Comments and answers for "Week 1 Task 5: Citizen science project - signs of dangers"
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The latest comments and answers for the question "Week 1 Task 5: Citizen science project - signs of dangers"Answer by Jackroyd
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<p>Two things particularly stood out to me in this list:</p><p>1. Ability attributes - for high school I attended a grammar school in the UK, which you needed to pass an examination to go to. I was the only one from my primary school to go there, everyone else went to one of the two local comprehensive mixed ability secondary (high) schools. At primary school I excelled at Maths. I raced my friends through the series of Maths cards, which we used rather than books, and was one of the few who completed the whole series before leaving the school. I asked for homework in Maths regularly and never struggled much with it from what I remember. I found it fun and exciting.</p><p>At secondary school we were put into ability sets after the first year, and I was put in one of the two equal status "bottom" sets for this subject. It was a small school, with only 4 ability sets in our year group, and in reality if I had gone to a different school I would have probably been placed in one of the top ability classes, however, as an adolescent this isn't something that occurs to you. I went through the whole of high school believing I was bad at Maths, got very low marks, and didn't enjoy it, which looking back seems really odd considering how much I loved it in my earlier childhood. At the beginning of my last year at the school, the year we took our GCSEs (leaving certificates) I got over 90% in a test, rediscovered my love of mathematics, and ended up getting an A grade for the GCSEs. I guess this is a long winded way of saying that being placed in group perceived as a "low ability" class, my confidence was destroyed and I believed my ability was gone and I assume that I must have stopped putting in the effort to learn that I had done previously until that confidence returned.</p><p>2. Can't see connections between math and other endeavours - after regaining my confidence in Maths and rediscovering my love for it, I studied it at a higher level for a while at college and then as a subsidiary at university, but was never shown any practical connection to a future career (other than Maths teacher or accountant!) or application of the subject, so I saw little point in continuing to study it. Later in life I worked in a job where I organized events for high school students to show them future university options, and what they could do with the different subjects that they were studying. I observed the speakers at the Maths day with fascination, suddenly seeing all the opportunities that studying Maths could open up, and how many applications it has, and wished so much that I had had such career / informative advice when I was at that age.</p><p>In my opinion schools spend far too much time getting people to learn the abstract concepts (times tables for example - an example of your "mnemonic complexes set in stone" point) and they neglect to inject any life or reality into the subject.</p>Sat, 26 Apr 2014 02:23:55 GMTJackroydComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/3292/view.html
<p><a rel="user" href="/users/2156/christym.html" nodeid="2156">@ChristyM</a>, your kid seems correct in the essence of the system, but uses an original way to symbolize. Many kids make up pet names for ideas that are new to them. So do all mathematicians, scientists, engineers, artists... </p><p>You COULD call the ten that follows twenty-somethings "twenty". You just need to make the rest of your naming system consistent. I am very curious about your kids' system - he's probably onto something! Let's make a little research.</p><p>Print out some visual hundred charts that do not have labels. Here is the elegant version our <a rel="user" href="/users/42/yelenam.html" nodeid="42">@yelenam</a> made, color-coded for primes:</p><p><img src="/storage/temp/633-100chartprintablevisual.png"></p><p>You can find many more 100-charts in the Week 3 activity called <a href="http://ask.moebiusnoodles.com/questions/3220/week-3-task-3-factorization-diagrams.html">Factorization Diagrams</a>. Say that you and Maria D. are very interested in your child's naming system, and ask to name numbers on the chart. Maybe not all, but enough to see how the system works. Focus on tens. Ask what comes after the last number in this chart (in our terms, ninety-nine). In other words, investigate the system your kid made up!</p>Wed, 23 Apr 2014 12:10:19 GMTMaria DroujkovaAnswer by ChristyM
https://naturalmath.com/community/answers/3282/view.html
<p>An issue I need to address with my child right now is an inability to understand 0 properly. I teach my children to count objects, not to count by rote. My child now believes that 20 is "10" that follows up the twenty-something's, 30 follows the thirty-something's etc. I would be interested in any other thoughts you have about exactly what kind of misunderstanding this is and how you would work against it.</p>Wed, 23 Apr 2014 02:48:32 GMTChristyMAnswer by Meera
https://naturalmath.com/community/answers/3274/view.html
<p>I took my time to read through all the activities and this one really struck a chord. </p><p>Personally, inspite of scoring well in all my classes, I still feel mathematically challenged as I think it was my memory that helped me through. Now, well into professional life, I dont "remember" any of the theorems or proofs. Basically, I dont think I ever appreciated or assimilated the concepts that I learnt. So again, during personal finance planning, some concepts, however simple seem complex :(</p><p>For my children, I exactly want the opposite. I want them to be able to appreciate the concept and internalize the beauty of playing with numbers. </p>Tue, 22 Apr 2014 22:08:42 GMTMeeraAnswer by lisa.koops
https://naturalmath.com/community/answers/3241/view.html
<p>Thanks for this list - separating it out makes it easier to think through! My 7-y-o daughter said last week she was tired of getting 100% on her math tests because the other kids made (unkind) comments about it. I wanted to shout "NO! Shine on!" or something because this was such a familiar story to my middle school and early high school years (it's not cool to be good at math). amightygirl.com has some great resources on this as well.</p>Mon, 21 Apr 2014 17:03:29 GMTlisa.koopsAnswer by alex73
https://naturalmath.com/community/answers/3187/view.html
<p>Try to think of an example that captures the key issues but has no 'fluff' that obscures them. Generalize progressively</p>Sat, 19 Apr 2014 13:24:42 GMTalex73Answer by classicalmama
https://naturalmath.com/community/answers/3146/view.html
<p>I have say homeschooling has opened my eyes to education.(even as a preschool teacher I was blind to things) I could pass a math class and I had math through 12th grade pre or calculus. I wanted my kids to really understand math. I still struggle with the multiplication tables.</p>Thu, 17 Apr 2014 23:11:05 GMTclassicalmamaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/3137/view.html
<p>Thank you for sharing your thoughts, which is sometimes harder if you mull over longer... I think we need to add a caveat about deep perfectionism to the essay about celebrating mistakes. When someone has a serious and systemic issue, as you describe, they need more serious and systemic help than a slogan a-la "Just do it!"</p><p>There is an exercise where the purpose is to make as many mistakes as possible in one math problem. With preparation and discussion, maybe you can do something like that? It probably won't be easy, but if you know a task is emotionally charged, you can prepare ahead with extra hugs, kittens in your lap, or whatever else makes you feel better...</p>Thu, 17 Apr 2014 17:17:38 GMTMaria DroujkovaAnswer by cleabz
