Stressed out? Try some math!
(Developing a case for using math to generate cognitive load distraction from the ‘news of the day’) Debra Plowman, PhD COEHD Curriculum, Instruction and Learning Sciences Texas A&M Corpus Christi On top of the demands of work and school, daily news of the COVID-19 Pandemic brings us has created a lot of stress in our everyday lives. When I am stressed my wonderful partner will give me a math problem. “What?” you say, “A math problem? That would make me more stressed!” Sadly, it is true that for many people the mere thought of solving a math problem stresses them out. But hang in with me for a few paragraphs and I’ll explain why that is not so for us and how math works as a de-stressor for me. Perhaps it can also help you, too. How many people have used counting sheep or even counting backwards to go to sleep? In my personal experience concentrating one thing, by counting backwards from 1000 for example, can occupy just enough brain power for a moment that I let other thoughts go. In other words, counting provides just enough “cognitive load” that I let go of thoughts that are preventing me from sleep. I have often wondered about why it works. While not much research has been conducted on this particular strategy, the idea aligns with what is known about concentrative meditation as a strategy to relieve stress. Research on these types of meditation have found increased theta wave activity -- an indicator of relaxation -- associated with attentional focus on simple cognitive tasks (e.g. Baijal & Srinivasan 2010; Cuthbert, et al 1981) . Using math to relax helps me forget about other things if the problem is “just right”. The cognitive load encountered in doing these “just right” problems allows an immersion in thought which can provide a needed break from the heavier challenges everyday life bombards us with. As an example of a ‘just right’ mathematical tasks, I like to play with 99 + anything. 99 + 3 is 102 because 99 and one more makes 100 and just two more is 102. 99 + 56 works the same way. Use any starting number you want to and you will find a pattern. Try 67 + anything. What patterns will arise? I encourage you to do these in your head. I also enjoy multiplication puzzles like 4 times anything or 50 times anything. For example, 50 times a number is the same as 100 times the number divided by 2. There are several problems that Number Theorists have yet to solve, but are easily studied by a common person. Here are a couple of examples of famous problems that make what I call “just right” for relieving stress. A “just right” problem begins with an easy idea which you can use simple mathematics to begin to explore. The Collatz[1] Conjecture is a famous unsolved problem in mathematics that anyone can explore using a sequence rules: 1) pick any whole number (1,2,3,4,5,6,7,….) 2) if your number is even, then divide by two and if the term is odd then multiply by 3 and add 1, 3) if the next result is an even number, divide by 2, if not multiply by 3 and add 1. Keep doing those steps until you get an answer of 1. The conjecture is that any whole number selected will always end at 1. Another interesting question related to these sequences is the predictability of the length of sequences given any number. Okay, so let’s give a number a try using the rules involved in the Collatz Conjecture. Let’s try starting with 10. Example with the Number 10
Number theorists record these sequences to look for patterns and the pattern we created here is: 10,5,16,8,4,2,1- a 7-number long sequence. Doing this in your head is fairly easy, the rules are simple and you have to concentrate just enough to keep the numbers straight, but it is complex enough so that other thoughts cannot intrude. This meets my criteria for a math stress reliever. Another thing to notice is if I had begun with the number 16 a larger number, my sequence would have been shorter at just 5 numbers in length (16, 8,4,2,1). Try a couple of numbers yourself to see. Try it while you are walking somewhere, or sitting on the couch trying to not check your phone for news alerts, or even, trying to fall asleep. Share the love! Play around with the conjecture with a friend so you both are not looking at the news! Another ‘Big Unsolved’ is the Goldbach Conjecture, that all even numbers can be written at the sum of two primes. This conjecture has been around since the early 1730s, and the largest number ever tested is 4 x 1014 (that is 4 with 14 zeros!).And yes, it worked. An interesting thing about working with this problem is that it takes a little more exploration, rather than the straight-forward, mechanical operation of Collatz. This can start with a conversation on an afternoon walk, and end with writing some simple calculations down and looking for patterns. So, I have shared a few examples of what works for me, and I hope that you will find solace in these ideas. I also want to share some other math diversions that can help you get through these times. The first two are YouTube channels which fall into the category of “Math-tainment” and the third is an invaluable resource to find fun and engaging math tasks to do with the whole family. I have presented more ideas and thoughts in this vein on my blog: Math Nerd Under Construction (http://debbieplowman.blogspot.com). Enjoy your Cognitive Load! Numberphile (https://www.numberphile.com) The host is video-journalist Brady Haran. He interviews mathematicians from around the world who are willing to explain in engaging ways about many topics often using plain brown paper and simple drawings and calculations. 