JackieB Thinking about what math students *really* need to know. How much do they need to be able to do by hand?

212 responses to this plurk.

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  July 04, 2008 at 02:07 mindelei thinks they need to learn how to do it all by hand...then the shortcuts. (I know, not making your job any easier.)
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  July 04, 2008 at 02:09 JackieB Why do they need to solve complex equations by hand?
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  July 04, 2008 at 02:10 budtheteacher Yeah, I'm curious, too.
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  July 04, 2008 at 02:10 budtheteacher How many of them will actually use that later? Like, for really real?
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  July 04, 2008 at 02:11 mindelei says I just think you should know how to go about doing it without the tech.
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  July 04, 2008 at 02:11 mindelei says It's the same way when I learn how to use a program, I want to know the long way around through it before learning the shortcuts.
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  July 04, 2008 at 02:11 mindelei says Then I know how to use it better. But, hey - that's just me.
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  July 04, 2008 at 02:12 JackieB Why? How is it "better" to be able to solve it by hand than using a graphing calculator or a CAS program?
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  July 04, 2008 at 02:13 mindelei says It's not really about "using the equations" it's about learning to think through a problem.
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  July 04, 2008 at 02:14 mindelei says I'm not an advocate for memorizing equations - they're in reference books. But I think kids need to learn to think their way through math.
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  July 04, 2008 at 02:16 JackieB How does using tools (CAS, graphing calc) prohibit problem solving? I think it makes higher math more accessible to more students.
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  July 04, 2008 at 02:16 JackieB I'm not trying to be contrary here, I really want to know.
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  July 04, 2008 at 02:18 mindelei says It's about building on the basic knowledge that can be expanded by the use of tools. Plus, these skills transfer to...
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  July 04, 2008 at 02:18 mindelei says higher thinking and reasoning skills.
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  July 04, 2008 at 02:18 mindelei says I'm not saying you shouldn't use the tools, I'm just saying they need to know how to think things through which will allow them to use...
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  July 04, 2008 at 02:19 mindelei says the tools to their advantage.
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  July 04, 2008 at 02:19 budtheteacher mindelei - Do they? I had someone push back at me on that lately, and it's put me in a weird place.
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  July 04, 2008 at 02:19 milobo says Students need to know basic skills as building blocks so they can understand higher concepts.
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  July 04, 2008 at 02:19 budtheteacher I'm all for lots of thinking - but there're plenty of practical and real problems requiring solutions.
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  July 04, 2008 at 02:19 milobo says We wouldn't say that because we have books on CD (or iPod) that kids don't need to have reading skills, right?
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  July 04, 2008 at 02:20 mindelei shares milobo opinion on higher concepts.
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  July 04, 2008 at 02:20 budtheteacher Math problems in a book that have already been answered might not be the best problems for our kids to be looking at.
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  July 04, 2008 at 02:20 mindelei asks budtheteacher to clarify "pushing back".
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  July 04, 2008 at 02:20 budtheteacher Would anybody here disagree that kids need "basics"? The trick is - what're the basic tools/concepts?
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  July 04, 2008 at 02:21 JackieB But if your "goal" is to solve an equation, why not just use a graph or CAS? If your goal is problem solving with math, then use the tools?
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  July 04, 2008 at 02:21 milobo says math needs to be a mix of fact based skills as well as understanding the how and why those skills are important.
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  July 04, 2008 at 02:21 milobo says yes, jackieb - it does come down to outcome!
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  July 04, 2008 at 02:21 mindelei just wants to say what a great conversation this is...this is the kind of stuff I have been waiting for!
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  July 04, 2008 at 02:22 budtheteacher mindelei - by "push back" I mean they asked me if that was really true, to think through it some more.
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  July 04, 2008 at 02:22 JackieB So, a student who has not "mastered" the basics can't have access to higher math?
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  July 04, 2008 at 02:22 mindelei thinks that purpose is very important. Kids need to see how these ideas relate to what is around them.
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  July 04, 2008 at 02:22 budtheteacher Push back meaning that there wasn't automatic agreements and head nodding.
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  July 04, 2008 at 02:23 budtheteacher jackieb - Uh oh, I smell Papert.
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  July 04, 2008 at 02:23 JackieB And wouldn't use of available tools help in that?
