tag:blogger.com,1999:blog-42500501470422364152017-07-29T01:21:21.626-07:00Don't Panic, The Answer is 42A journal on Teaching Math and my only hope for Professional DevelopmentLizzy-Senseinoreply@blogger.comBlogger51125tag:blogger.com,1999:blog-4250050147042236415.post-69028792436706155592015-07-20T12:03:00.003-07:002015-07-20T12:18:30.555-07:00Teaching AgainI've taken the last year off to have my son. <br />It's been quite a ride.<br />People have been asking me which is easier, parenting or teaching. Since he's only 11 months old, I don't really know yet what parenting consists of but I can say that this has been the most relaxing year of my adult life. What does that say about being a teacher? <br />I was able to take some classes for myself (abstract algebra, graph theory 1 and graph theory 2) and I reconnected with what I love and hate about being a student. It was really helpful for me to remember what not knowing math feels like and what a different persona I adopt as a student (super quiet, shy and uncertain) vs. who I am as a teacher (gregarious, adventurous, unashamed of making mistakes.) I spent a lot of time observing the other women in the classes (only about a quarter of the graduate students were women) and how much they participated compared to the men (about 90% of the comments made in class were by men.) None of the students were black or Latino. I'm still processing how these observations should influence my teaching but for now, it's clear that I need to do more for my female and minority students. Why don't women participate? Why don't I participate? My personal reasons are related to fear that at some point, I will hit a wall mathematically and just won't be able to understand something (even though I've overcome every wall so far), inherent shyness and introvertedness, fear of being wrong, math being so tied to my identity that I don't want to be revealed as a fraud (which I do feel like sometimes. What right do I have to be telling other people how to do math when I'm unsure I could have pursued math seriously.) I did have some sexist math teachers. I never felt encouraged in math. But these are my reasons. Does every woman in math share these misgivings? Or do we all have our own individual insecurities reinforced by our cultural context? My sample size was really tiny. And my shyness prevented me from sharing my observations with other women in the class. <br />Anyway. I'm going back to teaching. Algebra 1 and Japanese for next year. I'm excited and scared to go back but I'm looking forward to catching up with what everyone's been doing on the MTBoS while I've been away. I hope I can start contributing again and I'm so grateful I have this community to lean on when I'm in need of inspiration, which I always am! I hope someday I can contribute something useful in exchange for all this community has given me.Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-47282576514527718062014-07-15T15:37:00.000-07:002014-07-15T15:37:48.768-07:00Intro to Proofs in GeometryI wanted to blog about this a looooong loooong time ago but the school year got in the way along with moving across the country twice because of family health dramas (NY to California in the fall, now California to Oregon. I know Cali to Oregon doesn't seem that far, but it is over 1,000 miles from San Diego to Portland. California is freekishly big.) So though I know posting lesson plans in the summer is kind of silly, I want to get it out of my system before I forget what I did. <br /><br />The school in which I taught this past year was a high poverty school where 30% of our students had IEPs. It is a charter school so it's pretty small meaning I got to work very closely with my students, colleagues and parents but we did lack funding and our students were weak in a lot of basic skills. In fact, most of my geometry students this past year <i>hadn't even passed algebra 1 yet.</i> The previous algebra teacher found them so lacking in basic skills that she gave the entire algebra 1 class "incompletes" because they didn't finish the algebra 1 curriculum. My principal decided to enroll all these students in geometry anyway because she figured (rightly I think) that they needed a bit of a break from algebra and if they saw some algebra in a geometrical context it might make going back to algebra more meaningful (which it did. Every time algebra popped up in geometry the students were actually excited because it was familiar and wasn't too difficult. They really mastered equation solving, writing expressions and equations of lines by studying these topics through geometry.) All of this meant that when proofs came up I was super freaked out. I've always struggled with teaching them and I feel like I've done a very poor job in the past. I put a lot of thought into how to build proofs into our curriculum this past year and I feel like what I did was relatively successful. My students weren't scared of proofs for the first time in my teaching career. When they came up, the students knew at least where to start and always attempted them. So I want to lay down what I did just so I don't forget. <br /><br />So here is the description of how the unit flowed. It's definitely a more traditional approach to proofs and I stuck to two-column proofs. I tried to transition the students to paragraph proofs, but their skills and confidence were too low; they liked the organized nature of two-column proofs. First, I didn't nix the logic unit. Even though logic is not in Common Core anymore, I think that the reasoning done in the logic unit helps prepare students for proofs. <br /><br />Logic Unit Lesson 1: Intro to conditional statements.<br /><br /><ul><li>First I did Sam Shah's lesson <a href="http://samjshah.com/2012/08/12/if-students-learn-then-weve-accomplished-something-part-i/">introducing conditional statements</a>. I did the drawing activity and posted all their pictures on the wall. It went really well- I was surprised at how much trouble some students had following the directions precisely. A lot of them didn't know the geometric vocabulary (like what an isosceles triangle is) or were hesitant drawing so it was a great activity to do towards the beginning of the class.</li><li>Then I just did a mini-lecture on the notation of conditional statements, Euler diagrams and what a negation is.</li><li>Then I gave them this assignment: <a href="http://www.scribd.com/doc/232385975/Geo-Ch-2-Lesson-1-Hw">Logic Unit Lesson 1: Conditional Statements Intro</a> </li></ul><div>Logic Unit Lesson 2: Manipulations of Conditional statements</div><div><ul><li>I did the second part of Sam Shah's lesson on conditional statements</li><li>I did a mini lecture on the vocab: converse, inverse, contrapositive and negatin</li><li>Finally I gave them this assignment:<a href="http://www.scribd.com/doc/232387196/Geo-Ch-2-Lesson-2-Hw"> Logic Unit Lesson 2: Manipulations of Conditional Statements</a>. </li></ul><div>Logic Unit Lesson 3: Word Proofs</div></div><div><ul><li>First we did a <a href="http://www.scribd.com/doc/234015314/Syllogism-Cards">syllogism activity </a> where I just cut the cards apart and had them put the syllogism in the correct order and a mini lecture on syllogisms</li><li>Then we did word proofs. This is one of the most successful lessons I've ever taught on proofs. I totally stole it from another blogger, and of course forgot to save their name in the name of the file I downloaded like I normally do. When I figure out who made it I'll update this post. I reformatted the file I stole from that other blogger and did the lesson in kind of a workshop style. I did one or two of the word proofs on the board to demonstrate how to do it, then I gave them time to work on their own and then we compared answers. I had them come up and show different solutions they'd discovered and we talked about the fact that there's more than one way to do a proof correctly. Like I mentioned before, this was actually a pretty bright class but one lacking in discipline and both basic math and study skills. I had to <i>hold the kids back.</i> They were chomping at the bit to do more and more and more puzzles. I made the last two pages of the lesson optional and almost all students did them anyway. Whoever designed this lesson was brilliant because it really hooked students who are usually disengaged with math. You can't not want to solve one of these puzzles when they're presented to you. </li><li>There are two pages in the lesson to have the students make their own puzzles and switch papers with each other. We had to skip this part because we ran out of time.</li><li>Here's the lesson: <a href="http://www.scribd.com/doc/233399614/Geo-Ch-2-Lesson-3-Intro-to-proofs">Logic Unit Lesson 3: Word Proofs</a></li></ul>Logic Unit Lesson 4: Angle Proofs</div><div><ul><li>This lesson was a little less fun than the last one, but it was very effective. First we went over basic angle terminology: complementary and supplementary angles, vertical angles and linear pairs (they'd learned these before), and we also talked about the algebra properties of equality, transitivity and the substitution property. </li><li>Then I had them do the lesson below in pairs. It's structured exactly like the word proofs from the last lesson with 4 or so "rules" and space for them to use the rules to go from the given to the prove. The students were able to stumble their way through these proofs <i>without me doing any examples on the board </i>based on what they did in the last lesson. </li><li>Here's the lesson: <a href="http://www.scribd.com/doc/233400831/Geo-Ch-2-Lesson-4-Angle-Proofs">Logic Unit Lesson 4: Angle Proofs</a></li><li>Finally we concluded with a <a href="http://www.scribd.com/doc/233400392/Scrambled-Angles-Proofs">Scrambled Proofs Activity</a></li></ul>Here's where I made a mistake. I did parallel lines and transversals as the next unit because I liked flowing from points, to lines, to parallel lines to triangles to polygons. It seemed like the logical way to structure the course. Also, congruent triangle proofs are so much richer and more interesting if the students already know their parallel lines and transversals angle relationships. But if I were to do it again, I would do congruent triangles after the logic unit and then do parallel lines and transversals. The proofs for parallel lines and transversals are a little more abstract and involve more vocabulary than congruent triangle proofs so trying to launch from the intro to proofs unit straight into parallel lines and transversals was too big a jump. So here is how I built up proof using congruent triangles after I failed at teaching them proofs through parallel lines and transversal relationships. </div><div><br /></div><div>Congruent Triangles Lesson 1: Intro to Congruence</div><div><ul><li>For this lesson I just did a standard lecture over what congruence is, the notation for congruence and examples of using congruence to find missing parts. Nothing really exciting. The only thing about this lesson that I like is my warm-up. We define congruence as "identical in every way" and then I ask students if the two identical twins are congruent which leads to a great discussion of what "corresponding parts" means: <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-0bln_81DX1A/U8WdV5h5GfI/AAAAAAAAEYc/exleHVX00Dw/s1600/Identical+Twins.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-0bln_81DX1A/U8WdV5h5GfI/AAAAAAAAEYc/exleHVX00Dw/s1600/Identical+Twins.jpg" height="127" width="200" /></a></div></li><li><div class="separator" style="clear: both; text-align: left;">Also, I always get a laugh out of my student by choosing another teacher who's the same height as me (at this school I used the principal) and I use that teacher to discuss the difference between the "equal" symbol and the "congruence" symbol. Our heights can be equated, but if you accidentally use the congruence symbol you're saying I and this other teacher are identical in every way. </div></li><li><div class="separator" style="clear: both; text-align: left;">Here's the assignment for that lesson: <a href="http://www.scribd.com/doc/234005729/Geo-Chapter-4-Lesson-1-Homework-Intro-to-congruence">Congruent Triangle Lesson 1: Intro to Congruence</a></div></li></ul><div>Congruent Triangles Lesson 2: Triangle Congruence Theorems</div></div><div><ul><li>I used this <a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=4">cool illuminations app</a> on congruent triangle theorems along with <a href="http://www.scribd.com/doc/234025378/Triangle-Congruence-Illuminations-Activity">this worksheet </a>that I wrote to introduce SSS, SAS, ASA and AAS. I've done this activity twice, the first time I let the students pair up and use their own computers. A lot of the students wouldn't or couldn't follow the directions so I spent the whole period frantically running around trouble shooting. The more motivated students in the class were able to make the connections I wanted but the other students left class mostly confused. So the second time, we did each activity first as a whole class with computers closed, then I let them open their laptops and play with the simulation to confirm the results for themselves. This worked much better. All the students were successful on the homework without need for more instruction. </li><li>Then I totally stole a few worksheets <a href="http://mathteachermambo.blogspot.com/2010/11/puzzle-sheet.html">here</a> and <a href="http://mathteachermambo.blogspot.com/2011/01/cpctc.html">here</a> from <a href="http://mathteachermambo.blogspot.com/">Math Teacher Mambo</a> for the students to work on independently. Here's my mash-up of her brilliance: <a href="http://www.scribd.com/doc/234005726/Geo-Chapter-4-Lesson-2-Homework-Congruent-triangle-theorems">Congruent Triangle Lesson 2: Congruent Triangle Theorems</a></li></ul><div>Congruent Triangles Lesson 3: Using Theorems in Proofs</div></div><div><ul><li>This is a lesson I created that I'm exceptionally proud of. It was super boring though. I realized that students weren't really getting what theorems were for or how to use them. They were still struggling with problems like: if angle A and angle B are a linear pair and angle A measures 40 degrees, what is the measure of angle B. They weren't thinking about what "linear pair" means and how to connect that meaning to the problem. Even if they got this far they didn't understand that in a "proving" situation they needed to state how they know that angle A and angle B add to 180 (that they need to say: by def of linear pair or state the theorem).