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Everything posted by lewelma

  1. I know what it is to go super concrete. Although I do use algorithmic approaches with some students, others definitely need a conceptual approach. One student in particular comes to mind. I took on this student at 14. She was homeschooled and at that time had completed MUS alpha - delta (1st through 4th grade) and had started on epsilon (fractions). She had been taught completely algorithmically, as she would not watch the videos and her mom was basically mathematically illiterate (she would count 0.1 wrong if the answers said .1). This girl could very quickly and accurately add, subtract, multiply, and divide multidigit integers with traditional algorithms, but her mom had hired me because she was struggling with fractions. I very quickly realized that there was a WAY bigger problem than her mom realized. This 14 year old could not tell me the answer to questions like "you have 8 apples and I give you 4 more, how many apples do you have?" She had NO understanding of math, at all. NONE. She could manipulate the numbers as told, but had NO idea what she was doing. Luckily she was homeschooled so there were no exams looming. I put her back into the 1st grade book (alpha) and had her redo ALL the word problems. Basically, I thought that she must have dyscalculia and started making plans for making her just functional for life rather than any sort of university entrance. At the time she was interested in being a hair dresser, and there were no entrance requirements in math for this program. Over the period of a year, from age 14-15, I had her redo all the word problems in MUS from 1st through 4th grade, and when she was done, she still could not handle them. I figured that this was a core requirement for life -- to use math for very very basic functions -- so I had her do them AGAIN, but this time mixed up. Believe it or not, her mom rewrote all of them mixed up in a notebook -- we are talking like 400 word problems. During this word problem time, she and I worked on fractions. I did my standard pizza explanation only to find out that she did not understand circles. She could recognize the shape and call it a circle, but couldn't understand the idea of splitting a circle up into equal parts by going through the center. She also couldn't understand candy bars (rectangles) split up into pieces. In the end, the only way she could understand fractions was with liquids. So I had to draw tiny little measuring cups of water to explain all fractions, and had her work in the kitchen every week with water and measuring cups for months. By the time she had finished the 1st-4th grade word problems for a second time, we were through the fractions book (MUS 5th grade), and I felt that she might just be able to handle a very simplistic word problem. She was 15. And I felt we had made good progress. But then the very unexpected happened. Her dad stated that he wanted her to take the NZ national exams in Algebra, Statistics, Geometry, Trig, and Numeracy that year! I was like WHAT?!?!?! But he would not back down, would not talk to me, and the mom caved. I argued for an extra year to try to get her up to speed in those 5 subjects. Yes in ONE extra year. ONE!!! And you know what, we did it. She earned Merits on all 5 exams the following year in 11th grade - top 35% nationally. Just stop and think about that. It was simply incredible. I came to believe 2 things: 1) she had been VERY poorly taught with the focus on algorithmic knowledge at the expense of any conceptual knowledge, and 2) she needed more time than typical to develop mathematical maturity. For some reason, her brain just woke up to math at age 16. She is now working on Algebra 2 in 12th grade, and wants to be either an economist or a data scientist. She plans to delay university for a year, get through precalc in high school, and then do Calculus in University in either of these preferred majors. My point is that different students are different. She was poorly taught -- with algorithmic techniques to memorize, as if the goal was to be a human calculator. It was the going back to conceptual basics, way way WAY back, that allowed her to transition from 1st grade math to Algebra 2 in 4 years. Without going back, there would have been no way forward.
  2. Yes, I enjoyed that chat, but I still feel like I didn't express myself well because the systems are so different. Students in NZ take the national exams for a 3-year course of study (8th, 9th, and 10th grade) during the 10th grade year (8th grade is high school here). The exams concluding this 3 year course are in Numeracy, Statistics, Geometry, Algebra, and Trig. They must pass 3 of the 5 for university entrance. Kids here are *required* to take integrated math from 8th through 10th, so all kids get a standard american algebra 1 course over 3 years. 20% will fail the algebra exam by design (the test is curved with 50K students taking it). I think in the USA, kids just get a pass on algebra in some schools without there being any decently high standards. Really really depends on the kid. Some kids need to be concrete, some abstract, some practical. I've had 15 year olds in my kitchen pouring water in measuring cups, sent 13 year olds home with toddler's blocks to build 3D shapes to understand volume, played battleship with 14 years olds to help them with coordinates for graphing, and measured different sized plates with string with 15 year olds to explain pi. But yet I have also sent kids home with flashcards to *memorize* algebra. When working with kids in a school system, success breeds success. If I can get a kid to pass who has previously been failing, it creates momentum. I often use efficient memory-style techniques in the first month of tutoring to get these first passes, and then once I have the kids on board with these immediate successes, we switch to conceptual. So there are two reasons to use an algorithmic approach. 1) because that is how a student learns (like me), or 2) because it can be the more efficient approach in the beginning and can create immediate success and momentum.
