tag:blogger.com,1999:blog-47847168147972780122019-03-25T06:12:35.812-04:00MATHEMATICAL MUSINGS by mathteacher24These are some of my thoughts about teaching high school mathematics. The purpose of this blog is to help me reflect and become my best teaching self. #MTBoS #iteachmathLeigh Nataronoreply@blogger.comBlogger220125tag:blogger.com,1999:blog-4784716814797278012.post-51158293425513975542019-02-03T07:45:00.001-05:002019-02-03T07:45:16.634-05:00Even Tech Has Its Limits: Learning from StudentsOne of the things I absolutely love about my teaching situation this year is that I have students that are very curious. They ask "what if" questions, make connections between concepts and notice things that I, even after twenty-six years of teaching, haven't noticed. This type of noticing happened in Advanced Algebra on Friday and reminded me that math tech can have its limits.<br /><br /><table><tbody><tr><td><a href="https://1.bp.blogspot.com/-ZxVifYhxrJU/XFbW_xYIn_I/AAAAAAAAHuQ/b4dBqBjkNL4W-fQDIkWfgOv8Bg1WKL2vwCLcBGAs/s1600/Zoomed%2Bout.PNG" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="533" data-original-width="1148" height="185" src="https://1.bp.blogspot.com/-ZxVifYhxrJU/XFbW_xYIn_I/AAAAAAAAHuQ/b4dBqBjkNL4W-fQDIkWfgOv8Bg1WKL2vwCLcBGAs/s400/Zoomed%2Bout.PNG" width="400" /></a></td> <td>Here is what happened. Do students really need to know about end behavior or behavior at x-intercepts for polynomial functions? They can just plug the function into Desmos and see what it looks like. But because I let my students discuss ideas and they can share them freely without fear of "being wrong", one of my students commented the function <i><span style="color: blue;"><b>f(x) = x<sup>3</sup>(x+3)(x-4)</b></span></i> (shown here) seemed to have many </td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"></div><table><tbody><tr><td>x-intecepts around the origin. Because we had talked about behavior of x-intercepts of polynomials based on their factored form, she knew that there should only be x-intercepts at 0, -3 and 4. Yet Desmos was telling her something different. So, we decided to zoom in to verify that the function really only had one x-intercept at the origin. <br /><br /><table><tbody><tr><td>And this is what we saw, after zooming in and tracing along the graph. It looked as though my student was right! The graph showed that several of the values close to zero had y-coordinates of 0, but we knew from the factored form of the polynomial that this could not be. Desmos was lying to us!!! How could we trust it to do our math correctly? I could tell that several students confidence in Desmos was shook. </td><td><a href="https://1.bp.blogspot.com/-EEkR4hBzSlg/XFbaIXCUW6I/AAAAAAAAHuc/wsUC-8j00ikZdcQSvSK_360rctaaezzwgCLcBGAs/s1600/Even%2BDesmos%2Bhas%2BLimits.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="360" data-original-width="610" height="188" src="https://1.bp.blogspot.com/-EEkR4hBzSlg/XFbaIXCUW6I/AAAAAAAAHuc/wsUC-8j00ikZdcQSvSK_360rctaaezzwgCLcBGAs/s320/Even%2BDesmos%2Bhas%2BLimits.PNG" width="320" /></a> </td></tr></tbody></table><br />At this point, we decided that Desmos had to round the output to show it on the graph. And based on what rounding convention Desmos used, it rounded the result to zero. The students' understanding of polynomial behavior helped us understand what was really happening near the origin. Incidentally, f(-0.002) is not zero, but it's pretty darn close to it at 9.598 x 10<sup>-8</sup>.<br /><br />Letting students share what they notice and wonder can be risky. You don't know what they will say all the time and sometimes they may surprise you or throw you a curve ball, where you don't have a satisfying answer. But letting students see that you are on a learning journey with them can definitely have unintended positive consequences, where even you as a teacher can learn something new.</td></tr></tbody></table><div><br /><b><span style="color: red;"><i>Bonus tech tip: If students want to use the Desmos app on their phones, tell them to download the "Desmos Test Mode" app and use it when doing their math homework. If the select "Start Test", they can work in a distraction free way - no notifications of any kind will pop up on their phones!</i></span></b></div><div><br /></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com1tag:blogger.com,1999:blog-4784716814797278012.post-89746423989119844452019-01-03T10:02:00.000-05:002019-01-03T10:02:25.845-05:00Teacher-Centered vs Learner-Centered (The Post I Never Finished 'til Now)<h4><b><span style="color: blue;"><i>Note: I have not blogged in a while. It's been a little over 3 months since my last blog. When I went to blog today, I noticed I had a draft of a blog going. It was a blog that was to set the tone for the beginning of the school year. It will now set the tone for going back to school after Christmas Break. For what you will be reading below to make sense, you need to read my <a href="http://mathteacher24.blogspot.com/2018/09/new-school-weeks-2-3-more-they-talk.html">previous blog entry</a>. </i></span></b></h4><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-BrY3aS7ZYhA/XC4bOP00DXI/AAAAAAAAHtM/j0FH-o0IL8A8bBNTZoMqhyvG-qlhDq6dQCLcBGAs/s1600/Teacher%2BCentered.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="289" data-original-width="434" height="266" src="https://1.bp.blogspot.com/-BrY3aS7ZYhA/XC4bOP00DXI/AAAAAAAAHtM/j0FH-o0IL8A8bBNTZoMqhyvG-qlhDq6dQCLcBGAs/s400/Teacher%2BCentered.PNG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Image from https://blog.stetson.edu/faculty-engagement/2016/09/4849/</td></tr></tbody></table><h4><b><span style="color: red;"><u>Reviewing The Old Transcript: Teacher-Centered Learning</u></span></b></h4>1) Only the students who raised their hand are engaged in the lesson. Those who didn't raise their hand could be engaged, but I don't know that for certain. I have no sense of what those students know or understand about the lesson. It could be that over half the class is totally lost.<br /><br />2) The questions only allowed for one right answer. This leads students to think that math is only about getting right answers and perhaps explaining how you got your answer. To be honest, I used to think that math was only about getting the right answer until I had taught for many, many years and came to realize that learning math is more about the journey - the multiple ways to solve the problem and the connections between the concepts - than the destination of reaching "the answer".<br /><br />3) I was the creator of the knowledge. Students weren't constructing their own knowledge. I was seen as the source of right answers. Students were not expected to evaluate their own answers or thinking. Self-reflection was only reserved for the occasional student who had an inate sense of self-reflection as a key to deeper learning.<br /><br /><span style="color: red;"><b><u>Some Observations Relative to Becoming Learner-Centered</u></b></span><br /><br />Notice that the heading says "becoming learner-centered". Moving from a teacher-centered classroom to a learner-centered classroom is not like turning on a light switch. It takes time. It is easier for me to have a learner-centered classroom in Advanced Algebra and AP Statistics, courses that I have taught for much of my career. It is my first year teaching AP Calculus and I don't have the same sense of security in the content, in common misunderstandings, pacing of lessons, and in how the content is inter-related. In AP Calculus, it is very easy to, and often unintentionally, turn back to a more comfortable teacher-centered approach. <br /><br />So, what am I doing to get my classroom to be more learner-centered? I am allowing more time for students to talk. I truly believe that the more students talk about a concept the more they are engaged in learning. With increased engagement and teacher guidance, there are more opportunites for ALL students to understand what is being taught. Plus, it becomes easier for me to identify which students are struggling.<br /><br />I'll talk more about the strategies I have developed for creating a learner-centered classroom in future blog entries. But if you are ready for some now, I recommend reading <a href="https://saravanderwerf.com/2017/08/09/stand-talks-the-best-thing-i-ever-did-to-get-students-talking-to-one-another/">this blog by Sara VanDerWerf</a>. In it she discusses a strategy called "Stand and Talk", a valuable stragegy I have begun to use in my classroom.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-52830125766548794902018-09-22T12:42:00.001-04:002018-09-22T12:42:48.005-04:00New School Weeks 2 & 3: The More They Talk (Part 1)Now that I am into weeks 2 and 3 at my new school I am in the grading avoidance mode. Even first year teachers recognize this mode within a month or so of school. You look at your bag with the pile of papers in it and think of other things you would rather do. It's Saturday morning. I have plenty of time to attack those papers. Here's an idea. I'm going to blog instead!<br /><br /> <h4><b><span style="color: blue;"> <u>An Old Transcript: Teacher-Centered</u></span></b></h4><br />Are you a teacher who is afraid to have your students talk? Does your classroom dynamic go something like this?<br /><br /><b>Teacher: </b><i>Who can tell me what is f(6) for this function?</i><br /><i><br /></i>A few hands go up and then a few more. Teacher calls on one of the students who raised a hand.<br /><i><br /></i><b>Student: </b><i>Twenty-two.</i><br /><i><br /></i><b>Teacher: </b><i>Good. What about this question. What is x if f (x) is 12?</i><br /><i><br /></i>The teacher gives sufficient wait time for a few hands to go up, but notices that they are mostly the same students as before. Teacher calls on another student.<br /><br /><b>Student: </b><i>x is 8/3.</i><br /><i><br /></i><b>Teacher: </b><i>That is correct. Can you explain how you got your answer.</i><br /><i><br /></i><b>Student: </b><i>I wrote 3x + 4 = 12. Then subtracted 4 to get 3x = 8 and then divided by 3.</i><br /><i><br /></i>In my early years of teaching, I would have considered this a successful classroom discussion. I wasn't telling students the answers. I was asking questions and students were giving me correct answers. I look at this transcript now and cringe. This is my twenty-sixth year of teaching and even now my classroom looks different than it did five years ago. What does this look like for me in my classroom this year?<br /><br /><h4><b><span style="color: blue;"> <u>A New Transcript: Student-Centered</u></span></b></h4><div><b><span style="color: blue;"><u><br /></u></span></b></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-illHYp54lss/W6ZoiHfPOmI/AAAAAAAAHo4/RJWBdael-ckIn7YoHGtLwdrqen40f9hTACLcBGAs/s1600/Student%2BTalk.