One thing that stuck out to me in the video where Hewitt was teaching algebra to his class was his statement about not taking the role of the checker. As a teacher, we want our students to have confidence in their answer and check for themselves/with their classmates whether their answer makes sense. Along with this, in the video about awareness and fractions, Hewitt mentions a joke he heard about how a child learned adding 2 apples plus 3 apples is 5 apples, but is unable to answer what 2 bananas plus 3 bananas is. These two points stuck out to me because it reminded me that teachers should simple facilitate students' learning. We want our students to be independent learners, not dependent on teachers to tell them what the right answer is or if confirm whether their answer is right. In society, we won't have someone with us at all times to let us know how to do each step and if we're doing things correctly. Everyone can make mistakes, but what we want to foster in students is the ability to recognize them, identify why it happened and learn from them.
A third stopping point for me was when Hewitt was combining 2 pencils and 4 brushes together. In my head, I said "2 pencils and 4 brushes," but Hewitt stated we might have said "6 things" which made me think that as a teacher, students may not give the answer you expect and there's many approaches/answer to the same problem. In my mind, I was thinking in terms of collecting like terms and how since pencils and brushes were different, they did not collect and simply were "2 pencils and 4 brushes." However, to Hewitt, he was considering the problem in terms of fractions (and like denominators) and giving the items the "same names" so they were able to be added together. I thought that this was really interesting.
I really like the fractions problems Hewitt gives at the end because it starts off seeming simple (find a fraction between 5/7 and 3/4) because I've learned how to find equivalent fractions with like denominators. However, once I got 20/28 and 21/28, I actually really liked and appreciated how sneaky and tricky the problem actually is. To find a fraction between 20/28 and 21/28, you have to realize that there is no whole number between 20 and 21 and you can't put a decimal in the numerator. Students who understand the concept of fractions well will realize that you can actually simply divide further to get 40/56 and 42/56 and find 41/56 to be between 5/7 and 3/4. The following problems after also get more and more trickier and really makes you more aware of things that are true (and can discover are true) about fractions. I think teacher-created math problems are definitely more creative and interesting because it tests your thinking and understanding of a concept a lot more than the regular problems you'd get on assignments/tests. I actually have seen similar (I think) problems in a math class I volunteered for and I've always wanted to incorporate these types of questions in my own math classroom, perhaps as a warm-up thinking activity for students at the start of class or something to try and think about at the end of class.
Hi Lisa, thank you for your thoughtful response! I agree that encouraging students to verify their own answers and engage with their peers to confirm their understanding builds confidence and self-reliance, vital skills for real-world scenarios where constant guidance might not be available. Additionally, the anecdote about the child's understanding of adding apples versus bananas serves as a powerful reminder that teaching should aim to cultivate independent learners rather than dependency on teachers for correct answers.
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