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lewelma

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Everything posted by lewelma

  1. Oh, absolutely. Her mom was *clueless* in math. As in would mark all her dd's work, and if dd wrote 0.1 and the answer was .1, the mom would mark it wrong. Absolutely NO math skills what so ever. She has absolutely NO idea the success I have pulled off. None.
  2. She was and is fascinating. My approach with her in our first year together (age 14) was to get her into the kitchen. I gave her a list of things she needed to practice every week in the kitchen to build up conceptual understanding. Then for all fractional work, I had to draw little measuring cups in her notebook because she couldn't understand pies, like couldn't understand the idea of splitting a circle in half. It was crazy. I also had her mom rewrite Every. Single. Word. Problem in MUS books 1-4, and had her do them all again (she had done them all before while getting through the books the first time). And after she did them this second time, I realized she was still struggling. So I had her mom rewrite Every. Single. Word. Problem in books 1-4 all mixed up, and she did them again! At that point, she had a vague notion which I considered great progress! At this point she was 15. She is now 18 and doing calculus. The development of her mathematical maturity has been truly (as in literally) exponential. Very eye opening for me.
  3. I think you are right. I'm very good at what I do, but if you give me a younger student or an older student who is still doing primary school math, I'm often at a loss for how to help. I am practical, not theoretical; but the theory could help me solve some of these problems.
  4. Ugh is right. Then today he tells me he is thinking about dropping out of school because he has a job repairing cars, and LOVES it. And his boss doesn't have 10th grade math qualifications, so why should he? Well, maybe he doesn't need them. Not here. Don't know. sigh.
  5. I think this would work if we allowed for more differentiated math learning, but a lot of my students developed mathematical maturity very late. I have one student who at the age of 14, when I asked her "if you have 6 apples and I give you 3 more, how many apples would you have?" - she had NO idea. She had used MUS and had gotten through the first 4 books (up to 4th grade) and then just stalled. She could compute, but had NO idea what adding even was - at 14. 4 years later she is taking calculus. yes, believe it or not, I have turned this around. But I have come to believe that part of her brain just didn't mature until she was 16. She is *definitely* not my only student like this, just the most extreme case. I have come to believe that there is no way that you can guess at where a kid will top out, nor how long it will take for them to master content. For this girl, after working with her for a year, I planned to get her into some sort of consumer math, just basic life skills, because she just could not grasp anything I taught her. Wow, what a change!
  6. Aw, I didn't name this thread, "Mathematician and engineer opinions on math education." haha. Please feel free to post. If it helps at all, I'm a biologist and science teacher by training. 🙂
  7. So disaster with my tutor kid just now. He is taking an algebra exam on Monday and I didn't see him last week because of our working bee. So he comes today with the practice tests on quadratics and factoring/expanding. He can factor/expand, but only if it is exactly one kind (x-3)(x-2) variety. Any deviation from this format led to confusion. Then quadratic equations - oh my word, he couldn't understand them. I tried explaining them conceptually with these are 2 things multiplied together, if one is zero, they equal zero. I tried substituting the answers back in to prove to him what works and what doesn't. I tried comparing it to linear equations. I tried just procedural drill. Nothing. So I said, 'well, we can do this next year. Let's focus on your strengths.' This kid is NOT ready for this work, but on the school marches. I am SO glad that I had the privilege to homeschool my kids. This kind of thing just makes me sick. I rescued him from the 'cabbage' class this year, and now I am regretting my choice. sigh.
