A Continuing Conversation -> **Changing Math Education** - what should be part of the discussion

On 10/16/2010 1:44 PM, Linda Fahlberg-Stojanovska wrote on google groups: geogebra-na and mathfuture

I am sure that other people have thought much more deeply than I have and have done “proper” research – I have just my years of experience in the classroom. I have been thinking about Next Generation Learning Challenge grants and want to participate, but I am very concerned by the paragraph:

The employment of OER must be freed from current patterns of “not invented here,” so that institutions routinely employ best-of-breed OER no matter where it was created. Especially in the gatekeeper courses that pose the greatest impediments to at-risk student persistence and completion, institutions must become less concerned with the uniqueness of their curricular content than with its efficacy. This implies that the OER involved: Must be of the highest pedagogical //and //technical quality, making use, for example, of cutting-edge research in cognitive science, interactive/active learning, multiple learning paths, embedded assessment/frequent feedback, and scaffolded learning, as well as cutting-edge innovation in richly interactive media. The objective is to make the OER in question demonstrably superior to the instructional materials presently employed. These innovations must be disciplined by careful focus on //learning outcomes//: technical and pedagogical virtuosity must be put to the service of deeper student learning and engagement.

I am extremely logistical in my thinking. I don’t know educational research jargon, but I do not see how any of these sentences implies anything other than more of the same thinking: “Here is everything they must learn – find a way to get every student to get all of it and love it.” I put it to you that this is “mission impossible”. I ask 2 global questions and then give my personal comment/examples for these.

Global question 1: What has to be cut in school mathematics programs in order to add completion and rigor to the material taught (and some enjoyment would be good too).

Global question 2: What are the absolutely essential skills that a student must master in each grade so that he can master the absolutely essential skills of the next grade (and how can we test that he has mastered these skills and how can we make sure that what we teach in each successive grade requires ONLY mastery of these essential skills from the previous grade).

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Global question 1: What has to be cut in school mathematics programs in order to add completion and rigor to the material taught (and some enjoyment would be good too). Facts:

  • Kids are now taught significantly more “subjects” than in 1965-1970 when I was in middle school.
  • Even the “subjects” in mathematics have increased (e.g. data analysis, probability and statistics have been added to the curriculum – what was cut????).
  • We are now adding technology (applets or whatever) and time must be spent to work with these. Yes, yes – they increase understanding. But they take time.
  • We want to have them explore and have fun with mathematics – this takes time. So what has to be cut? Some things must go.

--- A specific “for example” from a first course in algebra: Why study the addition method of solving 2x2 systems of linear equations? Why study completing the square? First – think of all the time that could be saved if we didn’t study these two things. Second – I think they confuse kids. Remember, they already have solution methods that always work – namely the elimination method and the quadratic formula.
== An argument against addition method. Make them learn the elimination method backwards and forwards. The elimination method is the method by which the new is based on the old (reduce 2 equations in 2 unknowns to one equation in 1 unknown) The addition method requires that the equations of the lines stay in standard form. I am not at all convinced that kids understand that y=3/2 *x – 2 is the same as 3x-2y=-2 which is the same as -6x+4y=4? Always have them put in the form y=mx+b (think of all the skills learning to do this reinforces, think of the fact that they can easily see that this form is unique, think of how much easier it is to graph a line in y=mx+b form than in standard form NOT with slope-intercept but just by substituting points, think of the fact that it looks like a function so they can find x on the horizontal and then move to the vertical so it connects to y=mx+b, … Other than manipulation skills, what is being learned by the addition method? I have similar arguments for completing the square and for absolute value inequalities. I am sure each of you has something from the materials you teach …

Global question 2: What are the absolutely essential skills that a student must master in each grade so that he can master the absolutely essential skills of the next grade (and how can we test that he has mastered these skills and how can we make sure that what we teach in each successive grade requires ONLY mastery of these essential skills from the previous grade).