  3. For the kids who need to learn algorithmic skills before conceptual skills, they will work on the Merit-level problems only after mastering the Achieve-level material. But it would need to get done before the assessment. Very few of my kids can work towards excellence as abstract thinking in mathematics is quite difficult and only 10% of students nationally can earn these marks because the questions are that hard. I encourage all kids to work to the Merit level, explaining that if they only do Achieve level work then they will always be regurgitating and never do any interesting thinking. It is best to work at the Merit level in 10th grade, so that the transition to 11th grade merit level thinking is smooth. For students who struggle with algebra in 10th grade, NZ has a qualitative statistics course which encourages relational and insightful thinking in statistics without having to do abstract algebraic thinking. Students I have in 10th grade who want to work at a high level and can't do that in algebra, will choose to switch to statistics in 11th or 12th grade, so they aren't stuck at a low level all through highschool.
  4. Well, in NZ we are lucky. Grades are based on levels of thinking, not percent correct. This means that my kids have to get to an ah ha moment or a why, if they ever want to get higher than a C in math. So for Math in NZ, you get a: C (Achieve) if you can get 75% of algebraic manipulation correct, but you also get a C if you get 100% of this low level thinking correct, because you have only demonstrated low level thinking. B (Merit) if you can get 75% of word problems and relational thinking correct, but you also get a B if you get 100% correct, because your thinking is still only Merit level A (Excellence) if you show either insightful or abstract thinking in about 50% of problems on offer. And let's be clear 20-25% of students fail (Not achieve) any one national assessment by design, they are curved. These assessments are hard and require some serious study. So for my kids when I first get them, I am just trying to get them to 'Achieve' the assessments, but then as they get excited about doing better in maths, they target a Merit which means we have to go after more complex material, not just more accuracy in low level stuff. Because each assessment is separate for different units, some kids might choose to work towards an Excellence in Trig, but only a Merit in Complex Numbers, etc. They can pick and choose. But all kids I have ever met desire merits and excellences which drive our math studies beyond algorithmic thinking. Here is the link to examples of the internals assessments for 11th grade. (internals follow explicit requirements but can be adapted to the students. So context can be fertilizer if you are rural, and rugby if you are a sports school etc. Externals are all in November, done together nationally) Since many of you have commented on a lack of graphing skills, look at 2.2 Cv2 for the Graphing Exam example I am currently working on with some of my kids Here is the exemplars for Excellence, Merit, Achieve, and not Achieve student papers for this particular assessment. NZ does it very differently from the USA as you will see when you look at this test for Graphing. It is hard! Students study what you test, so if you want them to go beyond algorithmic material in algebra, you need to test it.
