May 6th, 2011
I had to get to an elementary session. I love elementary mathematics and especially fractions. The fraction strand for grades 3-5 in the Common Core has changed significantly.
We started by folding fraction strips into halves, fourths, eighths, thirds, sixths, and we left one whole. We discussed different ways that students might make these folds and how we can encourage accurate folding. We also discussed why we might want kids to fold thirds before you fold sixths. What connections can be made by students as they to fold these quantities. Then we might want the students to start comparing thirds and fourths, or other type of unit fractions. The Core places great emphasis on unit fractions.
Next we moved on to comparing 5/6 to 5/8 and talked about comparing like numerators. Getting students to line up their fraction strips is a nice strategy to help visualize the comparison. If we compare something like 4/6 and 2/3 we can start to get at equivalency. Comparing something like 7/8 to 5/6 helps us think about how far from 1 whole that we are. How do we get kids to see that eighths are smaller than sixths (unit fractions!) and help then transfer that to a problem like 7/8 compared to 5/6?
Consider the standard: Construct viable arguments and critique the reasoning of others. What does this really look like in the classroom? Students would be explaining their reasoning – not just telling you the steps that they went through. We need to listen to each other and very importantly teachers need to listen to what their students are saying. Not listening for what we want to hear, but listening to what the students are actually saying.
A key idea in these documents is the idea that although something may be taught in a certain place, those skills are expected to be used in later grades in different contexts. Not retaught, but spiraled through and built upon. This is going to require a great amount of collaboration across all grade levels to find those places and plan those units of instruction. Dr. Kepner mentioned that, for example, in third grade we should not even be touching computation but rather be building fraction sense in a hundred different ways. Then we start computation in fourth grade the students will have an idea of the concept of a fraction as a single number.
May 5th, 2011
Always nice to see Gail! As always, she started with a problem: How many handshakes are possible with 3 people? With 5? Then find a general rule for n people. Some of us drew a picture, others made a list. One smartie-pants worked out n pick 2. Gail mentioned the triangular numbers and talked about how this can lead in to the idea of discrete vs continuous and quadratic functions. A simple problem that can lead to some rich discussions. She’ll revisit the triangular numbers at the end of the session.
Gail emphasized two major ideas that may be new to us in the Common Core Standards:
- Words matter. You must read the standard and implement the thinking and reasoning behind it.
- Statistics for all students. For example, making inferences, or developing a margin of error, or deciding if differences between parameters are significant.
Make sense of problems and persevere in solving them. For example, a 5 oz steak is so much and an 8 oz steak is so much, how much for a 20 oz steak? We need to have a discussion with students about their solutions and go deeper. Do students make a decision based on just one of the data points, or do they use both? If they find a curve of best fit, what do the coefficients and constants mean in terms of the problem?
Construct viable arguments and critique the reasoning of others. So, make conjectures and evaluate arguments. For example, here is a pentagon. Make another pentagon with the same area. Students then look at six examples and decide which is correct and why. Something like area in the common core needs to be taught by the end of sixth grade and not taught again. It will be built upon and used in later grades but not taught again.
Use appropriate tools strategically. Technologies like Wallwisher or Linoit and Google Docs can be used to get kids involved when they are not at school. Gail also suggested that we think about why we use the technologies we do in class. Is it always necessary to use your interactive whiteboard? Does using it imply a front and center approach to teaching?
Reason abstractly and quantitatively. Think about a list of passing leaders in the NFL. This is kind of a Dan Meyer thing. Especially after you show the formula for quarterback passing ranking. What questions does this raise? (In Gail’s list the # of attempts for Fran Tarkenten was missing, btw.) Who’s the best? Why does the formula add 50? Why are some ratio’s weighted differently than others? Think about all the relationships that students could try and make sense of, for example, make a scatter plot of touchdowns vs. interceptions thrown. What does it look like and why?
Model with mathematics Think about the number of spaces from Go and the cost of the properties on a Monopoly board. Put a moveable line on the data and discuss what the slope means. Think about alternate scenarios like what if the board was a pentagon, or how much would Community Chest cost? In a high school setting, look at the regression line and notice that the line is pulled off course by the railroads. Maybe then we write two separate relationships: one for the railroads, and one for the rest of the properties. Henry Pollak has a book coming out from COMAP that will be awesome for everything you want to know about modeling.
Attend to precision Do we say reduce or simplify? Is there a difference? Do we use expression when its really an equation? Do we define our variables properties? Pay attention to equivalence!
Look for and make use of structure. 14/(2x-5) = 7/(3x+4) What do you do. Multiply the RHS by 2 right? Would our kids do that? Why not? We must be doing something wrong. Maybe we don’t choose our problems carefully enough…
Look for regularity in repeated reasoning Choose two whole numbers. Compute a^2 + b^2, then a^2 – b^2, then 2ab. What do you observe about the results? (they turn out to be sides of right triangles). Look at a spreadsheet where a goes from 2 on and b goes from 1 on. What do you notice? I’ll give you a hint: look for the triangular numbers. We want kids to see that math comes up in a lot of interesting places, and there may be unexpected connections.
In conclusion:
- Think deeply about simple things. -Ross
- Never say (do) anything a kid can say (do) -Reinhart
- If the class ends after the students have explained their work, there is no need for a teacher. -Takahashi
- When students don’t seem to understand something, my instinct is to consider how I can explain more clearly. A better way is to think “They can figure this out. I just need the right question.” -Kennedy
- I know what they have learned when I observe them in a place they have never been -Cuoco
