Once we believe that a pattern is established, we will state it as a conjecture about an entire class of objects. As we organize our data, we look for patterns and for ways of describing those patterns formally. Students learn about a variety of familiar patterns linear, exponential, periodic, etc. Techniques for identifying these patterns and activities that help them develop the habit of using these techniques should be central to this study. Problems can be so intellectually challenging or computationally demanding that we cannot solve them directly.
For example, a student reading Flatland wondered about the lengths of the diagonals of a hypercube. What were the longest segments that could fit in such a figure? She began by looking at the lower dimensional versions of the problem.
For a point, which is a 0-dimensional "cube", the length of the longest fitting segment is 0. For a unit segment the 1-dimensional "cube" , the length of the longest segment is 1. For a unit square, we can fit a diagonal of length and for a unit cube, the distance from one corner to the opposite one is figure Recognizing that 0 and 1 are both their own square roots, she extrapolated her pattern and decided that the diagonal of a unit, 4-dimensional hypercube must be or 2.
Her pattern did not constitute a proof, but the study of simpler related cases guided her to a solution and, ultimately, to a proof for any dimension. In the hypercube example above, the student used smaller cases because she could not, at first, visualize the situation that interested her. Using a smaller case is especially important if you find yourself attempting a brute force solution to a problem.
For example, instead of counting all of the possibilities for a five button simplex lock , one of the project hints is to first find a pattern for locks with fewer buttons. Problems can be made simpler using a smaller number, simpler shape, or more symmetric setting or shape e.
They can sometimes be made simpler by removing restrictions that seem to make them harder. For example, the problem "In how many different ways can an elevator leave the ground floor of a 20 story building, make 10 stops moving only upward, and arrive at the top floor? For a further discussion of methods for creating related problems, see Ways to Change a Problem in Problem Posing.
For a class activity that raises questions about how a simpler problem can be used to solve a more complex one, see the section Technology and Magnitude in the Numbers in Context chapter from the book Mathematical Modeling available from www. While we often seek to describe how some variable is changing , sometimes we want to show that a feature is unaffected by changes in a variable.
This fact is obvious once you study it, but can be a surprise to children when they first discover it. A property or quantity that does not change while other variables are changing is called an invariant. The notion of invariance is important throughout mathematics.
Even shapes much less constrained in their form than circles have invariant properties. Invariants often prove surprising and stimulate further investigation and explanation. When such consistency appears in the face of asymmetry and variability, we want to find out what accounts for this dependability. Figure For any quadrilateral, the blue segments are equal and perpendicular.
The identification of invariants can be a very powerful tool because it lets us find common properties of situations that look very different. The identification and use of invariants becomes more natural with experience. Here are two examples that provide a fuller picture of this concept:. We can use the quadratic formula to find the solutions:. There are many different ways to arrange these numbers in ascending order by exchanging pairs. You could start by exchanging the 9 and 2 to give you:.
Try this with some numbered tiles or scraps of paper and keep track of the number of exchanges that you make. The number of steps might vary, but you will always take an odd number of steps. The parity of a permutation is invariant with respect to the exchanges you perform to sort the permutation. Starting with a whole bar, a move consists of choosing a piece of the bar and breaking it along one of the horizontal or vertical lines separating the squares.
The two new pieces are then returned to the pile to be available for the next move. The challenge is to find the fewest number of moves needed to break the bar into all individual 1 by 1 squares. Introduce this situation to your class and have each pair of students pick bars of a particular size and keep track of the number of moves needed each time. They can carry out their investigations cutting pieces of graph paper, separating sections of an array of interlocked cubes, drawing segments on a grid, or breaking an actual chocolate bar.
Ask them to consider whether certain strategies for choosing which piece to break seemed more efficient than others. After several tries, it should become apparent that, for a given size bar, the number of moves required is always the same. Why is the outcome invariant with respect to the sequence of breaks? What does affect the number of moves needed? At this juncture, you can guide your class in one of two directions. Alternatively, and this is the faster route, you can ask them to keep track of the number of pieces after each cut.
You may even want to do both of these analyses in order. The key observation is that the number of pieces of candy is always one more than the number of moves because each move adds one more piece.
That is, the number of pieces is invariant with respect to the choice of breaks made. This activity provides a nice example of how we can use invariants as a tool for constructing proofs. You can also download Tackling Twisted Hoops , an article on invariants and knot theory from the former Quantum magazine. Although the most common kind of discovery for secondary students engaged in mathematics research to make is one about numerical patterns, there are other kinds of possible conjectures.
One type of observation could be that a pattern or arrangement that they are studying has been encountered in another context. Such an observation can lead to a conjecture that there is a common explanation for the two apparently dissimilar questions and to a way of showing that the two are related in some manner.
One such inquiry began when another student who was inspired by Flatland looked at the number of vertices in an n -dimensional cube a given distance away from a chosen vertex. The distance was measured by travelling only along the edges of the figure. For example, if a corner of a square is chosen, then there is one point 0 steps away the point itself , two vertices that can be reached by travelling along a single edge, and one vertex a distance of two edges away figure The distance along the edges from vertex A in a square and a cube.
When the student organized his data in a table figure 14 , he saw a familiar sight. The number of vertices distance d away from a vertex in each shape. For further discussion and settings that encourage the making of such connections, see Practice Activities in the Proof section. We seek to understand a conjecture at three levels: we want to determine its meaning, we want to identify reasons for why we might believe the claim to be true, and we want to understand how it fits within some larger set of ideas.
