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SusanJ
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| Posted: Jan 23 2008 at 1:29pm | IP Logged
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. . . and do you like it now?
Dh and I are having our semi-annual "spirited discussion" about how to teach math. Our oldest is only three but we like to plan ahead.
Dh loves math. Really loves it. He's a math genius--he likes to play with his numbers in his head, come up with systems, find patterns. It's amazing.
I hate math and I'm not very good at it. I got all A's in math until the end of seventh grade at which point I was put on the advanced track in school and took Algebra in eighth grade. I struggled and shed many tears for 4.5 years until I was finally allowed to drop out of AP Calculus halfway through 12th grade.
Dh argues that little kids (elementary age) need to drill math basics so that they have those basic math facts to call upon when they get to abstract math (algebra and beyond). The funny thing about his argument is that that is exactly how I learned math: I could always memorize the trick and the steps. I was the first kid in my class to learn the times tables. I can still produce that stuff from memory but I don't really know what any of it means. After I pushed dh he admitted that when he hears "seven times nine" he does not get his answer as a result of memorization but because he sees numbers differently than I do.
I am arguing that I had a bad foundational math education that failed me once I got to abstract math. Dh argues that my math education was fine until abstract math and then my first algebra teacher failed me. An interesting side note is that I did really well in ninth grade geometry when we used a "deductive approach" rather than proofs.
Any ideas, here? Those of you who like math and/or are good at it--how did you learn it? How are you all succeeding at teaching your kids math?
Susan
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Mom to Joseph-8, Margaret-6, William-4, Gregory-2, and new little one due 11/1
Life Together
[URL=http://thejohnstonkids.blogspot.com]The Kids' Blog[/UR
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hylabrook1
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| Posted: Jan 23 2008 at 2:14pm | IP Logged
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It seems to me that those basic facts have to be firmly in place before you can have *fun* playing with numbers in your head. It sounds like this is where your dh is coming from. I'm not so sure that the only/best way for every child to get those basic facts in place is by drill and on-purpose memorization. For many, maybe even most, it works that way. For others it may work in several ways, depending on which way it is introduced to the child. I think mostly my dc learned addition facts by just realizing they were *common sense*, from real life need to add, and from using manipulatives. They then pretty much intuited the subtraction facts. The way we approached the multiplication facts was a bit different. Lots of the facts came from skip-counting, which was done just fooling around. Then somewhere around late third grade I gave each of them a 12 x 12 multiplication table with the instruction to look at it 10-15 minutes each day and see whether they saw patterns. After maybe a week I asked them to show me some of the patterns they saw, and I mentioned a couple I saw. After a few weeks, the facts fell into place in their heads. But then I had one dd who didn't *get it* when I presented this method. Nor would she just memorize the facts. (There's one in every crowd...  ) I allowed this child to look up the facts as she needed them, referring to the 12 x 12 table. After a while, she began remembering the things she had looked up repeatedly, and then weaned herself off the chart.
I learned math in the most conventional of all methods; not a lot of imagination went into the approach. Once the basic facts were in place, other things just kind of clicked, like recognizing at about 12 how to convert Celcius to Fahrenheit and back. I loved Algebra, which seemed like learning a different language or breaking a code. When I hit Geometry, I was in an Honors section that used a *different* book, which didn't click for me at all. It also was impossible to go to any other resource to try and figure things out, because the book we were using didn't use the same terminology as conventional books. I am not kidding! (This was in 1969-70, when the educational world was really going wacko). So, to this day, I think of Geometry as being frustrating. I can do *higher math* than that, but generally don't have the patience to work through anything complex; it just doesn't interest me... (pitiful, huh?)
I don't know; this is an interesting topic to think about. I'm really interested to read what others experienced.
Peace,
Nancy
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SusanJ
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| Posted: Jan 23 2008 at 2:44pm | IP Logged
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Thanks, Nancy! I'm a lot younger than you, but I wonder if we used the same geometry book!