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<p>I've taken a couple of days to mull this over. My daughter, who is homeschooled by two people who enjoy and are "good at" math, struggles with math anxiety (eyes glaze over, shuts down around math), and I've been thinking all these thoughts through before responding. Part of her struggle would in school be called a learning disability. She's very kinesthetic, and also visual-spacial, so doing calculations on a black and white page get all jumbled up for her, and then this crashes into her perfectionism, and we end up with a dislike of math. We've done mnemonics, visualizing, manipulatives, movies (she used to like BrainPOPJr and we've used Math-U-See in the past, too), games, and plain old worksheets, in color and black and white plus one-on-one guidance. But when I say "Math" she says, "NOOOOOOO!" and gets worried. This shuts down any receptivity to exploration, so we start out on the wrong foot. As I type I realize I should just call it something else, like games or visual puzzles or something. I would love to find more visual (this class has been great) and kinesthetic ways not only to teach math concepts and skills, but also to bridge to paper computations. Celebrating mistakes is also part of what we do, but she just doesn't buy it. Like the other day we were making gluten free donuts in our new donut pan, and both batches came out tasty but not donut shaped. She cried! I was like - no, this is fine, it's our first attempt, we learned this recipe doesn't work, we learned this other recipe needs tweaking but we're closer, etc - and it just didn't make a difference. She is phobic of making mistakes (despite how she has been taught, I think it's hard wired, and hubby has the same fear, which is both transmitted to daughter - but oddly enough not son as much - and I think is partly just genetic hardwiring) in any feild, and since there is a "right" and a "wrong" in math, she has a lot of fear here.</p>Thu, 17 Apr 2014 15:53:43 GMTcleabzAnswer by shaunteaches
https://naturalmath.com/community/answers/3124/view.html
<p>The memorization and Mnemonic examples resonate with me. I think that memorization and Mnemonics can really impede number sense, even when they come easily to a child. The memorization and Mnemonics can work so well that you never stop to think about "why" certain things are happening. </p><p>I remember going over my multiplication facts and memorizing them (which certainly helped), but I never stepped back to think about the quantities I was dealing with. This never seemed to be a problem in school, but only now do I realize how much I was missing. </p>Thu, 17 Apr 2014 13:03:35 GMTshaunteachesAnswer by Shannon
https://naturalmath.com/community/answers/3117/view.html
<p>"Believes that success is largely due to abilities (rather than effort)" This can be troublesome for both those who believe they have innate ability and those who believe they lack it. I solidly believed this idea for most of my schooling years. I did see some who were capable of doing well because of lots of hard work, but because many things came easily to me I thought that was being successful. It wasn't until I was 16 or 17 and knee deep in AP/honors classes, varsity sports, Jr. engineering team, and all the rest that I realized I had to really apply myself. I was behind my peers in study skills, and it took me through my sophomore year of college to really get caught up in this area. But I failed to take advanced math (beyond calculus or statistics) because I felt overwhelmed, I didn't have the innate ability nor the study skills. My daughter was at the opposite end. Every day in 2nd and 3rd grade she was given timed drills of math facts. Every day she failed to complete any of those sheets. She disliked math, but worse she believed she was not good at it. After a year of homeschooling she had come 180 degrees about math. She loves math! She recognized, in the absence of pressure to remember facts or do dry calculations without context, that she had some ability, but also that she could learn all sorts of math by healthy studying. I am glad that she learned this lesson years before I did.</p>Thu, 17 Apr 2014 06:40:29 GMTShannonComment by sherylmorris on sherylmorris's answer
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<p>What does celebrating mistakes look like? I ask myself. OK, #1--"observe, mark, remember"</p><p>"celebrate verb 1) they were celebrating their wedding anniversary: commemorate, observe, mark, keep, honor, remember, memorialize. 2) let's all celebrate! enjoy oneself, have fun, have a good time, have a party, revel, roister, carouse, make merry; informal party, go out on the town, paint the town red, whoop it up, make whoopee, live it up, have a ball. 3) he was celebrated for his achievements: praise, extol, glorify, eulogize, reverence, honor, pay tribute to; formal laud."</p>Tue, 15 Apr 2014 14:27:45 GMTsherylmorrisComment by Maria Droujkova on Maria Droujkova's answer
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<p>9*6 is actually 6-1=5 and 9-5=4 so 54 - do you understand WHY this works? If you do, or if you figure it out, then it will turn from a mnemonic into a pattern!</p><p>This discussion about mistakes and measuring the world is very deep. Of course you would not celebrate a mistake in building a bridge, or getting the distance to Mars wrong and missing your landing window! There is another, more whimsical and free side to math. Mathematics can be not only about measuring this world where we live, but about making up other, fantastic worlds. In that, mathematics is closer to fairy tales, or science fiction, or fine arts where you can create rather freely. So when I talk about celebrating mistakes, I mean taking them from our world into one of those mathematical Wonderlands. </p><p>For example, my math circle kids kept confusing squares and cubes, or circles and spheres. But we played with it, and by now, they talk about three-dimensional squares - or even four-dimensional! From mistakes to abstractions.</p>Tue, 15 Apr 2014 00:56:48 GMTMaria DroujkovaAnswer by katying
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<p>This all makes a lot of sense to me. I definitely got hooked on mnemonics like the 9 times time table, i would and still subtract one and add the remainder so 9*6 is actually 6-1=5 and 9-5=4 so 54… so bad. </p><p>I am confused though, about 'celebrating mistakes' What do you mean by this? </p><p>I homeschool my son. He is only 5 right now. But one reason I like it already is that as his mother I know him so well, I know when he is tired, stressed, over worked, bored and silly and so I know when to push a bit further and when to pull back. We have fun and are silly at times, and he likes to make erroneous math sentences and have me correct them, or he likes when I make the mistake and he corrects them. This is fun because I know in a round about way, it confirms for me what he knows and doesn't know, which I am always keeping an unwritten log of. But I also think the goal of mathematics is to measure the world. In my homeschooling we have plenty of room for mistakes and practices, but I do intend for him to be a serious student of the world, and to have reverence and respect for teachers and adults and the thinkers who came before him… I guess I am not sure to what extent or in which context I agree with the idea of celebrating mistakes… Can you, any of you, please tell me more about this?</p>Tue, 15 Apr 2014 00:39:17 GMTkatyingAnswer by yileinei
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<p>I've always been good at maths but one of my biggest problems is related with the way I learnt the multiplication tables (I learnt them by heart). When I try to recall, for example, 8x7 I have to recall the full table to get there and sometimes I have to change the order of numbers, that is I can remember 6x8 but not 8x6 and it makes me feel kind of stupid! I'd love my daughters to learn this in any other way.</p>Mon, 14 Apr 2014 21:19:43 GMTyileineiAnswer by oxanavashina
