Math So Hard Redux

8/10/2019 Math So Hard Redux

1/77

Why Teaching Math Is So Hard

Dan KennedyBaylor School
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2/77

Teaching math is so hard because

Mathematics is not something you

learn.

Mathematics is something youdo.

This makes mathematics different

from virtually every other subject in

the traditional school curriculum.


3/77

Here are some things that some people can do:

Juggle.Waterski.

Play the violin.

Dunk a basketball.

Run a marathon.

Mathematics.

Other people cannot.
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4/77

People never say, I never could do

English, or I never could do History.

These are subjects you learn.

People do not readily excuse

themselves for not learning.
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5/77

And who defines what it means to do

math?

Math teachers.

This is a big dif ferencebetween the

ability to do mathematics and the ability

to read!


6/77

Someone who can read this sentence

knows how to read.How about this sentence:

Ontogeny recapitulates phylogeny.
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7/77

What doesit mean to do mathematics?

2. 24 6

3. Solve for : ( 2)(2 3) 49.x x x

4. Find cos( /3).

5. Find the product: 874539 374958.

6. Find 239121.7. What is ?ie

1. 5 2


8/77

Do we teach students to domathematics or to

understandmathematics?

2 1 1

1 4 2

1 1 0

2 1

1 4

1 1

+ + +

0+ 2 + 1 (4) (4) 0 = 11


9/77

Compare this to:

So how do we justify teaching a meaninglesscomputational trick that is ONLY good for

computing 3-by-3 determinants?

It does notgeneralize to higher orders.

It does noteven suggest anything important

about how determinants work!


10/77


The people who must identify which

students cannot domathematics are

the same people entrusted with

teaching them.

Their math teachers!


11/77

A sobering thought:

There are people walking the

streets of Buffalo right now who

became convinced years ago thatthey could not do math --

because they could not do

some things that we no longer

even teach today!


12/77

For example, here is what we were doing 45

years ago:

Theorem: (b+ c) + (c) = b

Statement Reason

1.

band care real numbers Hypothesis2.

b+ cis a real number Axiom of closure for addition

3.cis a real number Axiom of additive inverses

4. (b+ c) + (c) = b+ [c+ (c)] Associative axiom of addition

5. c+c= 0 Axiom of additive inverses

6.

b+ [c +(

c)] = b+ 0 Substitution principle7. b+ 0 = b Additive axiom of 0

8. b+ [c+ (c)] = b Transitive property of equality

9. (b+ c) + (c) = b Transitive property of equality


13/77

Teaching mathematics is so hard because:

This is America.

The problem is, this is such an

unusual country.

-- Jan de Lange, Director,

Freeudenthal I nsti tute for

Teaching and Learning,

Utrecht, The Netherlands
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14/77

My Ph.D. was in mathematics; by most

standards, I was very 'well trained.'Nonetheless, the education that I received

was in many ways impoverished.

-- Dr. Alan Schoenfeld,

Reflections on an Impoverished

Education, from Mathematics

and Democracy: The Case forQuanti tative L iteracy, NCED

2001


15/77


Our professional leaders cannotagree on what mathematics we ought

to be teaching!


16/77

In 1983, the College

Board published a

small booklet calledAcademic

Preparation for

College. It listed thefollowing basic

academic

competencies formathematics


17/77

The ability to perform, with reasonable accuracy, the computations of

addition, subtraction, multiplication, and division using natural numbers,

fractions, decimals, and integers.

The ability to make and use measurements in both traditional and metricunits.

The ability to use effectively the mathematics of:

integers, fractions, and decimals;

ratios, proportions, and percentages;

roots and powers; algebra;

geometry

The ability to make estimates and approximations, and to judge the

reasonableness of a result.

The ability to formulate and solve a problem in mathematical terms.

The ability to select and use appropriate approaches and tools in solving

problems (mental computation, trial and error, paper-and-pencil

techniques, calculator, and computer

The ability to use elementary concepts of probability and statistics.