Susan
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Mom to Joseph-8, Margaret-6, William-4, Gregory-2, and new little one due 11/1
Life Together
[URL=http://thejohnstonkids.blogspot.com]The Kids' Blog[/UR
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JuliaT
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| Posted: Jan 23 2008 at 8:21pm | IP Logged
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I learned math with the basic drill'n'kill worksheets. I don't remember ever using manipulatives. I just memorized everything. It worked okay for me until I got into highschool. I was in the advanced class of Algebra. I realize now that I had a really BAD teacher. At the time, I thought it was because I was stupid. I failed that class and then put down into, what the students called, 'the dumb kids math class." I had an amazing teacher for that class and I got A's until Gr. 12. But I never shed that feeling that I was stupid at math. Hence, my math phobia.
Due to my experience, I have decided to teach math differently. We are heavy on manipulatives, playing math games and reading living math books. It is important to me that my children know their math facts but also to understand why they get the answers that they do. This is why I like RightStart Math so much. It's great that they get the right answer but it is even better if they know how they got the right answer.
It is important that children know the math facts but I don't believe it has to be done with drill. I think it is more effective for it to be done with playing games. It has been my experience that this works well. My dd has a math phobia as well. I think I inadvertently passed it on to her. She is the reason why I have researched different methods to teach math. Doing drill with her would send her into a straight-jacket. She has learned her addition and multiplication facts (we are still learning subtraction) through playing math card games. So I know that they work.
Blessings,
Julia
mom of 3(8,7,5)
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mom3aut1not
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| Posted: Jan 23 2008 at 8:26pm | IP Logged
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Susan,
I learned math without much effort until calculus. I really didn't think about it except to be bored after the first few weeks of 4th grade. I did have excellent teachers in 7th-9th grades, but math was pretty obvious even then. (The reason I was so bored in 4th - 6th grades was that I caught on to the connections between fractions, decimals, and percents in the few weeks of 4th grade. I found spending three years on topics I thoroughly understood to be monumentally tedious.)
Actually, there is one thing I did have a little trouble with-- all the x facts. In 3rd grade we were supposed to get a perfect score on a 3 minute timed test on the x facts. Although I know them well, the time limit always made me goof up. (We were tested each week. When a child got 100/100, his or her name was posted above the chalkboard.) That teacher literally tried to frighten us into learning them. Looking back, I think she overstepped her bounds when she told us that she would call us up in the middle of the night about those x facts.
Actually, I disagree with your husband. Although learning math facts is a good thing, it is not my primary focus. I'm also not big on drill. I agree with you. I taught people who were taught math as a collection of recipes to be followed exactly with very little understanding. It was very frustrating. I want understanding in my students! If you really understand something, you don't need to do lots of brute memorization. You can derive much of what you need as you need it. I wonder if your husband might like to read about Japanese math education. They have the *best* math education in the world, and they focus on conceptual understanding. He might also like to read Liping Ma's book on math education. If you like, I can look up some of the titles on Japanese education (especially math education).
As for my actual students....
My oldest resisted math instruction after she went to school. (You wouldn't think that a few years of elementary math education would have such an effect.) She does not want to understand the math she learns. I used a variety of approaches with her, and the only things she enjoyed were stories. She learned her math facts without much trouble, but she has hated math since 2nd grade. She has resisted math as much as she could. She took math up to Algebra 2 with me and did Algebra 2 online. Later she took the same course at the community college because she only memorized what she had to and refused to understand. Even at the community college she didn't want to understand. Thus she retained virtually nothing. We'll see what happens next year if she takes another math course. (It's been 4 or 5 years since she last took a math course.) The next student was a very math-intuitive person who wants to understand what she does. She did struggle a bit in 2nd semester calculus. The third student struggles with memory issues among things although she is very bright. With her I treated things like x facts as a separate issue and worked on it gently but consistently for over a year. She has a good understanding in general but doesn't make connections as easily as her next older sister. She has done a semester of precalculus; she may do more, but right now we are focusing on finishing high school. My last student is severely delayed in language. I have been using RightStart with him -- I love RightStart -- and he is just about to finish level B. As far as anyone can tell, he understands and he has learned some + facts just by playing games (RightStart and other) and following patterns.
In Christ,
Deborah
Mom to four with special needs children (75% are on the autism spectrum)
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MichelleW
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| Posted: Jan 24 2008 at 2:38pm | IP Logged
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I attended a Special Ed for Homeschoolers conference last week. It was actually called a "Learning Efficiency Seminar" because they said the processes could be used for anyone.
