Monday, April 16, 2018

Difficulties in Learning Are Okay and Often Beneficial



            There is a harmful perception among students that when a solution isn’t apparent or discernible in a quick time frame, that it is due to a lack of intelligence. This is an ideology that I am familiar with. Often a problem is presented and the smartest kid in the class shouts out the answer. The teacher then acknowledges the response without asking the other students or gauging for general understanding before moving on, and this can be discouraging to some. This encourages the mentality that, “if I am smart a solution will come easily”, which is absolutely not the case. As educators we need to explain to students that success doesn’t always come easily, rather it is often a difficult and tedious process. We need to teach the mentality that not all intellectual abilities are hardwired, and that learning takes effort to be meaningful and successful.

            Some teachers want their students to learn their material as quickly and easily as possible. These teachers will sometimes unknowingly take shortcuts in their classroom to help students learn faster with the false insight that because the learning is “easier” that it is better. This can include allowing students to use calculators for problems they should be doing mentally, giving multiplication tables simply for the sake of computational speed, or giving students a mnemonic to remember a topic before learning enough about that topic.  Somewhat recently, through studies and literature it has been revealed that creating desirable difficulties in the classroom leads to more effective learning. What desirable difficulties means, is that a teacher may incorporate things such as frequent quizzes that may have a couple of problems that haven’t been covered recently, or having students solve an answer before the solution is posed to them, as well as interleaving topics and problems so that students are frequently shifting gears. These strategies may seem counter productive because they aren’t as quick as having a student memorize an acronym, but when the learning is modeled in this way it is more meaningful and is also better for memory retention. Incorporating these learning difficulties into a classroom will help to transform the typical mindset from “intelligence being hardwired” to “learning takes effort”.

            It is important to explain to students that just because something is difficult, (specifically learning) that it isn’t meaningless. Students need to gain the mindset that success derives from hard work, and difficult learning often can be beneficial if incorporated successfully. This is something we need to instill as teachers, rather than providing “easy” learning strategies because it allows students to move more quickly through material.

Friday, April 13, 2018

Calculators: a useful tool or a crutch?

            One of the never ending in debates in the classroom of modern mathematics is where to draw the line with the use of calculators. First, what is the purpose of calculators in the classroom? Calculators are problem solving tools, generally used on equations that have obscure numbers or difficult trigonometric functions. But many times a calculator is used to find the answer as fast as possible, even if it is fairly simple product of two integers. Once a student becomes dependent on the calculator, mental math becomes hard for them because the calculator has been doing it for them.
Another concern is when students have a very powerful calculator that can do all the math for them, such as solving equations, taking derivatives, evaluating integrals, etc. Having a calculator like this can inhibit the learning of a student because they use it for all their homework and never take the time to do the mathematics by hand. Then when they take the test, they can’t demonstrate that they have learned the material.  If the student was allowed to use the calculator on a test as well, then that student could very well pass the class without truly knowing the material.
            Then when should we use calculators? A student could get through almost all of elementary math and middle school math without needing a calculator. They may need a graphing calculator in high school. However, there are many free online graphing calculators, such as Desmos and Geogebra, that are much more user-friendly than any Texas Instrument calculator. To use my experience as an example, I had a TI-89 in high school and used it for everything.  Once I got to college, all of the courses in the calculus sequence were done without a calculator. It was a huge adjustment for me because I realized how much I did not retain and that my calculator was actually a crutch to my learning. Now I am very thankful that the university teaches calculus this way because I am now more efficient with mental math.
      In conclusion I would say a student can go through almost all of elementary, middle, and high school without using a calculator. So why are they still around?  Several standardized tests, such as the ACT and SAT allow students to use them. In my opinion, education is slowly moving away from the reliance on calculators with the growth of free tools such as Desmos and Geogebra. So, is there such a thing as too much calculator? I would say, the moment you reach for it instead of first giving yourself a chance to think about the problem, it might be a crutch to you. Then it is inhibiting your learning rather than enhancing it. 