</li><li>So I made this worksheet: <a href="http://www.scribd.com/doc/233401195/Using-Theorems-to-make-Deductions-Lesson">Congruent Triangles Lesson 3: Using Theorems to Make Deductions.</a> </li><li>In pairs I had them go through their notes and fill in the blanks for all the theorems we've covered. This was the boring part. I gave them a time limit though which helped keep them focused and this activity also reinforced the importance of taking notes. I refused to tell them any answers. If they didn't have it in their notes they needed to find someone who did. </li><li>The "classwork" part of the lesson is where it all really paid off. I did the first few examples with them- how to cite the correct theorem that had been used in each situation. As soon as I started doing these problems on the board, a bunch of "ohhhh so that's why we needed those stupid theorems" exclamations went through the room. It was especially rewarding to watch them do the last page where they have to think backwards- find the theorem that applies to the situation then figure out what deduction can be made. </li><li>Students guarded the list of theorems they made as the first part of the lesson fiercely and insisted that I do a similar "fill in the blank theorem review" at the end of every unit. </li></ul>Congruent Triangles Unit Lesson 4: Proving Triangles Congruent.</div><div><ul><li>This lesson went quite well although it's a very traditional lesson. I just did a few example proofs with them and talked about how to set up a two-column proof table (I know... but these students liked structure. I tried to show them a paragraph proof and their eyes all crossed and they started throwing paper.) </li><li>I gave them <a href="http://www.scribd.com/doc/234005728/Geo-Chapter-4-Lesson-3-Homework-Proofs-Involving-congruent-triangles">this packet</a> of problems from <a href="http://letspracticegeometry.com/">letspracticegeometry.com</a> (there are a lot of typos in this worksheet though. I would like to rewrite it) but without the first two pages. </li><li>Then I gave them this <a href="http://www.scribd.com/doc/234005723/Geo-Chapter-4-Lesson-3-Day-2-Homework-Simple-congruent-triangle-proofs">proof template worksheet thing</a> that I created. It has the students choose which proofs they want to try, the harder proofs being worth more points. They have to reach a certain number of points to get full credit. </li><li>This small spin on a worksheet created a night and day difference in students' attitudes about proofs. Every other time I've taught proofs students have been super whinny about them and would give up quickly. But when I handed out the above assignment I saw at least half the students immediately turning to the last page to do the harder proofs. A lot of them struggled on the proofs through a good chunk of the period without finishing and I kept suggesting they just go do more of the easier proofs, or build up to the harder ones but they said that they wanted to do hardest ones. The fact that there was a choice between easy and hard involved made them want to prove to themselves that they could do the hard. Students with less confidence started with the easy ones and were able to advance to the harder ones pretty smoothly. Everyone was engaged and no one was complaining that I was making them do proofs. </li></ul><div>Congruent Triangles Unit Lesson 5: CPCTC theorem proofs</div></div><div><ul><li>I taught this lesson the same as the last one. Examples then a "choose your own problems" proof worksheet. Here's <a href="http://www.scribd.com/doc/234005725/Geo-Chapter-4-Lesson-4-Homework-Proofs-Involving-CPCTC">the packet</a> from <a href="http://letspracticegeometry.com/">letspracticegeometry.com</a> and here's the <a href="http://www.scribd.com/doc/234005724/Geo-Chapter-4-Lesson-4-Homework-CPCTC-Proofs-Template">proof template worksheet</a> I gave them.</li><li>I've had trouble in the past with students using CPCTC inappropriately so I put the following message up on the projector in giant letters and made them recite it in unison a few times. I kept it up as they worked on the proofs and I didn't have students misusing CPCTC! </li></ul></div><div style="text-align: center;">To use CPCTC you MUST</div><div style="text-align: center;">FIRST: Prove triangles congruent</div><div style="text-align: center;">THEN: Say parts are congruent with CPCTC</div><div style="text-align: center;"><br /></div><div style="text-align: center;">CPCTC says that:</div><div style="text-align: center;">IF two triangles are congruent THEN their corresponding parts are congruent.</div><div style="text-align: center;">Prove the "IF" first, Only then can you use the "THEN"</div><div style="text-align: left;"><ul><li>Again, through this class period students were working on proofs <i>without complaint </i>and without giving up. If they started to have trouble they could persevere or choose a new problem and this flexibility eliminated a lot of the griping I've experienced in the past with proof practice. Boring but effective. </li></ul><div><br /></div><div>WHEW. That's all. I just wanted to catalog what I'd done because this was my most successful proof teaching experience so far. It still needs a lot of work though. And I know that under Common Core, it may not even be relevant anymore because I didn't work in any proving congruence with transformations. </div></div>Lizzy-Senseinoreply@blogger.com5tag:blogger.com,1999:blog-4250050147042236415.post-40362696833549119732014-06-30T09:19:00.001-07:002014-06-30T09:19:40.510-07:00Pedagogy vs. CompassionThis year has been the most tumultuous of my life which is why my posts this year have been so infrequent but this summer, my first summer off in my 5 years of full time teaching, I hope to spend some time reflecting on my teaching career so far. I also want to record my experiences in the school I most recently taught at in San Diego California before they grow cobwebs. I have a lot of lesson plans I want to share but first I want to think about how teaching at a disadvantaged, high poverty, high IEP percentage school was different from teaching at my relatively privileged charter school in Oregon and VERY privileged private school in New York. It wasn't that different.<br /><br /><b>I did have to change how I taught.</b> I used inquiry based approaches at the other two schools in which I've worked. The students had good study skills, were well organized and cared about their education so getting them engaged in self-discovery lessons wasn't that difficult. They knew how to accept challenges and persevere even if they didn't know how to do something. Boy did this bomb in my school in San Diego. In geometry class, if I gave the students rulers they were immediately put to use as either projectiles or weapons. If I asked them to spend 10 minutes completing an activity on their own, all the cell phones came out or hands went up asking for help. No one had the initiative to even attempt an activity on their own. Games descended into chaos. I quickly learned that these students needed a very firm hand and they would only behave under direct instruction. Maybe I should have persevered with inquiry based approaches and over time they would have gotten better but the standards hanging over my head made me too nervous to spend too much time on this classroom chaos. <br /><br />Their study skills were so weak that most didn't take notes, bring paper or pencils to class, and many didn't know their multiplication tables. I spent a lot of my time teaching them how to listen in class, how to take notes, how to use their notes effectively, how to show work and how to care. I did use a lot of questioning in my direct instruction lessons- I never actually completed a problem myself on the board, always asking for student input- but it was still direct instruction. At the end of the year though, as I was grading their final project and their final tests, I was astonished to realize that they'd mastered as much content as the students in my relatively privileged Oregon school and also exhibited the same enthusiasm for math that my Oregon students exhibited. Here's an excerpt from an e-mail a student sent me at the end of this school year- it's almost identical to letters I received from my Oregon students:<br /><br /><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 13px;">I'm not sure why, but it just recently dawned upon me that you will be leaving after this year and I'll probably never see you again, so I decided to write you a farewell letter. I've never really been compelled to write one to a teacher before so you'll have to bear with me here. I wanted to start off by thanking you for everything you've done, I can honestly say you're the best teacher I've ever had in my entire life. That being said, the support you've given me and the mentality of perseverance you have instilled in the classroom has really inspired me to work even harder and I wanted you to know you have made a big impact on my life. I want you to know that you'll always have a special place in my heart, even years from now, I'm sure I'll look back and be able to confidently say you helped me achieve my goals.</span><br /><span style="background-color: white; color: #222222; font-family: arial, sans-serif; font-size: 13px;"><br /></span>Without inquiry based learning, you-tube videos, gimmicks, games or technology my students in San Diego reached a similar mastery of content and a similar changing of attitudes about math that my students in Oregon attained. <br /><br />My pedagogy didn't matter. Or rather, I used the methods that I thought would work for my students. Method mattered much less than I would have thought. <br /><br />I don't want this post to sound boastful- I had the same number of failures and frustrations as other teachers but I did feel successful at the end of the year. I am left questioning the amount of time I've poured into thinking about my method- feeling guilty over not using more inquiry based approaches or not doing enough projects or relying too much on direct instruction or not letting learners of different styles shine since direct instruction caters to auditory and visual learners. Certainly method was important but it wasn't a question of "is direct instruction or inquiry instruction the correct way to teach," it was a question of "is direct instruction or inquiry instruction the correct way to teach <i>for my students</i>." Would my Oregon students have learned as well as they did had I used direct instruction on them? I don't know. Or was the method of instruction really not that important at all? <br /><br />I got numerous notes from students at the end of this school year and all of them cited my ability to listen to their difficulties, to work with them after school, and my stubborn refusal to let them give up that helped them succeed. (There were a few students that I failed though, don't get me wrong. And I felt like giving up on a lot of them sometimes.) None of them mentioned my lectures as being boring and a lot of them thanked me for teaching them how to listen and take notes because it helped them in their other classes. Where does this leave me in the pedagogy wars? I don't know... but maybe it's time I directed my guilt away from my methods of instruction and try to hone what does make me feel successful- treating each student with compassion and trying to be flexible in finding what works for them, regardless of what methods are fashionable in the larger ed-community.<br /><br />Lizzy-Senseinoreply@blogger.com1tag:blogger.com,1999:blog-4250050147042236415.post-56136897003817423812014-02-23T15:42:00.001-08:002014-03-08T17:17:30.625-08:00AbsenteeismRight now, I'm making lesson plans for my first period algebra 1 class. Here is the first slide I decided to add to my presentation for tomorrow.<br /><br />"Right now...<br /><br /><ul><li>4 of you have As</li><li>2 of you have Cs</li><li>9 of you have Fs</li></ul>It's really easy for me to tell though without looking at my gradebook who is passing and who is not: those who are here everyday are passing. Those of you who are absent two or more times a week are failing."