjectures as solutions. Anything less can not present the true nature of mathematics. Classrooms must empower students to make conjectures, create solutions, and justify the methods for attaining the results; instead of focusing on justifying their use of steps from teacher taught procedures.
See also:
- Outline of Goals and Outcomes for a Critical Thinking Curriculum and Specifications for Assessment
- Decision Making Process and Change
- Reasoning, logic, and reasoning errors
Proof as common sense
Proof - An explanation accepted by a community at a given time. The rigor of the proof is dependent on the members of the community. Communities can range from a kindergarten class as a community where proof is relative to what a particular child thinks; to a community of mathematicians, who require a formal logical proof.
Mathematical related definition
Proof is a sound, convincing, and unquestionable explanation, for anyone who understands the concepts, and for which no counterexample can be given. It can be provided for explanation, communication, verification, systematization, and discovery.
Types of proof
Visual representation as proof
Visual comparison is the first kind of proof offered to and demonstrated by children.
- Look! Is it a dog?
- Is it a circle?
- Is it a triangle?
- It's a square see.
Young children look at an object and compare it to their mental visual representations. If it looks like one of their previous minds-eye views, that is proof enough. They do not use properties as justification. It is a holistic visual comparison.
Comparison of properties as proof
The identification and use of properties appear as children begin to take closer looks at objects, images, or visual representations. They begin to compare properties or characteristics of the objects along with the images. For example, different and similar properties of dogs, cats, cow, pigs, squares, triangles, and other objects are added to their mental images. Images of dogs with four legs. Dogs with fur. That bark. The number of sides on squares, triangles, and circles. How the sides are curved or straight.
The mapping of properties from a person's mental representation of an object to a questionable object is an example of comparison being used as evidence to create a proof of equivalence or not. This use of matching properties or mapping properties from one to another to use as evidence to build a proof is rarely used by children. Children and even adults persist on using observations. If it looks like a cat, then it is a cat. for example teachers' attempts to teach children to use scientific classification as a way to identify living organisms often results with students' persistent claims that dolphins and whales are fish, snakes don't have bones, and humans aren't animals.
However, with continued experiences of having students identify properties, classify by properties, and organizing solutions based on properties, students will increasing use this as a strategy of proof. In geometry, for example, students can be asked to identify properties and use those properties to identify and organize figures and shapes by those properties. With these experiences students will learn that some properties are necessary for an object to belong to a group and others are not. They will learn that there are properties that are independent of the conditions they are exploring. They will learn that there must be a sufficient number of properties needed to describe or identify a particular object or groups of objects. And they will learn that if two objects have the same necessary and sufficient properties, then they are equivalent.
Stated another way - To prove one object as a member of a class or congruent with another object, then all of the necessary and sufficient properties of the object, class, or group must be identified for the questionable object.
Examples:
Lets explore these ideas for the class of objects known as triangles. What are the necessary conditions for a triangle?
Necessary properties/conditions/characteristics of a triangle
- Plane figure
- Closed figure
- Three sides
- All sides are straight lines
Not Necessary properties/conditions/characteristics of a triangle
- Three dimensions figure
- Open figure
- Four sides
- One side or more has a curve
Which of the following necessary conditions must be included to include all the sufficient conditions needed to describe a triangle or for a figure to have to be a triangle?
- A plane figure - Not sufficient, because there are plane figures that are not triangles
- A closed figure - Not sufficient, because there are closed figures that are not triangles
- Three sides - Not sufficient, because there are figures with three sides that are not triangles. In fact all polygons have three sides and more
- All sides are straight lines - Not sufficient, because there are other shapes with all sides being straight lines.
SUMMARY - Does this list include sufficient conditions or all the minimum number of necessary conditions to represent a triangle?
The list is sufficient And all are necessary for a plane figure to be a triangle or for the list to be used to define a triangle.
This comparison can be used to categorize or group objects by their common properties.
Or when considering if two things are equivalent or congruent. Triangles with identical necessary and sufficient properties are congruent.
Example of equivalent descriptions of numbers.
Two sets of properties that represent a whole number.