  8. This exam is for all 10th graders, and I would say about 80% take it. It is curved to create about 15% As, 25% Bs, 40% Cs, and 20% Fs. To earn a C, you need to get about 30% correct (you will notice the first 2 pieces of each question are very easy). In NZ, all students take the same exam whether regular, honors, or gifted courses, which means all students can be compared to the same benchmark. In NZ the exams are not based on percent correct, rather are marked based on the level of thinking demonstrated. So the test is carefully designed to have a balance of questions - regurgitation earns you a C, relational thinking earns you a B, generalization/abstraction/insight earns you an A. It is also a forgiving test, so that you only have to show generalizations/abstraction/insight on 30% of those advanced pieces to earn an A. This means you have more than one opportunity to show your level of thinking. So on this test you could have gotten an A if you only answered one question completely, and got only the moderately-difficult pieces of the other 2 questions. So you can shine in your strength. To get into university here you must pass 10th grade math, but only 3 of the 5 units you take during the year. And those 3 could be numeracy, measurement, and statistics if your school designs a program as such (there are 12 units available a la carte style for teachers to choose). But I will say that those exams are only slightly less hard. But a key difference to America -- you do not need to pass an algebra exam to get into university here. To move up to 11th grade, you typically need to pass 4 out of the 5 exams your school offers (although I have seen exceptions). So basically, they drop your lowest grade. In addition, NZ has a very high end qualitative statistics course for 11th and 12th graders who want to continue in mathematical thinking but have no interest in the algebra/calculus route (or who have failed the algebra unit). About as clear as mud. But I find it a very good system to get kids working on the higher level content. What you test is what kids try to learn.
  9. We basically did it on SAT scores. If ds blew those numbers away, we figured his transcript, ECs, and originality could hold its own. However, we may have been arrogant as we did consider CM and U-M to be safeties.
  10. I think the ideal would be to cover less content so that you could do a proper job with 1) building up the conceptual knowledge of pure math first like Square25 does, 2) then drill in the procedures, 3) then use the math to model complex problems. Personally, I would completely abandon geometry. I would also abandon the more esoteric topics of Algebra 2 and PreCalc. Calculus is wonderful for modelling complex problems so I would keep that. But if I also want to pull in Statistics, then I am sunk. Too much content to do it well and have kids actually understand the content well enough to use it. Most kids do math because they are told they have to, but they have no expectations they will actually use it. sigh Have to run and tutor some kids in MATH! Be back in a few hours.
  11. I agree with this too. For my best students, I ram them through the basic knowledge as fast as possible, and then give them separate more advanced workbooks to do IN CLASS while the teacher is still working on the basics. Insight, abstraction, and generalizations take a lot of time to master, and here in NZ they never ever leave enough time. We have a teacher shortage so currently have a number of foreign trained teachers from America, UK, and even Russia. Most of the programs there are much more driven by a pure-math approach rather than what NZ does which is much more big picture, modelling, use-in-life approach. What this has meant for my students with these foreign trained teachers is that too much of their class time is focuses on pure math that is of course important but not any more important that using math for complex modelling. The fact that these kids' class programs are misaligned with the NZ exams, is just one more thing. So I drill in the procedures fast and furious and then work for remaining 6 weeks to use them in a deep and meaningful way. In a holistic way.
  12. I completely agree. For students here who want to work at an excellence level, my best option is to get them a year early (9th grade) and start using their more basic graphing knowledge (lines only) to model complex problems. Basically, questions like the first one on the test, which only required line knowledge which is 9th grade knowledge. But when I have less time, my focus is on the modelling of the word problems rather than a deep conceptual understanding of linking the algebra to the graph. For most of my students, using math to model real life problems is more important than procedural or conceptual understanding of algebra on up. Most people will not use advanced algebra, but they would be well served by the ability to convert real life problems into mathematical language. And I am not talking about word problems. Those are pretty useless. NZ is now focusing on investigations, where often there is not ONE answer. Notice the name of the exam -- investigating tables equations and graphs. It is not called a mathematical understanding of algebraic graphing. This different focus requires different priorities from me.