<br /><br />What do I do? I can't teach students who don't show up. When I make this announcement, most likely at least half my class will be absent and won't even hear the message I'm trying to convey. <br /><br />Update: 3/8<br />The Monday after writing the above post I decided to try a new grading system in my class to see if that could help with attendance. I had been assigning homework and calling it homework, but I had been giving the students time to complete it in class. Only if they didn't finish it in class would they need to do it at home. I did this because I didn't want students to feel pressured to get the work done quickly- when I assign only classwork the slower and more careful students tend to get stressed out. The problem though was that students were not using class time well. When I asked them to work they said they would finish it at home, and then of course it (and the student) never came back. <br /><br />I thought that maybe if I made their grade entirely based on them showing up and using class time well, then I would have more luck with both attendance and with comprehension. Miraculously all the students did show up on Monday and I told my students that attendance was our biggest problem. That those who were failing were failing because they weren't here. I explained that I was going to make their grade based entirely on if they came to class, took notes and if they completed the work asked of them during class. Immediately, I saw relief wash through the classroom. I think because for the first time all year, they realized that they <i>could</i> pass. That they <i>could</i> do what I was asking them to do. The late homework, missed lessons and poor classwork completion had been weighing on them and had been causing them to avoid class. It was easy for them to not show up because this class is first period and at our school, only freshmen and sophomores have to come to first period. So my freshmen were hanging out with their Junior and Senior friends instead of coming to class. <br /><br />They <i>want</i> to do well and only their guilt and lack of confidence had been keeping them away from class. They constantly tell me that they like me as a teacher which is why I was so baffled by their poor attendance. Maybe the fact that they do seem to like me contributed to them not wanting to face me when they thought they'd let me down. <br /><br />Since changing my grading system two weeks ago, my attendance has sky rocketed. They're all completing class work, asking questions and performing well on quizzes. They still definitely lack initiative. Since I require them to turn in an exit ticket to receive credit for the day's work, the end of class has gotten awfully chaotic as students frantically try to get my help because they don't trust their own abilities. But they're trying and showing up now. We can work on initiative later. <br /><br />I am torn about this no- homework system. I have been following the homework vs. no homework debate and I'm more on the side of assigning homework because I've seen students grow so much from wresting with problems when they have no one around to help them. They take better notes, ask better questions, and demonstrate much more mastery over the material than when I don't assign homework. This experiment has reinforced my belief that homework does significantly contribute to learning because my other algebra 1 class to whom I still assign homework are demonstrating much more confidence with the material and are growing more rapidly. Both my first and fifth period Algebra 1 classes are composed of low-income students who have failed algebra at least once before. But my fifth period class has time earlier in the day (usually during lunch) to complete their homework so their homework turn in rate is good, their attendance is good and their learning is evident. But clearly when students <i>can't</i> do homework and the not doing it wears down their self confidence and causes them to avoid class, the homework needs to be nixed because it's doing much more harm than good. <br /><br />I guess this just reinforces my belief that there are no absolutes in education. Every thing about teaching needs to be modified depending on the composition of students sitting in your classroom. When students do homework it's good for them, but when they <i>can't</i> do it and are still expected to do it, it's bad for them. Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-82797597521436982342014-02-09T16:38:00.000-08:002014-02-09T16:38:28.281-08:00Asking for helpI seek help on-line constantly when it comes to lesson planning. I've grown used to the idea that anything I can think of, someone out there in the blogosphere has probably already perfected and I love that I can see kernels of lessons I've just dreamed come alive in others' hands. This doesn't even include the gazillions of ideas I've never thought of that are about a hundred times better than anything I can dream. <br /><br />But when it comes to actually teaching- implementing the lessons, getting my kids excited, supporting their growth, encouraging them to persevere, I've never received much help (administrators never pop in. I've been formally observed only once and that was by a coworker) and I feel like at this stage, I don't need much help. I have a thriving community of students coming during lunch to do math because they enjoy it, and I've watched the most recalcitrant math students slowly gain confidence and enthusiasm and I feel like this is what I'm good at. I'm good at patiently coaxing students into learning that they can learn math and over time, that they enjoy learning it. <br /><br />But this semester I have the most stubbornly anti math student I've ever taught. For three weeks, she was an angel in my advisory and a demon in my math classroom. She refuses to accept help saying that she doesn't need it, she'll do it at home. Then she proceeds to do nothing at all through the whole 80 minute block. When I try to help her she slides under her desk, covers her paper, refuses to look at the problem, gets up and walks away, or starts ranting about the uselessness of math. She's a wonderful student in advisory so I know she's bright and capable, but she refuses to cooperate in math (especially whenever division becomes involved. She says she never learned it and she never wants to learn it.) She slept through all of my math classes two weeks ago and refused to stir when I tried to rouse her. She got a 30% on her first test and even though I discussed with her the consequences of her actions through all of advisory that day she slept through math again the next day. I asked her if she wants to fail? It means she'll have to do it all again next year. She replied she doesn't but she can BS her way through the other tests. I said that learning to read is tedious, but once you do learn, it's magical what you can discover and that math is the same way. She replied that reading is vital but math is superfluous. I said that everyone needs help to learn math because it's several thousand years of accumulated knowledge that we're trying to impart in a few short years and that all I would like is for her to let me help her. Right now I don't even care about notes or homework or tests. I would just like her to allow me to talk to her about math without arguing. She wouldn't budge.<br /><br />I thought that I'd have to just wait her out. I'd need to stop nagging her and let her come around on her own. Maybe over time she'd start to feel left out. Or she'd realize that she couldn't BS her way on her own and she didn't want to fail. She was so obstinate that maybe just the fact that I was pushing was making her push against me and if I stopped pushing she'd stop fighting. I was worried though that she would get so far behind by the time she came round that it would be too late to learn what she needed to learn since she was already so far behind.<br /><br />So I turned to our vice principle, explained what was going on and what I'd tried and he said he'd talk to her. The next day she took notes, completed her homework and asked for help. I asked him what he said and he told me he'd talked about how many thousands of years of knowledge we were trying to teach her in a tiny span of time and that she could not learn without my help. He said that this will be maybe the only time in her life where she had a teacher who was willing to give her extra time, extra help and who really cared about her and if she waited, she would never get the help she needed. It was almost <i>exactly </i>the same logic I'd tried on her. Her efforts have continued through the week. <br /><br />I guess this just reinforces my belief that if I ever get to a place where I think I've figured it out- that means I've grown too complacent. Teaching will always and forever be something I'll need help with and that's the way it's supposed to be because it's a collaborative endeavor. I hope I'm always humble enough to ask for the help I need. Lizzy-Senseinoreply@blogger.com1tag:blogger.com,1999:blog-4250050147042236415.post-5439766108624413152014-02-01T13:38:00.002-08:002014-02-01T13:41:01.696-08:00Angleatron Failure and Distance Formula Game SuccessI taught the lesson on <a href="http://dontpanictheansweris42.blogspot.com/2014/01/angleatrons.html">angleatrons</a> that I previously posted about and it was not very successful at all. I'm reluctant to write about my failures because I'm already the type of person who doubts everything I do and even my most successful lessons leave me feeling like I'm not the teacher I wish I was. This is also why I'm a terrible blogger. In my most insecure moments, I can't help but compare my teaching to these fantastic teachers I so admire and aspire to me more like. I hope the fact that I am constantly striving to be better makes me a better teacher, but it also makes me very uncomfortable in my own skin much of the time. <br /><br />The lesson was unsuccessful for several reasons beyond my control. My speakers broke partway through showing the video so the students couldn't hear Vi Hart's narrations. I then tried to paraphrase what she was doing with paper folding but students grew bored watching a soundless video. This made me rush through the video to move on to the activity, but then students were confused about how to do the paper folding. Their confusion reinforced my reasoning behind doing the activity because if students couldn't grasp the idea that the corner of their paper can be used as a 90 degree angle, then they really did need to practice basic angle drawings. About half the class did take off doing drawings and folding angles. A couple of them produced really beautiful designs and I think all of them did grasp what 90 degree and 45 degree angles are supposed to look like. The other half of the class adamantly refused to draw, or refused to draw precisely (sloppily drawing 90 degree angles that looked more like they were 100 degrees because they refused to use the corner of their papers to guide their drawings.) Their reluctance and difficulty only convinced me that they <i>did</i> need to practice, but the activity didn't work for the students who <i>needed</i> the practice. <br /><br />I did try some other games this past week and they were much more successful. For me, the simpler the game, the easier it is for me to pull off because I have a very minimalist classroom (I have to buy all my own supplies, the students have tiny desks and we don't have a white board, only a smart board which allows only one student to write on it at a time.) I came up with a game to practice the distance formula which worked <i>beautifully</i> mostly because it was so simple. First, I had to bribe the students to play because playing games involves more thinking than taking notes and they actually wanted me to keep lecturing so that they could passively copy/ sleep. Then I asked them to group into threes and told them they were competing against their group members to convince them to work with people other than their best friends. Finally, I just displayed four numbers on the smart board. The students could rearrange the numbers into two ordered pairs however they wanted and could add negatives if they wanted. The person in their group that was able to organize the ordered pairs in such a way as to maximize distance won and earned a candy. I started with 0,0, 4, 12. Then gave them 2,3,4,5. Then started giving them bigger numbers. At first the students just paired the first two digits and the last two digits and used the distance formula. But after a round or two they started figuring out how to add negatives and rearrange the bigger numbers with smaller numbers to get larger distances. They also were doing a good job of checking each other's work because they only earned candy if they did the calculations correctly. By the end of the game, every student had figured out how to maximize distance and they were all tying and I was going bankrupt on Jolly Ranchers. My favorite part was when one person in a group announced their largest distance was 13.2 and students from a different group came over and clustered around asking the person from the first group how they'd gotten such a big distance. I think the game worked very nicely because it was simple, strategic, competitive but not so competitive that students who were "losing" became disheartened. By the end everyone was winning. <br /><br />I didn't like the game because I don't like the distance formula. I would much rather students use the Pythagorean theorem enough that they could then extrapolate the distance formula by picturing triangles on the coordinate plane without needing to graph. Unfortunately I just didn't have the time to reinforce this method of calculating distance so I caved and taught them the distance formula (but at least I did show them how it came from the Pythagorean theorem, though half my class fell asleep or glazed over when I tried to show the derivation to them. I've tried having them do the derivation themselves but their algebra skills are too weak.) At least though, they did do some critical thinking in terms of figuring out how to maximize distance. That was the saving grace of this game. Lizzy-Senseinoreply@blogger.com1tag:blogger.com,1999:blog-4250050147042236415.post-79040037724622561462014-01-25T09:57:00.000-08:002014-01-25T09:57:51.771-08:00AngleatronsI inherited a geometry class last semester that was already two months into the curriculum and it was very frustrating that their basic sense of shape hadn't been strengthened. I was supposed to start with congruent triangles, but many of them didn't even know what a right angle was supposed to look like. It was too late to go back and work on basic drawing skills but I've been thinking about how to help students with little practical drawing experience succeed in geometry. Gone are the days when all students had formal art classes and without these classes, their visualization and drawing skills are so weak that geometry can be really challenging and frustrating.<br /><br />With the new semester, I'm starting over with a new class and I'm working on building more drawing and visualization into my curriculum. I've just written up a lesson tied to <a href="http://www.youtube.com/watch?v=o6W6P8JZW0o">Vi Hart's angleatron video</a>. I want my students to be able to do rough sketches of all the basic angles so that their drawings, when we get to triangles and polygons can be at least a little bit accurate. <br /><br />First I'll show my students the video and have them try to explain how the different angleatrons were formed.<br /><br />Then I'll have the students make the different angleatrons, name their vertices, sides and the angles themselves.