https://naturalmath.com/community/answers/2993/view.html
<p>I didn't have problems with math at school or at the university, so when I saw the first 'glazed look' on my son's face, I was stunned. As you mentioned in one of your comments here, there is a danger of having it 'too easy', this was my case, so I have to kind of 'repeat my math' now to help my elder son with his math anxiety. </p>Mon, 14 Apr 2014 15:10:46 GMToxanavashinaAnswer by CHabq
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<p>"Panics about timed work"</p><p>Reading that brought back math anxiety I hadn't thought about in almost two decades. I always felt that I could do better at math if I had enough time to figure out a problem whether it was for a test or for homework. I could never understand why "good at math" was so tied to doing math fast.</p><p>I also feel a double stereotype threat about math in that I identify with two demographics that are perceived to be more successful in math--Asian American and scientist. So, not only am I expected to be good at math I am supposed to be incredibly good at math. Which I am not. I was taught math in traditional methods and it never made sense to me and it never came easily to me. When I hear or read, now, about other approaches to learning math I wish it was the way I had first learned it because I think I would have grasped mathematical concepts and ideas a lot better and quicker. I think that is why I am searching for multiple approaches to math to introduce to my child. So, that I can help him find the approach that makes the most sense to him.</p>Mon, 14 Apr 2014 05:20:05 GMTCHabqAnswer by mirandamiranda
https://naturalmath.com/community/answers/2982/view.html
<p>I can see several of the psychological maths dangers in my children. My oldest has a bit of a 'I hate maths' complex, stemming I think partly from forced timed tests in school (she is now homeschooled). She does really struggle with her 'math facts', she still needs to count on her fingers too (she is eight). But we are really working on her number sense and 'removing coercion' as suggested in the tips. Yesterday she was figuring out how much money she would need to save for seven of her friends to come to a party at $37 each (for her real need not as an exercise). It did not occur to her to use multiplication, she did repeated adding, but in a (to me) very convoluted way, breaking the numbers apart and adding bits here and bits there. She got to the answer in the end though!</p><p>I also give her a lot of input in what she does for maths, as well as providing lots of living maths books which she enjoys. When I see her struggling with something I try and change it up, approach a concept in a different way or take a break from that area.</p><p>My younger two (aged 6) are a bit different. One in particular is a real perfectionist and gets very anxious when she is confronted by something she does not recognise. She says she likes to do 'sums', worksheets, but if the problem is written differently she panics. She is also not very fond of open ended questions and explorations, she gets cross and frustrated almost immediately and refuses to participate. On the other hand when she is working on something she has decided she wants to do, she is insistent on getting to the end, even if it involves real struggle on her part. I would really like to involve her in more of the types of maths activities you discuss, but she does not respond well to adult-suggested activities on top of the other issues! So any tips would be most useful...</p><p>Her twin at least seems free of maths anxiety for now, and I think will really enjoy the activities.</p>Mon, 14 Apr 2014 04:35:17 GMTmirandamirandaComment by mirandamiranda on mirandamiranda's answer
https://naturalmath.com/community/comments/2981/view.html
<p>My mum used to tell me a story about a girl who named her doll 'fifty-six' to help her remember that difficult fact!</p>Mon, 14 Apr 2014 03:57:00 GMTmirandamirandaAnswer by annettehaddad
https://naturalmath.com/community/answers/2968/view.html
<p>My kids are young and don't have any of the math specific misconceptions yet, but I see a few of the math anxiety and stereotype threats. My son, age 7, is a perfectionist and gets very upset when he makes a mistake or thinks something is too hard. I see some of this coming up with my daughter, age 5, as well. We joined a math club but I'm not sure if this has been helpful or hurtful. They seem to like to see the friends but my daughter doesn't want to do the games in there since she sees she is behind the other girls (she is 1-2 yrs younger than most of the class). </p>Sun, 13 Apr 2014 21:11:13 GMTannettehaddadAnswer by CynthiaDadmun
https://naturalmath.com/community/answers/2967/view.html
<p>My kids are too young to be affected by many of the the higher level thinking problems listed. Instead, I worry that my 3yo daughter may come to feel that math isn't something "she does" because my 5yo son IS so engaged with it. She tries to differentiate herself from her older brother, and since he's very good with building (ie Legos) and math I worry that she will draw away from it. It takes the fun out of it for her when he barges in on an activity and is so much faster and better at it than she is. In fact I have decided to embrace the "girl Legos" because they at least make it quite clear that they are for HER too. She doesn't yet feel that "girls don't do math" -- instead it is more of a sibling rivalry and a difficulty that being the youngest, she is always "behind". I am trying to give her more one-on-one time away from Big Bro, and also to emphasize that these things are What Our Family Does, so they can be part of her identity without competing with her brother. But I would love to hear other advice on this!</p><p>Second issue -- my daughter's preschool over-emphasizes writing numerals and doesn't spend enough time with manipulatives. I see her resistance to (and difficulty with) writing numerals getting in the way of developing number sense. The school has many other redeeming qualities, but it is frustrating!</p>Sun, 13 Apr 2014 20:41:41 GMTCynthiaDadmunAnswer by corilewis
https://naturalmath.com/community/answers/2966/view.html
<p>Additive misconceptions: yup, my student will use addition instead of multiplication frequently.</p><p>Mmenomic complexes set in stone: yup, this includes myself and my student. I have to count on my fingers sometimes and I run through skip counting to get certain multiplication facts. I've told my daughter specifically the tricks I use to figure out stuff in my head. I realize, they are my tricks for my brain. And, by tricks, I kind of mean it. They take me a while and I feel ridiculous standing there thinking while someone else already got the answer. I work in STEM and I wear a calculator watch. I even feel embarassed writing it. I need to be correct, I'll check it a couple of times and I prefer my staff to double check me. I have said that I don't like to do math in public.</p><p>Collapse of scales: I thought this was a universal human trait! We don't see where we fit in time or space very well! :-)</p><p>Math anxiety: "I'd rather do anything than math" has been spoken. We definitely can't see the connection between math an other endeavors and can't model or draw concepts. At least, not if you are watching and have mentioned the word math in the past 30 minutes. My daughter will build something or stumble onto a shape playing on those peg boards with rubber bands but as soon as I mention something math-y about it, she doesn't want to hear it.</p><p>I guess the way we cope is by having humility. I very clearly remember that feeling of needing more time to get to where everyone else in the class seemed to be. Honestly, at some point my time was better spent doing something else. I struggle with the balance between work on your weaknesses vs. let you strengths shine. I feel like even by writing that I've hinted at that whole "some brains are better at math than others" taboo. If I'm being honest, some brains are better at math than others. We can all do it, we are born puzzle solvers, but why do some of us struggle more?</p>Sun, 13 Apr 2014 20:01:17 GMTcorilewisAnswer by pkouch