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  July 04, 2008 at 02:23 budtheteacher Must math be a linear progression from the basic to the higher order?
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  July 04, 2008 at 02:23 mindelei asks if someone hasn't mastered those basic skills how can they appreciate/learn the higher skills?
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  July 04, 2008 at 02:23 JackieB Papert has been reinforcing a lot of what I already do.
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  July 04, 2008 at 02:23 mindelei asks What is papert?
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  July 04, 2008 at 02:23 budtheteacher or can kids, when they see a need, explore the fundamentals after tackling, or attempting to tackle, a particular tough problem?
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  July 04, 2008 at 02:24 JackieB And extending my thinking. Yay!
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  July 04, 2008 at 02:24 milobo says it's funny to me (a math teacher by trade) that people argue against basic skills in math and not reading!
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  July 04, 2008 at 02:24 budtheteacher jackieb - Score.
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  July 04, 2008 at 02:24 mindelei wishes milobo would further clarify...
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  July 04, 2008 at 02:25 budtheteacher milobo - I'll push back about basic skills in reading. Kids need to be immersed in language. That's all.
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  July 04, 2008 at 02:26 budtheteacher Okay - it's more complicated than that - but learning to read isn't a linear process.
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  July 04, 2008 at 02:26 budtheteacher We just want it to be.
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  July 04, 2008 at 02:26 JackieB milobo don't we teach kids to read by having them actually read lit? Do we wait until they're "fluent" before they read.
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  July 04, 2008 at 02:26 budtheteacher mindelei - Seymour Papert. MIT Professor, mathematician. Smart dude.
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  July 04, 2008 at 02:26 milobo says but if they don't have the skills to interpret and appreciate the language, can they truly grow fully in that appreciation?
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  July 04, 2008 at 02:27 JackieB mindelei www.papert.org/ I'm reading Mindstorms now.
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  July 04, 2008 at 02:27 budtheteacher milobo - It's a recursive process. It's why kids read Hamlet six or seven times in school.
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  July 04, 2008 at 02:27 JackieB Yes, but you teach the skills in context, right?
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  July 04, 2008 at 02:27 milobo says agrees and thinks that kids that are kept out of higher math and reading because of a preceived lack of skills are being disadvantaged.
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  July 04, 2008 at 02:27 budtheteacher Louise Rosenblatt would be someone else for folks to look up here - she's a theorist who talks a lot about Reader Response theory.
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  July 04, 2008 at 02:27 mindelei Thanks for the link - I'll be sure to check him out!
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  July 04, 2008 at 02:28 milobo says (and having this conversation while also watching "History of the World" on TV isn't easy!)
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  July 04, 2008 at 02:28 mindelei Rosenblattt sounds good too!
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  July 04, 2008 at 02:29 budtheteacher It's way more complex than 140 characters - but basically she and others argue that reading a text is a conversation between a writer's . .
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  July 04, 2008 at 02:29 milobo says RE recursive - YES! and that should be the beauty of math - it's a spiral curriculum, not a linear one.
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  July 04, 2008 at 02:29 budtheteacher brain and the reader's brain. What's learned is what the reader takes away and blends with prior experiences.
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  July 04, 2008 at 02:30 budtheteacher So different readers will have different readings of the same text.
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  July 04, 2008 at 02:30 mindelei says milobo Plus, there's the intimidation factor in Math & Reading.
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  July 04, 2008 at 02:30 budtheteacher Depending on their backgrounds.
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  July 04, 2008 at 02:30 budtheteacher I think you could make a similar argument for math problems.
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  July 04, 2008 at 02:30 budtheteacher real problems.
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  July 04, 2008 at 02:30 mindelei says budtheteacher She sounds right up my alley!
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  July 04, 2008 at 02:30 mindelei feels that each person does come away with something different - particularly in relation to symbolism.
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  July 04, 2008 at 02:31 budtheteacher How interesting it would be to take a look at reader response theory and twitter or plurk - talk about different experiences.
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  July 04, 2008 at 02:31 budtheteacher Each network is its own text.
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  July 04, 2008 at 02:32 budtheteacher but anyway - back to math.
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  July 04, 2008 at 02:32 mindelei asks JackieB So - did you get more than you were looking for?
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  July 04, 2008 at 02:32 mindelei says That would be interesting!