- Set one - It is greater than one, it is less than three, it is a whole number. [2]
- Set two - It is the smallest positive whole number that is also an even number. [2]
To prove these are equivalent conditions we must show they are both necessary and sufficient. They are as they lead to the same conclusions. This would make it possible to write: the whole number greater than one and less than three is equal to the smallest positive even whole number. Further it would also be valid to set both equal to 2.
Take your time and make this information YOURS.
If you do, then you can use it to create questions to probe your students' depth of understanding. By having them consider what is necessary? What is sufficient? And what more needs to be added to make something sufficient that is not? What statements or groups of statements are equal or equivalent? What does it look and sound like if we put them together as equals?
Proof by disproving with a counterexample
We make a conjecture and find an example that doesn't fit the conjecture, then can claim we have disproved the conjecture by discovering a counterexample.
Example 1
- Conjecture - The squares of all numbers are odd.
- Example - Three squared is nine.
- Counterexample - Two squared is four.
Example 2
- Conjecture - The squares of all numbers are even.
- Example - Two squared is four.
- Counterexample - Three squared is nine.
Deductive proof
Deductive proof - uses a general definition to include a specific instance or a premise to reach a conclusion. Is generalizable to all instances with the same characteristics that satisfies the givens or represents the given class, definition, or law.
Example
Start with a set of assumptions and use them to derive a valid conclusion.
Assumptions
- It is a whole number
- It is a perfect square
- It is between 40 and 100
- It has a factor of three
What is it?
Specific instances
- Whole numbers are 1, 2, 3, 4, ..
- Perfect squares are 2, 4, 9, 16, 25, 36, 49, 64, 81, 100 ...
- Whole number perfect squares, between 40 and 100 are 49, 64, 81
Conclusion - The only one number of 49, 64, and 81 to have a factor of three is 81.
Inductive reasoning
Inductive reasoning - is using the specific to create the general or conclusion. Rolling two die hundreds of times to find the probability of the different sums. Inductive reasoning is the step child of deductive reasoning since it is based on experiment and variables. For example - statistical reasoning is not accepted in a courtroom.
Reasoning and proof - if you include ideas of sufficient, necessary, equivalent, and independent conditions, you have the tools for reasoning and proof.
Independent conditions are properties or conditions that vary independently of the other.
Example size and shape. It is possible for the shape of a square to have an infinite variety of sizes and still have the shape of a square. Similarly the properties necessary to make a shape a square do not include size.
Proof by analogy or metaphor
Proof by analogy or metaphor - mathematicians might distinguish analogies as a directly related comparison of two like arguments and metaphor as indirectly related.
Analogies:
- Feathers are to birds as scales are to fish
- Feathers are like scales (body coverings) and birds are like fish (animals).
- Feathers : bird :: scales : fish :: hair : mammals
- Den:fox :: hive:bees
- Bird :: nest:: burrow:rabbit
The relationships are obviously clearly definable relations of the respective parts or artifacts to animals as well as parts or artifacts to parts or artifacts and animals to animals.
Metaphors are not as tight - At night all cats are black. Can not substitute - like the previous analogies. There is a violation of syntactic structure. Night relates to black as light or dark and black relates to cats as to color - white, brown, yellow, black; while the figurative meaning of the metaphor is something unrelated.
Examples
Proof of the Commutative Property
To show students a specific example, can be a starting point of a proof. Specially for very young children. Like 3 + 5 = 5 + 3; can be demonstrated by gluing three beans to card stock and five beans to card stalk, then showing that either order of the cards has a cardinality of eight beans. However, if we stop here we must recognize it is not a proof. In fact it is recognized as not being a proof - by attempting to prove a specific example or a selective instance as equivalent for all possible examples.
To make it a proof the idea must be generalized for all possible cases or the infinite number of cases that might exist. This can be suggested by asking what happens if the number of beans are increased. Again, young children usually reach a limit because of their limited understanding of the number system. However, as students understanding of number systems increase toward a conceptualization of infinity, the generalization can be pushed toward infinite possibilities and the equivalence for all of them. For children to be able to do this they will need to have achieved all the conditions for conservation. When you believe this is so, the next example can be presented.
Prove that when you add any two numbers the sum will be the same no matter the order of the numbers. While all the students may be convinced of this it is necessary to push them to explain why it would work for addition of all numbers (infinity again) for the case of whole numbers. Have them create an explanation that they could give to another person (art, music, PE. teacher or principal).