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="800" data-original-width="1200" height="213" src="https://4.bp.blogspot.com/-illHYp54lss/W6ZoiHfPOmI/AAAAAAAAHo4/RJWBdael-ckIn7YoHGtLwdrqen40f9hTACLcBGAs/s320/Student%2BTalk.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Image from www.teachervision.com/blog/morning-announcements/tips-managing-chatty-class</td></tr></tbody></table><b><br />Teacher</b>: <i>Let's say that I wanted a function to make the input of 4 become an output of 3. We could do that by writing f(x) = x - 1. I want you to work with your partner or trio to come up with a different function that would contain the point (4, 3). Write your function somewhere on the board.</i><br /><i><br /></i>Students work together and talk in their groups for a minute or so with each group putting a function on the board. Here are several of the functions that were created.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-8pXufm5_jWU/W6Zv1iw2TOI/AAAAAAAAHpE/oeOYHt8iQKMer__x_GlRBLz1XWeT0rBWACLcBGAs/s1600/Capture.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="339" data-original-width="1050" height="103" src="https://3.bp.blogspot.com/-8pXufm5_jWU/W6Zv1iw2TOI/AAAAAAAAHpE/oeOYHt8iQKMer__x_GlRBLz1XWeT0rBWACLcBGAs/s320/Capture.PNG" width="320" /></a></div><br /><br /><b>Teacher</b>: <i>What do you notice about the functions that are written here?</i><br /><i><br /></i>Many students are raising their hands. Teacher calls on one of the students.<br /><i><br /></i><b>Student: </b><i>Several of the functions are the same.</i><br /><i><br /></i><b>Teacher: </b><i>That's true. Is there anything that all or most of the functions have in common?</i><br /><i><br /></i>Teacher gives time to think and then calls on a student who does not have her hand raised.<br /><br /><b>Student: </b><i>They all had something multipled to the x and then either added or subtracted a number.</i><br /><i><br /></i><b>Teacher: </b><i>Good observation. Which ones had addition and which ones had subtraction.</i><br /><i><br /></i>Teacher calls on student with hand raised.<br /><br /><b>Student:</b> <i>When you multiply by a number bigger than 1, it made the result get larger. So you needed to subtract. When you multiplied by 1/2, it made the result smaller and that meant that you needed to add a number.</i><br /><i><br /></i>Teacher sees some students nodding in agreement as she restates what the student just said.<br /><br /><b>Teacher:</b> <i>I would like each group to answer the following questions.</i> (Points to the board.)<i> What is f(6) and what value of x would make f(x) = 8? Write your answers and work under the function you wrote on the board.</i><br /><i><br /></i>Students work together and show their work. A few students finish quickly and the teacher asks those students to create a new function that is more complex, maybe involving exponents. Those students add their function to the board.<br /><br /><b>Teacher: </b><i>Talk in your trio or with your partner about the difference between how you answered these two quesions.</i><br /><i><br /></i>Students talk together for a minute or two.<br /><br /><b>Teacher: </b><i>Who would like to share what you just discussed?</i><br /><br />Teacher acknowledges a student near the windows.<br /><br /><b>Student: </b><i>The first question gives us x and we plug it in to get the answer. The second question gives us the answer and we have to use it to find x.</i><br /><i><br /></i><b>Teacher: </b><i>That's true. Would anyone like to add to what [student's name] just said.</i><br /><i><br /></i>Teacher acknowledges a student near the classroom door.<br /><br /><b>Student: </b><i>The first question gives us an input, or x and we find the output, or y. The second question is asking the opposite. We are told an output and we find the input.</i><br /><i><br /></i><b>Teacher</b>: <i>That's a good observation. I'd like everyone to take a minute to write that observation in their notes.</i><br /><i><br /></i>Teacher observes students writing and waits for almost all the students to be done.<br /><br /><b>Teacher: </b><i>Does anyone else have observations to make about what we just did?</i><br /><i><br /></i>Student raises her hand and teacher calls on student.<br /><br /><b>Student: </b>(Referencing the answers on the board.) <i>Why don't we all have the same answer?</i><br /><i><br /></i><b>Teacher: </b><i>That's a good question, [student's name]. Why don't we all have the same answer? Can anyone clarify this.</i><br /><i><br /></i>A few students raise their hand and teacher calls on one of them.<br /><br /><b>Student: </b><i>They all went through the point (4, 3), but that doesn't mean that they will all have the same answer for the other questions. They are different equations and will go through different points.</i><br /><b><i><br /></i></b><b>Teacher: </b><i>That's true. You all created equations of lines, but they aren't the same line for everyone. They have different slopes and different y-intercepts. Think back to geoemtry. If you have a single point, there can me many lines that go through it.</i><br /><br />Teacher looks at student who originally asked the question to see if this makes sense to that student. Student is nodding head in agreement.<br /><br /><h4> <span style="color: blue;"><u>Compare and Contrast: Old vs New </u></span> </h4><br />My blogs don't often invite comments, but for this one I definitely want to invite comments. I can see the difference between the old transcript and the new transcript; can you? Write what you notice in the comments below. I'll be blogging my compare and contrast thoughts about this lesson in a week or so. Now...do I attack those papers, clean the bathrooms or mow the lawn?<br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-37894450606027543842018-09-10T09:32:00.003-04:002018-09-10T09:33:37.150-04:00New School Week 1: The Zumba Observation<div class="separator" style="clear: both; text-align: left;">Note: Last year I did a #teach180 blog. Since I am at a new school this year, I know that there are times I will be in survival mode. Instead of blogging daily, I have decided to blog weekly. The entry below is week 1 of this school year.</div><br />This year marks my 26th year of teaching and my first (of hopefully many) year at Kent Place School. Friday was the fourth day of school and students met in the Great Room to learn about PE requirements and PE exemptions. Students can do morning PE and one of the options for morning PE is Zumba. This must be a popular option, because many of the girls screamed in excitement when they were told that they would be doing one of the Zumba dances right then and there. All three hundred and five girls stood and about a dozen of them went to the front of the room and showed how to do the Zumba routine. Many girls were moving and waving their hands from where they were standing. However, I noticed that very few, if any, of the ninth grade students were participating. In fact, they seemed quite uncomfortable about the whole thing. This reminded me of how classrooms can be passive for many students. A few are very excited about learning or sharing what they know and can do. Some watch from the sidelines and particpate minimally. Others watch in uncomfortable silence, not wanting to particpate for fear of making a mistake and looking stupid in front of their peers and the teacher. This is clearly not what I want for the students in my classroom, a primarily passive environment.<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-z1Z1vgKRYWU/W5ZxBX6QflI/AAAAAAAAHoU/irW3uujBp_ApT_dmPmeR2lCA64rh0c-ggCLcBGAs/s1600/active%2Bclassroom2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="363" data-original-width="1200" height="120" src="https://4.bp.blogspot.com/-z1Z1vgKRYWU/W5ZxBX6QflI/AAAAAAAAHoU/irW3uujBp_ApT_dmPmeR2lCA64rh0c-ggCLcBGAs/s400/active%2Bclassroom2.jpg" width="400" /></a> </div><div class="separator" style="clear: both; text-align: center;"><span style="font-size: x-small;">Note: Thanks to @aliceaspinall for the image here. This is what I plan to have my classroom look like this year.</span></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div>But the Zumba PE lesson didn't end there. The twelve students that were doing Zumba in the middle of the room spread around the room, forming small groups. They were in the center of each group and pulled other students in to form a circle of studnets about two or three deep around them. As I looked around the room, I realized that about 90% or more of the students were actively particpating <br /><div class="separator" style="clear: both; text-align: center;"></div>and having fun! Were they all getting the moves down perfectly? Absolutely not. Where the girls trying, risking, making mistakes, making improvements, helping each other and having a blast? Most definitely. This is what I want my classroom to be like. In order for it to happen, I need to help my students know that taking risks and making mistakes is a part of learning. I also need to figure out who my "Zumba leaders" are - those students who have a solid understanding of what is going on and use them as experts. My responsibility at that point would be to work with the 10% that are on the perimeter of particpating.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-66867421647678769262018-08-19T21:27:00.000-04:002018-08-19T21:27:04.678-04:00Ten Weeks of Summer: Big Changes (Weeks 9 & 10)<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-OEhG5ALbDCs/W3oQto7TR7I/AAAAAAAAHm8/XwrcBZFsmOQvymK4ly8ze3aDT59dRqBVACLcBGAs/s1600/0%2BDays.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="169" data-original-width="130" height="200" src="https://3.bp.blogspot.com/-OEhG5ALbDCs/W3oQto7TR7I/AAAAAAAAHm8/XwrcBZFsmOQvymK4ly8ze3aDT59dRqBVACLcBGAs/s200/0%2BDays.JPG" width="153" /></a></div>As my summer winds down to a close (my first day of in-service is in about 12 hours), I reflect on the fact that the summer was actually quite relaxing despite all of my travels and getting my daughter packed and off to college. (<span style="color: #38761d;"><b>Sic 'em, Bears!</b></span>) I decided that it mostly had to do with the fact that I wasn't going into school once or twice a week to deal with hiring of new faculty, parent meetings, administrator meetings or placement test and summer acceleration test scoring. In fact, I don't think I realized how much work it was to be department chair until I didn't have to do that work over the summer anymore.<br /><br />So, what have I done for the past two weeks?