Flash - At Least One Of Euler's Equations Does Fail!!!!!!

 For the past few centuries, mathematicians have been striving to create equations that model the motion of fluids.  Some of these equations such as the one that models the ripples that cross a pond, have been used to predict the weather, help design better airplanes, and help explain how blood circulates through the body.  

Although the equations can appear quite simple, the answers can be a lot harder to explain, in fact, almost ridiculously difficult. One of the oldest of those equations was formulated over 250 years ago by Leonhard Euler. It described the flow of a fluid with no viscosity, no internal friction, and can't be compressed into a smaller volume.  Practically all of the equations dealing with nonlinear fluids have been derived from this formula.

Unfortunately, no one is sure if the equation actually models ideal fluid flow.  Mathematicians have been looking to see if there are any points at which it fails, where the equation breaks down.  Now there are two mathematicians who have shown that one particular set of his equations sometimes does fail.  It does not solve the problems with the more general version but offers hope.

The 177 page proof is the result of 10 years of computer based research. This does make it hard for other mathematicians to check the proof and it has people looking at the question of what makes a proof along with the idea of how viable is it if the only way to prove something is with the use of a computer.  

As far as Euler's equations, if you know the location and velocity of each and every particles in the fluid, it should be able to predict how the fluid changes and evolves over time but mathematicians want to know if this is true for all cases. If at any time the values shoot up to infinity, this singularity then blows up at that point and fails.  Once that singularity is found, the equation no longer can calculate the fluid's flow.  Everything becomes more complicated if you try to model a fluid with viscosity. 

In addition, it is very hard to prove a singularity of this type because most computers are unable to compute infinite values. A computer is able to get close but cannot compute the actual values so it is not an actual proof. Instead, mathematicians have to go back to a previous point that gives them a self-similar solution.  Two mathematicians came up with a possible point but were unable to prove it so they went back, looked at things and developed a hybrid approach.  

They decided they were proving that if you took any set of values close to the approximate solution and put it into the equation, the results wouldn't be that far off.  So they had to define closeness before they could create a complex inequality using terms from rescaled equations and the approximate solution.  They also had to make sure that everything came out balanced to something small. 

They ended up breaking the inequality into two parts. The first part could be solved by hand using techniques from the 18th century but the second part required the assistance of a computer due to the number of calculations and precision needed. Using computers to help prove in this particular field is relatively new and will take a while before people are able to check their work.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If your orchard produced 87,250 pounds of pears and each pound contains 2.5 pears, how many pears did you harvest?

Warm-up

 

If each pear tree produces 1,550 pounds of pears, how many pounds does your 23 pear trees produce?

Two Quick Assessments

Today, we'll be looking at two quick assessments that are quick and do not need much preparation.  At the same time, you'll have a chance to check for student understanding.  One I mentioned briefly earlier this week and the other is something that can be used at any time including as an exit ticket.  

Earlier this week, I mentioned Stop and Jot but I wanted to go into more detail and talk about different variations one could use.  Stop and Jot is as it sounds, students stop and jot down their thoughts. When students write their thoughts down, it helps promote both learning and retention.

The way to implement Stop and Jot, begin by having students draw a rectangle box on their notes or worksheet.  At some point during the lesson, stop and ask students to respond to a question that you pose. Once everyone has a chance to write down their thoughts, ask for one or two volunteers to share their thoughts or read the responses later.  

The stop and jot can be used at any point during the lesson as a way of providing time to process the information and to help students with their note taking.  If it is used before the lesson, it helps activate a students prior knowledge, if it done in the middle of the lesson, it helps students make sense of the material, can be used to check for student understanding, and after the lesson, it helps students clarify their thoughts, make connections with previously taught materials, and find relevance. 

Best of all, there are several variations of this activity available.  One is Jot-Pair-Share.  In this one, students jot down their thoughts individually.  Then they break up into pairs to share their thought and the last step is to share with the whole class.  Another variation is the quick jot in which students have between a minute and a minute and a half to record their response to a specific prompt or question.  If students need to record important information from the textbook or from a video, use the stop and fill. Students are given a sheet with blanks so they can fill in the material.  One can always do the group jot which as students broken up into larger groups and they share their thoughts from the stop and jot activity. Students are expected to expand their own notes based on this discussion. Lastly is the jot survey where students write their responses on sticky notes and place the notes on a poster containing a question or topic.

The second activity is called Triangle - Square - Circle which you may have seen before with a different name. This one offers students to opportunity to reflect on their learning while they are processing information from the lessons. This is used at the end of the class period to close the lesson or as an exit ticket. It can also be used right before an assessment so the teacher knows what students need to review most. When the lesson is done, have the students draw a triangle on their paper where next to the triangle they will write down three things important points from the lesson or the reading. Then they will draw a square and next to it they will write down anything that "squares" with their thinking or understanding.  Finally, they draw a circle on the paper and next to the shape, they will write down anything that they have questions about.  