<br /><br />Then I want my students to try making 3 different geometric patterns using their angleatrons like Vi Hart did. <br />Finally, they'll each pick the pattern they like the best and we'll make a class quilt out of their different patterns. <br /><br />I'm a little nervous because a lot of my students really hate drawing, but I hope the structure of this activity and Vi Hart's beautiful examples will help. <br /><br />Here's the lesson sheet I'm planning to use: <br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/202182041/Angleatrons-Lab" style="text-decoration: underline;" title="View Angleatrons Lab on Scribd">Angleatrons Lab</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="undefined" data-auto-height="false" frameborder="0" height="600" id="doc_29452" scrolling="no" src="//www.scribd.com/embeds/202182041/content?start_page=1&view_mode=scroll&show_recommendations=true" width="100%"></iframe>Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-83963079719908691352013-12-01T11:10:00.000-08:002013-12-01T11:10:14.539-08:00Article on "Math People"Soo.... I wrote an article for Quartz magazine on why so many people identify themselves as "math people" or "non math people" and what we can do about it. It's the first time I've ever been officially published and I'm both terrified and excited. The magazine editors came up with the article title and the subheadings but the rest is mostly my work. <a href="http://qz.com/151460/how-i-get-all-my-students-to-be-good-at-math/">Here it is</a>.<br /><br />I don't know how many people will read it, but so far it's been well worth the two weeks I stressed over it because of some of the wonderful responses I've gotten from former students. I have to share them. If the article was terrible, it was worth writing just to get these wonderful words of encouragement from my students. <br /><br />Student<span style="background-color: white;"> 1: <span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #333333; font-size: 11px; line-height: 14px;">Congratulations on having your article published! I miss having you as my teacher so much! You were the best teacher I have ever had</span><span style="color: #333333; font-size: 11px; line-height: 14px;"> <3</span></span></span><br /><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 11px; line-height: 14px;"><br /></span><span style="background-color: white;">Student 2: <span style="color: #333333; font-size: 11px; line-height: 14px;"><span style="font-family: Arial, Helvetica, sans-serif;">You are so brilliant! I am so lucky I had you as my teacher. Not only were you a good teacher, but a super-cool one. I miss you!</span></span></span><br /><span style="background-color: #edeff4; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 11px; line-height: 14px;"><br /></span>Student 3 (this is the one that made me cry): <span style="background-color: white; color: #333333; line-height: 17px; white-space: pre-wrap;"><span style="font-family: Arial, Helvetica, sans-serif; font-size: xx-small;">I want to thank you for writing that article. I have been so scared to take another math class because of the last one I took. It was Math 110; basic college Algebra. I failed the class. I went to the math lab regularly, I participated in study groups, office hours, the works. I tried hard, but the teacher just could not explain things in a way that I could understand well and remember. After that class, I decided that I could not ever have a career in the math/science field despite my love for them because I just was not a "math person." A few weeks ago I was reading a Biology/Science textbook and realized that those were the only textbooks I had ever read for fun. I always have. But right then I nearly started crying thinking about how I just did not have the math skills to ever pursue it. Your article has given me hope... I now have the courage to try again. Thank you so much. I am so grateful for the time I had as your student. I know that all of your former students feel the same way. We love you. You are the best. Don't believe anyone who tells you otherwise</span></span>Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-78071593946184989602013-11-30T17:41:00.000-08:002013-11-30T17:41:46.972-08:00Why teach AlgebraI've been thinking about a response to the article published in the <i>New York Times, <a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?pagewanted=all&_r=1&">"</a></i><a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?pagewanted=all&_r=1&">Is Algebra Necessary?" </a> for over a year. I've started maybe 10 different draft posts and scrapped them. I've been following the follow-up debates in blogs (see Wiggins' post on <a href="http://we%20no%20longer%20make%20this%20spend-years-learning-inert-parts-out-of-context%20mistake%20in%20english.%20even%201st%20graders%20learn%20to%20%E2%80%98write%E2%80%99%20ideas%20right%20away.%20we%20appropriately%20do%20not%20now%20ask%20kids%20to%20first%20endlessly%20parse%20sentences%2C%20for%20example.%20so%2C%20why%20is%20algebra%20still%20doing%20the%20equivalent%2C%20with%20our%20official%20blessing/?%20Learning%20bits%20of%20algebra%20out%20of%20context%20doesn%E2%80%99t%20make%20you%20a%20better%20mathematical%20thinker%20or%20problem%20solver%20%E2%80%93%20the%20supposed%20goals%20of%20math%20courses%20%E2%80%93%20any%20more%20than%20merely%20trudging%20through%20brush%20lessons%20in%20art%20makes%20you%20a%20better%20painter%20or%20going%20through%20Warriner%E2%80%99s,%20page%20after%20page,%20can%20make%20you,%20by%20itself,%20a%20better%20writer.">algebra 1 as a poorly designed course</a> and <a href="http://mrhonner.com/archives/11406">Honner's response</a> to Wiggins) and thinking about related articles like<a href="https://app.box.com/s/p5zsk913anc4wk95j36z"> "Wrong Answer: The Case Against Algebra II"</a> and <a href="http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf">"The Mathematician's Lament"</a>.<br /><br />I've been so torn about how to respond because as a math teacher, of course I believe teaching math is vital. I became a teacher because I wanted to save the world and martyr myself with 80 hour work weeks and panicked sweats every Sunday night. And reading these articles seems to trivialize what I have poured sweat, tears and many many gallons of coffee into. Yet I see their side of things. I hate the idea of algebra 1 being the barrier between a talented artist and a career in art. I am now teaching students in algebra 1 who have failed it two or three times before and it broke my heart yesterday when I handed back a homework assignment I had given a 7/10 to a student and her face lit up as she said that she'd never gotten a passing score on a math assignment. <br /><br />Also, I've never used math in "the real world". I'm not an engineer, an economist, a physicist or a banker. I don't know how math gets used out there so who am I to tell students year after year that they're going to need these skills when I don't know that they will. The argument that math sharpens general cognitive skills and teaches students problem solving strategies that <i>will</i> be useful later in life, especially as it's backed up by <a href="http://www.economist.com/blogs/freeexchange/2009/08/how_to_get_smart">research</a>, holds water but that doesn't help us algebra teachers argue for teaching algebra. Why not teach statistics? Or a formal logic class? Should we defend the traditional math sequence, or should we branch out and give students who are failing at algebra alternative math options? <br /><br />But the other day I was talking with my husband and I realized why I love math, not why I teach it or how I use it, but why I <i>love</i> it. And I think the reason for my love is also the reason it needs to be taught. I am decidedly introverted, perhaps the queen of introverts. I can't handle phones- it's very <i>very</i> difficult for me to talk on the phone with those I love and even harder with those I don't know. I need to see eyes, to gauge reactions, to be able to comment on surroundings or engage my conversation partner in a task that removes the focus of conversation off of me. I've found the adult world intimidating and overwhelming and need frequent breaks from it. I like playing board games to escape because they have defined protocols. I know exactly what the object of the game is and how to get there. I can enjoy socializing while playing because of the game's comforting structure. <br /><br />The world is overwhelming for anyone- even those not so introverted as I am. There are complex political systems to understand, the natural world can be scary and confusing, bad things happen to good people inexplicably, we are born with deficiencies and insecurities that make socializing difficult or awkward. School is for this- to help our young students learn that <i>knowledge </i>will conquer their confusions and difficulties<i>. </i> When they understand how something works, they aren't as afraid of it and they know how to navigate it. Or when they understand how something works, they won't make a mess of it because of overconfidence or arrogance. Understanding our history and political systems is vital but impossible. We give our students the best analysis tools we can and hope that time and a love of learning will help guide them in making wise decisions for themselves. Learning science is fascinating and practical, but requires lots of field trips, labs, props and math to even begin understanding the basics of how our world works. Math is the <i>only</i> field where understanding can be <i>created</i> by the student with nothing more than a pencil, a paper and a system of logical rules- just like a board game. Yet unlike a board game, math helps us untangle the mysteries of how the world around us works. It gives us a sense of order and control over our own minds and our own environments. Isn't our job as educators to help students make sense of the world around them and to help them feel in control of their own lives? Math is instrumental in accomplishing these two goals but especially for helping students realize what their minds are capable of and that they don't have to go outside to conquer a small piece of their universe. <br /><br />So this is why we need <i>algebra</i> and not just statistics or logic. Algebra is about finding the unknowns. It's about looking at how the complex variables in our lives that affect each other and us. It has the further advantage of being the bedrock of higher level math so that if a student chose to pursue advanced math, she could. It's got an easily understood framework of logic so that when the basic properties of algebra are mastered, all the other results are easily provable by a 14 year old with a pencil. But most importantly, mastering algebra - especially because it can be such a difficult transition for many students- makes a student feel powerful and in control of her mind and world. Isn't this how we want students to feel when they go out to help shape our society? Lizzy-Senseinoreply@blogger.com1tag:blogger.com,1999:blog-4250050147042236415.post-68886294460675426812013-11-17T12:49:00.000-08:002013-11-17T12:49:37.463-08:00Congruent Triangles Review GameI played a review game with my geometry class a few weeks ago that they loved so I thought I'd share it. I think I stole this idea from a blog, but I can't for the life of me remember which blog, so if it's yours let me know. <br /><br />I stole the problems from the Pearson <i>Geometry Common Core Edition.</i> <br /><br />I printed the document below double sided but didn't staple it. Then I shuffled the pages and made 8 or so copies of all of them for the 8 groups in my class. The groups needed to start with the page that has "RP" at the top and do the proof. Then they hunt for the answer in the sheaf of papers. When they find the answer, they grade their proof against the answer key, turn the answer key over and work the problem on the back of the answer key. Then they hunt out the answer key to the new problem.<br /><br />If they keep track of the order in which they did the problems, they can write down all the letters from the upper right hand corners of the problems, unscramble them and a message appears. <br /><br />I had students for the first time actually paying close attention to every step of the proof, asking great questions about why different steps appeared, if they were necessary, and if/how order in the proof matters. They also really loved working out the code. <br /><br />I know it's just drill and kill two-column proofing, but it did a nice job of getting my students to compare different proving methods and getting them to analyze their own work.<br /><br />Here it is: <br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/184924755/Triangle-Congruence-Proofs-Review-Game" style="text-decoration: underline;" title="View Triangle Congruence Proofs Review Game on Scribd">Triangle Congruence Proofs Review Game</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="undefined" data-auto-height="false" frameborder="0" height="600" id="doc_68842" scrolling="no" src="//www.scribd.com/embeds/184924755/content?start_page=1&view_mode=scroll&show_recommendations=true" width="100%"></iframe>Lizzy-Senseinoreply@blogger.com1tag:blogger.com,1999:blog-4250050147042236415.post-91323880025002863312013-11-03T10:49:00.003-08:002013-11-03T10:49:36.474-08:00Awesome ArticleHere's an AWESOME article on that oft heard phase that crushes teachers' souls, "I'm not a math person." It's titled <a href="http://noahpinionblog.blogspot.com/2013/10/miles-kimball-and-noah-smith-on-fallacy.html">"Miles Kimball and Noah Smith on the fallacy of inborn math ability."</a><br /><br />Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-55941156358584847382013-10-13T18:33:00.001-07:002013-10-13T18:33:59.487-07:00I'm a real teacher again!My life has kind of been in turmoil for a while. After one cross country move last year, we've just driven back across the country. By car: Oregon-> New York-> California. By airplane: DC, Japan, Indiana, Ohio, and Arizona (and Oregon and New York a bunch). This is all in a 12 month time period. I wonder how many miles I've traveled. I think it's probably close to a world record. <br /><br />I'm also undergoing the teacher licensing process in my 4th state! Woot! Soon I'll collect all 50 :). I would totally go for national board certification, but I'm not wise enough or anywhere consistent enough to go for that yet. <br /><br />We moved back to the west coast in September because of a family illness. I left my job at the one-to-one private school abruptly and was actually looking forward to some time off. The private school was year-round so I was getting pretty tired. But of course, obsessive compulsive me couldn't stop checking craigslist and within 24 hours of arriving in San Diego I had a job interview at a charter school. I was really really nervous about not having a job plan when we decided to come spend some time in California so I was delighted at the prospect of a job so quickly. But it's in a classroom again with a lot of kids and I start tomorrow and I'm terrified. <br /><br />It's been a year since I worked in a classroom with more than one student and my pitiful classroom management skills have completely atrophied plus my obsessive compulsive work ethic already has me fruitlessly planning lesson after lesson even though I have no idea where the kids are and will have to scrap all this work and start over. <br /><br />Do the nerves ever go away? I wish they would. Everyone says I'm super lucky to have landed a job in late September/early October but right now I wish I could go back two weeks and kick my over eager, initiative grabbing, hopeful self and tell her to just CHILL! <br /><br />But I am happy to be teaching in a classroom again, I am. And I will try to post more because now I'm legitimate. Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-90670523186894779792013-08-31T10:08:00.000-07:002013-08-31T10:08:35.712-07:00Exponent Rules GAMEI've posted on this topic a bunch of times <a href="http://dontpanictheansweris42.blogspot.com/2012/01/i-feel-like-getting-up-onto-my-little.html">here</a>, <a href="http://dontpanictheansweris42.blogspot.com/2012/02/dont-break-product-rule-or-youll-be.html">here</a> and <a href="http://dontpanictheansweris42.blogspot.com/2012/03/dont-break-quotient-rule-or-our.html">here</a>, but I'm not tired yet of hammering more nails into this coffin. I think that correctly mastering exponent rules is a gateway skill. Maybe one of the most important gateway skills in algebra. Exponent rules:<br /><br /><ul><li>Formalize the meaning of multiplication and division for algebra.</li><li>Provide the first forum for students to effectively use reducing in an algebraic context.</li><li>Introduces students for the first time to how simple algebra definitions (i.e. the definition of an exponent) can be used to prove a multitude of other <i>cool</i> rules that make doing math easier. In other words exponent rules formalize the structure of mathematical logic and proof for students.</li><li>Is often the first time students see and manipulate algebraic rules represented purely with variables. If students can understand and use exponent rules, it prepares them for using and understanding other rules represented with abstract mathematical language. </li><li>Set the foundation for a student's understanding of polynomial functions, radical functions, exponential functions, and logarithmic functions. Without a solid understanding of exponents and their properties students will struggle with all of these types of functions later.</li></ul><div>And I think we can teach the exponent rules well because they're just not that hard to derive, but the level of abstraction is what makes it difficult for students. So in teaching exponent rules, I believe we should focus on teaching students the abstraction, and the rules get learned along the way. </div><div><br /></div><div>That being said, I don't know how to do it but I keep trying. I've updated my lesson on developing the exponent rules. You can find that <a href="http://www.scribd.com/doc/164479724/Exponent-Rules-Lesson">here</a> and also, I've developed a simple game that I hope helps students cement the rules and learn to play with them. The game involves both strategy, luck and understanding of exponents so I think it's pretty good but it's only had a few trial runs. <br /><br /><b>Materials:</b> You'll need a set of blue cards and a set of green cards. You can download the cards and the rules <a href="http://www.scribd.com/doc/164482628/Exponent-Game">here</a>. Lay the cards out like so:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-WMEE2rc_rrc/UiIgVuA8KlI/AAAAAAAADCs/LUfvPddbwqA/s1600/Exponent+Game+layout.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-WMEE2rc_rrc/UiIgVuA8KlI/AAAAAAAADCs/LUfvPddbwqA/s320/Exponent+Game+layout.png" width="281" /></a></div><div class="MsoNormal"><b>Object:</b> Combine your starting expression with green cards to create the target expression.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b>Rules:</b></div><div class="MsoNormal" style="text-indent: 0px;"><span style="text-indent: -0.25in;">(1)<span style="font-family: inherit;"> You may use as many green cards as you wish.</span></span></div><div class="MsoNormal" style="text-indent: 0px;"><span style="font-family: inherit;"><span style="text-indent: -0.25in;">(2) Cards that look like this: </span><span style="text-indent: -0.25in;"><span style="line-height: 17px;">( )^2</span></span><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">must be applied to your whole expression so far.</span><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">So if you start with the card â€ś</span><span style="text-indent: -0.25in;"><span style="line-height: 17px;">ab"</span></span><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">and you grab the green card </span><span style="text-indent: -0.25in;"><span style="line-height: 17px;">( )^2 </span></span><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">you will end up with </span><span style="text-indent: -0.25in;"><span style="line-height: 17px;">a^2 b^2</span></span></span></div><div class="MsoNormal" style="text-indent: 0px;"><span style="font-family: inherit;"><span style="text-indent: -0.25in;"><span style="line-height: 17px;">(3) </span>If neither player can find the right cards to create the target expression, three more green cards can be put down.</span><span style="text-indent: -0.25in;"> </span></span></div><div class="MsoNormal" style="text-indent: 0px;"><span style="font-family: inherit;"><span style="text-indent: -0.25in;">(4) Once the target expression is reached by a player, that player gets the blue card </span><i style="text-indent: -0.25in;">and all the green cards they used to make the winning expression</i><span style="text-indent: -0.25in;">.</span><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">A new blue card is then put down and green cards are added until there are 9 green cards again.</span></span></div><div class="MsoNormal" style="text-align: start; text-indent: 0px;"><span style="font-family: inherit;"><span style="text-align: center; text-indent: -24px;">(5) </span><span style="font-size: 12pt; line-height: 107%; text-align: center; text-indent: -24px;">Once all the green cards are gone, the game is ended and the player with the most green and blue cards wins.</span></span></div><div class="MsoNormal" style="text-align: start; text-indent: 0px;"><span style="font-family: inherit;"><span style="font-size: 12pt; line-height: 107%; text-align: center; text-indent: -24px;"><br /></span></span></div><div class="MsoNormal" style="text-align: start; text-indent: 0px;"><span style="font-family: inherit;"><span style="font-size: 12pt; line-height: 107%; text-align: center; text-indent: -24px;"><b>Examples:</b></span></span></div><div class="MsoNormal" style="text-align: start; text-indent: 0px;"><span style="font-family: inherit;"><span style="font-size: 12pt; line-height: 107%; text-align: center; text-indent: -24px;">Here are several examples of how the players in the set-up above could reach the target expression. </span></span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-xakdMePQVwg/UiIh-ETqb3I/AAAAAAAADC4/iMs9qnh6nb8/s1600/Exponent+Game+Examples.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-xakdMePQVwg/UiIh-ETqb3I/AAAAAAAADC4/iMs9qnh6nb8/s320/Exponent+Game+Examples.png" width="262" /></a></div><div class="MsoNormal" style="text-align: start; text-indent: 0px;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><!--[endif]--></div>Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-23523162830278553412013-08-03T14:16:00.000-07:002013-08-04T09:48:53.084-07:00Not Ratios again!Students either seem to "get" ratios or they don't. I don't know what to do about it. I made a really detailed ratio and proportion lesson based on beats per minute and the fastest guitar player in the world for my algebra 1 students. 3 students that I used it on loved it and understood everything just fine, 1 student could not get it no matter what I tried.<br /><br />I drew a picture of a person and said that he was 6 ft tall but a shrink-ray shrunk him to 2 ft. If his legs were 3 ft long originally how long are they now? My student thought for a second and then said "-1 feet?" I tried pictures of triangles, I tried explaining that scale factors worked with multiplication and division, not addition and subtraction, I tried just showing him the math steps based on fractions in a last ditch attempt to get him to walk out of the class with something. But none of it worked because he didn't have an internal sense for proportion. He could do the mechanism of cross-multiplication, but a "sense" for proportion just eluded him. <br /><br />I'm now working on a lesson for geometry introducing ratio and proportion and I'm getting a little cold and clammy because I have nothing. My experience is just kids see it or they don't and if they don't, I don't know what to do. Sheer perseverance and drill have helped these students eventually reach an "aha" moment, but it just seems to be based on time, not on cleverness of the activity (or I haven't found or thought of a sufficiently clever activity.) I've been sitting in a coffee shop for an hour now and so far, I just have a warm-up:<br /><br /><ol><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">Which pair of numbers is out of place?</span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"> </span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">Explain why you chose that pair.</span></li><ol><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">3 and 4</span></li><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">5 and 6</span></li><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">9 and 12</span></li><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">27 and 36</span></li></ol><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 18px; text-indent: -0.25in;">Which pair of numbers is out of place?</span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 18px; text-indent: -0.25in;"> </span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 18px; text-indent: -0.25in;">Explain why you chose that pair.</span></li><ol><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">9 and 12</span></li><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">12 and 15</span></li><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">20 and 25</span></li><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">32 and 40</span></li></ol><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 18px; text-indent: -0.25in;">You got a part time job at </span><span style="font-family: "Gill Sans MT","sans-serif"; font-size: 12.0pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-font-size: 13.0pt; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: "MS Mincho"; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-fareast;">The Pizza Hub. You just found out that your co-worker makes more money. Which statement would make you angrier? Why?</span></li><ol><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 18px; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><span style="font-family: 'Times New Roman'; font-size: 7pt; line-height: normal;"> </span></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">Your coworker makes $10 more than you.</span></li><li><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">Your coworker makes double what you make.</span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"> </span></li></ol></ol><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"> If there's anything out there to follow this warm up with, I'd love to hear about it.</span><br /><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;"><br /></span><span style="font-family: 'Gill Sans MT', sans-serif; font-size: 12pt; line-height: 115%; text-indent: -0.25in;">Update [8/4/2013] Here's the lesson I eventually came up with. I think it does a decent job. </span><br /><br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/158042611/Ratio-and-Proportion-Lesson-for-Geometry" style="text-decoration: underline;" title="View Ratio and Proportion Lesson for Geometry on Scribd">Ratio and Proportion Lesson for Geometry</a> by <a href="http://www.scribd.com/ezmoreldo" style="text-decoration: underline;" title="View ezmoreldo's profile on Scribd">ezmoreldo</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772922022279349" data-auto-height="false" frameborder="0" height="600" id="doc_49372" scrolling="no" src="http://www.scribd.com/embeds/158042611/content?start_page=1&view_mode=scroll&access_key=key-136sr5c053giohho9d03&show_recommendations=true" width="100%"></iframe>And here's an "I notice, I wonder" activity that could be used to get students thinking about ratio and proportion. Both of these are PDFs to preserve formatting, but if you go to my <a href="http://www.scribd.com/ezmoreldo">scribd profile</a> you can find the .docx versions.<br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/158045886/Ratio-and-Proportion-I-Notice-I-Wonder-PDF" style="text-decoration: underline;" title="View Ratio and Proportion- I Notice I Wonder PDF on Scribd">Ratio and Proportion- I Notice I Wonder PDF</a> by <a href="http://www.scribd.com/ezmoreldo" style="text-decoration: underline;" title="View ezmoreldo's profile on Scribd">ezmoreldo</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772922022279349" data-auto-height="false" frameborder="0" height="600" id="doc_2691" scrolling="no" src="http://www.scribd.com/embeds/158045886/content?start_page=1&view_mode=scroll&access_key=key-xy2ei44wrujnyzxh7b8&show_recommendations=true" width="100%"></iframe> <div class="MsoListParagraphCxSpMiddle" style="margin-bottom: .0001pt; margin-bottom: 0in; mso-add-space: auto; mso-list: l3 level1 lfo3; text-indent: -.25in;"><br /></div>Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-25246953172208860012013-07-29T07:57:00.001-07:002013-07-29T07:57:12.408-07:00Parallel Lines and Transversals gameHere's a simple game that helps students cement all the different vocabulary words for the angles formed by parallel lines and transversals. I teach at school specializing in one-to-one instruction, so unfortunately it's not very much fun, but it does work! I basically just made a geometry version of my <a href="http://dontpanictheansweris42.blogspot.com/2012/11/my-new-school-is-one-on-one-instruction.html">parallel and perpendicular lines game</a>.