Post on 02-Jun-2018
8/10/2019 Math So Hard Redux
1/77
Why Teaching Math Is So Hard
Dan KennedyBaylor School
http://rds.yahoo.com/_ylt=A0WTb_gDMb1L5w8A2hqJzbkF;_ylu=X3oDMTBqajcyaWVvBHBvcwM4MwRzZWMDc3IEdnRpZAM-/SIG=1k1665s6c/EXP=1270776451/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dfrustrated%252Bteacher%2526b%253D64%2526ni%253D21%2526ei%253Dutf-8%2526xargs%253D0%2526pstart%253D1%2526fr%253Dyfp-t-701%26w=500%26h=333%26imgurl=farm3.static.flickr.com%252F2486%252F4170724378_a7344fcfae.jpg%26rurl=http%253A%252F%252Fwww.flickr.com%252Fphotos%252Fjustinpoliachik%252F4170724378%252F%26size=105k%26name=Day%2b135%2b365%2b%2bWHY%26p=frustrated%2bteacher%26oid=45d928a5a8e60dfe%26fr2=%26fusr=jp3553%26no=83%26tt=532%26b=64%26ni=21%26sigr=11odhrhfh%26sigi=11mhlvjio%26sigb=13ji15cpo8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A0WTb_wrNb1LSxEAL7GjzbkF/SIG=129sfvupd/EXP=1270777515/**http%3a//www.flickr.com/photos/46945810@N07/4307344455/http://rds.yahoo.com/_ylt=A0WTb_m9NL1LMAEAJpWjzbkF/SIG=12k2497dc/EXP=1270777405/**http%3a//www.abc.net.au/science/news/img/health/juggling150506.jpg8/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A0WTb_ug0MBL_XMA89mJzbkF;_ylu=X3oDMTBpaWhqZmNtBHBvcwMzBHNlYwNzcgR2dGlkAw--/SIG=1hn5hvmkd/EXP=1271013920/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dabraham%252Blincoln%2526ei%253Dutf-8%26w=456%26h=599%26imgurl=www.teclasap.com.br%252Fblog%252Fwp-content%252Fuploads%252F2009%252F02%252Fabraham_lincoln.jpg%26rurl=http%253A%252F%252Fwww.teclasap.com.br%252Fblog%252F2009%252F02%252F23%252Ffrases-abraham-lincoln-3%26size=35k%26name=abraham%2blincoln%26p=abraham%2blincoln%26oid=5487f0d49ed4e21a%26fr2=%26no=3%26tt=394652%26sigr=123sn7i9o%26sigi=1279c73kc%26sigb=127753heg8/10/2019 Math So Hard Redux
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!
8/10/2019 Math So Hard Redux
6/77
Someone who can read this sentence
knows how to read.How about this sentence:
Ontogeny recapitulates phylogeny.
http://rds.yahoo.com/_ylt=A0WTefSlJ8FLPRgATbOjzbkF/SIG=12u29vvbv/EXP=1271036197/**http%3a//elementy.ru/images/eltbook/ontogeny_recapitulates_phylogeny_500.jpg8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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!
8/10/2019 Math So Hard Redux
10/77
Teaching math is so hard because
The people who must identify which
students cannot domathematics are
the same people entrusted with
teaching them.
Their math teachers!
8/10/2019 Math So Hard Redux
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!
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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
http://rds.yahoo.com/_ylt=A0WTefitvsBLx34Acs.JzbkF;_ylu=X3oDMTBqZDFlYmxzBHBvcwMxNgRzZWMDc3IEdnRpZAM-/SIG=1jhvvg629/EXP=1271009325/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Damerican%252Bflag%252Bclip%252Bart%2526rs%253D0ni%25253D21%26w=450%26h=395%26imgurl=images.clipartof.com%252Fsmall%252F37449-Clipart-Illustration-Of-A-Patriotic-American-Heart-Character-In-A-Hat-Flag-On-Independence-Day.jpg%26rurl=http%253A%252F%252Fwww.clipartof.com%252Fdetails%252Fclipart%252F37449.html%26size=75k%26name=37449%2bClipart%2bIl...%26p=american%2bflag%2bclip%2bart%26oid=193f12b33e8b59e4%26fr2=%26no=16%26tt=3082%26sigr=11j3j3tem%26sigi=143j6hhcl%26sigb=12hdvs1cnhttp://rds.yahoo.com/_ylt=A0WTefjivcBL0nQA8gCJzbkF;_ylu=X3oDMTBpdnJhMHUzBHBvcwMxBHNlYwNzcgR2dGlkAw--/SIG=1gv9q5hbt/EXP=1271009122/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Djan%252Bde%252Blange%252Bfreudenthal%2526ei%253Dutf-8%26w=165%26h=220%26imgurl=www.fi.uu.nl%252Fimages%252Fmedewerkers%252Fgroot%252Fjan.jpg%26rurl=http%253A%252F%252Fwww.fi.uu.nl%252Fnl%252Fmedewerkers%252Fmedewerkers%252Fmedewerker27.html%26size=55k%26name=jan%2bjpg%26p=jan%2bde%2blange%2bfreudenthal%26oid=df78aa31c1b70966%26fr2=%26no=1%26tt=2%26sigr=1202ahand%26sigi=11dugfua5%26sigb=12gaksaj08/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
15/77
Teaching math is so hard because
Our professional leaders cannotagree on what mathematics we ought
to be teaching!