The initial steps we take when exploring a conjecture are similar to those used to understand a definition :. The same steps help when we are familiarizing ourselves with a new theorem.
In the case of a theorem, we want to read and understand the proof as well. In the case of a conjecture, we are looking for evidence that would support a proof or provide a path to a disproof. The AAS theorem has many conditions. In order to see why each one is necessary, we need to remove each condition and then create a pair of triangles that satisfy the remaining conditions but are not congruent. For example, if we remove the condition that the figures are triangles, we can construct different quadrilaterals that share two congruent angles and an equal non-included side.
The table below figure 15 shows non-examples for three different conditions. As a class activity, present theorems and conjectures and ask students to first list all conditions of the statement and then produce non-examples for each.
Do not tell your class ahead of time that the first claim is a theorem while the second is a false conjecture. Students will get additional practice understanding the conjectures that their peers generate throughout the year. What are the possible characteristics of a conjecture and what makes one conjecture more interesting than others? Students should explicitly answer each of the following questions when they seek to evaluate a conjecture:.
We rarely have a definitive answer to this question right away, but our understandings of related results may guide our intuitions.
A study of examples and a search for counterexamples will further influence our belief in the truth of a conjecture. Students tend to be too prone to believe that a few examples constitute irrefutable evidence of a pattern.
If a conjecture immediately follows from a known result, then it may be less interesting than an unexpected conjecture. For example, a claim about squares may not be exciting if a student has already proven the same claim for a superset such as parallelograms.
A conjecture may be difficult to understand because of the way it is written or because the mathematics involved is inherently complicated. Students should re-write their conjectures if their mathematical language is unnecessarily confusing. A conjecture that, if true, applies to a broad range of objects or situations will be more significant than a limited claim. Does it suggest a connection between two different topics?
A second-grader was investigating which n by m boards could be tiled by the T-tetromino four squares arranged edge-to-edge to look like a capital T, figure Her ultimate conjecture, "If the sides of a board are even by even it may work, if not then not," left her dissatisfied. She correctly believed that any odd dimension made the tiling impossible, but she knew that evenness was not a precise enough condition to distinguish between all of the boards that did and did not tile.
She knew that if she could refine her conditions her conjecture would be stronger. See Necessary and Sufficient Conditions below.
People are attracted to different mathematics questions. It is important for students to begin to develop their own aesthetic for mathematical ideas and to understand that aesthetics play a role in the discipline. Conjectures that are unexpected or counter-intuitive, that reveal a complex pattern, or that would be useful in supporting other important conjectures are more likely to be appreciated by a wide audience. The first question that we face in evaluating a conjecture is gauging whether it is true or not.
While confirming examples may help to provide insight into why a conjecture is true, we must also actively search for counterexamples. When students believe a conjecture, they are not always rigorous in their search for examples that break the pattern that they have identified. We must help them develop the habit of being more skeptical.
How can students search for counterexamples? They should test cases between those that they have found to work. They should look at extreme cases at the far ends of the domains of their problems e. They should consider degenerate cases that do not have all of the complexity of a typical example. Degenerate cases often result from making some parameter zero. For instance, a point when the conjecture applies to circles the radius has been set to zero or a linear equation when the topic is quadratics the coefficient of the squared term is zero.
A quadrilateral is a degenerate pentagon in which two of the points are in the same location. Of course, some degenerate cases are not really relevant to a problem e. In geometry, a postulate is a statement that is assumed to be true based on basic geometric principles. A long time ago, postulates were the ideas that were thought to be so obviously true they did not require a proof.
A theorem is a mathematical statement that can and must be proven to be true. Euclid and His Accomplishments He lived lots of his life in Alexandria, Egypt, and developed many mathematical theories.
He is most famous for his works in geometry, inventing many of the ways we conceive of space, time, and shapes. Begin typing your search term above and press enter to search. Press ESC to cancel. Skip to content Home Philosophy How can you prove a conjecture is false?
Ben Davis September 21, How can you prove a conjecture is false? What is an example that shows a conjecture is not true? What is it called when a conjecture is not true? What is a conjecture that has been proven? What is conjecture law? It can happen that some statement is independent of particular logical system.
I fell there exist such statements for the axioms of reals, but I don't know one for sure. You do not prove it is true in objective axiom free universe. And we don't know something is true in an axiom-free objective universe. Take parallel lines. Real space is warped; parallel lines don't exist. In math; Euclidean, Spherical and Hyperbolic system all exist equally validly but separately in different differently axiomed "universes". In actual mathematics, there are many reasons we might be unable to prove a fact after seeing numerical evidence for it, but "the axioms might not be true" is not a reason for doubt.
Questioning the axioms is a possible reason for some very smart person to invent a new kind of mathematics with a new set of axioms; it has happened before in some fields of math, though I can't say how it might have happened in number theory. The existence of a proof hinges only on a ; question b determines whether such a proof establishes the conjecture's truth. The latter is a question more for mathematical philosophers than for mathematicians themselves, in my opinion, although there certainly is overlap.
Show 2 more comments. Active Oldest Votes. Brian Tung Brian Tung In fact, one point of my admittedly long-winded answer is that it doesn't really matter what the axiomatic system is; the proof is relative to that system, and not to some Platonic truth. All you need to know about ZFC is that it's currently our consensus axiomatic basis of mathematics. Add a comment. Q the Platypus Q the Platypus 4, 15 15 silver badges 39 39 bronze badges.
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