Stressed out? Try some math!

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

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 on 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 that 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 that you can use simple mathematics to begin to explore. The Collatz[†] Conjecture is a famous unsolved problem in mathematics that anyone can explore using a sequence rules: a) pick any whole number (1,2,3,4,5,6,7,….) b) if your number is even, then divide by two and if the number is odd then multiply by 3 and add 1, c) 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

·      So, 10 is even, divide by 2 and you get 5. 

·      Five is odd so multiply by 3 and add 1 and you get 16. 

·      16 is even and now divide by 2, that’s 8 so divide by 2 again and you get 4. 

·      Four is  even so divide by 2 again you get 2 and then 2 again and you get 1. Done!

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!

---

[†] Named after Luther Collatz, but also explored by  other mathematicians and  other names such as the Syracuse problem, and the hailstone sequence or numbers

This a cross posting of an item I wrote for the "Improving Your AIMM" blog for the Advancing Inquiry in Middle Mathematics project I am part of in East Texas.

Teaching is Improvisational

I want to share what I learned while writing a chapter about using improvisational games in educational settings. 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,

has been accepted and will soon be published in the Springer online publication, Encyclopedia

of Educational Innovation: Teaching and Learning Innovation Through Play.

I hope to have permission to share it directly with you when it is finalized. Until then, 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/). 

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.

What I learned this summer: Probability and Statistics

This summer I taught something new. Statistics for middle school mathematics. I had a blast! The approach we used was to use investigations of data to build statistical reasoning. Many curricular tasks focus in hard on the vocabulary and definitions statistics and then quickly move on to the procedural instruction on how to calculate these statistics and how to create graphical representations from that data. Very little time is spent designing and enacting investigations, creating good meaty questions, and then on the other side choosing representations and interpreting the results.

I quickly realized that the separation of probability from statistics is a mistake. This may seem obvious to others, but interpreting probabilities is directly linked to interpreting results from investigations. Trends and likelihoods are the same thing. Fair games are like designing an investigation that collects data equitably and without bias as well as choosing a statistical tool that fits the question. I don't know if teachers have been teaching these as linked ideas or not. My middle school teacher group reported that in their curriculum, probability tasks are separated from statistics tasks by months in the scope and sequence, and that they purposely teach statistics right before the state test to make sure kids will remember the procedures. **SIGH**

I also realize without a strong understanding of probability (e.g. certainly is rarely guaranteed, and that chance has no memory), the general population does not understand the responsibility of science to be transparent about their predictions, and honest about the level of certainty. For example, if a study on climate change shows a 99 percent chance that a cause is linked to the effect, this is pretty certain! The slightest chance of something being one way or the other make people doubt the predicted outcome. This happens in gambling or lotteries as well. People think "Well there is a one-percent chance that the prediction/correlation could not be true (we have human influenced climate change OR you will not win money), so I am going to believe the less likely outcome instead.

Last, I learned that I LOVE box and whisker graphs and dot plots. Honestly, it is true, until you teach something you don't really know it. Yes, I have used these tools in research, but it felt different teaching it, and teaching them conceptually, created from real messy data. Below are some photos taken from my group. We collected data on the first day about the groups favorite math topic, years of experience and certification. Teachers, the next day, created an investigation based in the data (yes, I know the investigation could have begun with a question, design of data collection, blah-blah-blah, but this was truly spontaneous). One question arose as they studied this data was, "Does alternate certification have a relationship with years of experience?" Here is the table of data and the results. Does certification type predict years of experience? Why might it be related?

Next, we wondered about the relationship of favorite topic to the other two questions. What do you think? Is this a good question? Why? Why not? What kind of representation could support answering this question?

In a couple of weeks, I am teaching another teacher group, and then after that we will be interviewing middle grades students to investigate their statistical reasoning and then teach a lesson based on what we learn about them. Can't wait!