To move to infinity students should be able to show with manipulatives and a drawing that represents additional or less numbers. You can question by asking for repeated larger or smaller numbers.
More examples with proof and more examples to prove
Geometric construct of a Proof. Prove that a construction of a triangle will yield an equilateral triangle. Figure at right shows how a compass can be used to demonstrate how three equal lines can be drawn to create a triangle with all three sides congruent.
Proof of when an Even number is added with another even number it will make an even number. See three different levels: concrete, semi-concrete, and formal proofs .
Proof of a specific and translate to general
Students found the mass of a dry sponge (98 g), then soaked it in water and found its mass (145g). When they were asked to find the mass of the water they wanted to subtract. The teacher asked them to prove it or convince her that it was an accurate solution.
The wet sponge's mass was 145 grams - the dry sponge's mass 98 grams. Students decide to use nice numbers and adjust to subtract. So they subtracted 100g. 145 - 100 = 45. Then they were not sure if they should subtract two more or add two more.
To represent the problem they drew a rectangular shape (sponge) with an enclosed blob to represent the water. They labeled the area outside the blob 98g for sponge and labeled the blob - water with a question mark. Then they wrote 145g beside the rectangular shape and said that represented the sponge and the water.
They proceeded to explain that taking 100g away was 2g too much so they would need to add two back.
To get to the generalization the teacher asked this follow up question: Would this always be true for subtraction?
What would the statement be? Or rule?
Like - If you take away smaller you end up bigger so you need to ... Or If you take away bigger you end up smaller so you need to ...
With the hopes to get to the idea of: How to argue for an infinite claim.
Think that problem is too complicated to start?
What if we use the same numbers only change the scenario?
Max has 145 collector cards and gives 98 to Chris. How many collector cards does Max have left?
This can also be solved with subtraction and the numbers are the same as the sponge problem. However, the representation of the problem isn't as complicated because of the type of subtraction problem it is.
Why is there such a difference? The sponge problem is stated as an addition problem with the start and finish numbers know, but not the change. This makes it more complicated than this second which is a start total known and the amount to remove or separate known with the unknown as what is left.
See samples of different addition and subtraction problems
The following are examples of specific instances that can use proof by analogy if the analogy is stated as to be infinite.
Prove why you can exchange numbers of equal value with equations of equal value.
- Start with two equal equations 4 + 4 = 2 * 4
- Add 3 to each (4 + 4) + 3 = (2 * 4) + 3
- Will this work for adding any number? If so why?
Prove the validity of this equation:
- 4X + 2Y - 3 = 0?
- Assuming X and Y are integers.
- Not valid, since no matter what value is assigned X and Y they will always (infinity) sum to an even number from which it is impossible to remove an odd integer to obtain 0.
Prove that two unknowns x and y when added together and squared are represented accurately with the following:
- (x + y)2 = (x + y)(x + y) = x2 + xy + yx + y2
- equivalent by analogy (12 * 34) = (10 + 2)(30 + 4) = 10*30 + 10*4 + 2*30 +2*4
- Sample explanation of whole number multiplication and take it to infinity.
Can prove the validity of the distributive property by analogy?
x2 - 1 = (x +1 )(x - 1)
Teaching suggestions
Arguments from children's representations (like the 3 + 5 = 5 + 3 ) and from other student stories where numbers are involved can be an effective way to move students toward a proof as a more general idea. The key is for the teacher to facilitate the inclusion of questions such as:
- Will this work for other number?
- Will it work for All numbers?
- If not, then is there a group of numbers for which it will work?
The purpose is to push thinking to generalize the explanation to include all possible examples that are true for all cases, or move to infinity.
Example
A = length * width
Does multiplying length and width determine area for every rectangle and square? Why?
We can use graph paper or tiles to represent a rectangular area. By modeling how every time a row of square tiles is add it is represented by multiplying the number of column by an increased row number of one.