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  July 04, 2008 at 02:33 mindelei is Impressed that there are now 72 responses to this plurk.
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  July 04, 2008 at 02:36 JackieB No, this is exactly what I'm looking for. Keep it coming! budtheteacher, thanks for the new addition to my reading list.
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  July 04, 2008 at 02:37 JackieB And regarding the intimidation factor, if we keep trying to "drill" the basics, isn't that more intimidating/frustrating than ...
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  July 04, 2008 at 02:37 mindelei says It goes back to dividing fractions. Rather than learning how to actually cut things into smaller pieces, we simply flipped and multiplied.
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  July 04, 2008 at 02:37 JackieB here, let's explore this problem.
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  July 04, 2008 at 02:38 JackieB So, if there are tools that help them explore this (in the context of a larger problem) why not use them?
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  July 04, 2008 at 02:38 mindelei I'm not really into "drilling." I do agree that we do to much of that - no real learning. But you do need to get basic understanding.
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  July 04, 2008 at 02:39 milobo says if you have to "drill" for understanding, then the students don't really understand and then it is frustrating for students.
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  July 04, 2008 at 02:39 mindelei But do they understand what the tools are doing? I've been in countless science labs where there is no real learning of why/how.
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  July 04, 2008 at 02:40 JackieB What is "basic understanding"? Estimation? Ability to use the skill in context? Ability to do 10 of the same problem? To extend the problem?
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  July 04, 2008 at 02:40 JackieB mindelei Then the teacher isn't asking the right questions.
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  July 04, 2008 at 02:40 mindelei Ability to apply the skill to a new problem.
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  July 04, 2008 at 02:41 mindelei Maybe not new...but different.
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  July 04, 2008 at 02:41 mindelei Ability to determine how to use what I already know on something I don't know.
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  July 04, 2008 at 02:42 mindelei Or at least move in that general direction.
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  July 04, 2008 at 02:42 mindelei I should have a general understanding to think my way through things that aren't completely familiar.
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  July 04, 2008 at 02:43 JackieB Can't we use larger problems to figure out smaller ones we need to figure out first? Gives us a real reason to learn the concept?
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  July 04, 2008 at 02:43 budtheteacher What about "an ability to see what it is you need to know that you don't know - and seek it out."
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  July 04, 2008 at 02:44 mindelei likes what budtheteacher has to say!
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  July 04, 2008 at 02:44 JackieB Thanks Bud, that's what I was *trying* to say.
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  July 04, 2008 at 02:44 mindelei thinks that's a better explanation of what I was trying to say.
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  July 04, 2008 at 02:44 mindelei asks Hey - how do you get italics?
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  July 04, 2008 at 02:45 JackieB One * on either side gives you italics, **two** on either side gives you bold.
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  July 04, 2008 at 02:46 mindelei thinks these little plurk tricks are very cool.
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  July 04, 2008 at 02:47 milobo says budtheteacher As long as students know enough to know what they don't know... and that's where some of my students struggled.
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  July 04, 2008 at 02:47 budtheteacher Okay - so we agree on what we want kids to be able to do. The argument comes when we try to figure out *just how much* context is needed.
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  July 04, 2008 at 02:48 JackieB We agreed? What do we want them to be able to do?
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  July 04, 2008 at 02:49 mindelei thinks she agrees. But also believes it comes down to problem solving skills.
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  July 04, 2008 at 02:49 JackieB Ah, so problem solving. Then why do they need to be able to do it by hand?
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  July 04, 2008 at 02:50 budtheteacher mindelei - That's a bit circular - we learn problem solving by learning problem solving skills.
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  July 04, 2008 at 02:50 budtheteacher Okay - how do we learn those skills?
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  July 04, 2008 at 02:50 budtheteacher *By actually engaging real problems with a skilled facilitator in our presence.*
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  July 04, 2008 at 02:52 mindelei feels that by knowing it by hand they will have a better understanding of what types of shortcuts work best and when to use them.
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  July 04, 2008 at 02:52 milobo says the best way to learn any skill is to apply it authentically.
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  July 04, 2008 at 02:52 mindelei feels This also helps to identify which tools are of most assistance when solving similar problems.
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  July 04, 2008 at 02:53 mindelei shares milobo concept of authenticity.