<br /><br />1) I learned how to create podcasts using Audacity. This required a little bit of work on my part, because I also had to install two different plug-ins to get it uploaded and downloaded in the form I needed. The Global Math Department had been posting their weekly <a href="http://globalmathdepartment.org/newsletters/">webinars as podcasts</a> for a while, but that was not happening any longer. A few people mentioned on twitter that they liked listening to the podcasts. So, I learned how to create them with Audacity thanks to the help of Carl Oliver's very clear directions.<br /><br />2) I worked on reviewing AB Calculus, because I will be teaching it for the first time this year. Since I had not used Khan Academy's courses before, I decided to give them a try. Most of the videos are well done and there are plenty of exercises for students to do. However, Sal Khan is not always careful with his use of math language. For example, he'll say "the slope of the point" when he really means "the slope of <u style="font-style: italic;">the tangent line at</u> the point". Points can't have slopes. Lines have slopes. If I use this resource with my students this year, I'll need to be a little cautious and preview any videos I would show my students.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-fzSp8Z2T6mA/W3oYb_ZXo6I/AAAAAAAAHnI/8nATSUzaXbIDTlfq0Iwkhg_ml8rtm7XNQCLcBGAs/s1600/TMC2018_SIHS.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="150" data-original-width="150" height="200" src="https://1.bp.blogspot.com/-fzSp8Z2T6mA/W3oYb_ZXo6I/AAAAAAAAHnI/8nATSUzaXbIDTlfq0Iwkhg_ml8rtm7XNQCLcBGAs/s200/TMC2018_SIHS.png" width="200" /></a></div>3) Today I watched <a href="https://www.youtube.com/watch?v=F_R59DlcfVo">Julie Reulbach's Twitter Math Camp 2018 talk on Teacher Leaders</a>. If you don't follow Julie on twitter, you should. She tweets @jreulbach. She gave me some thoughts to ponder as I start a new school year at a new school. At one point in my career, I thought being a leader meant that I needed to become a full time administrator. I applied to be Upper School Director at Moravian Academy in the spring of 2017 and I wasn't hired for that position. However, Julie's talk made me realize that I am a teacher leader and that I lead in so many other ways when she said "Teacher leadership is not a ladder to be climbed. It is what you do to support another teacher." And I support teachers in so many ways: leading AP Statistics workshops, sharing teaching ideas through talks at conferences, leading a book chat on twitter this summer, visiting classrooms of my colleagues and offering feedback, sharing teaching ideas through blogging, orgainizing hosts for the <a href="https://www.bigmarker.com/communities/GlobalMathDept/conferences">Global Math Department</a> and posting videos to help teachers and students on my <a href="http://www.youtube.com/mathteacher24">YouTube channel</a>. Whew! That's a long list. I was recently asked to serve as a teacher mentor for the National Math and Science Initiative, working specifically with teachers in New York City. I would visit their classroom twice in one school year, specifically giving feedback for teaching AP Statistics. Yet another opportunity to be a teacher leader.<br /><br />4) Last, but not least, my husband and I did shopping, shopping and more shopping, as we helped our daughter moved into her dorm. She is over 1500 miles away from home and we only packed 3 suitcases with her belongings. This meant two trips to Target, two trips to Bed, Bath and Beyond, a trip to the bookstore and a trip to CVS. Apparently we didn't get everything. Tonight I ordered more items for her online! We hope she will keep some of the items from one year to the next!!Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-76498439215009235962018-08-10T21:52:00.002-04:002018-08-10T21:52:52.801-04:00 Ten Weeks of Summer: An Alternative Multiple Choice (Week 8)I finally finished reading the book "Grading Smarter Not Harder". During the school year, I couldn't garner enough interest from my colleagues to have either a face-to-face or virtual book group. Then, I was able to lead a twitter chat over the past 5 weeks with each week focusing on a different chapter. This meant that I had to finish the book and really think about how I will implement some of these ideas in my classroom for the coming year.<br /><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-4Tauhp552OM/W25Ap9ozPPI/AAAAAAAAHmc/E54K7YnNeywBPnPrllK0DdKFVPuxEjjFQCLcBGAs/s1600/beaded-2137080_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="426" data-original-width="640" height="212" src="https://3.bp.blogspot.com/-4Tauhp552OM/W25Ap9ozPPI/AAAAAAAAHmc/E54K7YnNeywBPnPrllK0DdKFVPuxEjjFQCLcBGAs/s320/beaded-2137080_640.jpg" width="320" /></a></div>The last chapter was called "Creativity". At first I thought the chapter would be about the importance of creative projects in the classroom and how to grade the creativity of my students. Since the author, Myron Dueck, has a history background, I will admit that I had a bias against what he was going to say before even reading this chapter. Creative projects in math can happen and do have value, but they can potentially take too much instructional time. Plus, I wasn't comfortable with the subjective nature of grading student's creativity. I thought the book would give me tips on how to grade creativity and how to judge if one project is more creative than another. In fact, Myron Dueck did the opposite. He emphasized that projects should be grounded in learning targets and that it's perfectly fine to not grade creativity at all. In fact, just a written comment to a student about the unique way they did their project or displaying the more creative projects in class or the hallway is enough to "grade" creativity. What a relief! Grading a project that has a creative element based on learning targets, which are made known ahead of time, should be what we assess.<br /><br />However, the big take away for me from this chapter was being more creative on assesments. In AP Statistics, students need to be comfortable with multiple choice questions. In fact, multiple choice questions make up 50% of the AP exam for Statistics. Multiple choice questions are graded as right or wrong. A student that has a solid understanding of a concept could potentially narrow down the answer to one of two possibilities. This student could still get the question wrong, even though he or she knows much more than the student who got it wrong because they had no clue and simply guessed an answer.<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-sWBcQU5aedI/W25A6U-ev_I/AAAAAAAAHmk/4e1i6EngDMsiuLmrO_O01dZtcyPLt_-5wCLcBGAs/s1600/quiz-1373314_640.jpg" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" data-original-height="640" data-original-width="495" height="400" src="https://4.bp.blogspot.com/-sWBcQU5aedI/W25A6U-ev_I/AAAAAAAAHmk/4e1i6EngDMsiuLmrO_O01dZtcyPLt_-5wCLcBGAs/s400/quiz-1373314_640.jpg" width="308" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><br /></td></tr></tbody></table>What is the solution? <b style="color: red;">Strategy #5: Use the "I Know I Am Close" Multiple-Choice Response Format. </b>I have done something similar for multiple choice questions in the past. I called it scratch one, choose one. A student scratches a wrong answer and then chooses the right answer. One point is earned for scratching a wrong answer and three points are earned for selecting the right answer. However, the student might still narrow it down to two choices and get it wrong. In the end, I am not certain what the student was thinking that led to choosing the wrong answer. I may have a sense of what they are thinking, but I don't know for certain.<br /><br />Here is an alternative that I will try this year - "I Know I Am Close" Multiple-Choice Response Format. Here is how the directions are worded (as found on pg. 143 Figure 5.6): "<i>Write the letter that corresponds to the correct answer in the first space provided below. If you are unsure of your answer, write the letter that represents your second choice in the second blank.</i>" Then, under the spaces for the answers, there are some blank lines for writing an explantion for choosing them both. <br /><br />I typically have 8 multiple choice questions on my AP Stat tests and would limit studnets to 4 "I Know I Am Close" questions. So, why might this better than scratch one, choose one? Myron Dueck listed 7 reasons and here are the ones that resonated with me.<br /><br /><table><tbody><tr><td><b>1) Teachers gain insight into their students' answer-selection process. </b></td><td></td><td>Although I could probably guess which<br />questions my students will get wrong,<br />I still can't tell what they were thinking<br />by scratching a wrong answer and choosing another answer. This helps me to understand<br />where I may have fallen short in my teaching<br />and helps me to re-teach or work with<br />individual students.</td></tr><tr><td></td><td></td><td></td></tr><tr><td><b>2) Multiple-choice tests become more than just guessing games.</b></td><td></td><td>This encourages students to think more deeply<br />about what they know and understand. It helps<br />them to think about their thinking - metacognition<br />is a great tool for all students to develop.</td></tr><tr><td></td><td></td><td></td></tr><tr><td><b>3) The format can guide revision.</b></td><td></td><td>If I want students to revise their work before a<br />re-test (and I sometimes do give re-tests), this<br />format helps them to review what they were<br />thinking and can help them to see where their<br />thinking may have been in error.</td></tr><tr><td></td><td></td><td></td></tr><tr><td><b>4) Test anxiety and stress are reduced.</b></td><td></td><td>I was probably a strange child, but I enjoyed<br />test days. I saw tests as a challenge for me to<br />master and I loved being challenged. Plus, I<br />often finished early and then I could quietly<br />read whatever my latest book was for enjoyment.<br />Until I became a teacher, I had no idea that<br />students were stressed about tests. Increasing<br />student confidence and reducing stress will<br />ultimately help them to not only translate<br />into higher grades, but increased understanding.<br />And what teacher doesn't want his or her students<br />to understand the material on the test better?</td></tr></tbody></table></div><br />I'll be sure to post in the fall after I do this alternative method to multiple-choice on my first AP Statistics test. Stay tuned.