  5. Yes. He was fascinating! I was told he was a 'thicky' by a friend who knew him. Everything apparently just thought he was slow at everything. It all revolved around his dyslexia actually, because many dyslexics can't memorize their multiplication facts, so they have very little number sense for primary school math especially for fractions. I saw that he just couldn't see inside numbers, couldn't break them up, couldn't factor, couldn't reduce fractions etc. So we set out to memorize the multiplication facts at the age of 15. Took all summer with little flashcards I made him. I tried to convince his mom to hold him back a year in math and she said no. So 3 months after I started with a kid who did not understand fractions, decimals, or any prealgebra, he walked into the third year of the integrated math class (8th, 9th, 10th) and he was to take the algebra, geometry, trig, and statistics national exams at the end of the year. Wow, was that a year! But he did it, and moved on to algebra 2/precalc in 11th grade, and then the following year to calculus scoring the equivalent of a 4 on the AP Calculus exam. What happened with the fractions was fascinating. He simply could never leave the concrete stage, and beyond that he could not really understand the idea of a fraction at all. How is 1/6 half of 1/3? He could see it with the pizzas, but he could not understand why the smaller piece was only 1/6th. Basically, he couldn't grasp that it was 1 piece out of 6 equally sized pieces. It just made no sense. And he couldn't understand place value either, so I couldn't use decimals to explain fractions. In the end, I just abandoned the effort because we were walking into algebra the following week. So algebraic fractions were taught by rote with NO conceptual understanding at all. By removing the requirement to actually understand it conceptually, he could just memorize what to do for algebraic fractions and not get caught up in the understanding. And once he could do algebraic fractions, numerical fractions became clear. And when I say clear, it was just a instantaneous understanding once I simply pointed out the direct connection between the two.
  6. Well, all of algebra. Try to teach one of MY kid conceptually why 2x+3x = 5x and x^2 times x^2 = x^4 but 2x^2 +3x^2 = 5x^2 I actually draw the x^2 as squares, and the x's as lines to help them think 2 squares plus 3 squares is 5 squares, but they really don't have ANY conceptual idea what they are doing with ANY of algebra. It is all algorithms that they memorize. i can get out a excel spread sheet and have them work with cells and formulas, but some just don't get it at all. Or try to explain to them what a two bracket factor is actually doing. I can say that it is converting from addition of terms to multiplication. I can show them with a 2 by 2 grid how it expands out. I can do it with numbers and BEDMAS and then compare to the algebraic terms. But actually the best thing to do is to have them take ONE problem and factor it, then expand it, then factor it, then expand it, over and over without looking at the previous answer. For some of my kids, they will still struggle after 10 repetitions, even though the problem is the SAME and they are just converting it back and forth. Most of my kids who struggle like this don't have dyscalculia, and in fact all but one have continued through 12th grade statistics or calculus with me. So it is not like they have a LD, they just cannot intuitively grasp algebra for years, and then it is so beautiful to watch when one day they do. Algebra here is inside integrated math, so I teach it to them at age 13, 14, and 15. For the kids who struggle, their ah ha moment is usually at 16/17, in the beginning of 11th grade.
  7. See my above post about a student who memorized how to do algebraic fractions by rote, and only then could he understand numerical fractions.
  8. Some might be saying that, but I am NOT. I am walking evidence, as are many of my students, that intuition can be developed through algorithmic drill all the way through high school and beyond. I have a PhD in Mathematical Biology, from back before it had a name. My work has been listed in some of the core new books outlining mathematical and theoretical biology as an example of an intuitive approach. But I am *definitely* a parts to whole learner, who has learned all math and mathematical science through drill. And only *after* the drill, do I develop insight. Half of my students are like me. I have tutored many students with many different learning styles. Here is one interesting example. I took one boy aged 15 from not knowing 1/10=0.1 through calculus in 3 years. Let me be clear, that is prealgebra, algebra 1, geometry, statistics, algebra 2, precalc, and calculus in 3 years including the NZ National exams (this boy is now in university in fine arts). And by the end, he had great insight, into complex numbers in particular. Insights that I did not have. Great leaps of understanding that were a joy to witness. I got him to this point by drill, and followed up with conceptual knowledge. In fact, I taught him numerical fractions through algebra. He could just not understand how fractions work, no matter how many pizzas I drew or manipulables we played with. We spent the entire summer before his 10th grade year getting through prealgebra, and he just could NOT understand fractions. But once he had drilled algebraic fractions for 3 months in 10th grade, with basically no understanding, I looped back around and taught him numerical fractions. He understood because the rote learning had created structure in his mind which made it ready for deep understanding. I completely agree. To suggest that one method of learning is superior to another is false. They each have their strengths and weaknesses. And I use many approaches with my students, all equally valid.
  9. The chief censor has banned the video and manifesto. Hate speech is allowed, but content inciting violence is not. All copies must be destroy, and policy are starting to go after the people who have disseminated them.