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  July 04, 2008 at 02:53 budtheteacher mindelei: What's "it"? Basic math? Is that a problem solving skill?
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  July 04, 2008 at 02:53 milobo wishes I could stay up and hear the rest of this conversation! I'll check in again tomorrow.
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  July 04, 2008 at 02:53 budtheteacher Solving real problems isn't a linear process.
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  July 04, 2008 at 02:53 mindelei says It's like only knowing how to use Word. If I have the skills to use multiple word processing programs, I won't have issues when Word isn't
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  July 04, 2008 at 02:53 mindelei says available.
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  July 04, 2008 at 02:53 budtheteacher So why is math instruction?
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  July 04, 2008 at 02:53 mizminh mathematics is beautiful, engrossing & challenging - having a firm grasp of the basics promotes appreciation, engagement & creativity
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  July 04, 2008 at 02:54 mindelei loves what mizminh has to say about math!
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  July 04, 2008 at 02:55 JackieB mindelei So, if a student can't solve a multistep problem by hand, but can with tools, they shouldn't investigate further problems?
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  July 04, 2008 at 02:55 mindelei says she never said that.
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  July 04, 2008 at 02:56 mindelei says that understanding how to go about things by hand helps with the use of tools. Only knowing how to use the tools doesn't help the process.
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  July 04, 2008 at 02:56 JackieB mindelei Sorry if I misinterpreted. I'm just not following your thinking.
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  July 04, 2008 at 02:57 mindelei says No problemo. Clarification is always good!
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  July 04, 2008 at 02:57 mindelei says
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  July 04, 2008 at 02:57 JackieB Can't it work in reverse? Knowing how the tools work can help in understanding?
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  July 04, 2008 at 02:58 budtheteacher Wish I could stick around. Solve this one for me, y'all. I need some answers.
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  July 04, 2008 at 02:58 mindelei feels that it could - but what about the ones who don't really understand the tools - just know how to enter data?
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  July 04, 2008 at 02:59 mindelei thinks that many kids who have math disabilities go undiagnosed which is partially why tools seem to help so much more.
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  July 04, 2008 at 02:59 JackieB mizminh I agree with your statements about mathematics, but sometimes we focus so much on the basics that students never get to the beauty.
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  July 04, 2008 at 02:59 JackieB mindelei If they are just entering data w/o understanding then the teaching isn't facilitating the process correctly.
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  July 04, 2008 at 03:00 mindelei thinks that tools shouldn't be substituted for knowledge. They help the process, but don't take the place of the process.
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  July 04, 2008 at 03:01 mindelei agrees...
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  July 04, 2008 at 03:02 JackieB I agree. A tool without understanding is useless. I'm just questioning what the process is. (symbolic manipulation or problem solving?)
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  July 04, 2008 at 03:04 mindelei I think you need to know multiple routes to come to a solution. Providing the knowledge of how to do it by hand creates an additional route
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  July 04, 2008 at 03:05 mindelei I also think that allows you to have a better idea of what other routes are most worthwhile.
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  July 04, 2008 at 03:05 JackieB Okay. I agree it is a route. However should that route be a gatekeeper to other routes?
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  July 04, 2008 at 03:07 budtheteacher What about for problems that we don't know how to solve? Should students engage in those? Makes "by hand" or "with tools" irrelevant.
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  July 04, 2008 at 03:08 mindelei feels that learning "by hand" helps in solving unfamiliar problems.
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  July 04, 2008 at 03:08 JackieB Why would we work on problems we already know how to solve. That's boring. Unless it is to find a better/different way - then it's fun.
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  July 04, 2008 at 03:08 mindelei has a goal to get to that - using knowledge to engage in solving unknown problems.
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  July 04, 2008 at 03:09 mindelei thinks finding better/different way is where tools come in handy.
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  July 04, 2008 at 03:26 colleenk thinks that basic skilss need to be taught alongside authentic problem solving rather than before. (first plurk)
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  July 04, 2008 at 03:26 colleenk says skills not skilss
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  July 04, 2008 at 03:28 colleenk thinks a certain amount of fluency can make it easier to focus on the problem solving task especially in middle school and above.
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  July 04, 2008 at 03:30 JackieB I agree that fluency is a goal. But should lack of fluency stop further mathematical work?