<br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-36372554605920652432018-07-30T12:07:00.004-04:002018-07-30T12:07:55.807-04:00Ten Weeks of Summer: #gradesmarter (Weeks 6 and 7)<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-hFTwarxd1_g/W18seGf98II/AAAAAAAAHkM/V9zwTQTcjQgtGFQ-C8B0s03ybQ_fqD2xgCEwYBhgL/s1600/Screen%2BShot%2B2018-07-17%2Bat%2B3.05.57%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="322" data-original-width="251" height="320" src="https://3.bp.blogspot.com/-hFTwarxd1_g/W18seGf98II/AAAAAAAAHkM/V9zwTQTcjQgtGFQ-C8B0s03ybQ_fqD2xgCEwYBhgL/s320/Screen%2BShot%2B2018-07-17%2Bat%2B3.05.57%2BPM.png" width="249" /></a></div>Week 6: Last year I tried to lead a book group at my school around the book <i>Grading Smarter Not Harder</i> by Myron Dueck. Since all the teachers I know at my school are very busy, I decided to make it a virtual book group. I had everything set up in Google classroom. I had made prompts that would go live every 2 weeks and I was looking forward to having a lively discussion with my colleagues. Did they have a no zero policy? Did they have a re-testing policy? How did they view homework and grading of homework? There were two colleagues that posted once or twice and after that nothing happened despite the fact that about 20 teachers had joined the Google classroom for the book.<br /><br />Then at some point early this summer I noticed a few Twitter book chats by math teachers. I tweeted that I really had wanted to discuss this book with my colleagues, but my colleagues had not been interested. And so, the twitter chat #gradesmarter was born. Each week for the past 4 weeks, we have been discussing a chapter in this book. Week 1 was Chapter 1: Grading (July 11th) , week 2 was Chapter 2: Homework (July 18th) and week 3 was Chapter 3: Unit Plans (slow chat over July 22 and July 23rd). This week is Chapter 4: Re-Testing (July 31) and next week (August 7th) is Chapter 5: Creativity. I would encourage you to look up the hashtag #gradesmarter if you weren't able to join the chat to see what was discussed.<br /><br />What are my takeaways so far? First, I will not be giving a homework completion grade next year. Homework is practice. I'll make note of who is practicing and who is not, but I won't be awarding a grade for that. Grades should reflect the level of understanding of a student. Giving one student an A because they got it and a C to another because they didn't get it is fine on a test or quiz, but not for homework. Second, I hope to make my students more reflective about their own understading as they work through homework and after they get back assessments. I have tried test corrections in the past, but it ends up being more grading for me and I am not totally convinced that the student is always doing their own corrections. (It sometimes seems like the work of a friend/parent/tutor.) This year I will be starting a new job at Kent Place School and having the students be reflective learners is part of the culture of the school and math department. So, I will be looking to them for advice and guidance.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-NjLWNlm4-8I/W18137qvZFI/AAAAAAAAHkY/Ulqs6Xklyv4Y2STerdYhODEUFTFbK3PyQCLcBGAs/s1600/Screen%2BShot%2B2018-07-30%2Bat%2B11.53.24%2BAM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="360" data-original-width="557" height="256" src="https://2.bp.blogspot.com/-NjLWNlm4-8I/W18137qvZFI/AAAAAAAAHkY/Ulqs6Xklyv4Y2STerdYhODEUFTFbK3PyQCLcBGAs/s400/Screen%2BShot%2B2018-07-30%2Bat%2B11.53.24%2BAM.png" width="400" /></a></div>Are other people getting something out of this book chat? Here is what Kristen Fouss @fouss shared in the chat via a screenshot. One of the things I love about what she shared is that she has brief statements that are easy to implement. For example, retests "must be completed within two weeks of tests being returned". This is important to think about prior to the start of school and implementing a re-testing procedure or policy. However, it is also a work in progress. Notice the question mark at the end of 4.d.<br /><br />I am so glad that I was encouraged my my math twitter friends (a.k.a. #MTBoS) to lead the book chat for <i>Grading Smarter Not Harder</i> on Twitter. It has forced me to think and share and think some more. Plus @myrondueck (the author of the book) joined us during the chat! Hopefully, what I have taken from the book and the chat will make me a "Smarter Grader" and have a positive impact on my students learning.<br /><br />PostScript: Are you a new math teacher to twitter? Here are links to some helpful resources. They may be a little outdated, but can help you get started with your own Twitter Professional Learning Community. <a href="https://docs.google.com/document/d/1WnXw9_aoTlkDoWtf593NdIICnSKy5cC7TP4GEzaweBo/edit?usp=sharing">Twitter Chats for Math Teachers</a> and <a href="https://docs.google.com/document/d/1nHi_FFdDFLbZ8y95EOSG4Ljat155ucKUh-yGjOXSYsE/edit?usp=sharing">Math Teachers on Twitter</a><br /><div class="separator" style="clear: both; text-align: left;"></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-44051666064523014262018-07-16T14:07:00.002-04:002018-07-16T14:07:42.319-04:00Ten Weeks of Summer: Prepping for the Fall (Week 5)This week is a break between leading workshops for AP Statistics. My workshop group at Cabrini University last week was amazing. In addition to learning many ideas about teaching AP Statistics, the group collaborated and shared ideas related to teaching, grading, homework and classroom management. They even formed a Google Classroom group for everyone to continue sharing after the workshop was over!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-1DCnsPIYiD8/W0zcs7Q4a7I/AAAAAAAAHh0/0yNj1p4dHQQXESna8-xLrrdTGB2WZNeUQCLcBGAs/s1600/san-diego-79567_640.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="495" data-original-width="640" height="247" src="https://2.bp.blogspot.com/-1DCnsPIYiD8/W0zcs7Q4a7I/AAAAAAAAHh0/0yNj1p4dHQQXESna8-xLrrdTGB2WZNeUQCLcBGAs/s320/san-diego-79567_640.jpg" width="320" /></a></div>My next workshop will be in San Diego and there will be 30 people in attendance! I typically have workshops that range from 12 - 15 participants and 30 will be challenging. But if I always stayed in my comfort zone and never took on new challenges, I would never grow professionally. Here is to taking on a new challenge, seeing some sights in San Diego and learning from the experience!!<br /><br />In addition to prepping for my workshop for next week, I have been working on reviewing AP Calculus (AB) topics. This will be a new course for me in the fall. I know many of the topics in AP Calculus, because I have taught a non-AP calculus course over the past few years. However, there are some topics that are not as familiar. To help me prep for the course, I read the "Course and Exam Description" as found on the College Board's website. I worked through the 20 multiple choice questions and got 3 wrong. Two of them I was able to figure out what I did wrong, but the last one I had no idea how to approach the question. Specifically, it is to be done with a graphing calculator. Here is the question:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-RCQycyPgBRY/W0zd3KV4RdI/AAAAAAAAHh8/XgOoNbvlVJMc0wBvvPLPhkodEp6KEaoMgCLcBGAs/s1600/Screen%2BShot%2B2018-07-16%2Bat%2B1.55.57%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="198" data-original-width="700" height="180" src="https://3.bp.blogspot.com/-RCQycyPgBRY/W0zd3KV4RdI/AAAAAAAAHh8/XgOoNbvlVJMc0wBvvPLPhkodEp6KEaoMgCLcBGAs/s640/Screen%2BShot%2B2018-07-16%2Bat%2B1.55.57%2BPM.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;">If anyone wants to steer me in the right direction with this, feel free to post in the comments. However, I will likely be posting on the AP teacher community and will be posting to the AP Calculus Teacher facebook group, too. Back to doing some more prep work for fall. We are at the halfway point of summer this week!</div><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com1tag:blogger.com,1999:blog-4784716814797278012.post-88734171362685890092018-07-10T20:50:00.003-04:002018-07-10T20:50:39.538-04:00Ten Weeks of Summer: To Teach It or Not To Teach It (Week 4)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-jsMOFSUQTXA/W0VTeSyyknI/AAAAAAAAHX0/cQ8trq43SRswU6469MMSVD-GVRoUMekzQCLcBGAs/s1600/Screen%2BShot%2B2018-07-10%2Bat%2B8.46.24%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="579" data-original-width="482" height="320" src="https://4.bp.blogspot.com/-jsMOFSUQTXA/W0VTeSyyknI/AAAAAAAAHX0/cQ8trq43SRswU6469MMSVD-GVRoUMekzQCLcBGAs/s320/Screen%2BShot%2B2018-07-10%2Bat%2B8.46.24%2BPM.png" width="266" /></a></div><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif;">"To teach it or not to teach it. That is the quesiton. Whether 'tis nobler in the mind..." Whoops, slipped into Shakespeare's Hamlet for a second.</span><br /><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif;"><br /></span><span style="background-color: white; color: #222222; font-family: "arial" , sans-serif;">I am part of an email group of math teachers and last week there was a question going around about should we teach and/or why don't we teach concepts about alternate exterior angles formed by two parallel lines and a transversal. Some argued via email that it was important to teach that concept for future engineers. Others explained that they did teach the concept and that they taught it through discovery. I said that it is probably not seen in many textbooks, because interior angles have more importance, especially when considering quadrilaterals. </span><span style="background-color: white; color: #222222; font-family: arial, sans-serif;">So, to teach it or not to teach it. Here are two stories I shared with my email group.</span><br /><div style="color: #222222; font-family: arial, sans-serif;"><br /></div><div style="color: #222222; font-family: arial, sans-serif;">Story #1: I got my master's degree at Iowa State and for one of my projects, I used a theorem about secants and tangents to circles. It is a theorem based on similar triangles. The professor (with a doctorate degree) questioned me on the use of the theorem. Either he had never heard of it or he couldn't recall it. However, he had enough understanding of geometry that when I explained why it was true, he understood. </div><div style="color: #222222; font-family: arial, sans-serif;"><br /></div><div style="color: #222222; font-family: arial, sans-serif;">Story #2: Over the past 10 years as department chair, I have to speak to groups of prospective parents/students and describe what is offered in the mathematics department at my school. Do I speak about specific theorems or concepts? No - they assume that I am teaching the math content that is typical of a high school mathematics curriculum, and let's be honest, a list of topics would be yawn-inducing. Instead, I say that in math class students make and test conjectures, combine concepts in new ways, form logical arguments and critique the reasoning of others (verbally and in writing), notice and describe patterns, and make connections between different ways to represent a concept. What I have just described is something that ALL students need no matter what major or career they go into.