18/77

Ironically, it was that

very same year, 1983,

that another documentwas published, destined

to change the rules for

high school academic

preparation for years tocome

A Nation at Risk: The

Imperative forEducational reform


19/77

From A Nation at Risk:

If an unfriendly foreign

power had attempted to

impose on America the

mediocre educational

performance that exists

today, we might well have

viewed it as an act of war.
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20/77

The Math Wars:

Quantitative Literacy vs.

Calculus Preparation

Theory vs. Applications

Rote vs. Constructivism

Tracking vs. Mainstreaming

Etc.


21/77

No matter what mathematics we choose to

teach or how we choose to teach it, somepeople will believe that we have made the

wrong choice.

Some of them will say so.

And they might be right.
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22/77


Everybody recognizes the importance of

mathematics, even if they do not

understand it.
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23/77

There are only three R's:

Reading Riting Rithmetic

God help us; we are one of them.
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24/77

If Johnny can read and Johnny can

write, then the fate of Johnny will be

determined by whether Johnny can do'rithmetic.

The mathematics teacher is the

Gatekeeper.

Whether Johnny becomes a wealthy

CEO or a penniless beggar is entirely up

to you.

Have a nice day.
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25/77


Most people believe mathematics is

constant over time.

Unlike any other science.

Sort of like religion.

Thou shalt

factor.
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26/77

Parents realize that physics, chemistry,

biology, history, and geography are different

today from when they were in school.

So they allow those subjects to change.

Not so mathematics.
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27/77

The following things upset people:

Inability to do their child's homework.

Any appearance of the words NEW and

MATH in the same sentence.

Calculators.

The assertion that mathematics is for

everyone.


28/77

Some people seem to think that pre-college

mathematics is timeless.

If it was important for our parents, how

can it be unimportant today?

But technology has been rendering ourparents mathematics obsolete for decades.

For example,

consider log tables.


29/77

Here is a 1928

College Board

mathematics

achievement

exam.

It looks a lot like

todays college

placement tests.

But that is

another talk.


30/77

And speaking of logarithms


31/77


Assessment is out of control.

NYS

Regents
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32/77

What should we assess?

What we value

Learning
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33/77

I learned an important fact about classroom

assessment when I began teaching AP courses:

It changes the entire classroom dynamic when the

teacher honestly does not know what will be on the

test.

The teacher has no other option

but to teach the students how tothink for themselves!
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34/77

Why students dont think on tests:

Thinking takes time.

Thinking is only necessary when you cannot

do something without thinking.

If you can do something without thinking, you

can do it very well.

Students who can do something very well have

been well-prepared.

Therefore, if you prepare them well, your

students will proceed through your tests without

thinking!


35/77

This is how the educational game is played:

We show the students how to do math.

We let them practice at it for a while.

Then we give them a test to see how well they can

mimic what we did.

The game is won and lost for BOTH of us on test day.
http://rds.yahoo.com/_ylt=A0WTefepy8BLG0IAjhyjzbkF/SIG=12lq5iuhi/EXP=1271012649/**http%3a//www.nci.tafensw.edu.au/newsevents/news/images/DSC00015.JPG


36/77

We must value correct mathematics

more than we value correct answers.

We must let our students know it.

What we don't see can hurt our

students.


37/77

Good algebra:

2

2

2

3( 1) 12

( 1) 4

( 1) 4 0

( 1 2)( 1 2) 0

( 3)( 1) 03 0 or 1 0

3 or 1

x

x

x

x x

x x

x x

x x


38/77

Bad algebra:

2

2

3( 1) 12

3 3

( 1) 9

1 3

1 or 31 and 3 don't work.

1 and 3

x

x

x

x x

x x


39/77


Articulation is out of control.