  10. Square_25, It would be so interesting to sit down over a cup of tea. I work with kids on the other end of the spectrum from AoPS kids. 🙂
  11. As for AoPS, my ds said that there were a lot of students that could do the work only with hints, and once they had the hints all the problem solving was gone. I have come to believe that AoPS classes only work for students who struggle to find a way through without hints. There are only 10 or so problems each week. You can only gain mastery with so few problems if you have had to think deeply about them. If instead you turn them into drill because you have hints, then you need to do a more standard approach with way more practice to master the material. Ruth in NZ
  12. I agree with Farrar, I see algorithms throughout higher level math, and my post (copied below) was in reference to high school math. When I teach students about converting complex numbers between rectangular and polar coordinates, some kids have intuition and others require drill as I describe below. When we do the loci in complex plane, some kids need to drill the steps about 10 times before gaining enough insight that I can lead them through an intuitive understanding. Others get the concept first and can intuit a way to the answer in some messy way that I only later clean up with some standard algorithms.
  13. Tomorrow I wear a headscarf.
  14. They are banning both the purchase and the ownership of these weapons, and all parts to make them, and all larger magazine/clips. There will be a buy back program. There will also be carefully regulated exemptions to the ban -- I saw the example of a farmer who needs to cull his herd. We have a bunch of 20-something workmen (builders, joiners, electricians, etc) at my apartment complex this month, and they all support it.
  15. I think, however, that the essays are critically important. DS did the common app essay first, but then learned so much about himself after writing all the other essays over the period of 5 months. His story came out over time through self reflection. He and I knew that he needed to rewrite the common app essay, but he simply ran out of time. He did not get into the elite schools that read his common app essay. MIT doesn't use it, and their essay prompts really helped him share his story.
  16. Yes and Yes. The NZ national exams, in contrast to APs, are a series of 4-5 essay-style exams for each subject during the year. He earned 'excellences' on each one. Excellence = top 10%. The exams he took are for a 2nd year of the course, so like AP in that regard.
  17. My approach was to do just enough external validation to prove the rigor of the complete transcript, and then to focus on creating a unique program of study for my ds. His external validation: top marks in SAT, SAT2s, four AP equivalents; two DE courses (plus Music performance diploma and math competitions) Basically, he topped the test for every single external thing he did. This demonstrated that he was a top student. But notice that only a small subset of his high school courses were validated by external grades. The rest we did on our own and were very creative. Why take Economics with a textbook, when you can read Picketty's Capital? That course was called "the economics of inequality." Why memorize a textbook for AP Biology, when you can focus on statistical analysis of ecology data, study genetic engineering, and read Scientific American on a breadth of current topics? We were creative, and I made sure they knew that through my counselor's letter and course descriptions. You simply cannot stand out with a ton of APs. Plus APs are so rote. He wanted more from his education. He wanted to dig deep into things he was passionate about. He did a 50 hour research paper on the chemistry of Fracking. It was amazing. There was no way he could have had time to do it if he was working his way through the set curriculum of the AP world. And we also found that attending the local university was so time consuming - going up there 3 times a week - it just dug into his study time. Plus, he could do higher level work on his own. I discussed this in my counselor letter. Homeschooling allows you to stand out. Use it to your advantage. Ruth in NZ
