Explaining the "whys" of mathematics
An 8th grade exam from Bullitt County, Kentucky. I’m not sure about the other subject areas, but it seems to me that the standards for arithmetic for those 8th grades are approximately in line with what we expect of pre-algebra students today.
Of course, the students of 1912 didn’t have access to scientific calculators.
Source: http://www.bullittcountyhistory.com/bchistory/schoolexam1912.html
Solutions: http://www.bullittcountyhistory.com/bchistory/schoolexam1912ans.html
I just read a very entertaining article on the use of game theory for improving contestants’ odds of winning the various games on the long-running television game show “The Price Is Right.” Quoting from the article:
On a crisp November day eight years ago, I took the only sick day of my four years of high school. I was laid up with an awful fever, and annoyed that I was missing geometry class, which at the time was the highlight of my day. I flipped on the television in the hope of finding some distraction from my woes, but what I found only made me more upset: A contestant named Margie who was in the process of completely bungling her six chances of making it out of Contestants’ Row on The Price is Right.
Many contestants fail to win anything on The Price is Right, of course. But as I watched the venerable game show that morning, it quickly became clear to me that most contestants haven’t thought through the structure of the game they’re so excited to be playing. It didn’t bother me that Margie didn’t know how much a stainless steel oven range costs; that’s a relatively obscure fact. It bothered me, as a budding mathematician, that she failed to use basic game theory to help her advance. If she’d applied a few principles of game theory—the science of decision-making used by economists and generals—she could have planted a big kiss on Bob Barker’s cheek, and maybe have gone home with … a new car! Instead, she went home empty-handed…
To help future contestants avoid Margie’s fate, I decided to make a handy cheat sheet explaining how to win The Price Is Right—not just the Contestants’ Row segment, but all of its many pricing games. This guide, which conveniently fits on the front and back of an 8.5-by-11-inch piece of paper, does not rely on the prices of items.
The full article can be found at http://www.slate.com/articles/arts/culturebox/2013/11/winning_the_price_is_right_strategies_for_contestants_row_plinko_and_the.html.
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Kelsie Teague. Her topic, from Algebra I and II: factoring quadratic polynomials.
What interesting things can you say about the people who contributed to the discovery and/or the development of the topic?
In Renaissance times, polynomial factoring was a royal sport. Kings sponsored contests and the best mathematicians in Europe traveled from court to court to demonstrate their skills. Polynomial factoring techniques were closely guarded secrets.
http://www.ehow.com/info_8651462_history-polynomial-factoring.html
When reading this article, I found the fact that this topic was considered a royal sport very interesting. Students would also find that interesting because it would get their attention with the fact that kings thought this was very important. We could even have our own royal game for it. I think we could start off with a scavenger hunt to work on factoring just basic integers. Also, I think we could use the same idea to start the explore except to do it backwards and give them the polynomial already factored and have them FOIL it and get their polynomial. I want to see if they can see how to do it the other way around without being taught how. This game could show them that factoring is just the reverse of foiling.
How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?
I looked up factoring quadratic polynomials on Khan Academy and I found some really great videos. They have videos that show detail steps and also after a few videos they have parts where you can practice what you just watched and see if you understand it. This website is great for at home practice or in class practice because with the practice sections it tells you if you are correct or not and will also give you hints if you don’t know where to start. Also, if you don’t have a clue how to do the problem given, you can hit “show me solution” and it will redirect you to a similar problem in a video to help out. I think this website is a great tool to let students know about to learn and practice.
Also I found a great video on YouTube it’s a rap about factoring that would certainly get gets engaged.
Curriculum
Students first learn about the basic idea of factoring in elementary school and continue to learn and use this topic all the way through college. You need to factor polynomials in many different contexts in mathematics. It’s a fundamental skill for math in general and can make other calculations much easier. You use factoring for finding solutions of various equations, and such equations can come up in calculus when find maxima, minima, inflection points, solving improper integrals, limits, and partial fractions. Students will need to know factoring all the way up in to their higher-level math classes in college, and also be able to use it in a career that is related to engineering, physics, chemistry and computer science.