High School College
http://rds.yahoo.com/_ylt=A0WTb_zv5b9LkEsAO_.JzbkF;_ylu=X3oDMTBqamdoM3Q5BHBvcwMxMgRzZWMDc3IEdnRpZAM-/SIG=1ftf7grno/EXP=1270953839/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Duniversity%252Bof%252Brochester%2526ei%253Dutf-8%26w=490%26h=344%26imgurl=www.polisci.ucsd.edu%252F%257Esrickard%252Froc.jpeg%26rurl=http%253A%252F%252Fpolisci.ucsd.edu%252F%257Esrickard%252Flinks.html%26size=145k%26name=roc%2bjpeg%26p=university%2bof%2brochester%26oid=8a9f891b12ee5cba%26fr2=%26no=12%26tt=123362%26sigr=11cecfsdp%26sigi=117du0dvb%26sigb=12fak69tphttp://rds.yahoo.com/_ylt=A0WTb_yk5b9LnTAAZHqjzbkF/SIG=131um27i2/EXP=1270953764/**http%3a//family-images.com/misc/schools/Duquesne%2520PA%2520High%2520School.jpg


40/77


41/77

Currently, the greatest growth in the high

school curriculum is in courses that have

traditionally been taught in colleges.

The greatest growth in the college curriculum

is in courses that have traditionally been taught

in high schools.

It is not clear that either institution is serving

its clients very well.

--Dr. Bernard Madison, Chair of the

MAA Task Force on Articulation, 2002

350000


42/77

0

50000

100000

150000

200000

250000

300000

350000

1950 1960 1970 1980 1990 2000 2010 2020

u

u

u

u

uu

1955

285 exams

1967

10,703 exams

1986

51,273 exams

1993

101,945 exams

2003

212,794 exams

2008

276,004 exams

unofficial 2009 point


43/77

What once was designed to

be a program for validating

college-level study of a fewcourses by a small

percentage of exceptional

students for the purposes of

college placement

has gradually morphed into a

program for validating a

rigorous high-schoolcurriculum in a wide variety

of courses for the purposes

of college admissions.


44/77

What effect is this AP

scramble having on

the students?

On the one hand, they

are condensing or

skipping foundationalcourses, so they are

less prepared for

advanced courses.

On the other hand, they are taking more

advanced courses, assuring that their lack of

preparation will be exposed!


45/77

The Race to Begin Algebra I

8thGrade 7thGrade 6thGrade
http://www.pearsonschool.com/index.cfm?locator=PSZ16d&PMDBSUBCATEGORYID=&PMDBSITEID=2781&PMDBSUBSOLUTIONID=&PMDBSOLUTIONID=6724&PMDBSUBJECTAREAID=&PMDBCATEGORYID=806&PMDbProgramID=66782http://rds.yahoo.com/_ylt=A0WTefdwzMBLHk4AdFejzbkF/SIG=128ddkbd1/EXP=1271012848/**http%3a//www.performancecourse.com/images/runners1.jpg


46/77

College Mathematicians want our best high

school students to have:

Deeper understanding

More proofs

Harder problems

Challenges like the AMC Competitions

No rush through foundational courses

No courses in high school beyond

freshman calculus


47/77

What some colleges

are doing now:

Demanding 4s or

5s for credit

Changingfreshman calculus

to a course that assumes students have

already studied calculus


48/77

How students are reacting:

They are less likelyto seek credit.

They are more likely

to play it safe.

They re-take calculus in college

unless they opt out of collegemathematics and take the credit and

run.


49/77

Other secondary-to-college articulation problems:

Technology.

Pedagogy.

Placement tests.

Math wars continued.

Graduate students.

Etc.
http://rds.yahoo.com/_ylt=A0WTb_mTucBLRg8AEzOjzbkF/SIG=12582cnbg/EXP=1271008019/**http%3a//www.flickr.com/photos/torteloni/516199427/http://rds.yahoo.com/_ylt=A0WTb_u9tcBLhx0AJvSjzbkF/SIG=12p2j76se/EXP=1271007037/**http%3a//blog.spout.com/wp-content/uploads/the-paper-chase-houseman.jpg