  13. Edited to add: New topics to munch on: Page 2, post 12: What are the ramifications of the move to online math programs as a way of individualizing pacing? ++++++ Square25 and I completely overran a LC thread on math this morning, then we started talking by PM. I think that some people might be interested in our discussion as we both have opinions on math education. Basically we are discussing the ramifications of teaching procedural knowledge vs conceptual knowledge first. Both of us are high end educators with experience with a range of students, and we have each seen a lot and tried to learn from our successes and failures. I'm going to just dump you guys in the middle of the discussion with my last post to Square25 by PM. I am making the case for having no choice but to drill in procedures fast with little conceptual understanding to make time for more complex problems. ------ You would find the NZ exams very interesting because the test for insight, generalizations, abstractions etc. If you can only do what you have been shown, even if you get 100% correct, you will only get a C. So for algebraic graphing, I have to do basic drill on lines, parabolas, exponentials as fast as I can so that we can move onto *using* the math to actually model complex problems as this takes a LOT of time to build up skill in. Check out this exam. 1.5 hours for 10th graders. It requires very high level understanding completed in a very short period of time. This exam is typically given back to back with a 1.5 hour geometry exam so there is an endurance component too. https://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2018/91028-exm-2018.pdf If you sit down and actually try to do the whole exam in 1.5 hours including graphing (because to get full marks you need both a graph and an equation) and using a PEN (so no erasing) you will realize just how hard it is. Conceptual understanding for this exam is way way more than just *why* does this equation create this type of graph. So I have to budget my time. What is more important? Algebraic conceptual understanding or insight into modelling complex problems? This unit represents 20% of a 10th grade integrated math program of algebra, geometry, and statistics, so only about 7 weeks of time to build up this kind of skill.
  14. Have you seen my science fair threads? We did large scale investigations every year for 8 years.
  15. But come on, with a name like the "upside-down picnic table method" who could possibly forget it!?!?!?!?! Why bother to understand when it is so much easier to just drag out the picnic table upside-down.
  16. Isn't it great to be able to teach your dd how you wish all kids could be taught! What a great gift! My older boy has always said that math is his most creative subject. I think that 99.99% of the population would have absolutely NO idea how this could be the case. Mathematics instruction is awful in both the USA and NZ. I was a product of the American procedural math focus, and when I got to university and was asked to think conceptually, I failed. It has been a slow uphill battle for me to work conceptually because this is not how I was trained for a decade. I am working with a kid who I could barely squeeze in so the mom got a second math tutor for him. I just about had a heart attack when the kid showed me the 'upside-down picnic table method' for adding fractions with different denominators. Multiply this horizontally, cross multiply that, add it up. OMG! Horrible horrible horrible. This kid quite happily drew the picnic table crosses on the two fractions and added them up with no idea why it worked or, like you said, no idea that there was even a why to understand.
  17. Oh my, I have totally munted the quoting feature! yikes. Off to make my response in blue as I can't seem to fix it!
  18. Ah, yes, we had a great conversation about that one. 🙂 Sometimes I just find kids are so caught up in learning the procedural skill that they won't actually put brain power to the conceptual understanding until they master the procedural skill. So I drill in the skill without understanding, then back up when they are confident and work on the conceptual understanding. Not all kids, but there are definitely some. Partly, I think it is because they have been taught for so many years is that math is only procedural understanding. So they focus on that and simply can't focus on anything else until they can manipulate as required. Sad but true.
  19. yes, she is an unusual case. But what helped me more than anything to better understand the profound lack of understanding was that she considered 12-10=2 to be sleight of hand. It really felt like magic to her, a trick that I could implement to amaze an audience. Here is a kid who can read and interpret complex probability problems and code them correctly into a calculator, but then thinks subtracting 2 is an advanced skill. Oh to understand the human brain! But this situation taught me a few things. 1) there comes a point in time where you should abandon basic numeracy and switch to a calculator and more advanced work. It is hard to know exactly where that point is, but the decision must be made. 2) Many kids who are not as articulate as this girl, may have deep misunderstandings and simulate numeracy through memorizing tricks. I must always be vigilant to root out these types of computational-skills-without-understanding to insure kids can move forward. I have learned to ask very basic questions about how a student does something that they seem very confident and capable of doing, and at least 50% of the time, they have no idea. And I am talking very basic stuff for 16/17 year olds taking pre-calculus. Stuff like why is 0.1+0.01=0.11. or why is 0.95 = 95%. Shocking how many kids are using sleight of hand and don't recognized it as such like this girl did. Her metacognition was excellent.