<br /><br /><br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/156747315/Parallel-Lines-and-Transversals-Game" style="text-decoration: underline;" title="View Parallel Lines and Transversals Game on Scribd">Parallel Lines and Transversals Game</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="undefined" data-auto-height="false" frameborder="0" height="600" id="doc_11585" scrolling="no" src="http://www.scribd.com/embeds/156747315/content?start_page=1&view_mode=scroll&show_recommendations=true" width="100%"></iframe>Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-18336476027821166012013-07-21T12:30:00.000-07:002013-07-21T12:30:17.776-07:00Sharing and LazinessMy 5 year old nephew this morning had a bowl of blueberries. I had a devilish headache and was lying on my grandparent's porch swing. So when I wanted a blueberry, instead of going to get one for myself from the kitchen, I asked him if I could have one of his blueberries. He gave me three. And he fed them to me himself. Isn't sharing nice?<br /><br />A few weeks ago a new teacher was hired at my school. She's certified in English, but she'll need to teach some chemistry this summer (my school is kind of crazy) so I told her I'd share all my lessons with her so that she had somewhere to start. She was surprised and thanked me and this led to a larger discussion about sharing resources. She said, and I'm quoting almost verbatim, "I'm one of those teachers who gets up Saturday morning excited to lesson plan all weekend. I'm happy to share my resources with coworkers I like, but I won't share with teachers I don't know or don't like because I don't want to encourage them to be lazy!" <br /><br />A week later our school's curriculum writing team got together and our bosses told us not to make the lesson plans for the curriculum they're distributing across their various campuses too detailed. They don't want more than a page per lesson. Most of the teachers they hire aren't certified teachers because it's a private school so they told us they need a unified, regimented curriculum to support people who have never taught before. At the same time though, they said we could trust the teachers to know their subject matter and that they only needed an outline of content to be covered with no recommended instructional strategies or resources because "we don't want to stifle our teachers' creativity by giving them anything that's too detailed." <br />Is this a common view point held by teachers and administrators that providing rich, creative, and detailed lesson plans, activities and games promotes laziness and stifles creativity? It's hogwash! I am inspired by innovative and engaging ideas that I find in other people's lessons and the more detailed they are, the easier they are for me to adapt, to understand, or to use to help my students learn. The richer my materials, the more freedom I have to experiment. The more time I have to think about how to engage particular students. When I share my lessons I get feedback on what worked and what didn't and I become a better teacher. I can't believe how entrenched this miserly attitude is about sharing materials. This curriculum meeting was composed of dozens of teachers and administrators from all over California, New York and New Jersey and they accepted this rationale without question. That we should keep the good ideas secret to try to force other people to come up with their own good ideas. Don't they know that good ideas mate with other good ideas to produce litters of new, bouncing, energetic good ideas? <br /><br />And so what if you give a lazy teacher your lesson plan. Isn't that a good thing? Those kids in the lazy teacher's classroom maybe haven't encountered innovative teaching strategies before and if a lazy teacher uses someone else's lesson plan to good effect with their students, isn't that what we all want? That even kids who have "burned out" teachers have access to rich educational experiences? Why save the good teaching just for the students who happen to have been placed in your classroom? It's so silly to hoard! I guess I can understand not wanting someone to get credit for your idea, but in the end aren't we in this business to help students learn, not to accumulate "credit" for ourselves? I'm probably way too idealistic for this world. I guess I feel so strongly about this because I wouldn't be a sixth of the teacher I am today if I hadn't ripped, stolen, copied, and adapted every idea I could skim off of other teachers' blogs and websites. So I guess I am one of the lazy teachers but boy am I a better person and a better teacher for being willing to learn from all these amazing experts surrounding me. Lizzy-Senseinoreply@blogger.com1tag:blogger.com,1999:blog-4250050147042236415.post-86537755845437381292013-07-08T18:43:00.003-07:002013-08-04T08:55:19.852-07:00The Logic of GeometryI've noticed a distinct void in the Common Core where geometric logic used to have a home. This makes me really sad for three reasons:<br /><br /><ul><li> I think it's a really great way to show students how mathematical thinking can be applied to real life </li><li> I was skipped past geometry in high school and when I got to college, the lack of logic training was a real handicap</li><li>And I've developed a tiny bit of skepticism about Common Core. Not a lot, but I read this <a href="http://blogs.edweek.org/teachers/living-in-dialogue/2009/07/national_standards_process_ign.html">article</a> about how the Common Core was developed and it creeped me out a little. The fact that only one teacher helped develop the Common Core seems kind of terrible. I recently made friends with an economics blogger and he told me that every profession engages in "turf defending" where those in the profession reject outsiders' perspectives. That immediately made me not want to be a turf defender. I won't be the person who stomps on the school house floor and screams "MINE!" Of course I want outside input, but can I also add that it's maybe a little out of hand in education? I want to teach logic damn it! </li><li>Oh, also, I've found logic very challenging to teach and to give up on trying to find the perfect way to introduce it now just because Common Core gives me permission feels like a cop out. (so I am pretty selfish after all.) </li></ul>I've made a point of always teaching it thoroughly up until now. This year at least I'm stubbornly sticking to my guns and teaching a bit of logic (although I nixed conjunctions and disjunctions, mostly because I personally never used them in college)<br /><br />Below are my three logic lessons if you'd like to take a look. I'm writing lessons for the other math teachers in my school so the notes are kind of overly detailed. Also my school gives us only 50 hours a year (as opposed to the usual 150-180) to get through a year's worth of material, so it's all super condensed. I don't know if it's usable outside my school, but feel free to steal and I'd love feedback. I've uploaded the pdfs below to preserve formatting but the .docx are also available on Scribd. <br /><br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/152572249/Geometry-Lesson-9-V2-Introduction-to-Logic" style="text-decoration: underline;" title="View Geometry Lesson 9 V2-Introduction to Logic on Scribd">Geometry Lesson 9 V2-Introduction to Logic</a> by <a href="http://www.scribd.com/ezmoreldo" style="text-decoration: underline;" title="View ezmoreldo's profile on Scribd">ezmoreldo</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772922022279349" data-auto-height="false" frameborder="0" height="600" id="doc_13809" scrolling="no" src="http://www.scribd.com/embeds/152572249/content?start_page=1&view_mode=scroll&access_key=key-irlx25q3hexacwzru73&show_recommendations=true" width="100%"></iframe> <br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/152572263/Geometry-Lesson-10-V2-Converse-Inverse-Contra-Positive-and-Biconditional" style="text-decoration: underline;" title="View Geometry Lesson 10 V2- Converse, Inverse, Contra Positive and Biconditional on Scribd">Geometry Lesson 10 V2- Converse, Inverse, Contra Positive and Biconditional</a> by <a href="http://www.scribd.com/ezmoreldo" style="text-decoration: underline;" title="View ezmoreldo's profile on Scribd">ezmoreldo</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772922022279349" data-auto-height="false" frameborder="0" height="600" id="doc_42378" scrolling="no" src="http://www.scribd.com/embeds/152572263/content?start_page=1&view_mode=scroll&access_key=key-2483y4lyk03ixcflapg2&show_recommendations=true" width="100%"></iframe> <br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/152572279/Geometry-Lesson-11-V2-Proofs-Using-Logic" style="text-decoration: underline;" title="View Geometry Lesson 11 V2-Proofs Using Logic on Scribd">Geometry Lesson 11 V2-Proofs Using Logic</a> by <a href="http://www.scribd.com/ezmoreldo" style="text-decoration: underline;" title="View ezmoreldo's profile on Scribd">ezmoreldo</a></div><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772922022279349" data-auto-height="false" frameborder="0" height="600" id="doc_47588" scrolling="no" src="http://www.scribd.com/embeds/152572279/content?start_page=1&view_mode=scroll&access_key=key-pb4idcmgzsz1cn5la34&show_recommendations=true" width="100%"></iframe> Note: I borrowed some of the homework problems from Harold R. Jacob's <i>Geometry: Seeing, doing and understanding</i> 2nd ed. and from AMSCO's Geometry. I also totally stole the formatting from <a href="http://exponentialcurve.blogspot.com/">Dan Wekselgreene</a>.Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-72372078345907560842013-06-30T08:36:00.001-07:002013-06-30T08:36:57.944-07:00One-to-oneMy school teaches students in a one-to-one classroom; one teacher, one student. Doesn't this sound awesome? From the teaching perspective, you can customize every lesson to the student and be sure they're really learning. From the student perspective, you won't have to sit while the teacher's instruction outpaces your focus or interest. <br /><br />We had an IEP meeting for one of our students and we were trying to convince the local school board that this student was making dramatic progress with us and thus his needs were being met. The moderator said "who wouldn't make progress in a one-to-one classroom!" And this got me thinking. I wouldn't. I would have HATED a one-to-one environment as a student. I didn't like to be the object of too much intense teacher concentration (and really, we can be overly intense sometimes). I liked to sit back and evaluate what the teacher said before deciding to accept or reject it. I liked to have a little freedom and control over how I took my notes and whether or not my mind drifted during class. I liked the opportunity to work with other students- our collective energy was so much more powerful than mine alone. <br /><br />As a teacher, I'm not a huge fan of one-to-one either. First of all, while "classroom management" is easier (this is a pretty silly word to use for one-to-one) it can also be more difficult because there's no escaping personality conflicts. It's easier to get locked into battles of will (I avoid these like I would avoid Mexican food in NY- sorry guys, it's terrible) but I've noticed these types of intense, pride posturing battles in classrooms near mine. There are no other students near by to say "hey dude, chill out" and there's no one the teacher can turn to to exchange a sympathetic look with or to share a joke with to relieve the tension. This leads to my biggest complaint against one-to-one. It assumes that the teacher can teach all the student needs to learn. THIS ISN'T TRUE! Students learn so much more from each other. The teacher can present content and establish a respectful and fun classroom culture, but the students sustain the culture, teach each other how to act, give each other support and encouragement, joke to relieve tension, boredom or frustration, and reinterpret the content in different ways so that their classmates can see it from multiple perspectives. Learning is both a solitary <i>and</i> group endeavor. <br /><br />So I know not many people have one-to-one classroom environments so this doesn't really matter to many besides me BUT I've been reading a little bit about the advent of personalized learning software. It seems that since the whole constructivist approach to teaching math, where everyone had to learn everything in groups, hasn't really shown huge gains, there's a movement to go in the opposite direction. Personalized learning software seems like it's going somewhere. Knewton, a personalized learning software developer based in NY, and Pearson are <a href="http://tech.fortune.cnn.com/2011/11/01/the-race-for-education-tech-heats-up/">teaming</a> up to bring customized, competency-based learning to as many students as possible. I think their vision of the future will be students sitting at home, in coffee shops, in libraries or even in classrooms, logging in to their program and getting a wholly customized learning experience. The Knewton software amasses student data and has an algorithm that decides when a student is ready to move on to new material, and what that new material will be. This is exactly like what I'm doing right now, except without the teacher. <br /><br />Here's what I see happening:<br /><br /><ol><li>computers probably aren't smart enough to do this yet. I don't know if they ever will be. I have an Algebra student who asked me if numbers go on forever. We had a nice side discussion on the nature of number and infinity which he understood. His abstract reasoning capabilities have far outpaced his computational competence so he asks really good, deep math questions and understands the answers even when he struggles with adding fractions. I can satisfy his deeper curiosity while still drilling him on basic arithmetic. A program would decide that he's not ready for anything beyond basic computation. </li><li>I do believe that school is about more than learning content. I think that learning to work cooperatively is