50/77


Technology is out of control.
http://rds.yahoo.com/_ylt=A9iby6LaXtZF_04AtSCJzbkF;_ylu=X3oDMTBjcDR2NTN2BHBvcwM2BHNlYwNzcg--/SIG=1epqou3i3/EXP=1171763290/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dcasio%252Bfx-7000%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=136%26h=272%26imgurl=www.gearbits.com%252Fimages%252Ffx7000g.jpg%26rurl=http%253A%252F%252Fwww.gearbits.com%252Farchives%252Findustry%26size=16.2kB%26name=fx7000g.jpg%26p=casio%2Bfx-7000%26type=jpeg%26no=6%26tt=18%26oid=5c089d8786b657cc%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiXxXNZFb2cAIguJzbkF;_ylu=X3oDMTBkbzMyZHR2BHBvcwMzOQRzZWMDc3I-/SIG=1ehcs5jeq/EXP=1171762801/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dti-89%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526b%253D21%26w=100%26h=210%26imgurl=www.fisicachimica.it%252Fimages%252Fmat.TI-89%252FTI-89.jpg%26rurl=http%253A%252F%252Fwww.fisicachimica.it%252Fmatematica.htm%26size=12.2kB%26name=TI-89.jpg%26p=ti-89%26type=jpeg%26no=39%26tt=7,345%26oid=69fe518151b1e0fc%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMig6XNZFtvQAeWOJzbkF;_ylu=X3oDMTBkbWJybHJzBHBvcwMxOQRzZWMDc3I-/SIG=1e3a16u4n/EXP=1171762618/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dhp-48%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=100%26h=223%26imgurl=www.hpcc.org%252Fcalculators%252Fhp48gx.jpg%26rurl=http%253A%252F%252Fwww.hpcc.org%252Fcalculators%252Fhp48.html%26size=7.9kB%26name=hp48gx.jpg%26p=hp-48%26type=jpeg%26no=19%26tt=940%26oid=a1e20d78acc1f3dc%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiQNXNZFgHMBWn2JzbkF;_ylu=X3oDMTBjZGM1ZGE1BHBvcwM1BHNlYwNzcg--/SIG=1g3bs25pj/EXP=1171762573/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dti-86%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=176%26h=372%26imgurl=www.engineersupply.com%252Fimages%252Fproducts%252Fti_calc_ti86_med.jpg%26rurl=http%253A%252F%252Fwww.engineersupply.com%252FProductInfo.aspx%253Fproductid%253DES14%26size=19.9kB%26name=ti_calc_ti86_med.jpg%26p=ti-86%26type=jpeg%26no=5%26tt=1,495%26oid=ed6e91ffe4585d34%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiS9W9ZFqXMB1l2JzbkF;_ylu=X3oDMTBjdmNoOTVjBHBvcwMyBHNlYwNzcg--/SIG=1ek7tioh1/EXP=1171762493/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dti-84%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=314%26h=600%26imgurl=salestores1.com%252Fstores%252Fimages%252Fimages_747%252FTI84.jpg%26rurl=http%253A%252F%252Fsalestores1.com%252Fteintiplgrca.html%26size=30.6kB%26name=TI84.jpg%26p=ti-84%26type=jpeg%26no=2%26tt=1,529%26oid=6ca51ebe3663de7a%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiSTW9ZFgHMBAVSJzbkF;_ylu=X3oDMTBjcXBoZjEwBHBvcwMzBHNlYwNzcg--/SIG=1etda382t/EXP=1171762451/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dti-92%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=669%26h=380%26imgurl=www.datamath.org%252FGraphing%252FImages%252FTI-92_PLUS.jpg%26rurl=http%253A%252F%252Fwww.datamath.org%252FGraphing%252FTI-92PLUS.htm%26size=110.7kB%26name=TI-92_PLUS.jpg%26p=ti-92%26type=jpeg%26no=3%26tt=993%26oid=9cccc47d6ab85934%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiBTW9ZFI2wAkxuJzbkF;_ylu=X3oDMTBkN2lwNjljBHBvcwMxOARzZWMDc3I-/SIG=1e4thvgju/EXP=1171762387/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dti-85%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=114%26h=217%26imgurl=plaza.ufl.edu%252Fdrobeson%252FTI-85.jpg%26rurl=http%253A%252F%252Fplaza.ufl.edu%252Fdrobeson%252Fcalc_page1.html%26size=16.3kB%26name=TI-85.jpg%26p=ti-85%26type=jpeg%26no=18%26tt=583%26oid=f94a390c17b2c388%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiMsW9ZFXuUAHjWJzbkF;_ylu=X3oDMTBjcDR2NTN2BHBvcwM2BHNlYwNzcg--/SIG=1eenah9o9/EXP=1171762348/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dti-82%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=143%26h=284%26imgurl=www.yalosinstruments.com%252Fimages%252FTI82.jpg%26rurl=http%253A%252F%252Fwww.yalosinstruments.com%252Fgraficas%252FTI82.htm%26size=20.5kB%26name=TI82.jpg%26p=ti-82%26type=jpeg%26no=6%26tt=886%26oid=97a0ed93352c540a%26ei=UTF-8


