Do Math Diagrams Matter?

     Almost everyone has been in a math class where the teacher or professor has warned the class about the picture or diagram on a certain problem is not drawn to scale properly. For me, these problems mess with my head because I use visual aids to check if my answer makes sense. At the same time when doing problems such as those I look around the room and it seems many students could care less about the picture. After these observations, I wondered if diagrams make a difference for students in the classroom and why textbooks cannot have better pictures with the technology we have today.  

    When I started looking for information on this topic, I found that the use of proper diagrams in the math classroom is an understudied topic, and most of the articles you can find will tell you that. The article that aligned most with what I have observed is linked below the blog. This article talks about how all students, whether they have a learning disability or not, learn math in different ways. The focus of the article is on how students use diagrams and if they are helpful, whether they are making them or using one given to them. As one could guess, diagrams help students more than they probably realize. The part that I found interesting is that students seem to struggle if a diagram does not match what they have found. This is largely impart to figure #1 in the article (Sorry it would not post on the blog) that talks about strands of diagram proficiency. In one of the boxes for example it says one should be able to explain and justify how a diagram helped in a problem. If the diagram is not correct, how are we supposed to justify or explain how it helped solve the problem.  

    It is interesting that false diagrams occur with all the available technology. Technology has improved over the last decade, and there are several software programs capable of drawing diagrams for use. Over the past couple of semesters, I have been able to mess around with a website called GeoGebra. GeoGebra can construct just about any polygon or even polyhedron a teacher or textbook writer could ever want. Websites like this should be used to help students learn and understand. Whether that is having students do problems on those sites or the teacher taking the time to make a diagram that properly fits the questions. For more information on the impact of diagrams on student learning in mathematics, please see the article below. 

https://link.springer.com/article/10.1007/s11858-013-0519-1 

 

Mathematical “Common Law” of Order of Operations Divides the Masses

Are you ever scrolling through Instagram, Facebook, Twitter, or an equivalent social media platform and come across a post of an elementary math problem asking you to solve it? The post often provides viewers with an example of a problem with an answer found using basic order of operations and even sometimes give multiple choices for what the solution could be. When you work through it and are confident in your answer, you head to the comments to check to see if you are right. You find, though, to your dismay, that the responses are flooded with differing answers and agitated social media users. They note, “The result is infinite,” “12?,” “2,” “64,” “Both B and C are valid answers,” “to bad ur wrong,” “Isn’t it 1 no matter what you do” and even, “Why can’t anyone ever answer these math questions without a debate? Our education system is a FAILURE.”

Not only do these responses make you question your knowledge and retention of information from your math education, but they also might make you question the education system as a whole. Why are so many people getting these problems wrong and how can we actually solve these equations?

For problems containing information such as those on these controversial posts, the solution is most commonly found using order of operations. This is frequently referred to as PEMDAS, an acronym for parenthesis, exponents, multiplication, division, addition, and subtraction. These identify that the equation should be solved in this order, beginning with solving whatever operation comes first in PEMDAS before continuing to what comes next. With the pairs of multiplication and division, addition and subtraction, solutions are found by solving left to right. According to the South Dakota State Standards for Mathematics, this information is taught at the 3^rd-grade level and is a foundational step to learning mathematics. Remove this mathematical knowledge foundation and tell people that an answer they felt to be almost certainly true is wrong, and disagreements emerge.

What really, then, are the right answers to these equations? Well, as stated by Tara Haelle in her article “What Is the Answer to That Stupid Math Problem on Facebook,” “while the math itself lacks ambiguity… math has syntax just has language does – with the same potential for ambiguities.” Therefore, there are different ways to think about these problems, and while PEMDAS may seem like mathematical common law and a universal truth, it is more of a conventional way of solving equations to arrive at common answers.

In conclusion, though math equations on social media may divide audiences, causing those in comments to argue over what may be the “right” answer, these may not be the most beneficial discussions. Shouldn’t we be open to looking at various logical answers, rather than shutting down different ways of thinking? Personally, as a future math educator, I want to encourage students to think critically about problems they come across and explore possible alternate routes to solutions. Therefore, our goal for these problems doesn’t always have to be to find the right answer, but rather, to learn to use mathematics to express thoughts and reasoning in rational ways.

Links to articles mentioned:

https://doe.sd.gov/contentstandards/documents/0521-Math-Standards.pdf

https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html

Adding Artificial Intelligence as a topic in the STEM Technology Course

As we begin a new semester of Math 371, Technology for STEM Educators, I want to add information about how teachers can use AI in their classrooms. I know that many use it to create lesson plans, but I also want to include other ways that you can incorporate technology. Have any of you used AI with your preservice math teachers?  If so, how did you use it? I am looking for ideas and would love your help.

So, what did I do? I put the following prompt into Chat GPT (free version), “List some ways that AI can help preservice mathematics teachers.” And here is the response from ChatGPT:

AI can play a significant role in supporting preservice mathematics teachers in various ways. Here are some examples:

1. Personalized Learning: AI can analyze individual learning styles and preferences to tailor instructional materials, ensuring that preservice teachers receive personalized support and resources.
2. Adaptive Assessments: AI-driven assessments can identify specific areas of strength and weakness in preservice teachers' mathematical knowledge, allowing for targeted interventions and personalized professional development plans.
3. Virtual Classrooms: Simulated virtual classrooms powered by AI can provide preservice teachers with a platform to practice teaching strategies, classroom management, and lesson planning in a risk-free environment.
4. Feedback and Reflection: AI tools can offer real-time feedback on preservice teachers' teaching practices, helping them reflect on their performance and make improvements. This can include feedback on instructional techniques, communication skills, and classroom engagement.
5. Content Creation Assistance: AI can assist preservice teachers in creating engaging and effective instructional materials. This includes generating interactive lesson plans, creating multimedia content, and recommending relevant resources.
6. Data Analysis: AI can analyze student performance data to identify patterns and trends. Preservice teachers can use this information to adapt their teaching methods, address common misconceptions, and provide targeted support to students.
7. Professional Development Recommendations: AI algorithms can suggest personalized professional development opportunities based on the specific needs and goals of preservice teachers. This ensures that they receive training in areas that directly impact their teaching effectiveness.
8. Collaborative Learning Platforms: AI-powered platforms can facilitate collaborative learning among preservice teachers. These platforms can recommend study groups, learning communities, and forums for sharing experiences and insights.
9. Language Support: For preservice teachers who are not native speakers of the language of instruction, AI can assist in language development, providing pronunciation feedback, language comprehension exercises, and language support resources.
10. Accessibility Tools: AI can help in creating accessible materials for diverse learners, including those with special educational needs. This ensures that preservice teachers are equipped to address the needs of all students in their future classrooms.

These AI-driven tools can complement traditional teacher training methods, offering preservice mathematics teachers a more interactive, personalized, and adaptive learning experience.

There you have it--use AI to get ideas for how to use AI.