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  July 04, 2008 at 03:30 JackieB (and welcome coleenk!)
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  July 04, 2008 at 03:33 colleenk thinks how we teach math should be revised from the ground up. We don't teach math with any consideration of the way children (people) learn.
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  July 04, 2008 at 03:35 JackieB I agree. However I'm working on change with the students I have (h.s.) It's tough to change their (and their parents) idea of what math is.
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  July 04, 2008 at 03:35 colleenk thinks fluency should NOT get in the way. Fluency is not an indication of a students' ability to do higher level math and should not be a barrier.
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  July 04, 2008 at 03:36 colleenk shares that she is having the very same problem where she is.
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  July 04, 2008 at 03:37 JackieB Yay! I just had a conversation with a friend (different school) who said students can't get into alg. until they've mastered basics, so they
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  July 04, 2008 at 03:37 colleenk wishes Mindstorms was required reading for parents.
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  July 04, 2008 at 03:38 JackieB put them in a year long "basics" course. I asked if they were that much better after a year, she said no.
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  July 04, 2008 at 03:38 JackieB Heck, I wish it had been required reading in my math ed courses.
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  July 04, 2008 at 03:47 colleenk Are you trying to integrate higher level problem solving into you classes? Are people questioning your curriculum?
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  July 04, 2008 at 03:49 JackieB We use a "reform" curriculum (next year is our 4th year, so first year of offering all 4 years). Lots of solving using graphing calculator,
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  July 04, 2008 at 03:51 JackieB we're incorporating more Sketchpad. Everything is "taught" in context of a larger problem. Some staff still entrenched in old ways, some ...
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  July 04, 2008 at 03:52 JackieB parent confusion (though less now), still various levels of staff buy-in. I'm still questioning my own thinking (hard to undo they way I was
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  July 04, 2008 at 03:53 JackieB taught to think about what it means to "do math".
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  July 04, 2008 at 03:53 JackieB I'm also questioning what kids need to know to be "ready" for college. (I'm teaching the senior course - 1st time we're offering it).
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  July 04, 2008 at 04:14 mizminh the "by hand" usage is interesting. Mathematical skills are best developed by moving from the concrete to abstract -graspable to symbol
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  July 04, 2008 at 04:19 JackieB Hmm. What if the tool allows them to "grasp" the understanding. Isn't looking at a graph or a list or computer output more concrete than
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  July 04, 2008 at 04:19 JackieB symbolic manipuplation?
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  July 04, 2008 at 04:35 mizminh we speak of levels of abstraction maybe levels of concrete or "graspability" would be useful
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  July 04, 2008 at 04:38 JackieB How would you quantify that? Wouldn't it vary from student to student?
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  July 04, 2008 at 04:47 mizminh of course & that is the eternal pedagogical challenge - differentiation- catering for the individual learner
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  July 04, 2008 at 04:58 mindelei asks What are the final goals? Different majors require much different math skills in college.
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  July 04, 2008 at 04:59 mindelei says Earlier it was mentioned that "by hand" wasn't necessary because most of this stuff wouldn't be used after HS anyway.
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  July 04, 2008 at 05:00 mindelei wonders which aspects can be made more relevant. The process of thinking through the problems is what can be taken beyond the classroom.
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  July 04, 2008 at 05:00 mindelei asks Which other aspects can be taken beyond the classroom?
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  July 04, 2008 at 05:04 JackieB I want them to take away problem solving skills, the ability to evaluate the solutions of others, to communicate solutions (and process) to
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  July 04, 2008 at 05:05 JackieB others, to have the willingness to tackle big/new problems, to be able to make a mistake and keep going,... just to name a few.
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  July 04, 2008 at 05:06 JackieB And that was part of my original question. What do students need to be successful in college? (not that that is my only goal, but ...
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  July 04, 2008 at 05:07 JackieB it is a factor - preparing them for future studies).
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  July 04, 2008 at 05:08 mindelei asks Is this one of those instances where it's more helpful to work backwards from your end-goals? Or at least look at that in conjunction with
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  July 04, 2008 at 05:08 mindelei asks initial goals?
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  July 04, 2008 at 05:09 mindelei likes the goals that you're working with so far.