Anyway, basically what they said in regards to math was that math facts are most effectively stored in audio memory, and that those facts are not the same as understanding. They described math as actually several different brain functions. There is rote memory, there is computation, application and there is problem solving. Each of these is a separate skill and you use different parts of your brain for each component. They felt that in order to succeed in application and problem solving you need to have quick access to memorized math facts. In order to succeed in computation you need to have understanding.
They advocated teaching each math fact as a separate chunk (not as part of a song because then to access the fact they have to sing the whole song), independent of understanding.
Not sure that I communicated that very well...
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Michelle
Mom to 3 (dd 14, ds 15, and ds 16)
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cathhomeschool
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| Posted: Jan 24 2008 at 9:17pm | IP Logged
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I learned math without much effort until calculus, too, and was taught with standard textbooks and no manipulatives. I never had anything but the most ordinary, uninspiring math teachers. I just understood math.
When I teach my children, I put great emphasis on understanding *why* math works -- on the process and reason -- and make sure that they get it looong before letting them see the "trick" shortcuts. (And sometimes by then they've figured out the pattern or forumla on their own anyway.) I use manipulatives to teach whenever possible.
When it comes to math facts, we use manipulatives to "see" the facts way before trying to think of them abstractly. We verbalize as we work with blocks. After a lot of practice with this, I use flash cards and games to "memorize" addition and subtraction facts. We have used skip counting songs and worksheets (not timed drill sheets, but just a handful of facts a day along with other math work) to work on multiplication facts. It is more a matter of gaining enough familiarity by regularly using math facts than it is about sitting down to "memorize facts." I think of this in the same way that I think of learning to read. When a child first learns to read, they usually sound out many words. Eventually, they won't have to do that anymore. They will have "memorized" the look of the word and will know it without thinking. Math facts are the same. My kids build with blocks (all 4 operations), verbalize, and eventually write math problems/math facts out. At some point, they have worked with them so much that they remember them. I do agree with your dh that it is good to have math facts memorized (because it just makes it easier to do math later), but I don't think that this ought to be done until after the child thoroughly understands what he's memorizing. And at that point, I think that it ought to be done with as little drill (none??) as possible.
My older two children are very different. The oldest struggles through math while my second son finds patterns and "wacky" ways of solving problems all on his own. With both, I make sure that they use manipulatives until they really understand what they're doing. Sometimes they say that they understand and even think that they understand until they have to build heavily on a concept, then it all falls apart (dividing fractions by fractions is a big stumbling block here). When this happens, we go back to the simple stuff and back to the manipulatives and review and build on concepts again until we see where the thinking went wrong. Often it is overwhelming to my oldest to see a "big" problem with complicated fractions or requiring lots of steps to solve for an unknown. I emphasize that the "big" problems are worked in the same exact way as the "little" problems. If they don't understand the concepts, though, and why something works, the harder problems -- or any problem that doesn't fit the "formula you memorized" -- can seem impossible and frustrating.
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Janette (4 boys - 22, 21, 15, 14)
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Veronica's Veil
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Growing up, attended Catholic School, we were taught using MCP for elementary grades and Saxon for 4th-8th, then of course Algebra, etc... in High School. I learned to do Math well enough I didn't and still don't really struggle with Math, however that said I HATE MATH. It just automatically triggers a boredom reaction in me and I attribute this to the Zillions and Zillions of problems we had to do in Saxon, I always felt that if I understood the concept why did I have to do more than say 20 problems in it per day? Instead we had 50 or more! I loved MCP when I was little I enjoyed Math and learned it easily, when they transitioned us to Saxon it became a drag...just my opinion.
I am gearing my own children's mathematical education towards LIFE Math...we are of course covering the basics in fact and concept but I look for every opportunity to include life Math in their experiences.
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SusanMc
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| Posted: Jan 25 2008 at 8:38am | IP Logged
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My husband and I are still talking theory right now as our son is only 9 months old!  But DH is a mathematics professor and has seen the level of actual UNDERSTANDING of mathematics decline sharply. I think some form of memorization and drill'n'kill is good, particularly at the younger ages. He is very much an advocate of manipulatives, writing proofs (in english!), and discovery-based learning. In his college level courses, particularly those that have a smaller number of students, he likes to use the Moore Method. It is a discovery-based technique desgined by R.L. Moore at UT Austin in the 50s. You give the student some axioms and ask them to use them to prove or disprove a statement. The result is a student who can "think" mathematically and not just plug in numbers into a math recipe.