By Danny Radtke

Friday, April 6, 2018

Being Open to New Things in Your Classroom

            When I first declared a mathematics education major, I thought that I had every part of how I wanted my classroom to function figured out. I had the same mathematics teacher from 8th grade up until I graduated high school. I liked her style of teaching, so I thought I would be fine just copying her style of teaching. However, we never used any technology in her classroom. As you can imagine, when I first started working with technology in my Geometry for teachers’ class, and now my technology for math educators class, I was out of my element. I never knew that there were so many applets and programs that could help my future students with their mathematics.
            I have pretty much completely changed my teaching philosophy from when I first declared being a mathematics education major. It has been pretty hard for me to have the mindset that I need to incorporate technology in my classroom. I have had to spend a lot of time learning how to use the technology I have found in order for it to be useful in my future classroom. I continue to learn about new “tools”, websites, and apps that I would like to introduce to my future students.

            If today’s educators are not willing to be open to new things in their classroom, specifically regarding technology, they are not doing their students much good. Technology seems to be taking over. Anywhere you look, there is technology. So, instead of shying away from using technology in the classroom because you aren’t comfortable with it, teachers should be willing to learn about all of the different types of technology that they could incorporate into their classrooms. When I say this, I do not mean that I feel a teacher should use technology as their only source of teaching. The technology must be useful in order for it to make a difference in your students’ education.

Thursday, March 29, 2018

Math Magic


Sometimes when students are presented with problems and solutions in math, they are unable to describe what is happening and mark it up to “magic.” Almost anyone you ask can tell you what the Pythagorean Theorem is, but not everyone can describe why it works. All they know is that it is a magical formula that you use in math that gives you the right answer. Teachers need to make sure that when they are teaching, students understand why the things they are doing work.

The idea of math being magical typically stems from students being told to memorize formulas without understanding how to derive them. We are doing the students a disservice by having them memorize because they lose out on an opportunity to critically think. If students get in the pattern of believing that math is magic, they give up when they come across something they don’t understand because they can’t remember the magical formula they thought they had memorized. Another danger is that if the students do not understand why something works, they may apply it in incorrect ways. We need to teach students to think critically and one of the ways to do this is by showing them how to derive these formulas, so they are not stuck if their memory fails them and so they can properly apply the formulas.

Technology also plays a key role in making math seem like magic. Increasingly, teachers are having students use applets or other forms of technology to make discoveries. In these technologies, often critical features or steps go missing and students do not come to an understanding on why something works but instead rely on the magic of the applet. While these applets can be beneficial to visualizing the concept, they often need to be supplemented. If an applet or other technology-based tool is lacking, the teacher needs to help the students fill in the blanks. Coming up with extra questions not included or giving an explanation for missing information are a couple ways to supplement these tools. We, as teachers, need to use technology to enhance our lessons instead of having it take over every aspect of learning. Sometimes spoken word is needed to reach true understanding.

Even though math sometimes seems like magic, we need to work to make sure that students are firm in their understanding of the “whys” in mathematics.

Friday, March 23, 2018

Building Student Confidence

In the past few years, I have had my share of aiding students with their math homework. Anywhere from college to elementary, I have observed and compared the tendencies of these students. Overall, the students have no issue remembering what they learned, but, if anything, they lack the confidence in their efforts to finding the solution. They answer each solution with uncertainty, looking for any direction from the instructor.

How can we build confidence in our students? It is one thing to have them find the correct answer, but if they continually need our confirmation to find confidence, are we helping or harming at this point? Henry Ford once said “Whether you think that you can, or that you can’t, you are usually right.” This quote helps underlines the importance of having confidence in your own ability, especially in the case of our students. For the remainder of this post, I will discuss a few of the concepts that I have applied to my teaching methods.
Instead of providing an explanation of how to solve the problem, form questions that allow the student to describe the process, helping them consider the question in their own understanding. Asking questions may seem primitive, but good questions can allow students to comprehend and retain the information.

Good questions can also be based in a student’s understanding. For instance, one of the students I helped enjoyed cooking, yet struggled with fractions in math class. By basing my questions and examples on student interests, they were able to base their reasoning in their own terms.
Depending on the relationship one has with the student, it can be intimidating for the student to express thoughts on the subject. Another contributor is the fear of being wrong. More than not, students will mess up when trying something new. In my own experiences, encouraging students to give explanations, even wrong ones, helped them to approach their goal in learning and gain the confidence needed to perform on their own.

Although I have only worked with students one-on-one, I have still seen them grow in confidence as these concepts were applied. Rather than end with some impeccable conclusion, I will end asking for additional ideas, because I wish to continue learning more about this notion. What are other ways we can aid in building student confidence? How does building an individual’s confidence compare to building a class’s confidence? What possible roadblocks are in the way to achieving this goal?