  • if a skill is required, you must have a “completion and rigor” version for an A, B, C and D in that skill
  • since a D is a passing grade, a student who gets a D must have mastered the minimum skills in that grade to be able to learn the minimum skills in the next grade. correct answers, organization (readability) and rigor count; we cannot continue to give partial credit.
    (I am talking about Solve:3x-6=0 where we give partial credit for: (a) student writes 3x=6/3=2, (b) student writes 3x=-6, x=-2, (c) student writes 3x=6, 2) e.g. 5th grade: Being able to calculate 50% of $2 is essential (D); being able to answer “what percent of $2 is $1?” for a C; being able to calculate 25% of $2 for a B and being able to calculate “what percent of $2 is 50 cents?” for an A. 7th grade: Being able to calculate the third side of right triangle given the 2 other sides with Pythagoras’ theorem is essential; being able to bisect an angle using a compass and straightedge is not. In geography, knowing last year’s material is not important, but this is an essential feature of mathematics.

This is a major source of frustration among our kids. Too much material is being “covered” and there are no specifics about what are the absolutely essential skills and what it means to master them. (And having studied them in some depth, I will tell you that this conversation is not part of the new common core standards.)

Thanks for reading, Linda

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Maria Droujkova replies

Sat, 16 Oct 2010 17:22:24 -0400

Linda, Cool questions.
> This is a thread from the GeoGebra-NA group, but I thought it interesting enough to add it here. It regards new grant opportunities: Next Generation Learning Challenges - http://nextgenlearning.com/
> I am very concerned by the paragraph in the call: The employment of OER must be freed from current patterns of “not invented here,” so that institutions routinely employ best-of-breed OER no matter where it was created.
I choose to read it as, "Don't be parochial."
> Especially in the gatekeeper courses that pose the greatest impediments to at-risk student persistence and completion, institutions must become less concerned with the uniqueness of their curricular content than with its efficacy.
Again, this can be read either as a call for standards, or as a call for openness to ideas of others. I like openness more.
> This implies that the OER involved: Must be of the highest pedagogical and technical quality, making use, for example, of cutting- edge research in cognitive science, interactive/active learning, multiple learning paths, embedded assessment/frequent feedback, and scaffolded learning, as well as cutting-edge innovation in richly interactive media. The objective is to make the OER in question demonstrably superior to the instructional materials presently employed. These innovations must be disciplined by careful focus on learning outcomes: technical and pedagogical virtuosity must be put to the service of deeper student learning and engagement.
This paragraph is neat. I like the list. My favorites here are multiple learning paths, which means each student can choose her own pace, and richly interactive media.

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The caution about virtuosity in service of learning is important, too. I just spent an hour and a half folding squares of paper in four (in a certain tricky way), with a math club. We actually planned on spending about half an hour on that activity, but everybody (kids and adults) were extremely deeply engaged, and we all literally lost the track of time. I claim a lot of learning with very little technical virtuosity.

> Global question 1: What has to be cut in school mathematics programs in order to add completion and rigor to the material taught (and some enjoyment would be good too).
I loved what Gary Stager said last week in his event with Ihor Charischak: "Remove a half of the curriculum. Any half."
http://mathfuture.wikispaces.com/Gary+Stager

I have my list topics that would be " the first against the wall come the revolution," but I fully realize that other teachers can love these same topics and do wondrous things with them. With this caveat, here are a few examples:

- Remove counting in kindergarten
- Remove addition tables (as in, memorizing them) in k-2
- Replace long division with partial quotients in 4-5

So, what I do about it on my end: I don't design materials for these topics, and don't teach them. But I would not want to institute a global ban.

Global question 2: What are the absolutely essential skills that a student must master in each grade so that he can master the absolutely essential skills of the next grade (and how can we test that he has mastered these skills and how can we make sure that what we teach in each successive grade requires ONLY mastery of these essential skills from the previous grade).
In mathematics:
- Problem-solving
- Creating definitions of novel concepts
- Programming and making algorithms (in any language)
- Conjecturing and proof
- "Patterning" (defining similarities and differences, making taxonomies, discovering inductive and deductive patterns and formulas)
- Measurement (forming systems of units, power-based systems, reunitizing)
- Construction (2d and 3d) and deduction/logic surrounding construction
- Aesthetic math values (simplicity, symmetry, sparsity, completeness)

MariaD

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Reply from Linda ...

Firstly - for making the paragraph from the call make some sense. I read it to my sister and asked her what she thought it meant - we were completely bewildered. Reading your letter both translated it into "people language" and gave me hope that the call is "real".  (One of the things I like to emphasize to my college kids is that true mathematics is written in "people language" and that they shouldn't let themselves be intimidated by "math talk". I should take my own advice and not be intimidated by "eduspeak".)