A slightly different way to do this activity is to have students write down three things they need clarified next to the triangle and for the circle students can write down how the topic either connects with prior knowledge or with the real world. 

Let me know what you think, I'd love to hear.  Have a great weekend.

Evidence Based Math Instruction Part 2.

This past Monday, we explored the first two strategies recommended for evidence based math instruction. Today, we'll look at the other two.  I love learning new things.  Friday, we'll look at a couple of activities to help teachers do a quick assessment.  

The third strategy is schema based instruction.  This is considered one of the most effective strategies to help students learn to do word problems.  This also helps students who struggle in math. 

The idea behind schema based instruction is to teach students to recognize patterns in word problems rather than key words.  There are two types of schema for word problems.  The first is additive which includes addition and subtraction type problems while the second is multiplicative which includes multiplication and division problems.  They use the way the word problem is written to identify which scheme it is.  Schema based instruction helps students identify the pattern so they can connect it to the best way to solve said problem.  Once they've identified if it is additive or multiplicative, they then use either a diagram or an equation to represent the information. 

Research indicates that students who have been taught using schema based instruction are more likely to be able to solve both familiar and new multistep problems.  Students are taught to identify the pattern by looking for unique features.  They are also taught the vocabulary associated with each type of schema.  In addition, students are taught how to represent the information in the problem visually and show multiple ways to solve the problem.

The last strategy is by using peer interaction where you pair students up to work together and discuss the math.  Working together might happen after they've completed independent practice, or during.  These discussions help students develop student mathematical language and vocabulary. In addition, the discussions can help them become more aware of problem solving via the way they solved it or how others solved it.  

It is important to teach students how to conduct peer-to-peer discussions.  Take time to establish class rules for these discussions and establish some prompts to help students get started.  Encourage students to compare the ways they solved the problem while contrasting their approaches.  

If you've never used any of these techniques, start with one until it becomes part of the routine.  If you can implement all of these, your students should do better.  Let me know what you think, I'd love to hear.  Have a great day.

Evidence Based Math Instruction Part 1.

Evidence based math instruction is quite the same as data driven instruction even though it sounds like it.  Evidence based math instruction is defined as practices and strategies that have been identified as the most effective ones based on rigorous research.  Based on the results, it appears that students have positive math outcomes, provides data to show improvement, fewer wasted resources and less wasted time, and it is easier to convince students to use a program. 

There are four suggested strategies to use in class to help all students learn at all grade levels and abilities. The first element is to use explicit instruction with cumulative practice. This is because explicit instruction models a skill, has the teacher verbalize their thinking, and uses both guided and independent practice.  It should include both the new skill and previous skills learned.   It allows students to see how the process works.  

Research shows that using explicit instruction helps students improve their ability to perform operations and to solve word problems. One reason explicit instruction works is because students see exactly what they have to do and it keeps the older skills fresh in their minds.  In addition, it helps students to develop a working memory so they are able to quickly retrieve information.

Students should know exactly what the goal is for the lesson.  Teachers should include "Do now" at the beginning of class to revisit the skill they learned the previous day.  When teaching the skill or strategy, use crystal clear explanations and provide multiple examples showing more forms of the problems.  Instead of always teaching a linear equation as x + 2 = 7, show it as 7 = x + 2 or 2 + x = 7 so they understand they are all the same problem.  Always use think alouds so students understand the thinking behind solving the problems.  

Rather than always calling on one student, think about using choral responses, stop and jot, or thumbs up thumbs down.  If you 've never used stop and jot, it is a technique where you have students stop and jot down what they are learning at that point in the lesson to check for understanding.  Always, always, always, include problems dealing with a previous skill and finally, give students immediate feedback.

The next technique is to provide a visual representation of the skill or strategy because it allows the students to "see" the math.  A visual representation might be a number line, a tape or bar model, picture, graph, manipulatives or graphic organizer.  These help students understand abstract concepts better. One reason visual representations work is because they remove language barriers.  In addition, if students create the visual representations, they are showing their understanding of the skill or strategy.  In fact, research shows that students who are able to create accurate representations are six times more likely to solve word problems accurately.

Teachers need to teach students to use the different types of visual representations.  It is also important to encourage students to use visual representations to show their thinking.  In addition, the teacher should introduce and show the new skills using visual representations. Then model the concepts and skills using numbers, variables, and symbols. 

On Wednesday, I'll share the other two techniques.  I hope you find them useful.  Let me know what you think, I'd love to hear.  Have a great day.

Welcome To 2023