https://naturalmath.com/community/answers/2965/view.html
<p>This is a comprehensive list of problems and I have noticed quite
a few of them throughout years of working with students. However, I’d like to
focus on psychological and social dangers and among the items in that list, “thinks
that some people are born to be good at math and some can’t do it,” is my main
focus. I believe this is the most prohibitive factor for my students in
learning math concepts. In majority of cases the origin of this “I suck at math”
expression comes from home. It kills me when I hear a parent saying, “She/he is
like me; math wasn’t my strong subject either.” And I have heard that statement
too often! </p><p>Another danger, I believe, is the current memorization
phobia among both educators and parents. There is no doubt that children need
to explore and discover, try and make mistakes, find patterns and connections
on their own and in groups. However, I strongly believe that memorization has
its own merits and should not be condemned all together. What we should be careful with is
what to memorize and how to memorize. </p><p><strong>How I’d cope with the problem:</strong></p><p>I’d usually focus on confidence building more than on teaching math concepts for a long
period of time until I’d feel that the child has a decent perception of his/her
own ability. I’d break a concept into very smaller ones and walk my students through small steps with lots of encouragement in each step. We know that negative self-perceptions that come from home with frequent
confirmations in school environment grow deep roots in children and it is extremely
hard to change them. I just hang on to this hope that by focusing on giving
them a more positive perception of their abilities maybe some of them see some
light and save themselves.</p><p>For memorizing basic facts, I use games and games and games
with lots of encouragement.</p>Sun, 13 Apr 2014 19:43:59 GMTpkouchAnswer by sherylmorris
https://naturalmath.com/community/answers/2960/view.html
<p>Confessing here--I have always had math anxiety–understanding the problem, recognizing problems that I've seen before, and giving up. Had I been in a learning environment that helped me recognize the math in creative things we already where doing (like cutting snowflakes), celebrated mistakes, and allowed me to experience without coercion the math connections in a variety of lessons, e.g. life skills (sewing and cooking), language, science, art and music, and nature my whole-being would have felt more supported. I would have had a more comfortable relationship with math and learning overall! </p><p>I have craved more help with "visualizing" scaling, exponential systems. I would love it if a Montessori school with many classrooms (and therefore many sets of materials) would/could combine all their sets of gold bead materials to have a night or weekend when families could come and see firsthand what larger and larger numbers look like. There are logistical problems with this, of course; teachers are rightfully afraid of having their classroom materials get lost. </p><p>When I see the word "praise" I cringe because I know what the quick, take-away is for many people. You've explained it well here, however I'd suggest even more emphasis on the fact that the praise needs to be on the effort. I'm posting an article by Alfie Kohn that clarifies for me the differences between "praise" and "encouragement."</p><p><a rel="nofollow" href="http://www.alfiekohn.org/parenting/gj.htm">http://www.alfiekohn.org/parenting/gj.htm</a></p><p>Math anxiety: One of the features that had me quickly falling in love with a Montessori environment is that competition is really tamped down. Children may come into the classroom with a built-in competitive edge just by living in our culture, but the teachers or guides help them acclimate to an environment where they aren't to be compared, timed, and tested. Other points I want to share here is the fact that many Montessori "works" are self-correcting allowing the child to correct himself. When a Montessori guide does see a mistake she/he is trained not to swoop in and correct the child, rather she makes note of the error and knows that she needs to present the work again to the child(ren) the following day without talking about the mistakes she saw the day before. (I'm thinking that this is not exactly "celebrating" mistakes, however maybe preferable at the very youngest of ages.) </p><p>My own book is intended to help purposefully prepare, at home or classroom, an environment for the very youngest children that scaffolds or builds on what they already know, and generates curiosity and connections (mind hacking if you will), and help to eliminate math anxiety.</p><p>What does celebrating mistakes look like? I ask myself. OK, #1--"observe, mark, remember"</p><p>"celebrate verb 1) they were celebrating their wedding anniversary: commemorate, observe, mark, keep, honor, remember, memorialize. 2) let's all celebrate! enjoy oneself, have fun, have a good time, have a party, revel, roister, carouse, make merry; informal party, go out on the town, paint the town red, whoop it up, make whoopee, live it up, have a ball. 3) he was celebrated for his achievements: praise, extol, glorify, eulogize, reverence, honor, pay tribute to; formal laud."</p><p>I was introduced to modeling or drawing ideas, brainstorming, and lateral thinking as an adult in graphic design classes. Another problem-solving technique suggested was to take a break, sleep on it, let ideas "percolate"; don't be surprised if and when answers come to you when you least expect them (but, only when you've worked on them). Who would have guessed that these same techniques can be applied to mathematics?</p>Sun, 13 Apr 2014 17:02:04 GMTsherylmorrisAnswer by AGray
https://naturalmath.com/community/answers/2959/view.html
<p>This is a great list! I need to study it and the suggestions. The biggest obstacle I have noticed in my kids is math anxiety. So far I have handled it by sitting with them, working problems together, modeling making mistakes and trying different approaches. They seem to think they have to know the right answer right off the bat. I'm trying to get them to see it as a process .</p>Sun, 13 Apr 2014 16:18:44 GMTAGrayComment by Elizabeth02 on Elizabeth02's comment
https://naturalmath.com/community/comments/2953/view.html
<p>Absolutely! Everyone figures out things on their own and ends up memorizing certain facts that stand out to them for whatever reason. Why do we force everyone to do it the same way though, as if there is <em>one </em>right way? I really try to just teach the concepts, and practice. Everyone will develop their own personal connections to the material and thus learn it, hold on to it, and hopefully apply it their own way. </p>Sun, 13 Apr 2014 03:29:46 GMTElizabeth02Answer by Valerie
https://naturalmath.com/community/answers/2952/view.html
<p>My answer to this question really is the reason that I wanted to do this course. In primary school I loved playing with numbers and arithmetic. I had an enthusiasm for it, even up to 16 years old. Then, I lost my confidence in my ability in maths in the last two years of school when we had to tackle integration at school. My teacher would spend the lesson writing lines and lines of calculus on the blackboard and ask us to copy it down in our books without explanation. If we did ask a question, she would get very cross with us and we were too frightened to ask her questions. After that experience, I believed that I couldn’t learn maths. However, I needed maths for my biology degree and luckily my university lecturer was very different and made me start to believe that I could do it with effort and support. Now my job includes quite a lot of statistics and some maths and this wouldn’t have happened without the support of my uni lecturer.