3 Blue 1 Brown (https://www.3blue1brown.com) Grant Sanderson, author of this channel uses visualization to share big mathematical ideas. Math “eye-candy”, if you will, that allows you to see the mathematics even if you are not ready to make any calculations. NRich Maths (https://nrich.maths.org) I have used this website to do some math on my own as well as curate for lessons with my students as well as families. The description directly from the website explains it best: “NRICH is an innovative collaboration between the Faculties of Mathematics and Education at the University of Cambridge, part of the University’s Millennium Mathematics Project. NRICH provides thousands of free online mathematics resources for ages 3 to 18, covering all stages of early years, primary and secondary school education - completely free and available to all.” References Cuthbert, B., Kristeller, J., Simons, R., Hodes, R., & Lang, P. J. (1981). Strategies of arousal control: Biofeedback, meditation, and motivation. Journal of Experimental Psychology: General, 110(4), 518. Baijal, S., & Srinivasan, N. (2010). Theta activity and meditative states: spectral changes during concentrative meditation. Cognitive processing, 11(1), 31-38. Thanks Tony for finding my de-stress zone and for the counseling reference help! [1] Named after Luther Collatz, but also explored by other mathematicians and other names such as the Syracuse problem, and the hailstone sequence or numbers
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I want to share what I learned while writing a chapter about using improvisational games in educational settings (Plowman, 2018). Teaching in discussion-based classrooms is a lot like an improv performance. To make this comparison, in formal theater everyone has specific lines and has practiced until perfect. Everyone on stage knows how the performance will end. The audience’s task is to absorb. Contrasting that type of theater is improvisation. During improv there are rarely any props, no script and the acting team needs to listen and respond to each other to keep the act going. However, there are rules and guidelines that the acting team must follow, it’s not just a free-for-all. This comparison parallels the contrast between teaching as directive and teaching in a discussion-based lesson. And, like improv, the task and goals of a mathematics lesson form the rules that the team or class follows. However, as the AIMM team well knows, improvisation during math lessons is a demanding task. As we have taught your students while you watch, you have observed how we have to be ready to respond authentically to students. You have all seen us struggle at times to be open to an unexpected idea or an idea that is incorrect or partially formed without shutting down the student contributions. By the way, these struggles become an important part of our follow-up discussion of the lesson, as well and the unexpected student responses. The “yes-and…”/”yes-but…” improv game is one of my favorites for building understanding of what we need to be doing and not doing with our students during math conversations and discussions. When learning how to have these discussions, we might be tempted to respond to a student idea and say, “Yes but…” and go on to insert the correct term, or redirect the student to a more efficient way. But what if, instead, we said, “Yes, and what else are you thinking?” or “Yes, and can we hear from another student?” or “Yes and, I like how you have used what we have learned about graphs to explain the pattern…” The object of thinking “yes, and…” is to keep the conversation going, and to be inclusive of the other players’ (students’) ideas. This game can be played between students in a fun way first to help them develop better, more supportive communication skills between each other. In the beginning, students can play the game using a non-mathematical context, like planning a vacation together. It is important to play the “yes, but…” scenes too so students can have a discussion about what that feels like in comparison to the “yes and…” scenes. The Bridging Project was a Mathematics PD that used an improvisational framework along with content sessions to develop middle school teachers understanding of mathematics argumentation (conjecturing, explaining, justifying and