Reading Journal Week 6

1.         Reread Chapter 5 – carefully

2.         Design a lesson and mini-experiment for a Discover-a- Relationship objective from the Unit Draft, and modify the Unit Plan if there is a need. For more details consult Activity 5.7 p.202, Activity 5.8 p.202, Synthesis Activity #3 p.206.

Click here for discover-a-relationship lesson plan.

Click here for discover-a-relationship tasksheet.

Click here for discover-a-relationship miniexperiment (also found in "Assessment" section).
*Also found in "Cognition/Instructional Strategies/Planning" section.

3.       In your journal describe the stages of discover-a-relationship lesson.

The stages of discover-a-relationship lessons are the following:

Stage 1: Experimenting

In this stage, students are given some activity, tasksheet, etc., in which they are led to experiment with concepts related to the targeted relationship. The teacher’s role is to facilitate the activity, ask stimulating questions, and help guide students who are having trouble.

Stage 2: Reflecting and Explaining

In this stage, students will examine the outcomes of their experiments and attempt to draw meaningful conjectures based on their findings. Students will verbalize their ideas. The teacher’s role is to guide these discussions and ask probing questions.

Stage 3: Hypothesizing and Articulating

In this stage, students attempt to formulate and verbalize their ideas about possible relationships based on the analyses of their experiments. The teacher’s role is to guide students to further think about and test these ideas.

Stage 4: Verifying and Refining

In this stage, students will try to justify or disprove their ideas about relationships (this justification may be informal verification or formal proofs). If there is reason to believe a proposition isn’t correct, the class will attempt to formulate new propositions, and again test these. The teacher’s role is to guide this cycle of refining propositions until the class can agree upon a proposition about the relationship.

4.       Express anything that you find confusing about the construct-a-concept or the discover-a-relationship learning levels.  (Note that, as is customary in Mathematics, ``or" is inclusive.)

            The biggest confusion I have with both types of lessons is how to conduct the discussions. The learning levels make sense to me, and the stages of the lessons are logical, but I am a little bit overwhelmed with the thought of leading a discussion within these lessons. It is easy enough to say what you want to accomplish during the discussion, but an entirely different matter to actually conduct it.

5.       If you have not already, please: Find a middle or high schooler and engage them in an interview as described in Synthesis Activity #5 p. 171.

A.              If it were not a required subject, would you choose to study mathematics? Why or why not?

“Yes, because it was the easiest subject for me. Once you got it, you got it, and then you were done. It really depends on the teacher, though. When I have a bad teacher, I don’t enjoy it at all, so I wouldn’t choose to take it in that case. But when I have a good teacher, I love my math classes; they just make sense.”

B.              Is mathematics more or less interesting than other school subjects you take? Why?

“I wouldn’t say it’s incredibly interesting to me, but I enjoy learning it and solving problems because it makes sense and you can do it over and over again. I’ve never learned applied math really, I’ve just learned how to solve problems and deal with equations and things like that. I’ve never taken stats or logic because I’d rather take the classes that are numbers, numbers, numbers. I like things that are right or wrong.”

C.              Is mathematics more or less difficult to learn than other school subjects? Why?

“Less difficult, because once you’ve seen it done you can work through it on your own; it’s totally replication. I like that it builds and I can apply what I’ve learned to what I’ll learn next in the class – other classes are a lot of new concepts piled on and on. Once you know you can do something in math, you can do it over and over again.”

D.             From where do you think the mathematics we study in school comes? Explain why you believe what you just told me.

“I think it comes from mathematicians who spend their life trying to study a concept and then come up with a formula, and prove things that actually work. Just like science: it comes from people who devote their lives to studying it and then come up with a system of rules.  I think this way because I remember having to do a report about a mathematician and what things they were working with when they came up with equations and things. I’ve never really learned stuff like this in class though, besides this paper.”

E.              What do the words “mathematical research” bring to your mind?

“A lot of white boards, plugging and chugging, calculators, a lot of computers running, seeing if things work out. It makes me think of The Life of Pi when he just writes out all the digits of pi on a blackboard for hours. I picture people in white coats having a problem and siting in front of it and trying to apply the rules that they know figure out new things. I picture everything taking a lot of time. But I kind of feel like this doesn’t happen anymore because it’s all figured out.”

F.              Tell me what you know about any one of the following people: Pythagoras, Euclid, Hypatia, Rene Descartes, Maria Agnesi, Isaac Newton, or Andrew Wiles.

“I know a lot about Newton because I took physics. I know Newton’s three laws, but I’m not sure how those apply to math. I learned that Newton had an apple fall on his head and ‘what goes up must come down.’ Pythagorus goes with the Pythagorean theorem, dealing with triangles.”

G.              What is the easiest thing about learning mathematics?

“ Usually there’s only one answer, so you know you’re done when you have one answer. With writing a paper or something like that, you could go on and on. With math, you just follow the steps you’re given and get the answer; your hand is basically held, there’s not creativity. You just follow step one, step two, step three… If you learn how to problem solve, there are no curve balls because you know how to read a question and follow a pattern.”

H.             What is the hardest thing about learning mathematics?

“It takes time. If you really want it to be natural, you have to spend LOT of time. Also, when one of my teachers took away our calculators, it was SUCH a handicap. I second-guessed myself at everything and it took me ages to do by hand. It’s almost like relearning it because you have to start from the ground up again. The calculator just does it for you; you don’t have to remember rules or anything anymore. But I think that’s good because it makes you really have to learn the material on your own.”

I.               What is the most interesting thing about doing mathematics?

“I think it’s interesting how many different answers you can get when you make mistakes. Even a little mistake in the beginning can make a HUGE difference in the outcome of the number.  Little mistakes are incredibly important in math. Even things like radians and degrees, I still don’t get the difference. Math takes a lot of focus and attention to detail.”

J.               What is the most boring thing about doing mathematics?

“Not knowing how math is supposed to fit in. I enjoy doing it, but sometimes it gets to the point where I feel like I’ve been doing this exact same problems for several weeks, just with different numbers. I think there could be more mixing it in with history, or something to make it applicable. That isn’t my teacher’s strong point, it’s always like: this is the formula we’re using, and this is the type of problem you use it for.”

K.             Do you read your mathematics textbook differently from the way you read textbooks for your other subjects?

“Yes definitely. With other subjects I read everything because everything is necessary. With math textbooks, I only need the examples because I really don’t care about anything else; I just need to see the process to do it so I can replicate it. There’s a lot of skimming to try to find out what’s important.”

L.              How often and for what reasons do you use mathematics outside of school-related work?

“I use it a lot for cooking, doubling or halving recipes; I really like to cook and bake. I use a lot of simple math for change and tips and stuff. It’s all really simple, I never use any of the formulas. Sometimes I help people with stuff, like tutoring. I want to be a nurse, so maybe using statistics for doses and stuff would be applicable someday.”