Looking back, I think much of my school teaching was focused on rote-learning methods to derive the correct answer, rather than having an understanding of what that really means. So, for example, while I had no problem learning multiplication of integers, I remember getting muddled with multiplication involving numbers with decimal points, which wouldn’t have happened if I’d understood multiplication more fully. I also think that mathematical terms can strike fear into the hearts of the under-confident, particularly if you have had a bad experience in school, so I’d like for my daughter to learn and understand mathematical concepts first without the labels.
I would also like her to have a full understanding of the concepts, and not be taught primarily to learn a series of methods to get the right answer. I am grateful to you and this course for encouraging us all to find the love and enthusiasm of maths again - thank you.</p>Sun, 13 Apr 2014 01:54:52 GMTValerieComment by Maria Droujkova on Maria Droujkova's comment
https://naturalmath.com/community/comments/2950/view.html
<p>I consider all non-mathematical mnemonic devices dangerous, when it comes to patters, conjectures, results of computations - anything that has meaning to it. You can use a mnemonic to remember which one is sine and which one cosine, because these are rather random names. But 9*9=81 is not random. It is a part of several neat patterns, and it can be computed, so it's better for our minds to focus on these structures. To memorize times tables, if memory is desired, I would use a spaced repetition tool like Anki. You want your tool to serve you individual facts, straight up (no mnemonics), and only those you actually have trouble remembering. We'll work more on that stuff in Week 4.</p>Sat, 12 Apr 2014 23:47:46 GMTMaria DroujkovaComment by FoxFox on FoxFox's answer
https://naturalmath.com/community/comments/2948/view.html
<p>7*8 has always been one of the hardest for me to remember. I minored in math and still don't have my multiplication tables completely memorized. What are your thoughts on mnemonic devices like the Times Tales? I've always thought that mnemonic devices like the skip counting songs were good, but have also regarded ones like the 9*9=81 story with suspicion. Would you consider them to be helpful? Dangerous? Harmless?</p>Sat, 12 Apr 2014 20:29:31 GMTFoxFoxAnswer by zzzeee2000
https://naturalmath.com/community/answers/2938/view.html
<p>As a teenager I feel like math anxiety is something a lot of us teenagers face. Both in homeschooling and in public school.</p>Sat, 12 Apr 2014 19:03:15 GMTzzzeee2000Comment by sherylmorris on sherylmorris's comment
https://naturalmath.com/community/comments/2945/view.html
<p><a href="/storage/temp/465-m16-2.jpg">m16-2.jpg</a>Much like the Montessori, multiplication board.</p>Sat, 12 Apr 2014 17:46:37 GMTsherylmorrisComment by sherylmorris on sherylmorris's comment
https://naturalmath.com/community/comments/2940/view.html
<p>I would have thought that arithmetic is rather a part of mathematics. How do you, then, define arithmetic in relation to mathematics? (I do get and appreciate what you say, ". . . exclusive exposure to arithmetic only leads to conceptual shock later on!" TY!</p>Sat, 12 Apr 2014 15:41:18 GMTsherylmorrisComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2932/view.html
<p>How would you name the danger of what happened to you? People usurping and then abusing power over your learning?</p>Sat, 12 Apr 2014 11:35:48 GMTMaria DroujkovaAnswer by ramandada
https://naturalmath.com/community/answers/2918/view.html
<p>In 5th grade I was moved back up to the advanced math group because the middling group was too slow for me, according to the teachers. The first homework I was given in the advanced group were word problems where the teacher explicitly emphasized how we were to "write out" the problems completely. I spent the entire evening, weeping, late into the night, copying out longhand the entire question in my notebook before attempting to answer it. Math association with meaningless drudgery. Much later, in geometry, I invented new proofs for some problem that made sense given what we knew at the time. The teacher informed me that it would not work given what I did not yet know. Oh. The unknown will defeat you, there is no point trying. Unlike many subjects where I had an idea of the grade I would earn on an exam, I could never predict with any accuracy the grade I would get on a math exam, through college. It felt like the rules of math were held tight by some exclusive club of people who always knew the right answer. And there was *only* one answer. </p>Sat, 12 Apr 2014 11:33:10 GMTramandadaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2931/view.html
<p>Memorization can be meaningful, if you memorize something that is beautiful or useful TO YOU. I am afraid the danger you name is deep and insidious: it's not about the overall amount of memorization, but about forcing everyone to memorize the same content, whether they find it meaningful or not.</p>Sat, 12 Apr 2014 11:32:24 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2930/view.html
<p>I prefer to start "high concepts" early - even with toddlers. Arithmetic is not mathematics, and exclusive exposure to arithmetic only leads to conceptual shocks later on!</p>Sat, 12 Apr 2014 11:28:49 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2928/view.html
<p>This is a subtler danger for more successful kids: never running into math difficulties until something like calculus hits like a ton of bricks! I will add it to the list. Julia Brodsky of The Art of Inquiry talks about an algebra student she helped simply by explaining that some problems have more than one step and require taking notes for intermediate steps. Up to that poor kid's sixths grade, all the exercises were so easy he could leap to the answer in his head. When algebra happened, he thought he was missing some math recipes for solving those problems in one step, as before...</p>Sat, 12 Apr 2014 11:26:47 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2927/view.html
<p>This is a complex discussion: the danger of mentors not knowing the content. Overall, I am cautiously optimistic about "numerically challenged people" if they take steps to learn - as you do. I've seen very nice results, especially with support! For example, the leader of <a href="http://www.livingmath.net/">Living Math</a> Julie Brennan and my co-leader at Natural Math Yelena McManaman describe their journeys from rocky relationships with math to reading algebra and calculus for fun, with and for their kids. Many parents get their second chances at math with children! Leading big math projects is not for everyone, but helping kids and friends learn math well is doable and frequent among parents with past math challenges. So I am happy to see you here in the course!</p>Sat, 12 Apr 2014 11:22:51 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2926/view.html