8/10/2019 Math So Hard Redux
16/77
In 1983, the College
Board published a
small booklet calledAcademic
Preparation for
College. It listed thefollowing basic
academic
competencies formathematics
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A9G_bDsERTRK1VYBcDKJzbkF;_ylu=X3oDMTBxYjc2MGU1BHBvcwMzBHNlYwNzcgR2dGlkA0kxMTBfMTMx/SIG=1h3kthqfn/EXP=1245025924/**http%3A//images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Datomic%252Bbomb%2526ei%253Dutf-8%26w=360%26h=492%26imgurl=www.everyscreen.com%252Fphotos_03%252FHiroshimaAtomicBomb_02.jpg%26rurl=http%253A%252F%252Fforums.military.com%252Feve%252Fforums%252Fa%252Ftpc%252Ff%252F672198221%252Fm%252F4550042402001%26size=120k%26name=HiroshimaAtomicB...%26p=atomic%2Bbomb%26oid=9956df3a7f6902ce%26fr2=%26no=3%26tt=107315%26sigr=127g69jf5%26sigi=11om3ge41%26sigb=123mams4p8/10/2019 Math So Hard Redux
20/77
The Math Wars:
Quantitative Literacy vs.
Calculus Preparation
Theory vs. Applications
Rote vs. Constructivism
Tracking vs. Mainstreaming
Etc.
8/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A0WTb_tuh75L53kA3UejzbkF/SIG=12kvd4acf/EXP=1270864110/**http%3a//thetimefinder.com/wp-content/uploads/2009/02/scolding.jpg8/10/2019 Math So Hard Redux
22/77
Teaching mathematics is so hard because:
Everybody recognizes the importance of
mathematics, even if they do not
understand it.
http://rds.yahoo.com/_ylt=A0WTb_ss2cBLMEQAvdujzbkF/SIG=122uhbll4/EXP=1271016108/**http%3a//www.uh.edu/engines/isaacnewtonstamp.jpg8/10/2019 Math So Hard Redux
23/77
There are only three R's:
Reading Riting Rithmetic
God help us; we are one of them.
http://rds.yahoo.com/_ylt=A0WTb_46ib5LsGsAK1CJzbkF;_ylu=X3oDMTBpZm5udGl1BHBvcwM1BHNlYwNzcgR2dGlkAw--/SIG=1d3p6hb07/EXP=1270864570/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dmath%26w=376%26h=254%26imgurl=page.bps101.net%252Fweb%252Ft2370%252Fmath.gif%26rurl=http%253A%252F%252Fpage.bps101.net%252Fweb%252Ft2370%26size=5k%26name=math%2bgif%26p=math%26oid=945cde012d517eee%26fr2=%26no=5%26tt=7990459%26sigr=110l2dt8p%26sigi=1120u71cn%26sigb=11jrkltmihttp://rds.yahoo.com/_ylt=A9G_bF7TuC5KUosAHp2JzbkF;_ylu=X3oDMTByNjJjZTY3BHBvcwMxMQRzZWMDc3IEdnRpZANJMTEwXzEzMQ--/SIG=1f3tdtkso/EXP=1244662355/**http%3A//images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dquill%252Bpen%2526ei%253Dutf-8%2526fr%253Dyfp-t-501-s%26w=130%26h=98%26imgurl=www.criticalpages.com%252FImages%252FQuill%252520pen.jpg%26rurl=http%253A%252F%252Fwww.criticalpages.com%252F%26size=7k%26name=Quill%2Bpen%2Bjpg%26p=quill%2Bpen%26oid=e811938a31265616%26fr2=%26no=11%26tt=12297%26sigr=10t6dci49%26sigi=11c93vph4%26sigb=12ghengaihttp://rds.yahoo.com/_ylt=A9G_bF97uC5KDfwAhiWJzbkF;_ylu=X3oDMTBxdHZyM2NlBHBvcwM4BHNlYwNzcgR2dGlkA0kxMTBfMTMx/SIG=1j3olo518/EXP=1244662267/**http%3A//images.search.yahoo.com/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dold%252Bbook%2526ei%253Dutf-8%2526y%253DSearch%2526fr%253Dyfp-t-501-s%26w=111%26h=107%26imgurl=www.bankersoft.de%252FCMS%252Fimages%252Fstories%252Fold_book_small.jpg%26rurl=http%253A%252F%252Fwww.bankersoft.de%252FCMS%252Findex.php%253Foption%253Dcom_content%2526task%253Dview%2526id%253D12%2526Itemid%253D1%26size=2k%26name=old%2Bbook%2Bsmall%2Bj...