</div><div style="color: #222222; font-family: arial, sans-serif;"><br /></div><div style="color: #222222; font-family: arial, sans-serif;">Of course there are big ideas that matter in the teaching of mathematics and we can't gloss over those - function, inverse, transformations, proportionality, and rate of change are just a few. However, if we leave off some concepts, will our students be somehow lacking? Think back to Story #1. Was the Ph.D. mathematician lacking? Some might say yes and that he should have learned and remembered that concept. I would argue no, because he had a firm understanding of the big ideas and the ability to reason and apply his understanding. His ability to do this, and not recalling a specific theorem, is what makes him, and I hope my students, mathematicians.</div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-47615155030260106002018-07-03T14:02:00.000-04:002018-07-03T14:02:01.121-04:00Ten Weeks of Summer: Analyzing THE Exam (Week 3)Right before school ended, AP teachers at my school got an email from the person in charge of our AP program saying that the AP exam books from 2017 were available. They were put in my mailbox and I didn't have time to look at them at that point. I remember thinking that it probably would have been more helpful to get them before the students took the AP exam on May 17th. <div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-dfc0MQPXFZI/WzuyuVGfOhI/AAAAAAAAHXQ/M1_UY1en0mkvlI7v7aXoVBroH4qDJ_rbgCLcBGAs/s1600/Screen%2BShot%2B2018-07-03%2Bat%2B1.30.05%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="328" data-original-width="486" height="134" src="https://3.bp.blogspot.com/-dfc0MQPXFZI/WzuyuVGfOhI/AAAAAAAAHXQ/M1_UY1en0mkvlI7v7aXoVBroH4qDJ_rbgCLcBGAs/s200/Screen%2BShot%2B2018-07-03%2Bat%2B1.30.05%2BPM.png" width="200" /></a></div><div>At my most recent workshop, participants wondered how it worked to get AP exam books back. <a href="https://apcentral.collegeboard.org/ap-coordinators/after-exams/ordering-labels-booklets">Here is what I found</a>. It cost us about $300, because we got all the exam books back for all of our students for all our AP courses, and the exam books arrived at our school somewhere between October and December. This makes me wonder where they were from December through May. It is a bit frustrating, because I could have reviewed the material in the books and used what I learned with my current AP Stat students.</div><div><br /></div><div>Today I finally had a chance to look at my exam books and I used the rubrics to determine each student's score. I was happy to see that no student left a question blank and it appears that some students got the highest possible score for a few of the questions! But what were some of the big things I noticed? Do I need to modify my instruction or emphasize certain things more? Here is what I discovered.</div><div><br /></div><div>1) From question #1, I noticed that many students interpreted slope correctly, but did not mention a predicted change in the response variable. Using words the words on average, approximately or predicted, would have indicated the non-deterministic nature of a LSRL.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-PvWtU3w36Ic/WzuzKkJlJkI/AAAAAAAAHXY/Yif3ndDM59MBGY76T2kP3HTp5doLnuMywCEwYBhgL/s1600/Screen%2BShot%2B2018-07-03%2Bat%2B1.27.32%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="299" data-original-width="573" height="166" src="https://3.bp.blogspot.com/-PvWtU3w36Ic/WzuzKkJlJkI/AAAAAAAAHXY/Yif3ndDM59MBGY76T2kP3HTp5doLnuMywCEwYBhgL/s320/Screen%2BShot%2B2018-07-03%2Bat%2B1.27.32%2BPM.png" width="320" /></a></div><div>2) From question #2, almost all my students used their graphing calculator to do a one sample z-interval for a proportion. However, not all students named the procedure or checked to see if the conditions for doing the procedure were met. A few students did not have the correct confidence interval procedure at all, but all students gave a correct interpretation of the interval they found.</div><div><br /></div><div>3) From question #3, almost all of the students could do the normal distribution calculation correctly for part a. However, to get full credit for this, they needed to show boundary and direction. If the student had simply drawn a normal distribution and marked it appropriately, they would have received full credit for part a.</div><div><br /></div><div>4) In general, I noticed that students weren't showing their work for probability calculations. Students may get partial credit on probability calculations if you have a wrong answer, but the work must be shown. The directions on the exam even state, "Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations." I need to emphasize that more. </div><div><br /></div><div>5) From question #5, I had many students leaving off the expected counts condition entirely or comparing the sample size of 207 to 30. But on the plus side, almost all students were able to draw the correct conclusion in context and included linkage between their p-value and a chose alpha level.</div><div><br /></div><div><b><u>The Positives</u></b> - Students are using context, making correct inference conclusions and correctly interpreting confidence intervals!</div><div><br /></div><div><b><u>Areas to Work On</u></b> - I need to model drawing a normal distribution and require students to draw a well labeled/shaded normal distribution for any normal distribution calculation. We need to spend more time on mixed review of identifying inference procedures and their associated conditions.</div><div><br /></div><div><b><u>Consider</u></b> - I currently split up LSRL between the two semesters. LSRL, residuals, standard error, correlation and coefficient of determination are in the first semester and inference for LSRL is in the second semester. Interpreting slope (and y-intercept) with non-deterministic language can be challenging, because students don't really understand that one sample produced that one very specific LSRL and that it is not likely to be the true regression line for the population. In addition, they don't understand the concept of sampling variability early in the first semester. Perhaps moving all the LSRL topics to the second semester would make this better, I think. This is the approach used by some stats teachers I know and one that I may want to use myself.</div><div><br /></div><div>The remainder of this week I will be working on reading the AB Calculus Course Description from the College Board and identifying my areas of weakness relative to AB topics. For example, I remember learning about slope fields in differential equations in college, but that was in 1990 - twenty-eight years ago!! Since I haven't seen that topic since then, it is safe to assume that I am rusty in that area. Perhaps next week I'll blog about AB Calculus or I'll blog about meta-analysis of my teach 180 blog from last year. (Spoiler: I am about 1/3 of the way through and definite themes are emerging!!)</div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-69810387821945621062018-06-24T23:21:00.002-04:002018-06-24T23:21:35.706-04:00Ten Weeks of Summer: Math Love! (Week 2)Last week I took my daughter to freshmen college orientation and she signed up for classes. I knew she was considering elementary education, but then she changed her mind. Her newly declared major - middle school mathematics education! (Of course I can't jump up and down physically beside her, that would be embarrassing. But inside, I was doing a happy dance.) Of course this study by the <a href="https://www.insidehighered.com/news/2017/12/08/nearly-third-students-change-major-within-three-years-math-majors-most">Education Department's National Center for Education Statistics</a>, says that about 1/3 of students change their major at least once. If changing major is genetically linked, we may be ok. Cassie's father was always a chemistry major from the first day of college and I was always a math major. Neither of us changed our majors EVER.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-S-KUI7Kcsy0/WzBfDP-va3I/AAAAAAAAHWs/SX_78GFulbsHhfmJrbBm4ZPsEgwhnFDxgCLcBGAs/s1600/Screen%2BShot%2B2018-06-24%2Bat%2B11.18.08%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="466" data-original-width="617" height="241" src="https://3.bp.blogspot.com/-S-KUI7Kcsy0/WzBfDP-va3I/AAAAAAAAHWs/SX_78GFulbsHhfmJrbBm4ZPsEgwhnFDxgCLcBGAs/s320/Screen%2BShot%2B2018-06-24%2Bat%2B11.18.08%2BPM.png" width="320" /></a></div>Then, I got a tweet from a fellow AP Stat teacher saying that his first ever AP Stat class created a class shirt and my name was on it because they found my TI-84 videos so helpful. I am glad to know that my work is being used by others. I thanked the teacher and wished his students well on the AP scores that are set to come out next week. Of course, I am keeping my fingers crossed that my students did well, too.<br /><br />Starting tomorrow, I will be leading an four-day AP Summer Institute for seven AP Statistics teachers. I spent two full days last week finalizing my plans for the workshop. I am definitely well prepared and hope it goes well. Sharing teaching ideas and learning from other math teachers is what keeps me teaching. It may seem strange that I get renewed energy in the summer by doing more teaching, but I do! Sharing the math love in workshops, over twitter, on my <a href="https://www.youtube.com/user/mathteacher24">YouTube</a> channel and in this blog motivates me. Plus I am always learning new ideas from my virtual math colleagues - shout out to #MTBOS, #iteachmath and <a href="https://www.bigmarker.com/communities/GlobalMathDept/conferences">Global Math Department</a> folks!Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-52280382105626343412018-06-17T17:11:00.001-04:002018-06-17T17:11:25.841-04:00Ten Weeks of Summer: The AP Stat Reading (Week 1)During the school year, I did a teach 180 blog and I will be spending some time next week reviewing my successes and failures of the previous school year. If you are a new teacher and you are reading this, you may be thinking, "Wait. You have taught since 1992. You still fail in your teaching?" Fail, fall short, learn and succeed. If I want to continue to be the best teacher I can be for my students, I will need to try new things to keep them engaged and help them learn. I think I may have even said in a previous blog post that once I stop learning and growing as a teacher it will be time for me to leave teaching. But I am nowhere near ready for that yet.