I was very fortunate to have this sort of education as a child (almost by accident).
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mom3aut1not
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| Posted: Jan 25 2008 at 8:33pm | IP Logged
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Susan,
Someone else who has heard of the Moore Method! My dad used it with me when I was in college, and I really learned how to write a proof. Then I used it a year or so later teaching other students in college for a month-long winter term Apparently, this was a huge change for my students -- one student changed his major and decided to study math!
In Christ,
Deborah
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ALmom
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| Posted: Jan 28 2008 at 3:54pm | IP Logged
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Ok, I'm a maverick in math. I flunked math in elementary school and loved it in high school and went on to major in it in college. That being said, my number facts were never super solid in terms of off the top of my head and I often mentally added fingers in my head  . Math can have areas of specialization so it all depends. I had more trouble with the initial 2 years of calculus. I think it just took some down time for what limits were really all about to sink in. I was never much into advanced applied math (differential equations sort of lost me and I faked my way through that class with my memory).
I don't remember a whole heck of a lot about my early math instruction except that when everyone else was marching around the room reciting 6 X 8 is 48, I was still officially on my 2 times tables with the teacher because I stubbornly refused to memorize them. I kept adding in my head until the very end of the year when I finally convinced myself that it really did work the way the teacher said it did. Then I gladly memorized in one week what the class had been working on all year.
I loved manipulating numbers in equations, looking for patterns, factoring, ets. Algebra to me was a great blast, more like a numbers game than anything. It seemed to come naturally, as did balancing chemical equations and plugging in things to solve physics problems. (I generally didn't get the science concepts but loved the math part of it and since we basically memorized formulas, it was fun. The teacher did derive some of the formulas and I had fun seeing if I could do this too and basically minimized what had to be memorized  . I do distinctly remember my Geometry class in high school. We started the year with 3 postulates related to lines and planes and maybe one or two other givens. We proved everything else we ever used in that class as part of our homework. I absolutely loved this and found myself seeing how many different ways you could prove something or which way took the least amount of steps or... I also had fun with trig functions.
In college, Calculus I and II were a bit of a challenge. It took some time for things to sink in. I think some of it was that I'd never understood the physics concepts in high school and now was really thrown for a loop cause all those formulas I'd had so much fun with seemed to be superseded by all this mumble jumble. No one seemed to detect where I got stumped. I also had trouble picturing snowplows and pulleys and other such things - so I think my block was more science than math but the relationship or connection between things took a summer after my first year of college to sink in. Calculus III and IV - now that was sheer joy, back to all those trig. functions and playing number games. Higher level math was like philosophy with numbers. I loved it but I'd never be an engineer.
I have found that with tutoring, it is essential to find where the person is stumped and give them time to let ideas sink in where they can make connections. I've never tutored someone who wasn't quite capable at math, but many who were frozen because classes moved faster than they grasped something. If you ask the right questions, you find the confusion and help explain the gap. I love tutoring Algebra and Geometry. I'm not going to say everyone becomes a geek at math, but everyone can become comfortable and reasonably confident.
I do think knowing math facts make for better efficiency and such but understanding should come first so that the memory work is more efficient.
Janet
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Isa in Michigan
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| Posted: Feb 18 2008 at 7:53am | IP Logged
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For the most part, I learned math the conventional way. I was an excellent math student until I needed to really start understanding math and doing problem solving (algebra courses on up). At this point, I still did O.K., but I had to work really hard to get a B or an A-. Looking back, I realize now that I didn't really understand some of the math, I was just following formulas.
As a result, I have really focused on getting my sons to understand math. As they have come to understand math, the math facts have been learned naturally. A math program I really like and highly recommend is Miquon (1st-3rd grade). There is always something new and interesting to do. It has created a love of math in my children, and they understand math concepts several grade levels above theirs.
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4Real Forums : Living and Loving Numbers
Topic: How did YOU learn math . . .