Secondly -this is the first time I have heard anyone else say "something must go" - and how timely that it just came up on mathfuture 3 days previously. Every time I have mentioned it in the last 5 years I have been roundly shot down. But I also liked your caveat. Students learn much more when teachers do something that interests them.

Re: Essentials
- I definitely would add (my opinion of) essential skills to your list of essential abilities and of course there would be lots of discussion.
- Logistically I would want examples of e.g. problem solving ability for 6th grade* - grade of D. (ex. Make a drawing of a simple 10'x12' closet with a 7' ceiling. Assume that there are no windows and that the door is simply part of one wall. Put in the 3 dimensions. You need to paint the walls and the ceiling. What do you need to calculate and what is the calculation? Write your final answer in a full sentence using the correct units.)

Best, Linda

*Once I was making problems for 6th grade - I think Florida. It restricted us to using percentages: 5%, 10%, 25%, 50% and 100%. This was the first (and maybe even only) time I saw definitive guidelines like this. I thought this was great.

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Reply from Maria ...

Linda, thank you for writing!
>I should take my own advice and not be intimidated by “eduspeak”.)
For people who are in education, educators surely use too much non-transparent jargon :-) It's fine in places like Journal for Research in Mathematics Education, but...

Secondly –this is the first time I have heard anyone else say “something must go” – and how timely that it just came up on mathfuture 3 days previously.
The problem with many Western curricula is the pace of courses. In Eastern Europe, geometry is stretched over 5-6 years, not one year. You can't process math concepts faster than your personal speed, and the average speed of processing is simply slower than most US curricula require, for example.

> Every time I have mentioned it in the last 5 years I have been roundly shot down. But I also liked your caveat. Students learn much more when teachers do something that interests them.
> Re: Essentials
> -  I definitely would add (my opinion of) essential skills to your list of essential abilities and of course there would be lots of discussion.
I am collecting stories of unschoolers and multiplication:
http://naturalmath.wikispaces.com/Child-Led+Multiplication+Study You can see how WILDLY everything varies. This is an example of why I don't make lists of particular essential skills.
Give me almost any particular skill, and I will show you a person very successful in math and low on the skill. That's why my list is so general and consists of meta-skills like measurement and making of algorithms, rather than "metric lengths and the long division algorithms." If someone pressed me hard enough, I would probably name the skills of counting, splitting (inherently multiplicative operations), exponentiating, covarying (e.g. in tables or proportions), comparing, and linear measures as essential. But these are still more like general areas from my initial list.

> -  Logistically I would want examples of e.g. problem solving ability for 6th grade* - grade of D. (ex. Make a drawing of a simple 10’x12’ closet with a 7’ ceiling. Assume that there are no windows and that the door is simply part of one wall. Put in the 3 dimensions. You need to paint the walls and the ceiling. What do you need to calculate and what is the calculation? Write your final answer in a full sentence using the correct units.)
This is an excellent problem. But not everybody does grades. I don't do grades. The essential math unit for me is a topic. For example, right now two of my math clubs study the topic of stars. Mutually prime numbers are coming up for them. Since kids are young, they have not started that theme yet, which may become their next topic, an offshot of stars.
The same topic can have multiple levels within it. There is no telling how far will each person go within the topic, and at what age. However, I am all for leveled expert systems indexed by levels within topics. So, for example, the message to a student may be that "you are strongly advised to spend time in Prime Numbers up to the level PN3 before continuing in Stars beyond the level S5."

I am attaching a picture of how such a system may look like when it's programmed. I made it last year, using http://bubbl.us/,  for three topics (labeled A, B and C) related to fractions.

This is with the caveat that an expert system is NOT a list of skills essential to everybody, but a map for navigating a particular set of activities in a reasonable manner. I would not even claim such a "map" is necessary to follow within a given set of activities.

MariaD

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geogebra-na@googlegroups.com; on behalf of; Melanie [ms.translate@gmail.com]

Sun 10/17/2010 3:34 PM

Hi Linda and anyone interested,
I found your thoughts interesting, even though I didn't understand the document you were responding to.

Although this wouldn't be my major complaint about math teaching, I have certainly seen some middle school teachers make an overambitious plan for the year, resulting in a breakneck pace and not enough depth of understanding. I think it's fine to take a step back and think about the minimal set of knowledge and techniques needed for the future, i.e. look at your list of topics and put parentheses around the ones that could be included if time permits. You'll get larger, better apples from a young tree if you thin the new fruits in the spring.