generalizing). The most compelling finding was that students of teachers who learned how to use improv in math class had higher academic achievement that students of teachers who did not have the training in improv. These teachers also held substantive discussions more often in their classrooms. Both sets of teachers received the same PD in content. Interestingly, teachers did not have to teach the students improv games directly (some did and some did not) as that aspect did not influence the results at all (Knudsen and Shectman, 2016). This indicates that it may be as or more important for teachers to practice and build improvisational skills when learning these new and complex discussion practices. My chapter, Improv games in educational settings: Creative play and academic learning, Is published in the Springer online publication, Encyclopedia of Educational Innovation: Teaching and Learning Innovation Through Play. You can access it here: https://link.springer.com/referenceworkentry/10.1007/978-981-13-2262-4_16-1 The rules of the game I highlighted here in this post are described via this link: http://www.yesandyourbusiness.com/portfolio/yes-but-yes-and/). By. Dr. Debra Plowman Texas A&M Corpus Christi References Knudsen, J., & Shechtman, N. (2016). Professional development that bridges the gap between workshop and classroom through disciplined improvisation. Taking Design Thinking to School: How the Technology of Design Can Transform Teachers, Learners, and Classrooms, 163. Plowman D.L. (2019) Improv Games in Educational Settings, Creative Play and Academic Learning. In: Peters M., Heraud R. (eds) Encyclopedia of Educational Innovation. Springer, Singapore A primary component of the AIMM program is the use of Manipulatives in the teaching of Mathematics. My working definition of a Manipulative in Mathematics is any tool (whether physical or digital) that can be used to model the conceptual/concrete knowledge of a topic in mathematics. For example, a teacher could use a set of wooden rods (let’s call them Cuisenaire Rods J) that are cut to different centimeter lengths and each painted with a different color. Students could then explore the concept of fractions through part-whole relationships between the Rods.
I fully believe in Manipulatives in Mathematics, if they are used properly. When I say, “Properly”, I mean they are not just taught as another procedure in a mathematics classroom that students should memorize. Manipulatives should encourage discovery of concepts and inquiry into those concepts. Recently, I had a teacher ask about manipulatives and their effectiveness in the mathematics classroom, particularly the achievement of students. Over my time as both a teacher of mathematics and teacher of teachers of mathematics in higher education, I have read many articles and listened to many presentations related to manipulatives and their effect. I recently read through the following article that has an EXCELLENT review of the literature surrounding manipulatives (positive and negative results) and shares positive findings from their own research on the effectiveness of manipulatives in a geometry classroom (https://files.eric.ed.gov/fulltext/EJ1097429.pdf). Even though this research was conducted in Turkey, the researcher cited many American researchers that have found similar, positive results who have also shared warnings towards the perceived “Magical” properties of manipulatives(see for example Deborah Ball https://www.aft.org/sites/default/files/periodicals/ae_summer1992_ball.pdf) Ball stated in her article, “My main concern about the enormous faith in the power of manipulatives, in their almost magical ability to enlighten, is that we will be misled into thinking that mathematical knowledge will automatically arise from their use.” I fully agree with this statement. One cannot make the statement “If you use manipulatives, they will learn!” Teachers must focus on what Deborah Ball also stated in her article in that teachers must use tools and solicit student thinking through classroom discourse and exploration. If you are to IMPROVE YOUR AIMM, you must think about how you manipulate the concepts you teach. If you are simply teaching procedures (computational and/or physical) then students are not thinking, you are. However, if you are supplementing your procedural instruction with tools that challenge student thinking and encourage classroom discourse, then greater learning can take place in your mathematics classroom. Keep Improving your AIMM, John One of my graduate school professors would end almost every class discussion with one simple question, “so what?” At first, I found it a bit disturbing that he would spend an entire meeting teaching something, only to ask at the end if it was even important enough to study. Over time, this question has come to frame much of what I do in my own career, especially as I try to answer that question as it relates to how and what I teach.