<br /><br />This summer I will be blogging once each week to reflect on things I am doing over the summer relative to teaching. I might blog about something I learned from a participant at a workshop. Or I might blog about one of my summer goals, <a href="https://teacher-staging.desmos.com/computation-layer/documentation">learning the computational layer in Desmos</a>. Or I might blog about a book I am reading related to teaching.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dx-9sO7FMpZtq9g8VNNTqzSCQmeWZHYQboD98Vc0Csvppq5LFyAcB-R1OY-lZ4okFv3VY3wSokvJlTE4BPwpg' class='b-hbp-video b-uploaded' frameborder='0' /></div>Let's begin. Week 1: The AP Statistics Reading. Today is Day 7 - a.k.a. the last day of the AP Statistics Reading. It is one of the best professional development experiences of the year and this is my 9th year as an AP Reader. Each year I learn more about how to help my students become better statistical communicators and better statistical thinkers. I pick up teaching ideas, book recommendations and share stories with old friends and new friends. This year I was able to escape from my third Escape Room in Kansas City and a bunch of us tried ax throwing. (It's ok to throw axes in statistics, but you don't forget to label them.)<br /><br />This year at Best Practices night I presented about a teaching technique called "Stand and Talk". It was based off of <a href="https://saravanderwerf.com/2017/08/09/stand-talks-the-best-thing-i-ever-did-to-get-students-talking-to-one-another/">Sara VanDerWerf's blog</a> and talk that I saw at NCTM. (My blog posts about this technique can be found at <a href="http://mathteacher24.blogspot.com/2018/04/teach-180-stand-talk-some-success-day.html">Day 150</a> and <a href="http://mathteacher24.blogspot.com/2018/05/teach-180-stand-and-talk-again-day-151.html">Day 151</a> in my blog from the 2017-2018 school year.) I also picked up some ideas from other teachers at that session. Specifically, I want to try Kelly M. Spoon's (@kellymspoon) Desmos card sort activities. (As I was writing this blog, I tried to search for it and could not find it online.) So, I sent her a tweet and will modify my blog with the link once she sends it to me.<br /><br />Stay tuned for week 2 - prepping for some workshops and looking ahead to teaching at my new school.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-3125469194141105012018-06-08T17:14:00.002-04:002018-06-08T17:14:16.274-04:00Teach 180: The Last Day (Day 178)Today was the last day for all faculty. It was sad to think that this was my last time to be with this particular group of teachers and friends. I took pictures of my empty classroom to remind me of where I have taught for the past 12 years. I also took a picture of this one special frame I have hanging on my wall.<br /><br />My classroom has no real windows and has served as "the detention room" for many years. About a year or two after being at Moravian Academy, the photography teacher, Nancy, asked me "If you could look out a window, what would you see?" <br /><br />"A beach scene," I told her. "It doesn't need to be tropical, but it needs to be a beach scene." <br /><br />It was right before spring break and many of her photo students would soon be spending their days relaxing at a beach. So, Nancy held a contest for her students. The winning photo would be enlarged and framed behind this window that had been salvaged from Illick (a building on campus). Thank you Lauren Comes (class of 2009?) for giving me something beautiful and relaxing to look at each day. Thank you colleagues and friends; I will miss you greatly.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-zhfeaHkfy9o/WxrwJ1n4DpI/AAAAAAAAHVs/3abb11CbwdckT73uQQrqz4pnBiDooxXMwCLcBGAs/s1600/Screen%2BShot%2B2018-06-08%2Bat%2B5.07.21%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="517" data-original-width="543" height="304" src="https://3.bp.blogspot.com/-zhfeaHkfy9o/WxrwJ1n4DpI/AAAAAAAAHVs/3abb11CbwdckT73uQQrqz4pnBiDooxXMwCLcBGAs/s320/Screen%2BShot%2B2018-06-08%2Bat%2B5.07.21%2BPM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-8480860259848926272018-06-05T23:03:00.000-04:002018-06-05T23:03:12.331-04:00Teach 180: 26 Boxes and a Gem (Days 174 & 175)<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-M5hORexsMSI/WxdFtTulP8I/AAAAAAAAHTQ/gOT01AbRGoIkntCUS7JrpLupF3ldaRzGwCLcBGAs/s1600/Screen%2BShot%2B2018-06-05%2Bat%2B10.21.59%2BPM.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="678" data-original-width="552" height="400" src="https://2.bp.blogspot.com/-M5hORexsMSI/WxdFtTulP8I/AAAAAAAAHTQ/gOT01AbRGoIkntCUS7JrpLupF3ldaRzGwCLcBGAs/s400/Screen%2BShot%2B2018-06-05%2Bat%2B10.21.59%2BPM.png" width="325" /></a></div>Yesterday I finished organizing some files and I am down to my final "To Do" list as I prepare to move from my job at Moravian Academy to a job at Kent Place School. This afternoon I counted the number of boxes and bins in my basement and there was a whopping 26! As I sorted through one box today, I found this gem from an NCTM Math session from 2006.<br /><br />I would say that if you asked my students that it means to do mathematics they would say things like find examples, look for patterns and solve problems. I am not so sure they would say the ones I have highlighted - "take chances, make mistakes", "discover more elegant solutions" and "explain, validate, convince". Why would they not say these? Here are my thoughts.<br /><br /><span style="color: red;"><u style="font-weight: bold;">Take Chances, Make Mistakes</u> - Although I try to have a culture where mistakes are seen as part of the learning process, students often don't want to make mistakes for fear of "losing face" in front of their peers. However, taking chances and making mistakes are a huge part of learning. Desmos has helped my students to take chances, try things out, make mistakes, revise their answers. Desmos gives feedback in a non-judgemental way. It says your wrong, but without using the word "wrong". I don't use the word "wrong", but what do I say specifically? I try to say things like "Why do you think that?" and "Can you explain your thinking?" Do I really value mistakes? I'll need to be more cognizant of this next year.</span><br /><span style="color: red;"><br /></span><span style="color: blue;"><u style="font-weight: bold;">Discover More Elegant Solutions</u> - Rarely do we take the time to have students discover more solutions, let alone more elegant solutions. This is especially true in the public school system. At times this has also been true in my classroom. To have more elegant solutions, you need to have problems that are interesting and can be solved by multiple perspectives. As I look to next year, can I make this part of my classroom culture? As I plan for my classes for the fall, my goal will be to find one or two problems per unit that lend themselves to discovering more elegant solutions.</span><br /><span style="color: blue;"><br /></span><u style="font-weight: bold;">Explain, Validate, Convince</u> - This is something that should be happening in every classroom (not just math classrooms) daily. "Convince us." and "How can you tell if your answer is right/reasonable?" needs to also happen more frequently in my classroom. In the name of "covering" content, we often rush past this part of a lesson to do more examples. The answer is "right" and for many teachers that are pressured by state testing, a "right" answer is all that matters. However, I believe that explain, validate and convince will have a stronger and more lasting impact on understanding than drill and kill in the name of "right" answers.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-8417489516880299832018-06-03T13:39:00.000-04:002018-06-03T13:38:59.984-04:00Teach 180: Celebrating the End (Day 173)Friday was the beginning of the end. Although we still have underclass exams for three more school days and two days of teacher meetings, we celebrated the graduation of this year's senior class on Friday evening with a baccalaureate service. It is especially bittersweet this year, because my own daughter is in the graduating class. After the service, we were milling on the lawn outside of the church with many pictures being taken. This one was taken of me and my first period AP Stat students. I will especially miss this group. Teaching them was a joy. They asked questions, made me laugh and hopefully, they will think more critically about any statistics headlines they see in the news. Congratulations to the class of 2018!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-HDza6w_zXj8/WxQg3WxDEpI/AAAAAAAAHS4/OkRONJL8_Qs9A6RZBzEb_6OtftkC2r-GwCLcBGAs/s1600/Screen%2BShot%2B2018-06-03%2Bat%2B1.09.00%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="514" data-original-width="824" height="199" src="https://2.bp.blogspot.com/-HDza6w_zXj8/WxQg3WxDEpI/AAAAAAAAHS4/OkRONJL8_Qs9A6RZBzEb_6OtftkC2r-GwCLcBGAs/s320/Screen%2BShot%2B2018-06-03%2Bat%2B1.09.00%2BPM.png" width="320" /></a></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-50672239874025680452018-05-31T19:04:00.000-04:002018-05-31T19:04:40.028-04:00Teach 180: Last Day of Classes (Day 171 & 172)<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Ah9FCFQBT7k/WxB-TTDqsUI/AAAAAAAAHSs/gkZrBroY78wUtFJlWMz2Eyz0iSSjtjmrQCEwYBhgL/s1600/Screen%2BShot%2B2018-05-31%2Bat%2B6.57.58%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="308" data-original-width="822" height="119" src="https://2.bp.blogspot.com/-Ah9FCFQBT7k/WxB-TTDqsUI/AAAAAAAAHSs/gkZrBroY78wUtFJlWMz2Eyz0iSSjtjmrQCEwYBhgL/s320/Screen%2BShot%2B2018-05-31%2Bat%2B6.57.58%2BPM.png" width="320" /></a></div>I don't normally combine two days of teaching into one blog post, but here are two posts for the price of one. Yesterday was not a good teaching day. About an hour into the school day, I got a call from my daughter, Cassie, saying that my dad (Pop-pop to her) was in ICU with chest pains. I wan't teaching at the time and had the next hour to make phone calls and figure out what was happening. He is home now and fine (tests came back negative), but the scare gave me time to think about how precious time is with family. After school, I spent time with Cassie using her camera to take a "photo shoot" of her at the local park in her white dress for graduation. We had fun and not once during that time did I think about teaching/school. Plus I have some new photos for the background on my computer and phone to remind me of her when she goes to college.