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I think the choice of which fruits to keep should be somewhat based on the interests of the particular students and the particular teacher. I've been thinking about what you said about absolutely essential skills. I've noticed that sometimes what students get out of an academic experience isn't exactly what the teacher might have planned.
This past summer my tenth grader took a college freshman writing class, to prepare for graduating early from high school. I'm sure the instructor had an ambitious set of minimal goals, lots of important things he thought the students absolutely needed to get out of the class. I'm not sure how many of those goals my son managed; none of the essays were anything I would want to put in the filing cabinet. But he did learn something I consider important. He learned to make an outline before starting to write. He learned this the hard way. Week after week, essay after essay, he jumped in writing without making an outline, and suffered miserably through the revision process, having to restructure and then rewrite to a ridiculous extent. But he didn't want any meddling (pardon me, I should say 'help') until he got the last assignment, a big project. I said sure, I'd help, but first I needed to see an outline of the points he wanted to put in his essay. Well, that was one painful outline we made together, but then he found out that with a good outline the essay writes itself. This year I've been seeing him writing outlines before he starts his essays. So I would say the freshman writing class was a big success.
What you say about math building on last year's skills and understanding is very true.I sat down to think about what would be important for children to get out of elementary math. Here's what I came up with. They need enough arithmetic to expand a recipe, and to function in the world of shopping and discounts; they need to get in the habit of doing some math on a regular basis; they need to feel adventurous and communicative about problem solving; they need to have experienced the pleasures of abstraction and discovery; and most importantly they all need to believe that they are good at math.

My main complaints about the elementary math teaching I've seen in Ithaca:
- Kindergarten and first grade teachers sometimes use worksheets handed down from teacher to teacher over the years, with no systematic building of skills. - Teachers move the class through the textbook, Everyday Math, in lockstep, with no adjustment of the pace for slower or faster students. - The teacher's guide should indicate which parts of the student book may be omitted for lack of time. What teachers end up doing is skipping nothing, but sending the last quarter of the workbook home "to be finished over the summer". - Counting by 2's, 3's, etc., should be incorporated into everything. Then multiplication will be as natural as breathing. (Currently, it is too painful to describe.) My main complaints about the middle school math teaching I've seen in Ithaca: - Hypnotic homework assignments containing 20 problems all of exactly the same type. - Again, lockstep teaching. The teacher needs to have a variety of challenge sheets ready to hand out to students who finish their classwork early. - Many teachers have understood the ability to work cooperatively (i.e. in small groups) as a do-or-die goal for all students, rather than as an individual learning style that some students possess. (I've seen this at the elementary level, too.)
- Not enough review of what was done a month ago, and three months ago. Students get very good at doing something one month, but a couple of months later, it's gone.
- Too many algebra problems containing messy fractions.
- Classroom dynamics that tend to favor certain quick, show-offy boys and leave most girls moldering in passivity.

Thanks for a thought-provoking letter, and for allowing me to kvetch! -Melanie

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geogebra-na@googlegroups.com; on behalf of; Tony [abcalc09@gmail.com]

Sun 10/17/2010 7:52 PM

I agree with Melanie, there is no well thought out process by elementary, middle, or high school teaching program that allow students to learn that mathematics is a useful tool for everyday life. In the elementary school there are teachers that became elementary teachers assuming that they would not need to do a lot of mathematics. {I have even had them tell me that this was a reason that they chose to teach elementary school.} more...

Many will say that there are well designed, research based teaching programs for mathematics as is evidenced by the national educational programs adopted by 48 of the 50 states, Texas has their TEKS, the NCTM's National Standards, etc. All of these programs are based on what the subject area experts consider to be needed, some even think that they consider the students feelings or desires. Do students really know what they need?...No, they don't, but the lessons and attitudes provided by many instructors do not consider student skills and needs, they are lessons from the teacher's past.
Do I know the solution? Sorry, I wish I did. I do believe that there will be a renaissance coming in education within the next 30-50 years, e-Learning is in its infancy and it will be unrecognizable by the time it fully matures. Whenever, technology can detect the various learning styles of a single student from day to day. I feel that no one student has one specific learning style, each student may have many different learning styles, a style that can vary with in the same subject. Those that are defining learning styles at this time probably find it easier to talk of solitary styles for each student, but I have noticed that a student's learning style can vary depending on the subject and the student's background within the subject.
I need to expound on this as it is a pet peeve of mine.