In our time together, we have focused a lot on teaching through problem solving and providing students opportunities to engage with and understand the mathematics. So now, I ask you: So what? What is the purpose? Is this even important enough to study? Consider this video as you ponder the question: https://youtu.be/kibaFBgaPx4 Considering the video, how would you answer, “so what?” Why is problem solving so important to study? Several years ago, I was doing a number sense presentation to a group of my online students. I had the class chorally tell me how to add two numbers, such as 19 + 37. The class, together in one voice, told me to set up the problem in the standard algorithm formation (19 on top, 37 on bottom). Next, they stated that nine plus seven is sixteen. Put down the six carry the one. One plus one plus three is five. Put down the five. The answer is 56. Then I shared with them how students with number sense would see this (I used the same standard formation): nine plus seven is sixteen. Sixteen is 10 + 6. Put the six in the ones column and carry the ten to the tens. Ten plus ten plus thirty is fifty. Fifty plus six is 56. After the presentation, one student came up to me and shared how she never saw the 16 as a single number. It never occurred to her that the “put down the six carry the one,” actually represented a number, rather it was just a step she had to do in order to figure out the answer. A very basic concept in mathematics finally made sense to her as a college student. This was an exciting aha moment for her! So again, I ask, so what? How does your teaching help to answer that question for your own students? How can it frame what you do every day? Here’s to all the “so what” questions in hopes that asking it of ourselves continues our growth as educators! I’ve always liked the month of January. That’s probably because my birthday is in the middle of the month. Some years, I get lucky and have the day off thanks to the holiday honoring Dr. Martin Luther King, Jr. On other years, I end up teaching on my birthday.
This year, my birthday landed on the first day of the semester for two of my classes. I’m always excited to begin a new class, learn about my students, and help them understand mathematics better. I try to be careful and introduce concepts in a way that makes sense to them, and I resist the urge to introduce an algorithm before it makes sense to the students. For an illustration of what may happen when algorithms are introduced too early, I encourage you to watch the six-minute video of Rachel (http://www.sci.sdsu.edu/CRMSE/sdsu-pdc/nickerson/imap/files/clips/Rachel.mov) as she explains how to change a mixed number into an improper fraction. Her comments from 1:10 to 2:00 remind me that my students need to be the ones making sense of the mathematics, not me. Finally, I would like to share the work of Dr. Crystal Kalinec-Craig at the University of Texas at San Antonio. She wrote an influential piece in Democracy & Education about the Rights of the Learner to promote equity in mathematics education (https://democracyeducationjournal.org/home/vol25/iss2/5/) . Briefly, they are as follows:
Thank you for all that you do for your students. I appreciate your efforts. Sincerely, Dusty Jones Sam Houston State University [email protected] Twitter: @jonesmathed Happy New Year! Tony Robbins tells a story about a UPS worker who never made more than $14,000 a year and yet became a millionaire—he was disciplined and saved money to invest on a consistent basis. Robbins argues that having effective daily habits builds rewards over time. If I knew how to make you a millionaire, I wouldn’t be here, but let’s apply this concept of discipline to become successful math teachers.
You have learned a lot about how math can be taught from our workshops. You have the ideas, now it is imperative to apply these ideas on a consistent basis. What is something that you would like to implement? Would you like to include a BURST activity once a week or for a few minutes every day? Would you like to spend more time engaging those students who think that they are not successful in math? Go ahead and set a goal. It is effective to have a long-term goal and a short-term related goal to help you achieve that longer goal. I am sharing a personal goal and professional goal on our Twitter page. Post your goal/s at https://twitter.com/AIMM4ETX More on goal setting by Tony Robbins: https://www.success.com/tony-robbins-goals/ We look forward to seeing you at SFA on February 22 and 23. Jim Ewing, Ph.D. |
AuthorsJohn Lamb, Ph.D. Archives |
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