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Think about your personal history as a STEM learner.
From your history and personal experiences, what does it mean to think mathematically/scientifically?
I love mathematics. It is satisfying to be presented with a problem and be able to think it through. It is the language of many other fields especially science, technology, and engineering.
It’s interesting how connections form between what I’m learning in several different classes. For instance, this question about thinking mathematically/scientifically came up this week in Project-Based learning while engaging in a micro lab. During the class discussion portion, a fellow student made a remark that prompted me to confirm she was in the “math track” of UFTeach. I commented that I could usually tell because the math people and the science people seem to have their different “go-to” ways of looking at the world. Although I am in the science track, I would consider this observation and the connection making between courses mathematical thinking. Mathematical thinking involves consistently looking for patterns and relationships. To do this one needs reasoning skills, persistence, and a foundation of heuristic strategies. Building on those points leads to the ability to make predictions about results. The most significant aspect of mathematical thinking is consistent pattern finding. What I also find is to think mathematically one needs a few other important traits, too. The first is recognizing the utility of math in everyday life. Instead of woefully asking, “When am I ever going to use this?” doing math becomes more about enjoying the journey of discovering the patterns and making the predictions and wondering positively, “Where can I use this?” Other traits useful for mathematical thinking are risk-taking and persistence. Math might scare some people because of the risk of being wrong. We live in a results-driven world. This begins in elementary school with standardized testing and taking tests in general. One is bound to be wrong at least some of the time when confronted with new math problems. Thinking mathematically means having the willingness to be wrong, but persistence saves the day. It is so easy to give up and proclaim, “I can’t do math!” However, mathematical thinkers don’t seem too nonplussed about being wrong. They confront the problem using different strategies. In my mind’s eye, I always picture Einstein’s blackboard with a yet to be solved problem that lingered for years. One just needs to keep trying.
Scientific thinking is generally thought to revolve around using the scientific method. The mindset relies on being observant and asking questions. It has similarities to mathematical reasoning in that one must take data and compare, sort and organize it in a meaningful way. Also like mathematical thinking scientific thinking leads to the ability to make predictions. Questions and speculations lead to testing hypotheses and designing experiments. Experiments generate data. The final aspect of using the scientific method consists of evaluating the results. Further, thinking scientificlly goes beyond the results to the application. To think scientifically one needs the ability to interpret, explain, and predict outcomes, but also to draw on previous work by other people. Asking the question, “Why does X happen?” is the first part of the process of thinking scientifically. It might be less frightening to think scientifically. When doing so, we expect to not know the answer. That is the purpose of creating experiments. We realize from the outset that there is no right or wrong answer because we don’t know yet why the thing is happening. Thinking scientifically involves making observations, comparisons, looking for patterns, making predictions, designing falsifiable experiments, interpreting results, drawing conclusions, and finding applications.
Examples of some of my past mathematical and scientific thinking along with a photo representation of my mindset.
Explore
What lessons might we draw from the Fish is Fish story in relation to human learning?
"Fish is Fish" relates to human learning in that we each have our own perspective from which we see the world. This idea relates back to last week's topics of learning theories especially constructivism. Constructivism takes the stance that we each construct our own perspective of the world. It reminds me of the quote often attributed to Einstein (which I cannot verify, but I've tried numerous time to track down who originally said it) "Everyone is a genius. But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid." To me, this quote reinforces the fact that we are all individuals with our strengths and weaknesses. I link it to this story because of the frog's declaration that fish is fish and frog is frog. We each have to be ourselves and embrace the fact that our perspective helps to shape our learning and how we behave. Twins can sit next to each other in class for years but walk away with differing amounts of retained and understood information just because, as individuals, they will have different perspectives.
lifehacker.com
Your Ideas about the Seasons - Without Googling or looking up the correct answer, think about how you would respond to the following two questions (Explain and justify your responses):
What causes the seasons?
In a nutshell, the change in seasons is due to the fact that the earth is tipped on its axis at an angle of 23 degrees. It takes the Earth one year to revolve around the sun and one day to fully rotate on its axis. The angle of tilt does not change, but which hemisphere receives direct sunlight as opposed to light that sort of skims across the surface changes based on where our planet is in its revolution around the sun. During the northern hemisphere's summer, that area is tilted towards the sun more than the southern hemisphere. That angle of tilt is why the two hemisphere's experience opposite seasons throughout the year. New York's summer is Sydney, Australia's winter.
sciencing.com
What causes the phases of the moon?
I just had a conversation with a friend in my physics class about this recently. A really interesting fact about the moon is that it is locked in synchronous rotation with the Earth. This is called tidal locking, and it is also what causes high and low tides. This synchronicity means that we don't notice the rotation of the moon. The moon orbits the earth and is in a different position for each day of its 29-day cycle. The phase is determined by which portion of the moon is lit up by the sun. When the moon is between the Earth and the sun we call it a new moon. Because of the position of the moon, it looks dark to us on Earth. So, the sun is hitting the "back side" of the moon, the part we can't see. Just like when half of our planet is facing the sun, we call it day and the "back half" of the planet experiences night. A full moon looks big and bright because the Earth is between the moon and sun. This positioning results in us seeing the entire face of the moon lit up from the sun. I read these explanations aloud to my husband to check for coherency. He said he still didn't quite understand the moon phases. We grabbed two rocks and a flashlight, and I shut off the light. Once he saw the moon rock in between the Earth rock and the flashlight sun, a lightbulb went off in his head. Very exciting for both of us!
metro.co.uk
Consider the following:
In the responses of these students, what struck you as interesting or important?