Tony

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geogebra-na@googlegroups.com; on behalf of; Joseph Malkevitch [malkevitch@york.cuny.edu]
Sun 10/17/2010 8:18 PM

In conjunction with these recent general discussions on k-12 mathematics education you may find the following web page of interest:
http://www.mathismore.net/

Regards, Joe

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Reply from Maria
What are the main activities of Math is More?
Cheers, Maria Droujkova

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Reply from Joe ...

Dear Maria,
Probably the most important thing it was involved with was this Conference:
http://www.education.umd.edu/MathEd/conference/index.html
Materials from the conference are available on the Math is More Web site, and the web pages above.
It is difficult, given the very broad views of the "members," to do much more than push "our point of view" on the importance of teaching via contexts, applicability of mathematics, etc.
I am taking the liberty of linking an essay of mine that "summarizes" my educational philosophy.
Best, Joe

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Revathi Narasimhan

I very much agree with you on this. My 6th grade son is taking a prealgebra course that is extremely difficult to understand. The math supervisor tells me that they have "raised their standards" so that *everyone* will be able to take calculus by 12th grade. That means all algebra, all the time, from 6 through 9 th grade. With all sorts of useless topics - glad I'm not the only one who thinks completing the square is a waste of time.

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The parents want their kids to have calculus on the hs transcript so that they will impress college admissions officers. I'd be willing to bet that their hs calculus is simply glorified algebra.
This attitude is now prevalent in many school districts - more algebra supposedly means better chance in college, even though the statistics show no measurable gain in better math placement. Algebra is now pushed down to 8th, and even 7th grade. And what's this obsession with data analysis, arbitrarily dropped somewhere in hs algebra books?
Meanwhile, so many students fail to understand even basic math - and percentages, proportions, areas, etc. They place into dev math courses in college to just repeat the same fiasco, only twice as fast. Even college math profs teaching lower level math courses cling to every useless topic as well (my textbook would have only been half as thick if I didn't have to include stupid topics to please everyone...).
I too am disturbed to not see this as part of the conversation.
Regards, ...

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From: Jacqueline Barbour
Local: Tues, Oct 19 2010 6:10 pm
Excellent ideas. However, I am worried about which ones to eliminate. Even though I enjoy the multiplying polynomials, I have not find a real life application that uses the objective.

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Reply from Linda

Oh I understand Jacqueline. I so love absolute value inequalities, but I have not found a real-life application and by golly I looked and looked.
However, perhaps there is some way we can find a reasonable set of AESA (absolutely essential skills and abilities), make that a requirement and then let individual teachers decide what else they will teach.
I agree with Maria (and Melanie) here (I think I have this correct Maria/Melanie - if not please write). If a teacher loves something, the kids will love it and learn all different things (including mathematics) from this experience.
Best, Linda

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Edward Cherlin: Saturday, October 23, 2010 6:39 PM

On Tue, Oct 19, 2010 at 12:37, LFS <aliceteaparty@gmail.com> wrote:
> Oh I understand Jacqueline. I so love absolute value inequalities, but I have not found a real-life application and by golly I looked and looked.

Do satellite orbits count as real life in the age of GPS? How about statistical variance in sporting averages, industrial quality control, or political analyses?

Abstractly, of course, one has meromorphic functions defined on an annulus in the complex plane, of the form a>|z|>b>0, with applications to electromagnetism.

> However, perhaps there is some way we can find a reasonable set of AESA (absolutely essential skills and abilities), make that a requirement and then let individual teachers decide what else they will teach.

In reality, AESA should include logic, set theory, combinatorics, number theory, and other topics that we can't discuss due to thefailure of The New Math.

I have had the idea that doing probability and statistics in the context of sports would be adequately "real world", as long as each student gets to choose a sport. The book Money Ball is highly motivating to many. It describes shifting from the statistics that the public likes to follow to those that actually predict winning games, and the major impact that had on baseball economics. Every other team sport has been similarly affected.

I would be interested in following up on the idea of creating lists of essential topics with meaningful real-world examples, and then working out the dependencies among them. In addition, I am interested in using Etoys to build visual models of each topic.