<p>Kris, thank you for sharing these powerful stories. I held my breath reading. We will add to the list of dangers assumptions and presumptions about abilities (and morals, as in that cheating story). Expecting kids to cheat or dislike mathematics can be very toxic.</p><p>There is a famous experiment about assuming ability, the <a href="https://en.wikipedia.org/wiki/Pygmalion_effect#Rosenthal.E2.80.93Jacobson_study">Rosenthal-Jacobson study</a>.</p>Sat, 12 Apr 2014 11:06:30 GMTMaria DroujkovaAnswer by Kris
https://naturalmath.com/community/answers/2920/view.html
<p>I've always found math relatively easy; however, two experiences stand out for me. 1) My brother in about grade 9 was given the task of finding the perimeter and area of three triangles (he was quite behind in math according to his teachers and has a learning disability). My mother didn't know how to help him with the questions, so she asked me to work with him. He proceeded to ask me to do it because that's what the teachers did when he didn't know. I sat there for almost an hour waiting him out. I had a very strong gut feeling that the problems were within his ability. I will never forget the look on his face when he said FINE!, put his head-down, completed the questions (only making one error), and walked out of the house to go play hockey. Fifteen years he still refers to that single experience, as one of the only times that he felt that someone believed in his academic ability and didn't let him "play" the system. 2) In grade six, my math teacher called a parent teacher interview to discuss why I was "cheating" on my math tests. In the meeting the teacher pulled out my exam and showed my mother that I had gotten 100% on the exam (long division, algebra, etc) and that I had completed the test without showing any work-his conclusion I must have cheated. I sat there confused, thinking I was getting accused of cheating because doing work in your head and not showing your work was cheating (I didn't realise that he thought I'd copied the answers). For the next exam, I "showed" my work and again a meeting was called. This time because I didn't show the correct work and again he thought I was cheating (I did long division by estimating the answer and using multiplication to find the answer). My teacher didn't understand the method I used and said it wasn't possible. At the end of that meeting my mom suggested that he give me a test right then because she believe me. I completed the test my way without "cheating".</p><p>Dangers making assumptions about a students ability or knowledge based on their performance. But I feel both these examples illustrate the protective power of believing in someone's ability.</p>Sat, 12 Apr 2014 07:21:09 GMTKrisAnswer by PruSmith
https://naturalmath.com/community/answers/2916/view.html
<p>Man, some of the stories you hear about math trauma...makes you wonder why so many unhappy people decide to teach math. Anyway, I don't really know math instruction (I was "unschooled" as a kid) but if it's anything like art, just sloppy language on the part of a well meaning person can stunt creativity and stifle genius. For example, be sure you REALLY know something about color before you try to teach a bunch of preschoolers their colors. Take a poor or incomplete understanding, couple it with authority and you've got a recipe for creating some tremendous road blocks in their brains. I can only imagine what us numerically challenged people must do to kids with our imprecise and fumbling talk about mathematical concepts!</p>Sat, 12 Apr 2014 03:32:52 GMTPruSmithComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2905/view.html
<p>Some perfectionist and/or quick kids like to work with reference tables, handy at all times. Like the Pythagorean table for multiplication: </p><p><img src="/storage/temp/460-multiplication tablespythagorean.jpg"></p><p>It's different from using a calculator, because you navigate a chart by patterns. People usually end up color-coding, or otherwise tweaking their charts. </p>Sat, 12 Apr 2014 00:25:20 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2904/view.html
<p>This should be added to the list of dangers: a mistake that gets repeated a lot of times because of lack of feedback! It's okay if problems are self-checking or connected to other problems, because mistakes have consequences kids can notice. In worksheets... it's a danger.</p>Sat, 12 Apr 2014 00:21:07 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2902/view.html
<p>One wonders what real math is, and who decides!</p><p>I am copying my comment about embodied math:</p><p>It's okay to count on fingers. Some people who think a lot with their bodies - dancers, mechanical engineers, crafters - need to move and touch to think. Other people need to speak (maybe in their heads), others to doodle...</p>Sat, 12 Apr 2014 00:15:20 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2901/view.html
<p>The world looks different through math goggles! Thank you for sharing this feeling!</p><p>^ stands for power. So (a+b)^2 reads "a plus b to the second power." Many parents and teachers say the fear of mistakes (or misconceptions) stops their creativity... So I would focus on the natural view and creativity first.</p><p>It's okay to count on fingers. Some people who think a lot with their bodies - dancers, mechanical engineers, crafters - need to move and touch to think. Other people need to speak (maybe in their heads), others to doodle... </p>Sat, 12 Apr 2014 00:13:13 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's comment
https://naturalmath.com/community/comments/2900/view.html
<p>If you like to calculate, you can do 8th by doubling three times. It's easier than skip-counting seven times! There is a whole neuroscience-based explanation why 7*8 is the hardest multiplication fact in Dehaene's "The Number Sense." The short version is that both numbers are outside of the subitizing range, outside of the 10- and the 9-patterns, and it's not a full square (which are easier to remember).</p>Sat, 12 Apr 2014 00:02:56 GMTMaria DroujkovaAnswer by TraceySeier
https://naturalmath.com/community/answers/2899/view.html