important. Learning to share a joke to relieve stress is important. Learning how to speak up for yourself to an authority figure is important. Socialization is important. Maybe this just makes me old fashioned. But having people by your side makes learning more fun too. </li><li>Peers push you to accomplish more than you can by yourself. This is a problem I've seen in one-to-one teaching- that it's so much harder to motivate a student to study independently or to work on projects. When there's no one to share it with, no peer audience, teenagers shut down and disengage. Duh! Even teenagers with special needs (we teach a lot of students with severe social anxiety) need peers to support them. Even if you place kids in the same classrooms to use these educational technologies, they won't be learning the same content so won't be able to support or motivate each other effectively. </li><li>Finally, if you allow students to do the work at home, I think cheating will become the norm. I think it already is. A coworker of mine (a teacher!) admitted to me that she takes all of her sisters' online tests for her because her sister is an athlete and doesn't need school. </li></ol><div>Can't we find a middle ground? There is never going to be one model that works for everyone, but we all deserve to experience different types of learning. I may not like one-to-one learning because it makes me uncomfortable, but I went in to see my professors to get help on papers and it was good for me. We all need a mix of different experiences. We need technology <i>and</i> classrooms <i>and</i> individualized instruction <i>and </i>games <i>and </i>textbooks <i>and</i> teachers. Why can't technology make our classrooms richer and more full of different kinds of learning? Why is it a choice between classrooms or personalized learning? Can't we recognized that there are good things about classrooms? Why are we so obsessed with "or"? Why not <a href="http://www.adweek.com/news/advertising-branding/ad-day-coke-zero-134588">and</a>? Knewton and Pearson might say though that they are offering an "and" option. That this is meant to enrich teachers' curricula not replace them, but I'm suspicious that this is just another way to devalue teachers' professionalism. I read an article in Newsweek (I think but of course I can't find it now) where the head of one of these personalized software companies admitted that he didn't employ any educators on staff. And Pearson, based on what I've seen of their textbooks, doesn't know that much about teaching either. </div>Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-88051907811455631632013-02-02T08:55:00.001-08:002013-02-02T08:55:50.702-08:00Logarithm DominoesI'm currently planning my pre-calc unit on logarithms and I just can't seem to find enough fun ways to drill logarithms. It really is just about practice I think, but I HATE giving students worksheets full of problems. I groan every time I see a <a href="http://www.kutasoftware.com/">Kuta software</a> worksheet. I know that games basically just do the same thing in a dressed up form, but at least there's a measure of competition or strategy that gives students a focus. I just read Amy Gruen's <a href="http://squarerootofnegativeoneteachmath.blogspot.com/2011/03/logarithm-love.html">post</a> on practicing logarithms and almost threw out some expletives because I saw her link to logarithm dominoes and I spent a nice chunk of time last weekend to making my own logarithm dominoes. Its both wonderful and incredibly frustrating to spend hours on something then find that someone got there first and did it better. At least you know your idea was good, but you could have saved yourself so much time! (this happened to me last year with Kate Nowak's <a href="http://function-of-time.blogspot.com/2010/12/log-laws.html">logarithm laws worksheet </a>. I'd made one for myself then stumbled upon hers and hers was so much better!) But fortunately for me, Amy Gruen's logarithm dominoes are very different from mine, so I thought I'd share what I came up with. <br /><br />I threw together the following logarithm property dominoes. I haven't had a chance to try them out yet, so I'm not sure the ratios are correct. But I figured it was a start and I don't know when I'll have the initiative or time to post them again. Here are the dominoes:<br /><br /><div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><a href="http://www.scribd.com/doc/123496167/Logarithm-Property-Dominoes" style="text-decoration: underline;" title="View Logarithm Property Dominoes on Scribd">Logarithm Property Dominoes</a> by <a href="http://www.scribd.com/ezmoreldo" style="text-decoration: underline;" title="View ezmoreldo's profile on Scribd">ezmoreldo</a> </div><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772727272727273" data-auto-height="false" frameborder="0" height="600" id="doc_86211" scrolling="no" src="http://www.scribd.com/embeds/123496167/content?start_page=1&view_mode=scroll&access_key=key-1l6zvcr7ogf1svqly8aj" width="100%"></iframe> You should be able to play any domino game with these cards that you want but I planned to play the game "all threes". Here are the rules:<br /><br /><div class="MsoNormal"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://www.domino-games.com/domino-rules/img/AllThrees.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="http://www.domino-games.com/domino-rules/img/AllThrees.jpg" width="188" /></a></div><ul><li>Each player draws 5 dominoes. You start by playing a double domino (one end equals the same number as the other end: 2-2 for example)</li><li>You can build off the double domino in 4 directions- above, down, left and right. To play off a domino, you must match ends that have the same value.</li><li>Players take turns placing dominoes.</li><li>If a player canâ€™t place a domino, they must draw more dominoes until they can play. </li><li>After each player places a domino, count up the total of all loose ends. If the total is a multiple of 3, the player gets that many points. </li><li>The game ends when either a player runs out of dominoes in their hand or when a player reaches 100 points. If a player runs out of dominoes in their hand, the player with the most points wins. </li></ul><br /> <div class="MsoNormal">Since these are logarithm expressions and not straight forward pips, players must make the conversions in their head or on paper.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Now I just need to figure out a fun way to teach solving logarithm and exponential equations.... </div><br /><br />Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-14550148498459940372012-12-30T22:17:00.000-08:002012-12-30T22:17:01.769-08:00Sometimes Teaching is the BEST Job in the WorldI received this message today from a student I taught for 3 years back in Oregon.<br /><br /><span style="background-color: white; color: #222222; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13px;">I wanted to thank you. Even though you are not my teacher anymore, you still help me all the time. You wrote in my yearbook to remember that I am good at math, and I always go back to that and it actually helps me when I am stressed about algebra. Whenever I think about it, I feel as though I can push through and actually do it. I am doing pretty well in it so far and I owe part of that to you.</span><br /><span style="background-color: white; color: #222222; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13px;"><br /></span>Sometimes teaching is the best job in the world. Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-1582991025918136622012-12-19T09:51:00.001-08:002012-12-20T09:17:20.253-08:00Common Core vs. Regents?This being my first year teaching in New York, navigating the Regents has been a challenge. I feel so torn in different directions that I've ended up in a state of complete and utter indecision. Especially about geometry. Here are the facts:<br /><br /><ul><li>I'm teaching at a private school so technically, we don't have to do the Regents but our parents want us to offer Regents prep courses.</li><li>The private school has its own curriculum imported from its California model that isn't correlated either to New York State or to the Common Core.</li><li>We are restricted to 50 total sessions with the students per year rather than the 150 classroom hours you normally get at public school. If we need to go over 50, the parents have to pay more so we try very hard not to do that.</li><li>I love all the ideas the blogging community has for geometry, but everyone seems to be pushing Common Core and the geometry Regents exam doesn't seem to be there yet. </li><li>I have my own inclinations for teaching geometry that I'm having trouble shoving to the side to adhere to any standards. </li><li>Two months ago my boss asked me to look at our boxed curriculum from California and compare it to the New York State Standards and the Regents exam and make sure they were aligned. I discovered that they couldn't be more different and she has asked me to come up with a Regents friendly curriculum map. </li></ul><div>I LOVE the way <a href="http://drawingonmath.blogspot.com/2012/12/geometry-unit-2-lines-and-angles.html">Drawing on Math</a> has organized her geometry class, but I'm really torn. I was also very inclined to do parallel lines and transverals right at the beginning but a Regents aligned textbook, <a href="http://www.herricks.org/highschool.cfm?subpage=11368">AMSCO-Geometry</a>, puts it more than half-way through the course. Why did they make this decision? Is there some profound reason students should do congruent triangles and transformations first? They've split up all the points of concurrency in triangles into different chapters too, whereas I was inclined to put them all together. Which way is best? A lot of the organization seems strange to me, but I've only learned geometry through teaching it over the past two years (I was skipped through it in High School and my college didn't offer any college level geometry courses) and I'm unsure whether or not to trust myself on what seems logical to me vs. how the book organizes material.</div><div><br /></div><div>In the same Drawing on Math post, she also mentions scrapping most of the logic unit and only teaching converses. But the NYS standards have LOTS of logic material including converses, negations, contrapositives, direct and indirect proofs, truth tables and Law of Detachment. BUT, combing through old Regents exams reveals that they only ever seem to ask questions about negations, and the Common Core doesn't have much logic at all... Yet I love teaching it and when I got to college and took college level math courses, the fact that I'd been skipped through geometry became a real handicap in the more advanced proof based classes because I'd never been exposed to logic before. So I'm inclined to teach logic because knowing just high school level geo-logic would have really helped me. <i>BUT</i> we only have 50 sessions and I can't waste time on material not on the Regents exam. <i>BUT </i>everyone's saying the Common Core is better anyway so shouldn't I align our curriculum to the Common core and not to a standardized test? <i>BUT </i>our kids <i>need</i> to pass the Regents because our parents care about it so much. </div><div><br /></div><div>My heart tells me that I should just teach it in a way that feels right to me and if the kids really internalize the material they will pass the Regents. Yet the Regents has <i>such</i> specific types of questions covering specific topics that I'm worried if I don't teach them with the Regents in mind, they'll get to the exam and it will use vocabulary they're not used to and ask types of questions we haven't covered. I wish the State would just trust me a little. I can help the students navigate this material but I want to let them enjoy it and I want to let them explore and I feel like I can't do that with this ticking bomb hanging over my head. I guess I just have to try something and hope. Teaching is about experimenting however nervous this makes me. I hate the idea of an experiment failing at the detriment to a student's enjoyment of math. But we learn by making mistakes right?<br /><br />[12/20/12 edited to add the following paragraph] I'm still struggling with the geo curriculum and I decided to trust the book and do triangles before parallel lines and transverals but I'm running into difficulties. If you don't do parallel lines and transversals first, then you can't do the proof that there are 180 degrees in a triangle (or at least you can't do my favorite one) and trying to do all the triangle stuff without this is pretty crippling. In fact talking about angles at all becomes a little sticky. We're supposed to do exterior angles in the triangle unit, but how do you prove any of the exterior angle theorems without knowing there are 180 degrees in a triangle? And what about AAS triangle congruence? They've thrown that in much later in the course 3 units after doing all the other triangle congruence theorems. I wish textbooks provided a justification for how they organize their content because I always start by trying to follow a book (they know best right? Tons of experts and trials in classrooms and thousands of dollars.) and then <i>always</i> scrap the book a quarter of the way in because their sequencing just doesn't make sense to me. I wish I could squelch my internal sense of logic and just trust a textbook... my life would be so much easier.