I found it interesting that so many students held the same misconception. To me, it indicates that somewhere in our educational system, something went left where it should have gone right. I also noticed that even after creating a drawing to help the students make sense of the concept, the students continued to cling to their misinformation even if one could clearly see from their drawing that their explanations could not possibly make sense. One young lady drew the suns rays directly hitting the Earth and other rays bouncing off some unseen object and reflecting back without explanation as to where the rays bounced. Another interesting thing I noticed was that the age of the student did not seem to impact the common incorrect beliefs- even faculty and staff erred in their thinking.
In each of the cases, how might what the students say make (at least) some sense?
Why, for example, would they say that seasons are caused by the closeness to the sun?
The ideas make sense in light of common knowledge about heat sources. People know that if you get closer to a campfire the air feels hotter than at a distance. Couple that with the fact that we are taught that the Earth revolves around the sun in a slightly elliptical orbit. If the word slightly is removed, it changes how this information is interpreted. Now, if one imagines the Earth with an elliptical orbit the idea of the Earth moving closer to the sun and thus getting hotter is not so far fetched.
Where might the students have acquired the idea about the clouds and the phases of the moon?
The clouds causing the phases of the moon is such an unfamiliar explanation to me that it is hard to speculate from where the idea originated. I can only guess that the appearance of the surface of the moon looks like it is covered by clouds or that the student believes Earth's clouds occlude the moon. Maybe this student watched werewolf movies and the full moon was covered by clouds allowing the human to revert back to its normal form. This covering/uncovering of the moon might be what the student thinks are the phases of the moon.
Because of my unfamiliarity with this misconception, I did some digging. I found the website Lunar and Planetary Institute at
https://www.lpi.usra.edu/education/pre_service_edu/PhasesMisconceptions.shtml that has a page titled Moon Phases: Misconceptions and Educational Research. According to this page, there are several other commonly held incorrect ideas.
Common misconceptions include:
The Moon can only be seen at night
The Moon makes its own light, instead of reflecting sunlight
The Moon’s phases are caused by the Earth’s shadow
The Moon’s phases are caused by clouds
The Moon’s phases are caused by Earth’s rotation on its axis
The Moon’s phases are caused by the Moon’s rotation on its axis
The Moon takes one day to orbit the Earth
The Moon orbits the Sun instead of the Earth
(Lunar and Planetary Institute, 2018)
The webpage contains the video from class, that I also linked on this page, titled "Private Universe".What I like most about this site are the resources it provides to educators. Following the links, the virtual lab includes a 5-question survey that encourages teachers to also take to address their own misconceptions.
Explain
Some questions to ponder and write about:
What does it mean to know and/or learn?
Learning and knowing are not easily defined concepts. If I combine the three learning theories we've covered in class- Behavior, Cognitive, and Constructivist- then, learning is a process that can be elicited through a conditioned response, as a result of direct instruction, or from environmental factors such as social interactions. Learning will be evidenced as a change in behavior, an assimilation of knowledge to previously understood information, and result in a product. After all of this learning about learning and about preconceptions, I still hold to my previous belief that learning and knowing is an acquisition of information. However, I have expanded my definition to include the integration of knowledge with other bits of information stored in my brain. Evidence of learning can include new behaviors, but I also see an increase in confidence that one has been exposed to new ideas, remembers them, and can generalize this knowledge to other areas.
What does it mean to be a competent mathematics or science student?
Competency in mathematics and science stems (<---See what I did there?) from persistence. An effort is required on the part of the student to look for patterns, ask open-ended questions, to test hypotheses, and a willingness to go back to the drawing board. Competency is a result of making many mistakes but not stopping until a reasonable and rational explanation has been uncovered. It means learning what past mathematicians and scientists that came before did. It is useful to learn from others and attempt to replicate their results to verify for oneself the validity of their results and to give a sense of the process of "doing" math and science. Eventually, it leads to an accumulation of knowledge and critical thinking skills. That, to me, is competency.
What does learning mean? How is learning different from or similar to understanding?
Learning is assimilating new information into one's memory for later recall and application. Understanding relies on making connections. Learning is facilitated by understanding. Without understanding, misconceptions persist and impede learning.
How can you get at what students are thinking? How can you assess knowing and learning?
You can get at what students are thinking by asking them open-ended questions, have them draw diagrams, or even have them sort pictures or concepts. There are several methods for assessing a student's knowing and learning beyond quizzes, tests, and presentations. One of my favorites is the use of wait time. This gives the entire class an opportunity to reflect and contribute to the classroom discussion. There are also many wonderful informal ways to assess learning such as playing Kahoot!, using Plickers, entrance and exit tickets, journaling, classroom micro labs, etc.
harrisoncatalyst.com
Extend
What does it mean to be an expert?
Expert vs. Novice: The Beatles White Album
This is a question I researched on my own over the years. I remember reading that it takes at least 10,000 hours of practicing (perfect practice) an activity coupled with studying relevant content to become an expert. For example, to become an expert guitar player I need to practice 10,000 hours. However, playing "Twinkle, Twinkle, Little Star" for 10,000 hours does not make on an expert. It takes learning scales, timing, precision, studying different genres of music. Being an expert guitar player means practicing as close to every day as possible, but not just any sloppy practice. “Practice does not make perfect. Only perfect practice makes perfect.” (often credited to the legendary football coach Vince Lombardi). What this looks like for playing the guitar is, when working on a new piece of music, if I make a mistake I stop and go back to correct it. I do not want to practice it incorrectly because practicing builds muscle memory and brain memory. I do not want to memorize it wrong. This method generalizes to most any skill. It boils down to time and concerted effort. The result is an accumulation of knowledge and skill that can be shared with others.
References
Lunar and Planetary Institute. (2018). Phases Misconceptions. Retrieved from
https://www.lpi.usra.edu/education/pre_service_edu/PhasesMisconceptions.shtml
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