<p>Haven't had any of these things happen to me... except the eyes glaze over part... that has happened with anything past Calculus ... ie Fourier series. But it is helping me to understand what has happened to so many people around me, who seem to struggle with math. I just hadn't thought of all the ways in which they don't get it, or how/when they fell off the boat. I was lucky: I was always so far ahead of my (working-class town) school mates, that I just taught myself from books for fun, and so I never had pressure to perform, and I could always take a playful approach. Maybe the reason I struggled with the "harder topics" was that I ran into them in a high- pressure context... in college.</p>Fri, 11 Apr 2014 23:59:44 GMTTraceySeierAnswer by dnamkrane
https://naturalmath.com/community/answers/2897/view.html
<p>When I was seven years old I remember crying in frustration as I did my subtraction homework. The concepts of carrying and borrowing didn't make sense to me. By the time I was in the third grade though I was one of the "good in math" kids though. I didn't feel math anxiety after that until I hit calculus. After two months of scratching my head, I finally opened my text book and started at the very beginning until I figured out where I got tripped up. Not surprisingly, it's the momentary suspension of disbelief: "imagine you *could* do something an infinite number of times". Once I realized what it was, I was back in the saddle :-)
My son with math anxiety is a perfectionist. If he can't get it immediately, he doesn't know how to cope because he gets so much very quickly. He doesn't use mnemonics, but he still has to count on his fingers and "reinvent the wheel" every time he needs to do simple multiplication. I'm torn between letting him work this way and feeling like he *should* have certain facts at the ready.</p>Fri, 11 Apr 2014 23:22:36 GMTdnamkraneAnswer by James
https://naturalmath.com/community/answers/2896/view.html
<p>I still remember when I was in kindergarden and wanted to move to first grade early. The class was in the middle of learning subtraction, so the teacher gave me a long worksheet and quickly went over the two practice problems with me so I'd be good to get the rest. It just so happened the practice problems were both something minus zero, so I though subtraction was just picking the bigger of the two numbers. I got every problem on the worksheet wrong except when it was something minus zero, and felt stupid. My daughter will be unlikely to experience a similar problem as I'll be guiding her throughout her education, giving her tons of attention. If she makes a silly mistake, we can laugh together over it.</p>Fri, 11 Apr 2014 23:21:15 GMTJamesAnswer by Elizabeth02
https://naturalmath.com/community/answers/2891/view.html
<p>This is a tricky one. The only time I felt math anxiety is when I got moved to the more difficult math halfway through the year and I thought I would not be able to catch up. My son's biggest problems with math occured when he was still in the classroom and not progressing at all in math. He was so bored that he was really starting to hate math time. There was also a large emphasis on memorization and fact recall, as opposed to playing with concepts, which was disheartening in that they were only 6 year olds. Since he has been homeschooled he has really bloomed and I haven't seen any signs of math anxiety. He abhors memorization and solves problems on his own. Since math was very sterile when I went to school I really want to help him see connections more, so we are all working on that together. </p>Fri, 11 Apr 2014 20:48:37 GMTElizabeth02Answer by Hascoorats
https://naturalmath.com/community/answers/2889/view.html
<p>WOW! Can't wait to scour through the links :) My DD is 7 and was told by her grade 1 teacher that she wasn't allowed to count on her fingers. After that moment, she asked to go to the bathroom every time they did math and stayed there until the recess bell rang. We homeschool now and she still has an aeration to math... But when I play subitization games, cards, dice, pattern play, she is quite bright and enjoys *math* as long as it doesn't look, feel, smell like *real* math...</p>Fri, 11 Apr 2014 19:45:55 GMTHascooratsAnswer by Sblair
https://naturalmath.com/community/answers/2887/view.html
<p>Since I started this class, I have been stretching my brain to see ordinary activities in a math sense. My children and I are probably torn between Loss of creativity and misconceptions. I am wondering what the (^) means in the algebraic equation in the misconception portion? </p><p>My first F in the 6th grade was in math. From then on I thought I didn't have geometric brain function. I had just coasted by and am pretty anxious about teaching my children the upper level math classes for credit. Although I have confidence now, only because I feel I have to just let it go and face my fear, to do the math that is needed to show completion of work I want to STOP COUNTING ON MY FINGERS! It rolls off my tongue when the kids are doing worksheets or a timed drill. </p><p>As I literally look around me, I am starting to visualize the math concepts around me. I feel a fuller concept of natural view ahead.</p>Fri, 11 Apr 2014 19:18:31 GMTSblairAnswer by nancy
https://naturalmath.com/community/answers/2886/view.html
<p>I was usually fine with multiplication...memorized the tables by third grade. My challenges came when the concepts became more abstract. I identify with my students who find it difficult to represent or explain their process for solving a math problem.</p>Fri, 11 Apr 2014 18:03:54 GMTnancyComment by babyhclimber on babyhclimber's answer
https://naturalmath.com/community/comments/2885/view.html
<p>Exactly. And I've seen that link on FB.</p>Fri, 11 Apr 2014 16:53:49 GMTbabyhclimberComment by sherylmorris on sherylmorris's comment
https://naturalmath.com/community/comments/2883/view.html
<p>With these comments I wanted to share </p><p>http://www.upworthy.com/a-science-icon-died-17-years-ago-in-his-last-interview-he-made-a-warning-that-gives-me-goosebumps-5?c=ufb2</p>Fri, 11 Apr 2014 16:22:45 GMTsherylmorrisComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2882/view.html
<p>Interesting, annboyd! I once interviewed kids and parents from a math circle about well-being during math. Everybody said YES to having a dog or a cat nearby. But the group split very neatly in half on the question of relatives. Half of the people said that a family member in the room made them feel loved and supported. But other people strongly preferred not to be with family - or at least, not with this or that particular family member they named. I think your focus on your daughter's well-being, rather than just math values, made the difference. Math with love!</p><p>Here's <a href="http://growingleaders.com/blog/what-parents-should-say-as-their-kids-perform/">a quote about sports</a> that seems relevant:</p><p>College athletes were asked what their parents said that made them feel great, that amplified their joy during and after a ballgame. Their overwhelming response:
“I love to watch you play.”