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest: but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.Thus, a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.– George Pólya, from How To Solve It
I just read an interesting article about math education in The Atlantic: http://www.theatlantic.com/education/archive/2013/11/the-stereotypes-about-math-that-hold-americans-back/281303/. Among the great quotes:
Here’s the most shocking statistic I have read in recent years: 60 percent of the 13 million two-year college students in the U.S. are currently placed into remedial math courses; 75 percent of them fail or drop the courses and leave college with no degree…
[W]hen mathematics is opened up and broader math is taught—math that includes problem solving, reasoning, representing ideas in multiple forms, and question asking—students perform at higher levels, more students take advanced mathematics, and achievement is more equitable…
When all aspects of mathematics are encouraged, rather than procedure execution alone, many more students contribute and feel valued. For example, some students are good at procedure execution, but may be less good at connecting methods, explaining their thinking, or representing ideas visually. All of these ways of working are critical in mathematical work and when they are taught and valued, many more students contribute, leading to higher achievement.
If I had written this article, I would have been less effusive in praising the Common Core. But I am absolutely in sync with the author that there’s a whole lot more to grade-school mathematics than completing drill-and-kill procedures as quickly as possible.
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Kelsie Teague. Her topic, from Pre-Algebra: powers and exponents.
What interesting word problem using this topic can your students do now?
For the topic of powers and exponents I want to bring in the idea of money, and doubling a salary. The word problem I would give them to start with and to get them thinking would be this:
Two companies were offering you a job. Company A is offering you a salary of $1,000 a day for 30 days and Company B is offering you a salary of $2 the first day and it doubles each day after that for 30 days. Which job is the better offer?
Since this is just my engage problem I’m not expecting them to be able to tell me that the answer is Company B because the answer is 
How can this topic be used in your students’ future courses in Mathematics or Science?
I believe powers and exponents are important knowledge because students will be using them for the rest of their math career. This comes up when teaching functions, learning the graphs of functions, trig, pre-calculus, Calculus and etc. Powers and exponents are used extensively in algebra and it is important that students have a strong understand of how and why they work before continuing onto those higher classes. For example, when you have 
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
The earliest exponents came from the Babylonians. The number system was extremely different from modern mathematics. The earliest known mention of Babylon was mentioned on a tablet found around 23rd century BC. Even then they were messing with the concept of exponents.
I would show my students this picture and explain to them what the symbols mean and ask them if they feel any better about doing math in modern times rather than working with these symbols to add, subtract, divide, exponents, power and doing equations. This also shows that this concept has been around for many thousands of years and something that is obviously very important if we still use it in modern math. I might also bring up the website least below that talks about modern exponents and works backwards and talks about where they came from to give the students more depth in this knowledge.
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Derek Skipworth. His topic, from Geometry: introducing the two-column, statement-reason paradigm of geometric proofs.
A. Applications – How could you as a teacher create an activity or project that involves your topic?
I would have the students get in groups and come up with 5-8 statements on one sheet of paper, numbering each one. This could be a statement about the weather, something that happened the day before, anything. My example would be “I wore a long sleeve shirt today”. After coming up with these statements, I would then have the students create reasons behind these statements on a separate sheet. For each statement, the students would have to ask “why…”. For my example, it might be that it was laundry day and it was my only clean shirt, or that it was cold outside. Upon generating all reasons behind each statement, I would then introduce the proof model.