<br /><br />Today was the last day of classes and final exams start tomorrow. My PreCalc students requested a Kahoot to review for their last 35 minutes with me. They chose to review logarithmic and exponential functions and based on the Kahoot, they did need more work in this area. The fact that the students realized that they need more work in a certain area is good. Hopefully, this will help them target their review. I encouraged them to send me emails with questions they might have over the next few days. As several of them left, they shook my hand and told me to "have a nice summer" and "good luck next year". All my students have been wonderful this year and I will miss them.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-10517221721869365052018-05-30T00:11:00.002-04:002018-05-30T00:11:53.562-04:00Teach 180: Grow or Stagnate (Day 170)Beginning in the fall, I will be teaching at Kent Place School in Summit, New Jersey and I spent a good portion of my day continuing to pack and organize. The last day of school will be here soon! My school day ended with an "exit interview" conversation with the Headmaster, reflecting on what Moravian Academy does well and where Moravian Academy could improve. As I reflected on my twelve years of teaching at Moravian Academy, I realized that I grew as a teacher as a result of my desire to learn and continue to improve. It was not because of anything the school, administration or Board of Trustees did. The systems for faculty growth were missing when I started at MA and sadly, they are still absent now. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-oObNXgJcaHQ/Ww4jy4FfoEI/AAAAAAAAHSc/ZEPbFh1UkAIiS7524Gr7XiCLRmt4btssQCLcBGAs/s1600/Screen%2BShot%2B2018-05-30%2Bat%2B12.06.36%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="265" data-original-width="667" height="127" src="https://3.bp.blogspot.com/-oObNXgJcaHQ/Ww4jy4FfoEI/AAAAAAAAHSc/ZEPbFh1UkAIiS7524Gr7XiCLRmt4btssQCLcBGAs/s320/Screen%2BShot%2B2018-05-30%2Bat%2B12.06.36%2BAM.png" width="320" /></a></div><br />During my tenure at MA, I renewed my National Board Certification and helped three other teachers at Moravian (and a few at other schools) obtain their National Board Certificates. (The teachers taught middle school math, PE and Spanish.) I lead workshops on Desmos, Smartboard, plickers, Google Hangouts, flipping the classroom and twitter. Last year I visited the classroom of about a dozen teachers, many outside my department, to provide feedback to the teachers and consider how I might incorporate their teaching strategies in my own classroom. I tried to lead a group called "Faculty Fellows" where teachers would get together to discuss teaching strategies or articles we had read. When that failed (teachers did not want to give up their lunch time nor did they have time meet outside the school day), I tried to start a virtual book group on the book "Grade Smarter, Not Harder". Although twenty signed up for the group, only two have participated sporadically. I continued to grow and lead despite being in an environment that does not invest in growth.<br /><br />I am sure I will continue to lead in some capacity at my new school. However, I want to - no, need to - be surrounded by like-minded teachers who want to risk, share, and learn from each other daily. I hope that Kent Place School is a place where faculty professional growth is the norm and not the exception. I hope it is a school where faculty evaluation is primarily seen as an opportunity for professional growth, instead of a way to remove ineffective teachers. I need to be a part of an organization that values people as resources and invests in their continued journey to becoming better teachers, because I am not done learning or growing yet.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-38168390911139873272018-05-28T09:32:00.001-04:002018-05-28T09:32:15.445-04:00Teach 180: Exploring the Desmos CL (Day 169)Because I only had two classes on Friday and I was tired of sorting papers and packing up my classroom, I decided to spend some time playing around with the computational layer in Desmos. I had learned about the CL when I went to San Francisco last summer as a Desmos fellow Cohort #2. I had fully planned to dive into the CL when I returned from that weekend, but we still had two math teachers to hire with six weeks until the start of school. And when I go on vacation in Maine, we go to a place with limited WiFi.<br /><br />Thanks to @mrchowmath I was able to begin to dabble in the CL in an exploratory way through an Activity Builder he created. He posted his videos on his website <a href="https://www.blogger.com/">here<span id="goog_367410068"></span></a>. (I am now also noticing that he has some "<a href="http://www.mrchowmath.com/breakout.html">Breakout Room</a>" activities that he created with Desmos AB, too. I'll definitely need to look at these this summer.)<br /><br />What did I learn from Jay Chow's videos? First, that the computational layer is not as scary as the syntax makes it look. Second, that the built in de-bugging suggestions are very helpful. Jay created a google sheet for people to share their "homework" and now I have about a week and a half to try out one of the following.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-babCkVnsjWo/WwwD4ZanyrI/AAAAAAAAHSQ/-1vv7V0R7MsBac-wLXq_pxUrQEflAfbYwCLcBGAs/s1600/Screen%2BShot%2B2018-05-28%2Bat%2B9.17.43%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="701" data-original-width="1027" height="218" src="https://4.bp.blogspot.com/-babCkVnsjWo/WwwD4ZanyrI/AAAAAAAAHSQ/-1vv7V0R7MsBac-wLXq_pxUrQEflAfbYwCLcBGAs/s320/Screen%2BShot%2B2018-05-28%2Bat%2B9.17.43%2BAM.png" width="320" /></a></div><br />I am thinking of trying #1 where a student enters a parabola in standard form and the vertex is reported back in a note along with the x-intercepts. It would be interesting to see if students notice that the x-coordinate of the vertex is the average of the x-intercepts. Right now my students don't notice this. It may be because I only have them focus on the ordered pairs or I only have them focus on the graph. Perhaps having these together will help them to see the relationship better.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-64419346774235877802018-05-25T08:44:00.001-04:002018-05-25T08:44:06.574-04:00Teach 180: Inverse Trig Functions (Day 168)Note: After flooding the lawnmower multiple times, doing ironing, making dinner and spending time with my daughter, I went to bed and realized that I had not written in my blog for the day. The events I am about to describe happened yesterday.<br /><br />Most of my day was spent copying final exams for PreCalculus and organizing files in my classroom. The one class I had today had only one student. (The other student in that class was at his sister's college graduation.) We spent about 45 minutes of class time reviewing inverse trig functions. We looked at the graph of f(x) = sin x and g(x) = sin<sup>-1</sup> x and realized that we would need to restrict f(x) so that it would be one-to-one. Desmos made this very easy to see and very easy to check the validity of our chosen restriction for the domain of f(x) = sin x. The graph of f(x) = sin x is in purple and the graph of its inverse is in black. The dotted line is the line y = x to show that f(x) and its inverse are reflections of each other over the line y = x. I added the points to the "ends" of the pieces to show that x and y coordinates are swapped when graphing a function and its inverse.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ops3VXmin5M/Wwf4-BDEEwI/AAAAAAAAHRs/EmR74aUN2zwsDN3RWGjc7JCKHmPBjBZKACLcBGAs/s1600/Screen%2BShot%2B2018-05-25%2Bat%2B7.46.11%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="571" data-original-width="858" height="212" src="https://2.bp.blogspot.com/-ops3VXmin5M/Wwf4-BDEEwI/AAAAAAAAHRs/EmR74aUN2zwsDN3RWGjc7JCKHmPBjBZKACLcBGAs/s320/Screen%2BShot%2B2018-05-25%2Bat%2B7.46.11%2BAM.png" width="320" /></a></div><br />What about f(x) = cos x and g(x) = cos<sup>-1</sup> x? Could we use the same restriction of -π/2 <u><</u> x <u><</u> π/2? We used Desmos to see what would happen if we did this.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Qtin0yBLDS4/WwgCeWjwcxI/AAAAAAAAHR4/_N-ECbvS5_sVai-l8yqwTlGO8vMaDSySQCLcBGAs/s1600/Screen%2BShot%2B2018-05-25%2Bat%2B7.55.12%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="441" data-original-width="637" height="221" src="https://4.bp.blogspot.com/-Qtin0yBLDS4/WwgCeWjwcxI/AAAAAAAAHR4/_N-ECbvS5_sVai-l8yqwTlGO8vMaDSySQCLcBGAs/s320/Screen%2BShot%2B2018-05-25%2Bat%2B7.55.12%2BAM.png" width="320" /></a></div><br />We could easily see that the restriction of -π/2 <u><</u> x <u><</u> π/2 was not correct. This was visible in two ways. First, the restricted cosine function did not pass the horizontal line test. Second, the red graph f(x) = cos x and the blue graph g(x) = cos<sup>-1</sup> x were not reflections of each other over the line y = x. We decided to change our restriction to 0 <u><</u> x <u><</u> π and it worked!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-YTiDVAXxJoE/WwgEOdKbF1I/AAAAAAAAHSE/Lv97AGOfRiQFW5YRQKyArAgUuB1hqhcAQCLcBGAs/s1600/Screen%2BShot%2B2018-05-25%2Bat%2B8.40.11%2BAM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="580" data-original-width="772" height="240" src="https://1.bp.blogspot.com/-YTiDVAXxJoE/WwgEOdKbF1I/AAAAAAAAHSE/Lv97AGOfRiQFW5YRQKyArAgUuB1hqhcAQCLcBGAs/s320/Screen%2BShot%2B2018-05-25%2Bat%2B8.40.11%2BAM.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>I love how Desmos makes it easy to construct understanding and to play with the mathematics. If we had chosen to do this discover by hand or with a graphing calculator, we would have been bogged down by the computations or button pushing. Thank you Desmos for making discovery more accessible to everyone!<br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-44774980538890231062018-05-23T21:28:00.005-04:002018-05-23T21:28:58.728-04:00Teach 180: Hello, Anyone Out There? (Day 167)<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-KCN-xVWbkEw/WwYPRBTCvCI/AAAAAAAAHQ4/kRtr9i_LGcwXfOiGQgus29nzDJQqqs-vQCLcBGAs/s1600/Screen%2BShot%2B2018-05-23%2Bat%2B9.02.11%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="252" data-original-width="821" height="121" src="https://4.bp.blogspot.com/-KCN-xVWbkEw/WwYPRBTCvCI/AAAAAAAAHQ4/kRtr9i_LGcwXfOiGQgus29nzDJQqqs-vQCLcBGAs/s400/Screen%2BShot%2B2018-05-23%2Bat%2B9.02.11%2BPM.