  18. My son takes private music lessons from MIT and he is not a music major.
  19. My son wears Keen sandals down to -10 degrees in Boston. 🙂
  20. Most of the students I have taught fall into one of two camps: 1) Those that naturally have insight, and then need to be taught the algorithm to clarify their thinking. 2) Those that use the repetition of an algorithm to develop mathematical insight. My younger son is the first and I am the second, so I know this first hand (plus having taught 30+ students one-on-one for 1-3 years each). The problem with teaching in a classroom is that you have both types of students and the approach I use for each type is completely different. For the insight kids, they need to take their web-like thinking and try to understand how to overlay linear, logical mathematical statements. This is often *very* hard to do, especially because they already have the answer and have no interest in backing up to figure out how to write it up properly. The problem is that at some point their insights will not move them forward, and they have no systematic mathematical knowledge to fall back on. At that point they drop out of math, because they don't like feeling stupid. For kids like this, I have to *train* their brain, and it is a slow process that requires a ton of encouragement. Insight it fun and fast, and linear, logical, mathematical thinking takes work and time. For these kids, I let them jump a ton of steps as long as they get the first one right - writing the problem into mathematical language. I don't care if they do computation in their head, with an algorithm, or with a calculator, it is the official steps that I want to see. I don't want to see the unorganized crap out of their head, and I tell them so. I have had very great success with these students and it takes them only a few months to appreciate what I have taught them. For the kids who gain insight through repetition (like me), they just need a ton of practice. The main problem these students have is that they must be redirected to the *goal* of the problem to make sure that they do gain insight, and don't just memorize their way through math. Don't try to teach me concepts without algorithms. This will never never work. It is too esoteric. Give me 10 problems to do with an algorithm, and once I have it like the back of my hand, I can then assign theory to my practice. Students like me must have the practice, but then also be encouraged/forced to apply it in a conceptual or theoretical way. Many traditional math programs are designed to work for students like these, but they spend way too much time on the drill, and not nearly enough time on the application to conceptual/theoretical practice. This must be designed into the program, and carefully managed or it will not happen. The longer I work with students, the more effective I become. This is because each student has a natural approach, and then I train them up on the other way. So by the time they hit pre-calc, we can use both approaches. Sometimes we drill to gain insight, sometimes we do concepts first and only then learn algorithms. But by having both tools in the tool chest, we can use what is most appropriate given the problem at hand. Ruth in NZ
  21. There is a consultancy group in NZ that is incredibly successful getting kids in. They start early in high school and mold them into what Ivy's want. We know one kid who has amazing stats, but the consultants also got him organized to create his own company to do non-profit work with under-priviledged kids. Would he have gotten into Harvard without this extra, who is to say, but clearly they know what they are about. I'm sure that the guarantee includes money back, so if they get 80% in, and give back 20% of the money, they could still be profitable. I did notice that they don't have any success with MIT, so that made me feel better about the school my ds attends.
  22. I wrote about our grading approach a couple of days ago and it seemed like an interesting addition to this thread: I never gave grades and never considered grading anything, ever. There were three reasons for this. 1) I just wanted to teach the love of learning and was very unstructured in my approach to the point of no clear cut courses even in high school, 2) NZ is an exam based entrance university system, so homeschool courses would NOT count for anything so there was no reason to give grades, and 3) he did not decide to apply to American universities until April of his Junior year. So as I went into making an American style transcript of our homeschool journey, I had to both create courses from what he read and wrote about, and I had to create grades out of thin air for courses that were years in the past. I will always remember the generosity of some members on this board for taking the time to sort through my often belabored descriptions of what we had studied over the prior 3 years, make sense of it, and recommend how to organize it into courses that admissions folk would understand. Basically for grades, I retro-fitted what seemed appropriate given all his standardized testing. 1) Excellences in NZ national math exams, NZ IMO math team for 3 years = all prior math courses earned As 2) Excellences in NZ national writing exams, 780 in SAT verbal, 20 on SAT essay = all prior English and humanities style writing courses earned As 3) Excellences in NZ national physics and chemistry exams = all prior Science courses (including Bio) earned As 4) ABRSM distinctions on music exams - all prior music courses earned As 5) Courses created from his 3000+ hours of reading (Contemporary World Problems, Philosophy, Comparative Government, World History, and Economics) -- All As because he put in way more hours than required for a standard Carnegie credit and read content above high school level (War and Peace, Capital, Godel Esher and Bach, etc). I made it very clear in my course descriptions that grades were based on readings and discussion (no output whatsoever for 2 of them (stated clearly on course descriptions), and for 3 of them that had a small amount of writing, grades were also based off the of the composition exams he took #2 above). Basically, they required grades, and I had none. I did what I could to make clear the effort put forth and the knowledge and skills learned, and I made this clear in the only way they could understand which was grades. No school questioned the grades I gave. My counselor's letter discussed how these unstructured courses emerged over time through his own interests. Hope this helps, Ruth in NZ
  23. I had a friend who told me that all she feels is guilt. Not anger, not horror, just guilt that we let this happen to our migrant and refugee community.
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