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  July 04, 2008 at 05:11 JackieB Yes, but I'm trying to figure out exactly what the end goals are (I know mine, I don't know what colleges want. Want to make sure I'm taking
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  July 04, 2008 at 05:11 JackieB all things into account when I'm setting my goals).
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  July 04, 2008 at 05:11 mindelei has a suggestion: Check w/ some prof friends or local unis to see what they feel their freshmen are lacking.
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  July 04, 2008 at 05:11 mindelei That should be another good perspective.
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  July 04, 2008 at 05:12 mindelei I think that would be an excellent plurk - an alert to math profs.
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  July 04, 2008 at 05:14 mindelei I do have an opinion on what part of the problem is: not enough good math teachers at the elementary level.
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  July 04, 2008 at 05:14 mindelei I was an Elem Ed major, A LOT of the ed students had difficulties with the math courses we had to take (they only go through 8th grade math)
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  July 04, 2008 at 05:15 mindelei There are a few people I know who had to switch to Secondary Ed because they couldn't pass the math courses for Elem Ed.
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  July 04, 2008 at 05:15 mindelei Many had to take extra prep courses before math to get them ready.
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  July 04, 2008 at 05:16 JackieB And that is just indicative of the larger problem.
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  July 04, 2008 at 05:16 mindelei I think if we had more teachers who had a better grasp of math in elem and could better understand the problems the students are having
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  July 04, 2008 at 05:16 mindelei you would have fewer issues by the time those kids get to you.
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  July 04, 2008 at 05:17 mindelei Agrees on that point.
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  July 04, 2008 at 05:18 JackieB Yep, but blaming the el ed teachers isn't going to solve anything. We need to figure out how to teach the students we have now.
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  July 04, 2008 at 05:19 JackieB (and fix the system)
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  July 04, 2008 at 05:20 mindelei Oh - I don't want to blame them...I just think they need to be better prepared too. It's a vicious cycle.
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  July 04, 2008 at 05:20 mizminh surely the goal is to have autonomous learners exploring their own meaningful learning environments.
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  July 04, 2008 at 05:22 JackieB Well, that's one of my goals. Will that help them in a college math class? With college admissions tests?
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  July 04, 2008 at 05:23 mindelei thinks mizminh has a way with bringing everything back to center.
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  July 04, 2008 at 05:23 mindelei says when you teach them well, the tests will take care of themselves.
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  July 04, 2008 at 05:35 JackieB Ah, but what if they can't use the tools on the tests? I'm still trying to figure this all out.
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  July 04, 2008 at 05:37 mindelei feels like we need a big white board or something. Do you ever watch NUMB3RS? I feel like we need to be making a big chart.
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  July 04, 2008 at 05:38 JackieB I thought this *was* the chart.
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  July 04, 2008 at 05:50 mindelei You could say that....but at the moment it has no sense of organization (although there is a definite flow).
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  July 04, 2008 at 06:17 budtheteacher To be continued, right? Not finished?
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  July 04, 2008 at 06:19 JackieB Oh no, not finished by a long shot. Just my initial thoughts.
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  July 04, 2008 at 06:21 JackieB And I'd love for others to keep adding tot he conversation.
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  July 04, 2008 at 15:02 milobo thinks part of the problem when students get to HS is they see "doing math
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  July 04, 2008 at 15:02 milobo thinks "doing math" as adding/subtracting/mult/div whole numbers, fractions, and decimals.
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  July 04, 2008 at 15:05 milobo thinks and they "do math" with no connection to anything in the real world.
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  July 04, 2008 at 15:06 milobo thinks that's why they come to hate math in MS and HS. We need them to see that math is a form of language used to communicate and create.
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  July 04, 2008 at 15:09 milobo thinks the hand calculations are less important than the facility with understanding why and how you use each calculation to answer a question.
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  July 04, 2008 at 15:11 milobo loves that JackieB started this plurk and can't wait to see what others think!
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  July 04, 2008 at 21:55 mizminh what milobo said" math is a form of language used to communicate and create."
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  July 05, 2008 at 03:48 JackieB Thanks to everyone for their thoughts. I'm still processing what's here so far.
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  July 05, 2008 at 03:50 mindelei says I can't wait for the next conversation. That was a lot of fun for me and very interesting.
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  July 05, 2008 at 03:53 mindelei thinks JackieB needs to check out this link www.plurk.com/p/11ah0