Elementary students will need to act this out multiple times. Talking about how the rows are increasing and the columns are increasing and how that can be represented in the product of the length and width. Then ultimately they should recognize that it doesn't matter what the number of rows or columns are. Therefore, the expression L * W will determine the area. To be able to do so students need to be familiar with all the necessary conditions or characteristics for conservation. Specifically in the case of conservation of area.
Proof by Exhaustion
Identify all examples and justify their inclusion as well as reasons for the elimination and identification of all possibilities.
Example
What are all of the possible four cube high towers that can be made with red and yellow cubes?
Proof by exhaustion using systematic process of recursion.
Students must take time to build meaning, which must be from concrete experiences. Time to notice patterns between the physical entities and retain these ideas in short term memory long enough to act on them to create relationships and communicate these relationships.
It is the communication element that pushes us to identify the limits, systematically explain all possibilities even if they extend to infinity.
There are several ways to systematically try to insure the discovery of all possibilities of the cube towers. Some examples follow.
First towers can be made to show the number of red and yellow cubes that could be to make a four cube tall tower. Below illustrates that if four cubes are used for each tower it would be possible to build a tower with 4 red, 3 red and 1 yellow, 2 red and 2 yellow, 1 red and 3 yellow or 4 yellow. There is no other combination of reds and yellows for a 4 cube tower.
However, that doesn't provide for the different placements of the two colors in different combinations. For example: There is only one way to arrange a tower with all red. Like wise there is only one to arrange a tower with four yellow. However, there are several ways to arrange three red cubes and one yellow cubes. Like wise for two red and two yellow and three yellow and one red.
These can be found by rearranging the cubes for each of the above combinations of colors and summing the totals 1 + __ + __ + __ + 1 = __ .
Another way is to use a tree. Start with R and Y and put a R Y pair below it and continue till there is a row for each of the four rows in the towers. Then find all the different paths through the maze of connections and label each.
- r, r, r, r
- r, r, r, y
- r, r, y, y
- r, y, r, r
- r, y, y, r
- r, y, y, y
- --------
- y, r, r, r
- y, y, r, r
- y, y, y, r
- y, y, r, y
- y, r, r, y
- y, y, y, y
Another example by exhaustion or discovery is all the examples to find,
How many squares in a 5 X 5 square?
Systematic solution could find the number of
- 1 X 1 squares
- 2 X 2 squares
- 3 X 3 squares
- 4 X 4 squares
- 5 X 5 squares
Then total the five groups.
See example of card trick explanation using elimination and identification.
Prove the handshake problem
How many handshakes will there be for everyone to shake hands with everyone else in your classroom.
Systematically - if there are five people, the first person shakes hands with the four other people in the classroom. Since the first person has shaken everyones hand, the first person can relax and sit down. The second person has shaken the first person's hand, so the second person begins by shaking the third person's hand. When the second person has shaken everyones hands the person may relax and sit down. The third person now has shaken hands with person one and two, so has to shake the fourth and fifth persons' hands. Then the fourth person only has to shake the fifth person's hand and when it becomes the fifth persons turn, by golly the fifth person has shaken all the hands and is done. To summarize.
Person five - 4, person four - 3, person three - 2, person four - 1, person five - done - o.
Add them and get: 4 + 3 + 2 + 1 = __ .
Misconceptions and fallacies
- Use of the word proof as a synonym for evidence.
- Think of proof as verification with a single case
- Think solving a problem is the same as a proof
- Can prove by checking single cases
- See this checking of a few cases as the most convincing argument.
- Often don't see the power of generality
- Lack understanding of logic
- Lack problem solving heuristic, strategies, and habits of mind
- Imply a solution is correct if the algorithmic procedure was followed correctly.
- Use authority to justify - My teacher said so. - Lampert (1993) found that students believe: doing mathematics means following the rules of the teacher and knowing mathematics means remembering and applying those rules when the teacher provides a problem and mathematical truth is determined when the answer is judged correct by the teacher.
Ecological fallacy
Ecological fallacy is thinking relationships observed for groups necessarily hold for individuals.
Examples:
- A correlation describes a measured relationship. Correlations of group averages can be much higher than correlations of individual scores. Thinking both are equal is an example of ecological fallacy.
