 understand the nature of evidence and justification in mathematics. Yet, cognitive science research has revealed surprising strengths in children’s abilities to reason inferentially in non-mathematical contexts. In particular, there is growing evidence that children are capable of developing sophisticated causal theories, and of using powerful strategies of inductive inference when reasoning about the natural world. This contrast raises a potential paradox: Why do children seem relatively adept at reasoning in non-mathematical contexts, yet seemingly poor at reasoning in mathematical contexts? The objective of this project was to examine this paradox. In particular, this project sought to understand both the strengths and weaknesses of students’ reasoning in and out of mathematics and, to examine ways in which students’ reasoning in non-mathematical domains may provide an important bridge to improving their ways of reasoning in mathematics.
Mathematics education researchers have suggested that students’ overreliance on examples as a means of conviction and justification is a primary reason for their difficulties in learning to prove. Thus examples-based reasoning is typically viewed as a stumbling block to learning to prove and researchers have advocated that instructional approaches be designed to help students learn the limitations of examples. In contrast, we view examples-based reasoning as an important object of study and contend that examples play both a foundational and essential role in the development, exploration, and understanding of conjectures, as well as in subsequent attempts to develop proofs of those conjectures. The goals of the project were twofold: (a) investigate the nature of middle school and high school students’, undergraduate students’, and mathematicians’ thinking about the examples they use when developing, exploring, and proving conjectures; and (b) investigate ways in which thinking about and analyzing examples may facilitate the development of students’ learning to prove.
The objective of this project is to better understand the nature of the opportunities present in middle school curricular materials to engage students in productive uses of examples. The overarching goals of the project are to (a) examine the curricular treatment of examples in proving-related activities, (b) examine cross-cultural differences with regard to the curricular treatment of examples in proving-related activities given differences in students’ productive example use, and (c) provide insight into the design of curricular materials and associated teacher guidance with regard to example use in proving-related activities.
Knuth, E., Kim, H., Zaslavsky, O., Vinsonhaler, R., Gaddis, D., & Fernandez, L. (Accepted). Teachers’ views about the role of examples in proving-related activities. To appear in Journal of Educational Research in Mathematics.
Ellis, A.B., Bieda, K., & Knuth, E. (2012). Essential understandings project: Reasoning and Proving in High School Mathematics (Gr. 9-12). Reston, VA: National Council of Teachers of Mathematics.
Knuth, E., Choppin, J., & Bieda, K. (2009). Proof in middle school: Moving beyond examples. Mathematics Teaching in the Middle School, 15(4), 206-211.
Stylianou, D., Blanton, M., & Knuth, E. (2009). Teaching and learning proof across the grades: A K-16 perspective. New York, NY: Routledge.
Knuth, E. (2002). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486-490.
Have you ever been to Vancouver?  Instead of just not stopping at stop signs, you just go a little bit out of the way, and they make it obvious when a car is around, and it especially slow down the cars or make them avoid those neighborhoods. There is a bunch on like 10th Street: [https://www.google.com/maps/@49.2611072,-123.0846916,284m/data=!3m1!1e3!5m1!1e4!10m2!1e3!2e3](https://www.google.com/maps/@49.2611072,-123.0846916,284m/data=!3m1!1e3!5m1!1e4!10m2!1e3!2e3).Over the last 12 years, the Rococo menu evolved as chef Bruce Rinehart honed his ideas of East Coast-style fine dining.
Rococo Northpark, owned by Rinehart and business partner/executive chef Jason Bustamante, celebrated its sixth anniversary Nov. 9.
In 2017, they’re planning massive revisions to the restaurants’ fan-favorite menu.
“It’s time,” Rinehart said.
It’s a season of change for both the original venue at 2824 N. Pennsylvania Ave. and the larger second location at 12252 N. May Ave. Rinehart recently parted ways with chef Josh Partain after working together eight years, and Rinehart and Bustamante are planning a new bar concept, The Manhattan, which is expected to open sometime in 2017 at 210 Park Ave., Suite 150.
Rococo won’t deprive guests of the restaurant’s famous crabcakes or Bruce’s Love Salad for Two, which marries a classic iceberg lettuce salad with an antipasto plate of Italian cheese and meats. But when he began asking customers about potential changes, he found them surprisingly willing to reply.
Anyone who visits either location knows the owners take a personal approach with diners, and Rinehart and Bustamante have been exploring how to best introduce new items and pare down menus without alienating loyal customers.
“We’re passionate about people,” Rinehart said, “but we have finally come to a place where Rococo will still be Rococo” if the menus begin to diverge.
That means the Northpark and Penn locations will grow into their own identities, yet remain bonded by philosophy and a select core of menu items.
“A lot of the New England flair will stay, and our seafood is still a big focus that’s important to us,” Rinehart said.
Everything that could delay The Manhattan’s opening has, he said. Bar construction is at least a month behind schedule. Though the slow pace of progress is frustrating, Rinehart still eagerly discusses the new concept, which he plans to open in early 2017.
“The place is going to be wicked sexy,” he said.
The Manhattan is going in at the base of Oklahoma Tower, and the bar’s signature drink will be the potent cocktail for which it is named, a favorite of partner Ted Streuli, editor of The Journal Record.
But all of it revolves around the drink, with tiered Manhattan offerings with varieties of vermouths, ryes and Bordeaux cherries.
As he does at Rococo, Rinehart said he will take his cues from his customers, start with a small menu and see how they respond to daily features.
One issue at The Manhattan is the bar doesn’t have a gas range, which he said just makes cooking a more creative process.
Print headline: Changing courses, Rococo owner Bruce Rinehart plans some radical menu alterations.
When something breaks, it’s so easy in the modern world to replace it. Fast fashion and inexpensive consumer goods mean that we can buy something new to replace anything that is no longer usable. However, it wasn’t that long ago when it would have been the norm to try and repair something first so as not to create unnecessary waste and to save money. If you want to save some cash and perhaps be a little more eco-friendly, you could also try to make repairs instead of replacing broken items. It helps if you invest in quality goods, but even cheaper things can be fixed instead of thrown away.
Let’s take a look at some of the things that you could be repairing instead of tossing out.
Your home appliances can experience a range of problems, which might sometimes make you think that it’s time for a new one. But just because it’s stopped working, it doesn’t mean you have to replace it right away. Getting it looked at by a professional is often the best idea if you’re not sure whether it can be salvaged or not. They can tell you if it can be replaced or if it makes more sense to get a new one instead. You might also be able to learn a few DIY skills for repairing appliances, but you have to be careful with what you tinker with.
Fast fashion has made it cheap and easy to buy new clothes and accessories all the time. If something is no longer wearable, you can just throw it out and get something new. But if you would rather avoid creating unnecessary waste and maybe keep some favourite items, you can focus on repairing clothing instead. A few basic sewing skills can go a long way. And if you can’t repair something, you can always get someone else to do it for you. Paying to have clothes, shoes, or other accessories repaired could save you money, and it will keep your things going for longer.
When you need new glasses, you’re often encouraged to buy new frames for your new lenses. But what if your current frames are just fine and you want to keep them? If you need a new prescription or you have broken the lenses, you could replace them instead of buying completely new frames. Replacement lenses can be put into your existing frames so that you get to keep the glasses that you love. Considering that frames can be the most expensive part of getting new glasses, you might save a lot of money by just getting new lenses.
Electronic items are also easy to throw out if they no longer work. The low cost of electronics compared to years ago means it