51/77

Technology has transformed our classrooms

Graphing calculators Computers Math software

TI Navigator Smart Boards TI Smart View
http://www.tech-line-inc.com/ProductImages/navigator/navgeobndl.jpghttp://rds.yahoo.com/_ylt=A0WTb_wNmoFLlBQAF_CJzbkF;_ylu=X3oDMTBpZm5udGl1BHBvcwM1BHNlYwNzcgR2dGlkAw--/SIG=1is559kum/EXP=1266871181/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dcomputer%2526ei%253DUTF-8%2526fr%253Dyfp-t-701%26w=400%26h=400%26imgurl=static.howstuffworks.com%252Fgif%252FproductImages%252F5%252F1%252F00000115951-DellXPSM2010SeriesPortableComputer-large.jpeg%26rurl=http%253A%252F%252Fproducts.howstuffworks.com%252Fdell-xps-m2010-series-portable-computer-review.htm%26size=16k%26name=00000115951%2bDell...%26p=computer%26oid=89dad5af51710dfe%26fr2=%26no=5%26tt=36558542%26sigr=12kmk7dii%26sigi=1388fbdjs%26sigb=12d87663e


52/77

We must honestly assess every advance in

technology for its appropriate usesin theclassroom.

As noted before, we must also determine

what is meant by important mathematics.

2

42

b b aca

Important?

Expendable?

Th Sk d 2020


53/77

The Skandu 2020:

It has the potential to scan any

standard algebra textbookproblem directly into its

memory for an analysis of key

instructional words, solve itwith CAS, and display all

possible solutions.

It will do the same forstandard geometry textbook

proofs.

The Skandu 2020

Not its real name)
http://rds.yahoo.com/_ylt=A9ibyGXFAP9FCJYATXuJzbkF;_ylu=X3oDMTBjdmNoOTVjBHBvcwMyBHNlYwNzcg--/SIG=1eb7r4br3/EXP=1174426181/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dsmiley%252Bface%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=600%26h=600%26imgurl=www.theportalcafe.com%252Fimages%252FSMILEY.png%26rurl=http%253A%252F%252Fwww.theportalcafe.com%252F%26size=23.3kB%26name=SMILEY.png%26p=smiley%2Bface%26type=png%26no=2%26tt=28,390%26oid=5a237a9d3f9189c6%26ei=UTF-8


54/77

HA HA! Im only kidding.

At least for now.

If there is no Skandu 2020 in our classrooms

in five years, I doubt it will be because thedesign is impossible.

It will be because teachers do not feel that it

would improve the teaching and learning ofimportant mathematics.
http://rds.yahoo.com/_ylt=A9ibyGXFAP9FCJYATXuJzbkF;_ylu=X3oDMTBjdmNoOTVjBHBvcwMyBHNlYwNzcg--/SIG=1eb7r4br3/EXP=1174426181/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dsmiley%252Bface%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=600%26h=600%26imgurl=www.theportalcafe.com%252Fimages%252FSMILEY.png%26rurl=http%253A%252F%252Fwww.theportalcafe.com%252F%26size=23.3kB%26name=SMILEY.png%26p=smiley%2Bface%26type=png%26no=2%26tt=28,390%26oid=5a237a9d3f9189c6%26ei=UTF-8


55/77

Obviously, the CAS conversations continue.

It just might be time for another change!

They are not just about technology, nor

should they be. They are about the teaching

and learning of mathematics.Stay tuned. Be informed. Join the

conversation.


56/77

Now it may be time to change this


57/77

Imaginethe SchoolPad 2 12
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58/77


There is so much more to teachingmathematics than just teaching

mathematics.