Friday, March 16, 2018

Gamification in the Classroom


            There are over 155 million Americans playing video games regularly. Chances are many of your students will be among these 155 million. To many parents, video games have a negative connotation that seem to have no value. The truth is that video games, in moderation, have many benefits. The gamers use many facts, tools, and information given to them to move on throughout the game. This makes it no surprise that some games can also be beneficial in your classroom.
            In our technology class, one of our favorite type of game is one that sets the students up against each other. If you have competitive students, as we do in Math 371, these types of games can be very beneficial. They can also be used for short quizzes, or exit tickets. One thing that we must be carful with when playing these is that we do not forget what we want our students to learn. Many of these games can teach the students speed, but is speed always important in math?
            Another type of gamification you can incorporate into your lesson plans are quests. During these, the students are given different questions, tasks, or projects to earn points. These are usually done throughout a semester or a month. They allow the students to work at their own pace but also allow your high-achieving students to work ahead. You can even create a fun tradition if you do it year in and year out.
            You can also use badges to reward your student’s learning. One advantage of this could be that not all badges mean the same for the students. A badge might signify an increase of a student’s grade, or an increased level of mastery. The badges will not only encourage the students in the classroom but will also allow the students to see where they are as far as understanding the content.

            In conclusion the main purpose of gamification in the classroom is to engage your students and create a deeper learning while having fun. Not only does it make it enjoyable for the students, it also makes it more fun to teach the content. I encourage every teacher to find a place in their lesson plans for gamification.

Thursday, March 1, 2018

How Twitter Became My PLN

Greetings students of Math 371!

I'm very excited to once again add to this blog.  Dr. Vestal asked me two years ago to post as a guest blogger.  I skimmed through my previous post and am happy that it's still relevant.  The revolution is as strong as ever.

I want to talk briefly about how Twitter has helped me become a better teacher. 

I first joined Twitter back in September of 2011.  It wasn't until the spring of 2014 that I started being active on Twitter and using it professionally.  Over the past four years, my use of Twitter has evolved into what I now consider my Personal Learning Network (PLN).  

I first started following some of the all-stars of mathematics education -- Dan Meyer, Andrew Stadel, and Dr. Vestal to name a few.  Over time, I've followed more and more people who I've either met, read about, or heard of via re-tweets and likes.

Each day, I spend between 5-10 minutes skimming through my Twitter feed, fishing for new ideas and resources.  Tonight's catch was okay, with a couple of potential keepers:




If there is a resource or idea that I like, I will simply send myself a direct message of the tweet.  If it is something really great, I'll implement the resource or idea as possible into my classroom.

Also, I will often times search #MTBoS and #iteachmath to see what I can find.  MTBoS stands for Math-Twitter-Blog-o-Sphere, and it is a collection of math teachers who love to share, collaborate through Twitter, and meet in the summer at Twitter Math Camp (TMC).  I have never been to TMC, but I hope to go someday.  Only about 85 people can attend each year; there is a lottery and waiting list for those who don't get chosen to attend.

In addition to providing ideas and resources, Twitter can also provide answers to questions you might have.  For example, just tonight this gentleman had a question he posed to #MTBoS:


Within four hours, there were more than ten replies and a pedagogical conversation taking place about this particular topic.  These conversations can be extremely beneficial to teachers who are in small districts and have no other math teachers in their building to bounce ideas off of.

Over the past four years, I have borrowed dozens of ideas and resources from people on Twitter.  I'm always looking for ways to improve my teaching and being active on Twitter has helped me grow as a educator.  

If you're not on Twitter, I challenge you to sign up today and begin to grow your own network.  I'd recommend these people as a great place to start:
  • Dan Meyer (@ddmeyer)
  • Andrew Stadel (@mr_stadel)
  • Jon Orr (@MrOrr_geek)
  • Fawn Nguyen (@fawnpnguyen)
  • Michael Fenton (@mjfenton)
  • Kyle Pearce (@MathletePearce)
  • I Teach Math (@iteachmathAll)
  • Classroom Chef (@classroomchef)
  • Open Middle (@openmiddle)
  • Mark Kreie ;-)  (@kreiem)
There are many, many more very good people to follow out there.
Best of luck on your journey.  I'll be seeing some of you around my classroom!

Mark Kreie
Brookings High School Math Teacher
mark.kreie@k12.sd.us
https://markkreie.blogspot.com/