  20. Over the years I have tutored, I have been absolutely fascinated to see how kids think about things. I am so very appreciative when a kid asks me a 'stupid' question because it clarifies deep misunderstandings, and can be so helpful in teaching them. I also taught Chemistry to the girl I described above, and she could not tell me the number of electrons in the outer-most shell because she could not count up in the way you are talking about. For example, you have 12 electrons to put in shells: 2 for the first shell, 8 for the second shell, and 2 for the final shell. She would have to draw all three shells around the nucleus and added in the electrons one by one counting up to finally tell me that there were 2 electrons in the outer shell. I tried many times to convince her that you could add 2+8 to get 10 and then subtract 12-10. That if the number of total electrons was more than 10, that all you had to do was subtract off 10 to get the electrons in the outer shell. She could not understand this, and felt *very* strongly that it was almost sleight-of-hand and a trick. I tried place value, I tried blocks, I tried drill, it was just not to be. She was 17. And yes, I did get her to pass 11th and 12th grade chemistry. haha. I taught her to love her calculator. 🙂
  21. OP, Just doing some more thinking, does your dh and ds like movies? My younger boy and I have been doing an analysis of how comedies are created. He loves loves loves humor. So we watch TV shows like Gilligan's Island and movies like Legally Blond, and discuss what makes them funny and how this was done with the cinematography, costumes, script, etc. We look stuff up about humor, and then try to find the elements in different shows. It has been great fun and has led to lots of questions about the cultural nature of humor and how it changes over time. Fascinating stuff.
  22. This I think is key. I believe that gifted kids crave discussion and interaction. I have worked hard to come to a level where I can offer this kind of support in most subjects. But in ones where I could not, we found creative solutions. For my older ds's math, once I did my last big co-learning push when he was 12 that I described above, we put him into the AoPS classes not because he needed to be taught, but because he wanted the discussion. But by 15 he was out of classes to take there. We tried having him take classes at the university but that was a huge bust because even at age 15 in 2nd year classes he topped the class by a 40 point margin. There was no discussion happening there. So I researched and bought him a pile of math books that he could work through on his own, and I listened to him talk and talk and talk about math (although I never knew what he was saying). We never could find a mentor, although we had a couple of wonderful computer science people who would talk to him about discrete math occasionally, but I am talking like 3 times a year. And he had his math camp 1 time per year. But in the end, he made his own reality like 8filltheheart's dd did with French. He organized fortnightly study sessions with the other 2 IMO kids in town to work through those tough problems he loves. Then he organized and taught his own competition-math after-school class for gifted kids. He convinced one of the other IMO kids to help him run the class, so they worked together to plan a year-long program, create lecture notes and worksheets, and run the class in a collaborative manner. This course gave him the interaction he craved by being a mentor rather than having a mentor. And in the end he walked into MIT and took a grad level math class as a freshman. This outcome did not happen by chance. I *managed* his high school math program, as varied and non-schoolish as it was. The solution for him was not to be taught by a person or by an online class, the solution was for him to battle through the material on his own with a ton of support from us and collaboration with his peers. And then to share his passion and knowledge by teaching others.
  23. Yes, she did. When I first got her, she told me that using a calculator made her feel slow and stupid. And it took me many many months to convince her that the goal was to get the question right in a reasonable amount of time, and that the calculator was a tool she could use to accomplish this. We focused on interpreting statistics using a qualitative program (although she had to do some mathematical statistics as these was the NZ national exams she was doing). I kept talking to her over and over about how: yes, there are people who *do* the math, and then there are people who *use* the math. She could be in the latter group and math could still be her forte. Understanding errors and assumptions in surveys, experiments, observations, inferences, etc, this would serve her well in political science (the field she was interested in). She did not have to DO the math for it to still be a useful skill. I convinced her that she wanted to do what computers couldn't - design and interpret statistics; and that she shouldn't try to be a human calculator because computation is a low level skill. Statistics was never easy for her, but she pushed through because I convinced her that 1) she could do it as long as she had her calculator and 2) she was learning valuable skills that she would actually use in the future.
  24. I'm a tutor, and I just wanted to let you know that I have taught a kid who at the age of 17 could not subtract 9-6. When I asked her to do it, it took her 2 full minutes with a tally chart to get 4. She had been born at 24 weeks and had obviously been diagnosed with dyscalculia. I got her through 11th and 12th grade statistics with the use of a calculator.
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