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Week 18

 Welcome back to the Alge-blog! This week, we read chapter 5 of the textbook, which focused on the justification of conjectures. This chapter highlights the importance of thinking beyond WHAT is true, but WHY it is true. 


What Stood Out and What I Learned

For this week, I was the discussion leader for my group. I summarized the main points of the chapter and created questions to help guide our discussion. This chapter focused on the justification of conjectures, shifting from ‘what’ to ‘why’. Justification is often the most challenging part of the mathematical process because a pattern alone does not serve as proof. This reminds me of my previous linear algebra courses, which placed a strong emphasis on solving proofs of theorems. I struggled a lot with that because while I could see the patterns, I couldn’t figure out how to explain the generalization. 

Furthermore, this chapter highlights that justification is challenging, but it can be even harder to convince someone else. There are 3 levels of convincing: yourself, a friend, and a sceptic. This helps to strengthen arguments and justification skills. The author suggests developing an “internal sceptic” who questions everything. This internal sceptic could play an important role in the mathematical thinking process because it encourages you to rethink and confirm conjectures. It describes this internal sceptic as “your own best enemy”, as it pushes you to be a deep mathematical thinker. 


The following were my discussion questions:


  1. How can we help students shift from noticing patterns through examples to providing convincing justifications?

  2. With the current class culture that often prioritizes correct answers, how can we intentionally shift students' mindset to focus on justification and reasoning over being “right”.

  3. How might the levels of convincing and the idea of an internal sceptic be implemented in your classroom, and how does this influence how you design math tasks or assessments?

  4. What challenges may arise when assessing justification and reasoning in your classroom? What strategies and practices would you implement?


Here is a summary of the main points of our discussion:

  • Give students the opportunity to explore mathematics in a guided setting

  • Move away from procedural questions and memorization and shift to open-ended questions that encourage exploration and reasoning

  • Give students credit for showing their thought processes and procedures

  • Use positive language, encouraging words, and celebrate mistakes

  • Low floor-high ceiling questions

  • Incorporate collaboration in the classroom



I think we had a very engaging and thoughtful discussion on this subject. We recognize that justification may be challenging for students, but through careful planning of instruction and activities, we can help to create a space where students can build their justification skills. 


In Class:

In class, we looked at typical lesson types and what makes a lesson effective. We compared a typical US lesson to a typical Japanese lesson and noticed that the Japanese lesson is a lot more student-centred by focusing on student problem-solving and collaboration. Allowing students to explore math problems on their own and share their findings can help support student understanding. This helps to build their mathematical thinking skills and build a deeper understanding compared to simply listening to a lecture. For this reason, the 3-part  lesson structure has been emphasized throughout our teacher education journey. It is common to think that incorporating collaboration and student-centred learning is more challenging in math classrooms, but it is indeed possible. We conducted a math problem in class using blocks, and each group worked on a solution. Then, we did a gallery walk to see everyone's processes. I really like these kinds of activities because they can enforce mathematical learning in an interactive way.


Key Takeaways

Overall, this week highlights the importance of shifting from a standard classroom that focuses on procedural mathematics and encourages creativity and exploration. I hope to create a classroom in the future that pushes students to ask questions and justify their reasoning. By focusing on exploration and justification, I can support students in building critical thinking and problem-solving skills that are applicable outside of the classroom. 

I also really like the structure of a 3-part lesson and plan to incorporate minds-on activities, interactive actions, and effective conclusions so that students are engaged throughout the lesson. I think student-led exploration and collaboration are so important to their development and allow for deeper understanding. 

I think using hands-on activities as well as incorporating technology, including graphing calculators and online manipulatives, are effective tools in exploration and give students the opportunity to make predictions, conjecture, and justify their ideas.


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