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: Times, Times New Roman, serif;">At the end of February, twenty faculty members signed up to be a member of the google classroom "Grading Smarter, Not Harder". I was excited by the possibility of sharing ideas with my colleagues around a topic that really interests me - assessment. My initial post said, "<span style="background-color: white; color: rgba(0, 0, 0, 0.87);">This is a virtual book group that is open to all upper school faculty. Discussion prompts will be based on the book "Grading Smarter, Not Harder" by Myron Dueck. Reading the book is not required to participate. Prompts will be posted weekly or bi-weekly." I have created prompts on each of the chapters for the past two months. However, </span>only nine posts have been made since the beginning of March and six of them were made by me. If a post happens in a discussion forum and no one responds to it, did the post happen? </span><br /><span style="font-family: Times, Times New Roman, serif;"><br /></span><span style="font-family: Times, Times New Roman, serif;">I know that my blog is essentially me talking to myself and I feel like that in the google classroom, too. But the purpose of a google classroom forum is to have interaction with other participants. Being a part of communities like #MTBoS, the Global Math Department and the 2018 AP Statistics Reader Facebook Group help me to grow as a <u>math</u> teacher. However, I want to have discussions about more general education topics that all teachers face. We don't have those conversations at lunch or in the faculty room. Discussions tend to focus on trying to understand the bell schedule, when grades are due, what to watch on Netflix and rating the school coffee. Conversations like this are a nice diversion. However, I want to learn from my math and non-math colleagues and grow in my profession. Setting up face-to-face meetings during lunch did not work last year (only 2 or 3 people attended). So, I thought this might work, because it would give teachers more flexibility with participation. But with 2/3 of the posts being written by me, it appears I am mostly talking to myself. Hello, anyone out there?</span><br /><span style="font-family: Times, Times New Roman, serif;"><br /></span><span style="font-family: Times, Times New Roman, serif;">Here is the post I did today about Chapter 4 - ReTesting. Maybe someone out there will read it and comment on it.</span><br /><span style="font-family: Times, Times New Roman, serif;"><br /></span><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-gPcueAbEayw/WwYUueqo56I/AAAAAAAAHRc/fjQUffAEx7Ybqhzhzj71R-boZtyZcUKawCLcBGAs/s1600/Screen%2BShot%2B2018-05-23%2Bat%2B9.25.52%2BPM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="403" data-original-width="755" height="341" src="https://4.bp.blogspot.com/-gPcueAbEayw/WwYUueqo56I/AAAAAAAAHRc/fjQUffAEx7Ybqhzhzj71R-boZtyZcUKawCLcBGAs/s640/Screen%2BShot%2B2018-05-23%2Bat%2B9.25.52%2BPM.png" width="640" /></a></div><span style="font-family: Times, Times New Roman, serif;"><br /></span><span style="font-family: Times, Times New Roman, serif;"><br /></span><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><span style="font-family: Times, Times New Roman, serif;"><br /></span>Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-27516325868300075792018-05-22T16:47:00.000-04:002018-05-22T16:47:07.111-04:00Teach 180: Transforming Trig Graphs (Day 166)As students learn new concepts or skills, it is important for them to get immediate feedback. Saying "Just check your answer against the back of the book." is NOT feedback. Sure, a student can see if they got the right answer, but it doesn't let them know why they got it wrong. And if they get it wrong, the student really should have a chance to try it again.<br /><br />In Calculus today, we reviewed transforming graphs of trig functions. The students were able to get immediate feedback and targeted practice through DeltaMath. The short video below shows a little about how we used DeltaMath for this feedback and practice. (Note: It does not have any sound.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/kpwpL655CYU/0.jpg" src="https://www.youtube.com/embed/kpwpL655CYU?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><div class="separator" style="clear: both; text-align: center;"><br /></div>Students worked for a full 30 minutes and completed about 25 problems each. I was able to help students, as they needed it, but students also were also able to figure out many of their errors for themselves.Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-74842093175527128272018-05-21T15:15:00.001-04:002018-05-21T15:15:54.350-04:00Teach 180: The Tower of Hanoi (Day 165)<table><tbody><tr><td><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-UAM-7ABe2Co/WwMX9bDIi6I/AAAAAAAAHQs/hIn_Ta8e2KwZrWsaXRDKmPD4yrGSPyTIACEwYBhgL/s1600/Screen%2BShot%2B2018-05-21%2Bat%2B3.02.46%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="349" data-original-width="517" height="216" src="https://4.bp.blogspot.com/-UAM-7ABe2Co/WwMX9bDIi6I/AAAAAAAAHQs/hIn_Ta8e2KwZrWsaXRDKmPD4yrGSPyTIACEwYBhgL/s320/Screen%2BShot%2B2018-05-21%2Bat%2B3.02.46%2BPM.png" width="320" /></a></div><br /></td><td></td><td>There are certain math ideas or puzzles that I believe every student should experience. The Tower of Hanoi is one of those puzzles. After we reviewed the graphs of the six basic trig functions in Calculus today, we had some time to play with this puzzle. Our goal was to move the disks from the left peg to the right peg in as few moves as possible. The rule is that you can't place a larger disk on top of a smaller disk. Rather than starting with 7 disks, we started with 1 and then moved to 2 and then 3, looking for a pattern as we increased the number of disks. It only took about 10 minutes to see that the number of moves, M, based on the number of disks D, is given by the function M(D) = 2<sup>D</sup> - 1.<br /><br /></td></tr></tbody></table>If you are looking for an animation of the tower of Hanoi, there is a nice 2-D and 3-D simulation at <a href="http://towersofhanoi.info/Animate.aspx">Towers of Hanoi Animation</a>. I will probably show this tomorrow and we may even create <a href="http://www.jamesrahn.com/pages/geometry/pop_up_cards.htm">a pop-up card based on the towers of Hanoi</a>. <br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-87437802646473107792018-05-19T11:37:00.001-04:002018-05-19T11:37:22.379-04:00Teach 180: Dance Factor Dance (Day 164)We are continuing to look at patterns in PreCalculus. As students walked in the room today, I had the website <a href="http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/">www.datapointed.net/visualizations/math/factorization/animated-diagrams/</a> open. The short screencast here you through the first 30 seconds of what they saw. (Music is called "Sunglasses" courtesy of "Loyalty Freak Music".)<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/vJC9OKiZY8M/0.jpg" src="https://www.youtube.com/embed/vJC9OKiZY8M?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br />I asked them "What's going on with the dots?" At first they noticed the pattern of colors. Something that I didn't even notice. But soon after that they noticed things like prime numbers forming a circle, factors of 3 forming triangles, factors of 4 forming squares, factors of 5 forming pentagons, etc. Although we weren't talking about factors explicitly today, it helped them to continue to think about patterns and was a nice way to get their brains thinking about math.<br /><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-28117235392073662202018-05-17T23:13:00.002-04:002018-05-17T23:13:50.710-04:00Teach 180: My Geeky Lightbulb Moment (Day 163)In PreCalculus today, we were doing some problems involving summation. One was the sum of n<sup>2 </sup> from n = 1 to 4. I knew Desmos could do this one and I showed my students how this worked in Desmos.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-UWk2_5R1wX8/Wv5BQZEwTdI/AAAAAAAAHPw/529UawjbsnsEqrZb4GAcCNSv_zErTycwwCLcBGAs/s1600/Screen%2BShot%2B2018-05-17%2Bat%2B10.45.26%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="180" data-original-width="418" height="136" src="https://3.bp.blogspot.com/-UWk2_5R1wX8/Wv5BQZEwTdI/AAAAAAAAHPw/529UawjbsnsEqrZb4GAcCNSv_zErTycwwCLcBGAs/s320/Screen%2BShot%2B2018-05-17%2Bat%2B10.45.26%2BPM.png" width="320" /></a></div><br />Then, we did the following "by hand".<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-KblwIRa6a_I/Wv5CGviGjyI/AAAAAAAAHQE/LxmDfKZ8nF4dEfyMbOMZeG4F89IA_o10wCLcBGAs/s1600/Screen%2BShot%2B2018-05-17%2Bat%2B11.01.18%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="250" data-original-width="318" height="156" src="https://2.bp.blogspot.com/-KblwIRa6a_I/Wv5CGviGjyI/AAAAAAAAHQE/LxmDfKZ8nF4dEfyMbOMZeG4F89IA_o10wCLcBGAs/s200/Screen%2BShot%2B2018-05-17%2Bat%2B11.01.18%2BPM.png" width="200" /></a></div><br />I asked my students, "what would Desomos do with this problem". I wasn't really sure. When I entered the expression, it didn't give me "an answer", it gave me a graph. "Why is it doing that?" I wondered.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-FDcUQv8AlnU/Wv5Cl082z0I/AAAAAAAAHQQ/wUpmzX31N9o5Fvm9rclxofyKkP8VhBh_gCLcBGAs/s1600/Screen%2BShot%2B2018-05-17%2Bat%2B11.03.17%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="475" data-original-width="735" height="257" src="https://4.bp.blogspot.com/-FDcUQv8AlnU/Wv5Cl082z0I/AAAAAAAAHQQ/wUpmzX31N9o5Fvm9rclxofyKkP8VhBh_gCLcBGAs/s400/Screen%2BShot%2B2018-05-17%2Bat%2B11.03.17%2BPM.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">And then it dawned on me! Desmos was graphing the polynomial that resulted from the summation. We confirmed this as a class by entering the expression 1 + x + x<sup>2</sup>/2 + x<sup>3</sup>/6 + x<sup>4</sup>/24 in line 2 and it graphed over the graph that was already there. As I was thinking aloud and having my lightbulb moment, my students asked me, "Wait, you mean you haven't done this in Desmos before?" They thought I was acting surprised and had done this before. I do that sometimes. But not this time, my surprise was real. About an hour later, I shared my discovery with the colleagues in my math department. They were also impressed. This is definitely something that can't happen on a TI-whatever.</div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0tag:blogger.com,1999:blog-4784716814797278012.post-21638000577114856802018-05-16T09:30:00.002-04:002018-05-16T09:30:57.699-04:00Teach 180: The Law of Sines Video (Day 162)Today I am reviewing Law of Sines with my two students in Calculus. As I was looking through my folder, I came across this video I created using my smartboard software and Animoto. Since I made the video back in 2010, it is not the best quality. However, it is a quick explanation as to why the Law of Sines work. Perhaps I should work on one for Law of Cosines to add to my YouTube collection.<br /><br /><div class="separator" style="clear: both; text-align: center;"><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/gJRaJxEZHrE/0.jpg" src="https://www.youtube.com/embed/gJRaJxEZHrE?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div><br />Leigh Natarohttps://plus.google.com/100447604897737206366noreply@blogger.com0