B. Curriculum – How can this topic be used in your students’ future courses in mathematics or science?
The two-column, statement-reason paradigm is a system that can actually be used in all subjects. The idea behind it, giving a statement on the right and a reason on the right, can be applied to almost everything. For problem solving, you can work through an entire problem step by step and explain why you think that is the correct process. In a class such as Calculus, this could be used to help them memorize derivatives by doing the problem on the left and listing what “tool” they used for each step of the process. Even for something like social studies, this process could be adapted into a tool similar to the Cornell Notes (http://coe.jmu.edu/LearningToolbox/cornellnotes.html). In this process, you use the two-column approach. On the left, you list your main ideas, while the right column “explains” what you know about the idea.
E. Technology – How can Technology be used to effectively engage students with this topic?
I had issues tying this topic to a third question. While it’s a good topic, its more of a process than an actual concept. This would actually qualify as an engage activity, slightly different to the one mentioned above, but I would see it working better as a take-home assignment than an in-class one. The assignment would be to use YouTube and pull up a video that piques their interest. Obviously, it needs to be school-appropriate. This could be their favorite music video, a funny video of cats, whatever it is would work. They would write the name of the video at the top and provide a link to the video if possible. Then, they would take the paper and fold it in half, hot-dog style. On the left, they list the names of videos on the suggested pane to the right, in order as they appear. On the right, they would add comments about how that video was related to the video they chose and why it was in that order. The idea is that this should take a bit of thinking since often times the videos appear to be randomly added to that queue. This would reinforce the model while hopefully developing a better idea of how a website they are familiar with operates. Though this could be done with any search engine as well, I feel those are just too similar to offer any “investigative” work for the students.
Seen above, one would likely suspect that a Florence and the Machine video would pull up various other Florence videos in the top 9; however, the snapshot shows that this is not the case. We see that most results have nothing to do with Florence. We see that there are a few matches based on the KEXP live performance. When listening to some others, it might be reason to believe they were in the queue not only based on a performance, but also because the genres are very similar. That would be my main conclusion about Gotye’s music video being included in the list: it’s a very popular song and is in the same genre as Florence.
Source: http://www.xkcd.com/184/
P.S. In case you don’t get the joke… and are wondering why the answer isn’t ![[a_2, -a_1]^T](https://s0.wp.com/latex.php?latex=%5Ba_2%2C+-a_1%5D%5ET&bg=ffffff&fg=000000&s=0&c=20201002)
where 
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Kelsie Teague. Her topic, from Geometry: congruence.
Application.
Many students in high school go to the county or state fair yearly. I would start off by giving students a picture of a ferris wheel and having them find as many triangles in that ferris wheel that have what seem to be the same sides and angles and see how many different answers I get. After defining congruence, I would continue to ask the students if they thought this ferris wheel could be constructed without the idea of congruence. If the shapes in this ferris wheel were different sizes would it still work properly? I would then use this as a basis of what people need the idea of congruence to do their job.
Culture.
Congruence shows up in art work all over the place. It can show up in photography with taking picture of identical twins. Those twins are congruent but they are not the same person therefore they are not equal. I would post some pictures of art work and talk about the differences and have the student explain to me what they see. The bottom piece is made using the exact same shape and the idea of congruence. I would show my students some pictures and how the lesson for that day can be related to art work in real life.
Technology
http://www.khanacademy.org/math/geometry/congruent-triangles
The above website is a great hands on activity. It lets the students move triangles around to see if they can form triangles that aren’t the same. It also uses previous knowledge to guide them into the idea of congruence. Khanacademy.org also has other activities that can help with previous knowledge and then activities that take the concept of congruence and build on it. The activity I did was really good, it let me drop and stretch triangles to try and make them non congruent. It also gives one where you can’t lengthen the side but you can move it around and try to make a triangle out of it. I think this activity could show students about congruence in a different kind of way.
These lists should be required reading for faculty and staff who wish to understand the perspective of today’s college students (with some applicability to today’s high school students also). And it’s a little scary how the Mindset Lists themselves have changed, contrasting the one for the Class of 2002 differs to that of the Class of 2017.
Sources: http://themindsetlist.org and http://www.beloit.edu/mindset
Mean Green Math 