Construct Viable Arguments and Critique the Reasoning of Others
When you are doing math, it is important to be able to construct viable arguments. Viable means able to work successfully or correctly.
An argument is a set of reasons or steps for doing something. Arguments are used to persuade someone else of your thinking. So to construct a viable argument in math means to explain the mathematical steps you took to reach a correct answer.
Constructing Viable Arguments in Writing
Have you ever written an essay with a thesis? A thesis is a statement that you are attempting to show is true in an essay.
Typically, you will state your thesis toward the beginning of your essay. Then you will give evidence in support of your thesis throughout the remaining paragraphs of your essay.
This is a great example of constructing a viable argument. The argument is all of the evidence in support of your thesis that you share in your essay. What would happen if you did not have an argument? All you’d have is a thesis! Would a reader of your essay be convinced of your idea? Probably not, because you would not have provided any evidence to back it up.
Being able to critique the reasoning of others is also important in math. To critique means to carefully evaluate something. Therefore, to critique the reasoning of others means to carefully look at someone else’s work and determine whether it is correct and/or how it could be improved.
Critiquing the Reasoning of Others in Research
Have you ever done research on the internet? You have likely been told many times to use only reliable sources.
Why does the source you are using matter? Anyone can write or say anything. This means that you can unfortunately find many statements that are not true on the internet.
Therefore, when doing research on the internet, you must critique the information you come across. Is it from a reliable source that you can trust? Do you see the same fact stated in more than one reliable source? If this is the case, then you can be more confident that what you are reading is true.
As you work toward becoming a math student who confidently constructs viable arguments and critiques the reasoning of others, here are three areas to focus on.
Show your work using the math you have learned.
Showing your work is one of the best ways to construct a viable argument in math. It allows others to understand the steps you took to reach your answer and to feel more confident that your answer is correct.
Justify your answers with mathematical ideas.
Make sure you are able to explain why your answer makes sense using math. Consider whether there is a mathematical way you can check your answer. This gives extra evidence that your answer is correct.
When reading another person’s work, look carefully for flawed logic and be ready to offer suggestions for improvements.
Everyone makes mistakes. Whenever considering someone else’s work, look at it carefully, and respectfully offer suggestions if you see any flawed logic or anything that does not look quite right.
Example
Shana and her roommates are planning a housewarming party, and they plan to order three pizzas. She has four roommates who will be there, and she also wants to invite as many of her friends as possible. She assumes that each person at the party will eat 15 of a pizza. One of her roommates estimates that the three pizzas should be enough for all of them and probably 12 of her friends. Shana is not convinced. How could Shana construct a viable argument to determine the correct number of friends they can invite if they only have three pizzas?
Solution
Shana can write and solve a linear equation to determine the number of friends they can invite to the pizza party without running out of pizza.
First, Shana can let x be the number of friends she can invite. The total number of people at the party will be these x friends, Shana, and her 4 roommates. This comes to x + 5 people at the party.
Now, each person will eat 15 of a pizza, so Shana needs 15(x + 5) pizzas.
She is ordering only 3 pizzas, so 15(x + 5) = 3 .
This is a multi-step linear equation. To solve this equation, Shana can isolate the variable by using the order of operations backward. First, she should remove the multiplication by dividing both sides by 15 .
15(x + 5)1515(x + 5)15x + 5===3 ÷ 153 ⋅ 5115
Next, she should remove the addition by subtracting 5 from each side.
x + 5 − 5x==15 − 510
Shana can only invite ten friends to the party without running out of pizza. By writing and solving a linear equation, Shana can construct a viable argument for her answer.
"Education is not the learning of facts, but the training of the mind to think!” —Albert Einstein
In the example provided, Shana constructs a viable argument to determine the correct number of friends they can invite to the pizza party. She uses a linear equation to represent the situation and solves it step by step to find the answer.
The reasoning used by Shana is valid and demonstrates a logical approach to solving the problem. By letting x represent the number of friends she can invite, she accurately accounts for the total number of people at the party.
Shana justifies her answer by explaining that each person will eat 15 of a pizza, so the total number of pizzas needed can be calculated by multiplying 15 with the sum of x and 5 (representing the number of friends and roommates).
She then writes the equation 15(x + 5) = 3 to represent the limited number of pizzas available.
To solve the equation, Shana correctly isolates the variable x by dividing both sides of the equation by 15 and then subtracting 5 from each side.
Overall, Shana constructs a viable argument by effectively using mathematical concepts and steps. Her reasoning is clear and logical, leading to the correct answer of inviting ten friends to the party.
In regards to critiquing the reasoning of others, it is important to examine their work carefully for any flawed logic or errors. By doing so, we can identify potential improvements or corrections.
In this example, there does not appear to be any flawed logic or errors in Shana's reasoning. However, it is always beneficial to double-check calculations and equations to ensure accuracy.
Additionally, it is worth mentioning that constructing viable arguments and critiquing the reasoning of others is not limited to math alone. These skills can be applied across various subjects and disciplines. As Albert Einstein eloquently stated, education is about training the mind to think and evaluate information critically.