</div>Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-77363633392015969192012-12-09T07:45:00.004-08:002012-12-09T07:47:31.874-08:00Where are the history teacher bloggers?I have a confession to make. I majored in history. I loved doing research and piecing together an argument out of scraps. I loved analyzing bias and wondering about how people's perceptions of history, true or false, shape how they act. But teaching history was a whole different world. The litany of timelines, facts, dates, and vocab words I was supposed to shove into students' heads while the clock was ticking left me with a sense of hopelessness. I switched to teaching math. In college I'd always taken a math class on the side because compared to studying history where nothing can be certain, the logical certainty of math kept my head from exploding. <br /><br />My boss asked me recently, because of my history background, to help reshape the 8th grade history curriculum for our school. We needed to take their curriculum that had been designed for California state standards and adapt it to fit into New York State standards. Whenever I'm about to plan a lesson for math I consult my friendly math blogging community. Sometimes I search specific blogs, sometimes I just google "system of equations activity" and scroll through the first few entries until I find one published by a blogger. I've used curricula published by textbooks and by for-profit internet companies and visited the teacher stores and bought the workbooks. None of the published material out there can even come close to matching the creativity of what math bloggers produce. The lessons published by math teacher bloggers are adaptable, easy to implement, enjoyable and thought provoking. I've been relying on this wonderful community for the last three years and I can't imagine teaching without it. So when I needed to help develop curriculum for history, with joy I started googling to find fellow history teachers who could help me with this project. Crickets. Silence. Page after page of historical info sites, or lessons published by for-profit companies. Museum published curricula or government sponsored curricula abounded. PBS has a wealth of nice lesson plans. But where are the bloggers? Maybe they're out there but they're much harder to find than their math teacher counterparts. In fact, even while math teacher blogging is rich and prolific, none of the math teachers I've run across in real life know about this community and while I give them lists of my favorite blogs and tell them that it really is worth their time, none of them have followed up. <br /><br />Reading math teacher blogs has revolutionized the way I think about teaching. It has made me humble and insecure at times (because I feel like there's no way I'll be as awesome as the teachers I read about,) but that has pushed me to try more ideas, to keep pushing myself, to try to come up with lessons worthy enough to share. When I feel overwhelmed or terrified by the responsibilities I've assumed the blogging community shows me others who push through difficulties with humor and humility and this gives me strength. I guess I'm just trying to give a post Thanksgiving thanks. My two month foray into history has made me so appreciative that there are math teachers out there taking care of each other. I'm not a very good blogger yet, but I will keep striving to give back to this community that has given me so much. Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-79059222767186529312012-11-21T08:21:00.000-08:002012-11-21T08:21:48.757-08:00The Math Teacher Anthem We had a workshop yesterday where each teacher at our school showcased a lesson to all the students and the other teachers. In the morning, the two music teachers had an awesome song writing workshop. Our kids busted out the most heartfelt, funny, tuneful ballads. The other math and science teacher, the history teacher and I got together in group to write a song which none of us had ever done before. It turned into more of a poem and most of the clever bits were thought up by the history teacher, but I'm proud to say that the original idea and some of couplets were mine. I think that this may need to be the official math teacher anthem:<br /><div style="text-align: center;">Doesn't matter if its black or light</div><div style="text-align: center;">Fill my cup and you fill my life</div><div style="text-align: center;"><br /></div><div style="text-align: center;">Sandy knocked out gasoline</div><div style="text-align: center;">But please don't limit my caffeine</div><div style="text-align: center;"><br /></div><div style="text-align: center;">Cup of Joe</div><div style="text-align: center;">Sweet and low</div><div style="text-align: center;"><br /></div><div style="text-align: center;">Paper work stacking up</div><div style="text-align: center;">Please oh god just fill my cup</div><div style="text-align: center;"><br /></div><div style="text-align: center;">You can keep your weak green tea</div><div style="text-align: center;">I think that Dunkin' runs on me</div><div style="text-align: center;"><br /></div><div style="text-align: center;">Thoughts are sluggish, head aches</div><div style="text-align: center;">Pump me up till fingers shake</div><div style="text-align: center;"><br /></div><div style="text-align: left;">We only had about 10 minutes to write, so I don't think it's done yet. We need a few more couplets (is that the proper literary term? I'm not sure...) Any suggestions?</div>Lizzy-Senseinoreply@blogger.com0tag:blogger.com,1999:blog-4250050147042236415.post-71472899555318911472012-11-18T11:01:00.001-08:002012-11-18T11:03:15.335-08:00Equations of Vertical, Horizontal, Parallel and Perpendicular LinesMy new school is one-on-one instruction. Just a teacher and a student. In some ways this is AMAZING. We can cover so much material, I can gear my explanations specifically to that student and take their learning styles into account, I can really see if they get it or if they're just faking it so as not to stand out. It is not amazing in terms of games though. None of my old games will really work. A lot of them are team based, or competition based or communication/discussion based. I can play some of the competition games with the student, but any of the games that are based on knowledge or practice are not too much fun because I'll always either beat the student or the student will know I'm going easy on them. One of my boys got very upset with me when he realized I was "letting" him win. I don't enjoy games where winning is based on chance (i.e. board games where you roll a die and answer the problem you land on.) Or where math is just a hurdle to play the game, not the focus of it. <br /><br />I've been writing a lesson plan on equations of horizontal, vertical, parallel and perpendicular lines and I came up with a game that I think will be good. Winning takes strategy combined with luck and the strategy is independent of, yet still related to knowledge of the material. This means that hopefully, the student will have a chance of beating me while still practicing equation writing skills. I have NO idea if this game will work, but I thought I'd share it. <a href="http://www.scribd.com/doc/113696743/Horizontal-Vertical-Parallel-and-Perpendicular-Lines-Game" style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto; text-decoration: underline;" title="View Horizontal, Vertical, Parallel and Perpendicular Lines Game on Scribd">Horizontal, Vertical, Parallel and Perpendicular Lines Game</a><iframe class="scribd_iframe_embed" data-aspect-ratio="0.772727272727273" data-auto-height="true" frameborder="0" height="600" id="doc_21886" scrolling="no" src="http://www.scribd.com/embeds/113696743/content?start_page=1&view_mode=scroll&access_key=key-295p7ywgfk7cl641afqx" width="100%"></iframe>Lizzy-Senseinoreply@blogger.com2tag:blogger.com,1999:blog-4250050147042236415.post-81726561328738082252012-11-08T10:25:00.000-08:002013-08-31T10:30:46.176-07:00Standards and Pre-AlgebraMy husband moved us out to NY so that he could get a physics PhD (I know, I couldn't bring him over to the much more beautiful and elegant world of math.) He has an Iranian classmate that we've started hanging out with. The other night he invited us over for dinner with his roommates and friends all of whom are Iranian and all of whom are either studying physics, mechanical engineering or computer science. Because most of them have TA-ships and are teaching undergraduate courses, when they found out I was a math teacher they all turned to me and asked a ton of questions along the vein of "why don't American undergrads know any math?! What DO they learn in high school?" My husband's friend had been struggling with his undergrads in a physics lab because they couldn't make a simple algebraic substitution (I can't remember what the problem was, but something like if a=b/c and d=2a, then d=2(b/c). Of course instead of a, b, c, and d they had maybe q with subscripts.) I asked him if maybe the subscripts had confused them, and he said he went back to simple a, b, c and d variables and they were still stumped. It took him 2 hours to explain this substitution to these students. He said they had no sense of variable at all. They could solve equations by rote, and they had bits and pieces of algebraic techniques, but no logical understanding of what algebra is and why they need to know it for physics. The other Iranian PhD students chimed in with their own anecdotes of students who have come to college to study the hard sciences with very little mathematical aptitude. They spent a while discussing how the Iranian education system is much more rigorous compared to what we have in the US. <br /><br />This is not a low ranked college. The students who come to Stony Brook University should know their algebra, especially those who want to study the hard sciences and math because it has very competitive science and math departments. And New York has the Regents. How can students who passed the grueling Algebra 1, Geometry and Algebra 2/trig Regents exams not know simple substitutions (and not be able to grasp them even when a physics TA comes over and personally explains the process for over an hour?) With such a small sample and only anecdotes from overworked TAs who aren't trained to teach math this is not a fair base from which to judge the New York high school math curriculum, but I'm feeling a little judgy at the moment especially after wrestling with the New York math standards and the regents for the first time this year. <br /><br />Pre-algebra was a sacred class for me at my old school because it creates the base the rest of students' algebra understandings must rest on. For this reason I went really slowly and carefully in my pre-algebra class and made sure students were really understanding the jump from concrete to abstract mathematics. I strongly believe that pre-algebra should spend as much time as possible on cementing the ideas of what variables are, how to write expressions, and how equations and formulas are linked to variables and expressions. These are DIFFICULT ideas. Students need time to process them. They need the freedom to explore them in their own ways. They need to see how variables aren't just unknown numbers- that they're so much richer and more flexible than number- that's why they're so useful in algebra. Students should spend time observing patterns in variables (specifically, combining like terms and the exponent rules are a great way to do this) and how we can generalize number patterns using variables in simple and elegant ways. I believe this is what pre-algebra is for. It's NOT for statistics! It's NOT for quadratics and FOIL. It's NOT for re-drilling fractions, decimals, ratios and percents again for the 50th time. The New York (and Oregon for that matter) standards cram so much into each school year that students don't cement their knowledge or have time to make meaningful connections. This means that each topic appears in the math standards for at least four years in a row because students have to constantly review stuff they should have learned last year but only "covered" because there wasn't time to go into it in depth. (i.e. adding and subtracting fractions appears from 5th-9th grades.) Each topic gets "covered" each year but not taught each year. So quick students have to relearn the same content year after year, while students who struggle never properly learn it at all. <br /><br />I know this argument doesn't necessarily have traction. Students need to review no matter how deeply you taught the material the year before, but I do know that I spent a month on developing variable sense and then another month showing students the usefulness of variables and expressions in writing out general number patterns placing specific emphasis on exponent rules and geometric patterns at my previous school and when the students needed the exponent rules again in algebra 1, we only needed a half-hour review and ALL my students were fluid with using them in very complex situations. I'm getting algebra 2, pre-calc, and calc students now who don't understand their exponent rules and their eyes glaze over every time I try to show them the logic behind the rules because to them, they're just a random assortment of letters to be memorized when needed and forgotten the rest of the time. You can't learn differentiation in calc without being able to turn roots and rationals into exponential expressions instead. This inability to understand that this one seemingly random technique (exponent rules) is rooted deeply in mathematical logic and needs to be understood logically because it is a foundational piece of the structure of algebra I believe is a symptom of the standards push for breadth over depth. Students have memorized math techniques as a history student memorizes dates. They may sort of have a sense of order, but no sense of significance. <br /><br />Variable sense is important and deserves time. If given time in pre-algebra, students will be much more successful in their higher math classes. It does not deserve a week a year spread over 4 years. So to answer the question posed by our Iranian friends on what is wrong with American education, I think it's the standards. And more specifically, that no one seems to know what should be shoved into pre-algebra so they make it a hodgepodge of random techniques they think will be useful for algebra 1 rather than spending that year to really develop variable sense. And I am a part of the problem too because I'm correlating my lesson plans to NY state standards so that my students will be able to pass the Regents. I'm scared of going off in the direction I feel is right because it doesn't cover the "standards" I'm supposed to cover. I think pre-algebra is the problem and I wish I could go shake the people who put "determine if a relation is a function" and "describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations)" on the PRE-ALGEBRA standards. There's a reason we have a whole year of highschool geometry and two years of algebra. Give them time to get used to the idea of variable BEFORE rushing them into function translations!Lizzy-Senseinoreply@blogger.com1