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Week 17

 Welcome back to the Alge-blog! This week, we looked at chapter 4 of the textbook, which focused on conjecturing. Conjecturing is an important part of the mathematical thinking process, in which students take the time to explore a concept, notice patterns, and make their own predictions. 


What Stood Out and What I Learned

This week, I was the implications detective, so I highlighted the main idea of the chapter and the potential implications and challenges in the classroom. This chapter encourages educators to implement conjecturing as a focal point in the math classroom, and encourages students to take the time to write down conjectures and test them through the attack phase of solving problems. Challenges I foresee include students giving up too quickly or avoiding sharing their conjectures out of fear of being wrong. 

During our group discussion, we highlighted potential challenges students may face and how teachers can best support them in conjecturing. We think using guiding questions and prompts can help to encourage students to conjecture because it reduces the likelihood that they get stuck and give up. Also, creating an environment where making mistakes is viewed positively can help students get more comfortable with making conjectures, even if they are incorrect. 

We also discussed that conjecturing can be a very valuable skill to teach because it encourages critical thinking, shifting the emphasis from procedural memorization. Educators should challenge students by asking them why a procedure works, and have students explain their thought processes and rationale. 

It is also important to teach students the value of making incorrect conjectures. Disproving conjectures can create a more meaningful learning experience by teaching students that learning is not linear and that making mistakes and trying again is a normal process. 


In Class:

We tried the following problem in groups:



While specializing in this problem, the first thing I tried was writing down examples of sums starting at 1+2, then 2+3, 3+4, and so on. I looked for a pattern by focusing on even numbers, odd numbers, prime numbers, and the squares of numbers. One of the first things I noticed was that 1 and 2 don't work. From there, I wondered if 2 had something to do with the pattern, so I looked at products of 2 and powers of 2. Once I found that 2, 4, and 8 didn't work, I came up with the conjecture that 2^x doesn’t follow this property. I tested up to 34 on paper and 64 using a calculator, and I was not able to disprove this conjecture; therefore, it was our conclusion. 


I mentally conjectured a lot while starting this problem, but I did not take the time to write them down until later. I think I rushed into the problem without realizing that my thoughts were conjectures, and I should practice writing down my ideas before testing them. 


In class, we looked at what a typical lesson looks like and compared 2 lesson types, a US lesson and a Japanese lesson. I liked how student-based the Japanese lesson was by emphasizing discussion and student-led instruction, compared to the US lesson, which is majority teacher-led. The standard teacher teaches, student listens and copies has several limitations to learning. Students who benefit from hands-on learning, discussions, and exploration may struggle in a standard US classroom but thrive in a classroom like the Japanese layout. 


Key Takeaways

The main idea I took from this week was that conjecturing is an essential skill within the mathematical thinking process. In my future classroom, I will ensure that I give students time to explore mathematical concepts and come up with conjectures. I will also encourage students to write down their thought process to help them navigate and show visual evidence of conjecturing. I will also commit to creating a classroom environment that values mistakes and making false conjectures as a learning opportunity rather than a failure. Through using positive language and providing examples of this, I can help student shift their focus from being “correct” to a growth mindset.


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