</p>Fri, 11 Apr 2014 16:16:54 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2881/view.html
<p>Yay for tasty math! What a great example of spontaneous multiplication fun.</p><p>Here are a few more ideas from our <a href="http://www.pinterest.com/naturalmath/eat-your-math/">Eat Your Math</a> collection.<img src="http://media-cache-ec0.pinimg.com/736x/f7/35/44/f7354418b3831134cea1d410617ed0eb.jpg"></p>Fri, 11 Apr 2014 16:07:11 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2880/view.html
<p>That danger of not valuing science (as a society) should be added to our list. Some groups of people don't have formal math and science, historically - and some have lost their scientific renaissances to social pressures. I lived through the event where a country lost 30-40% of its STEM publications within some 5 years, due to emigration and social disruptions of academic life...</p>Fri, 11 Apr 2014 16:02:41 GMTMaria DroujkovaComment by Eogruen on Eogruen's answer
https://naturalmath.com/community/comments/2877/view.html
<p>Thanks for the clarification, that helps a lot. Yes, I can skip count by sevens and eights but not very far, hence my perennial inability to answer 7x 8 etc. without serious thought. Am planning to finally do this right when I do it with my kids! </p>Fri, 11 Apr 2014 15:51:37 GMTEogruenAnswer by babyhclimber
https://naturalmath.com/community/answers/2876/view.html
<p>There is a danger in our society of not valuing math and science. The various fields/careers that involve math and science are often portrayed as geeky, nerdy, boring, etc. Politicians and parts of the media bashing science and math don't help. Anti-intellectualism doesn't help. Our country needs more scientists. But first we need more teachers who understand math & science to help pass on the love of math & science. And our society as a whole need to realize how much math is everywhere instead of the dissing of math with the "I don't need to know that" or "when will I ever need that." Combine with the culture attitudes with a student struggling in middle school or high school and then quitting on math is a problem. Sadly schools are so rigid that they don't realize if they changed the presentation or math or looked more globally at the problem, the student might be able to get the concept. Also there is a problem of holding kids back. If a kid can grasp higher level math, let them move on. Sometimes the repeated drills ending up killing the love or curiousness of math.</p><p>Ways to change: expose kids to algebra thinking earlier on, discussing how natural math can be, showing how math is all around us, getting & keeping kids curious, and connecting math with art & music.</p>Fri, 11 Apr 2014 15:50:45 GMTbabyhclimberComment by Maria Droujkova on Maria Droujkova's answer
https://naturalmath.com/community/comments/2874/view.html
<p>Skip-counting is not mnemonics, it's more a calculation technique. Do you skip-count by 7s or 8s? That's very impressive!</p><p>But if you memorize the skip sequence as a song, that's another story. For example, you can sing "four eight twelve sixteen twenty" to the tune of "Old MacDonald had a farm" - and it will help to memorize the skip sequence, but only as a whole. It may interfere with calculations and with algebra in some cases, since you will always need the whole song to get to 20.</p><p>Mnemonics are non-mathematical (non-calculation) memory techniques, for example:</p><p>"Five, six, seven, eight - fifty-six is seven times eight!" This is non-mathematical, because the fact that 56=7*8 has consecutive numbers is a coincidence. </p><p>Here is a visual mnemonic for 9*9 from Times Tales. The images have no connection with quantities or numbers or the mathematics of multiplication 9*9=81. So they don't support number sense or algebra.</p><p><img src="/storage/temp/447-9x9mnemonic.jpeg"></p>Fri, 11 Apr 2014 15:16:33 GMTMaria DroujkovaAnswer by fcogan
https://naturalmath.com/community/answers/2873/view.html
<p>I see that many students in high school "give up" on math and science because they believe it is "too difficult" and therefore they won't get marks high enough to allow entrance to university or college. I see many cases of students who enter university hoping to get into the sciences or medicine but give up after a year or less because they cannot tolerate the mathematics courses. This is a shame. I still think that the best way to set a foundation in multiplication is to learn the basic multiplication tables from 1 x 1 to 9 x 9 and the rest of multiplication is basically derived from there. Yesterday, my 4 year old grandson was having a sandwich which was cut in halves and he asked if it could be cut in "fours". This was a perfect opportunity to learn the concepts of 2 halves, 4 quarters make a whole, 2 quarters make a half, etc.. He really enjoyed that and the sandwich too! I think too much stress is put on students with testing, especially in math. Students have to be able to experiment and make mistakes in order to truly learn the concepts and applications of mutliplication. Testing and the expectation of perfection is very stressful for students, especially in math, where it is generally expected that there are only right or wrong answers.</p>Fri, 11 Apr 2014 14:46:00 GMTfcoganAnswer by Eogruen
https://naturalmath.com/community/answers/2869/view.html
<p>Uh-oh, I am totally busted. I still, at the age of 38, have to go through large portions of the multiplication tables to answer something like 7x 8. Not sure exactly what is meant by "not using mnemonics" to learn them though, as I always thought most people learned them by essentially "skip counting" their way through. Hope my girls can learn them another way., maybe more securely? I am re learning conceptual math with my girls and find, to my surprise that I do like puzzling over mathematical concepts once I get started, after a lifetime of hating to do equations on paper. </p>Fri, 11 Apr 2014 13:31:53 GMTEogruenAnswer by annboyd
https://naturalmath.com/community/answers/2868/view.html
<p>We've been using "xtramath.org" to do some math drills on basic addition facts. I thought it would help my daughter if I walked away while she worked on it so she wouldn't feel the pressure of me watching her every move. But I found that she actually performed worse this way. So instead, I decided to try "math with love," meaning that I sat next to her and smiled and encouraged her while she practiced her drills. Her performance improved a bit, but even better was her attitude about the practice sessions.</p>Fri, 11 Apr 2014 13:06:29 GMTannboyd