- It is an ecological fallacy when conclusions are made about individuals based only on analyses of group data. Assume you know the average math scores of a specific classroom is the highest in the district. If you meet a student from the class and think she is a math whiz, then that is an ecological fallacy. As her belonging to the class with the highest average score does not mean she has a high-score in math. She could be the lowest math in a class of math geniuses.
- Another ecological correlation fallacy is when people generalize that if something is good for an individual, then it is good for everyone. for example: High scores on tests grant entry to elite universities, therefore high scores must be right for the nation as a whole. Most parents would probably like their children to have high scores. However, looking at our nation as a whole we should come to a different conclusion. In that situation we would want a combination of people that are able to be leaders, followers, heroes, creators, producers, so that the society could function for the benefit of everyone.
Exception fallacy
Exception fallacy is sort of the reverse of the ecological fallacy. It is when a conclusion about a group is made on the basis of a specific case. This fallacious reasoning is often at the core of sexism and racism. The stereotypical response when seeing a woman make a driving error of: "women are bad drivers." Is wrong. Fallacy...
Scoring Guide for proof
Proof Production Level 0
Students at this level appear unaware of the need to provide a mathematical justification to demonstrate the truth of a proposition or statement. For example, a student might accept a proposition as true because a teacher, parent, or text "says" it's true (cf. Harel & Sowder, 1998); in this case, the justification is "non-mathematical." In other cases, a student might simply state a proposition is true without any reference to why the proposition is true (e.g., "the sum of two even numbers is even because that is just the way it is," "yes the numbers will be equal because they will always be equal").
Proof Production Level 1
Students at this level appear to be aware of the need to provide a mathematical justification, but their justifications are not general; in the majority of cases, students' justifications are empirically based. Among the empirically based justifications, we recognize distinctions (at sub-levels) among students who consider checking a few cases, students who consider systematically checking a few cases (e.g., even and odd numbers), students who consider checking extreme cases or "random" cases, and students who consider the use of a generic example (proof for a class of objects) (cf. Balacheff, 1987).
Proof Production Level 2
Students at this level appear to be aware of the need for a general argument, and often attempt to produce such arguments themselves; the arguments, however, fall short of being acceptable proofs. "Falling short" may happen in one of two ways: (1) Students express recognition of the need to provide a general argument and attempt to produce such an argument, however, the argument provided is not a viable argument (i.e., the argument is either incorrect mathematically or it would not lead to an acceptable proof). (2) Students express recognition of the need to provide a general argument and attempt to produce such an argument, however, the argument is incomplete (if completed, the argument would be an acceptable proof). In both situations, the point is that students are attempting to treat the general case. In addition. Level 2 justifications also include responses from those students who demonstrate an awareness that empirical evidence does not suffice as proof‚Äîby either expressing recognition of the need to deal with all cases or expressing recognition of the limitation of examples as proof‚Äîbut who are unable to produce (or attempt) a general argument.
Proof Production Level 3
Students at this level appear to be aware of the need for a general argument, and are able to successfully produce such arguments themselves. We consider the arguments students produce at this level to be acceptable proofs; that is, their arguments demonstrate that a proposition or statement is true in all cases. Arguments categorized at this level typically involve a reference to any assumptions or givens, a chain of deductions used to build the argument, and finally an explicit concluding statement. Although the arguments students produce may lack the rigor or formality typically associated with a proof, their arguments, nonetheless, do prove the general case.
See the original article for explanations and examples of the levels.
From Middle School Students' Production of Mathematical Justifications. Eric J. Knuth, Jeffrey M. Choppin, and Kristen N. Bieda. In Teaching and Learning Proof Across the Grades. A K-16 Perspective (2009).
Summary
Research suggests current practice is insufficient and only change can improve students abilities to reason and use proofs. Goswami, and Hatano and Sakakibara (2004, ISBN 0-8058-4945-9.) observed children in natural settings using everyday language while they were attempting to use the language of mathematics while reasoning mathematical. That language was mostly absent of spontaneous analogies. The authors attribute these findings to limitations in a child's mathematical knowledge as well as to the norms and practices in the students' instructional setting.
A mathematical community or culture must include many opportunities with experiences that emphasize problem solving, representation, reasoning, and communication, where students make conjectures and develop skill in using these tools of reasoning and proof through systematic representation to justify their ideas.