59/77

WHAT MATHEMATICS TEACHERS HAD TO DO

FOR HIGH SCHOOL STUDENTS IN THE PAST

1. Prepare the college-bound for calculus.2. Prepare the non-college-bound for employment.

3. Identify which students were which.


60/77

WHAT MATHEMATICS TEACHERS HAVE TODO FOR HIGH SCHOOL STUDENTS TODAY

1. Prepare them all for calculus.

2. Prepare them all for employment.

3. Prepare them all for a life dominated by

computer technology.

4. Prepare them all to pass state-mandatedcompetency tests.


61/77

WHAT HIGH SCHOOL MATH TEACHERS

HAD TO KNOW IN ORDER TO TEACHPROBLEM SOLVING IN THE PAST

1. Algebra.

2. Geometry.3. Probability.

4. Trigonometry.


62/77

WHAT HIGH SCHOOL MATH TEACHERS

HAVE TO KNOW IN ORDER TO TEACH

PROBLEM SOLVING TODAY

1. Mathematics.

2. Astronomy.3. Economics.

4. Statistics.

5. Physics

6. Biology.7. Etc.

Th f


63/77

The former

paradigm:

Learn themathematics in a

context-free

setting, thenapply it to a

section of word

problems at theend of the

chapter.


64/77

I can do the math

I just cant do

the stupid WORDPROBLEMS!

G hi l l t h d d
http://www.free-graphics.com/clipart/Education/Classroom_And_Activities/thumbnails3.shtml


65/77

Graphing calculators have made word

problems more accessible to students. The

emphasis has shifted much more towardmodeling.


66/77

An example of a problem that used to be

hard for students but that now is easy:

Three families order lunch at a fast food restaurant. The Jacksonspay $19.40 for 5 hamburgers, 3 small fries, and 5 soft drinks. TheGarcias pay $11.05 for 3 hamburgers, 2 small fries, and 2 softdrinks. The Lorenzos pay $21.25 for 6 hamburgers, 4 small fries,

and 3 soft drinks. How much would a person pay at this restaurantfor one burger, one small order of fries, and one soft drink?

5 3 5 19.40

3 2 2 11.05

6 4 3 21.25

h f d

h f d

h f d


67/77

After modeling the problem, there are two

easy methods of solving it:


68/77

For teachers, these changes have not come

easily.

We havemade changes, hopefully for the

better.

You might think we could pause, reflect, andenjoy what we have accomplished.

But that is not how technology works!

Here are a few changes we have yet to

make


69/77

We need to stop thinking of a students

mathematics education as a linearprogression of skills that must be mastered.

Arithmetic Fractions Factoring

Equations Inequalities Radicals

Geometry Trigonometry

FunctionsCalculus Statistics

Proofs


70/77

If students who have not mastered our

traditional mathematics skills can solve

problems with technology, should it be ourrole as mathematics teachersto prevent them,

or even discourage them, from doing so?

Dr. Retro, Ive got it!

That does not count,

Miss Nouveau. Put

that thing away.
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71/77

We ALL must teach fundamental

mathemics skills to our students, whoprobably will not have mastered them.

Patiently. Casually. As a matter of course.

Mr. Oiler, if there are twice as many dogs as

cats, doesnt that mean that 2d= c?

M J if th t i ll l d d
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72/77

Mr. Jones, if that is all you learned

last year, you had better drop this

course before it drops you.

Good question, Mr. Jones. Lets

see what would happen if there

were 4 cats
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73/77

We must honestly confront the goals of our

current mathematics curricula.

Just because it is good mathematics does

not mean that we have to keep teaching it.

Nor is it necessary,

advisable, or perhaps even

possibleto teach

everything that is in your

textbook.
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74/77

We should treat every mathematics course

as a history courseat least in part.We will probably always teach some topics

for their historical value.
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75/77

MATHEMATICS

Culture

Quantitative

Literacy

Technology

Mental

Discipline

Research

College

Prep)

But whatever you do have fun!


76/77

But whatever you do, have fun!

Learning is enjoyable.

So is teaching.

Any other philosophy of teaching

and learning is counterproductive.

Even in mathematics!


77/77

[email protected]

Documents

Math So Hard Redux