%26p=old%2Bbook%26oid=b1151b2420f96d26%26fr2=%26no=8%26tt=676589%26sigr=12i9255au%26sigi=11n4rg9er%26sigb=12olj5nc68/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A0WTb_yGi75L7lMArteJzbkF;_ylu=X3oDMTBpc2ozM2gzBHBvcwM0BHNlYwNzcgR2dGlkAw--/SIG=1f7teb4b2/EXP=1270865158/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dmath%252Bteacher%2526js%253D1%2526ei%253Dutf-8%26w=648%26h=344%26imgurl=www.phillipmartin.info%252Fclipart%252Fmath_teacher.gif%26rurl=http%253A%252F%252Fmy.uen.org%252Fmyuen%252F94326%26size=31k%26name=math%2bteacher%2bgif%26p=math%2bteacher%26oid=cf2dd6548c5939e8%26fr2=%26no=4%26tt=101710%26sigr=10t3bilrn%26sigi=11f4ujd5a%26sigb=129j4i4vdhttp://rds.yahoo.com/_ylt=A0WTb_4Di75L7QIAvQaJzbkF;_ylu=X3oDMTBpY2Y5NXNiBHBvcwM2BHNlYwNzcgR2dGlkAw--/SIG=1ff2c793j/EXP=1270865027/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dyoung%252Bboy%2526ei%253Dutf-8%26w=338%26h=506%26imgurl=www.homeaidhouston.org%252Fimages%252Fyoung_boy_smiling.jpg%26rurl=http%253A%252F%252Fwww.homeaidhouston.org%252Findependence_place.html%26size=11k%26name=young%2bboy%2bsmilin...%26p=young%2bboy%26oid=bfbd535206258ac6%26fr2=%26no=6%26tt=318198%26sigr=11l8llc9s%26sigi=11jr3jhfo%26sigb=121cgpr2t8/10/2019 Math So Hard Redux
25/77
Teaching mathematics is so hard because:
Most people believe mathematics is
constant over time.
Unlike any other science.
Sort of like religion.
Thou shalt
factor.
http://rds.yahoo.com/_ylt=A0WTefiP279LhCEA_32jzbkF/SIG=134ds4rro/EXP=1270951183/**http%3a//myfascinatingyear.files.wordpress.com/2008/07/mosesheston2703_468x611.jpg8/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A0WTefVz3b9LcEoAOEKjzbkF/SIG=12bs1iq9p/EXP=1270951667/**http%3a//www.cglapocatiere.qc.ca/sciences/images/atom.jpghttp://rds.yahoo.com/_ylt=A0WTefgL3b9LsigAS0qjzbkF/SIG=12cld2mcg/EXP=1270951563/**http%3a//mikeely.files.wordpress.com/2008/11/alchemist.jpg8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
30/77
And speaking of logarithms
8/10/2019 Math So Hard Redux
31/77
Teaching mathematics is so hard because:
Assessment is out of control.
NYS
Regents
http://rds.yahoo.com/_ylt=A0WTeff04L9LUXkA0KmjzbkF/SIG=12dob0oh5/EXP=1270952564/**http%3a//geology.com/state-map/maps/new-york-county-map.gifhttp://rds.yahoo.com/_ylt=A0WTeff04L9LUXkA0KmjzbkF/SIG=12dob0oh5/EXP=1270952564/**http%3a//geology.com/state-map/maps/new-york-county-map.gifhttp://rds.yahoo.com/_ylt=A0WTefen4b9LcnwArDyJzbkF;_ylu=X3oDMTBpdDZuNzZrBHBvcwM5BHNlYwNzcgR2dGlkAw--/SIG=1h5em0rv5/EXP=1270952743/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dno%252Bchild%252Bleft%252Bbehind%2526ei%253Dutf-8%26w=120%26h=120%26imgurl=www.srvrwf.org%252FProjects%252Fimages%252Fno%252520child%252520left%252520behind.gif%26rurl=http%253A%252F%252Fwww.srvrwf.org%252FProjects%252Fprojects.htm%26size=4k%26name=no%2bchild%2bleft%2bbe...%26p=no%2bchild%2bleft%2bbehind%26oid=5c517e078bcbdc60%26fr2=%26no=9%26tt=34003%26sigr=11bidvih2%26sigi=11taucnf7%26sigb=12c50le5jhttp://rds.yahoo.com/_ylt=A0WTeff04L9LUXkA0KmjzbkF/SIG=12dob0oh5/EXP=1270952564/**http%3a//geology.com/state-map/maps/new-york-county-map.gifhttp://rds.yahoo.com/_ylt=A0WTefc94L9Li30APr.jzbkF/SIG=12t9pu3a0/EXP=1270952381/**http%3a//cdn.newsone.blackplanet.com/wp-content/uploads/2009/06/testing.jpghttp://rds.yahoo.com/_ylt=A0WTefaj379LfAMAql2JzbkF;_ylu=X3oDMTBqbjZwOGU3BHBvcwMzNgRzZWMDc3IEdnRpZAM-/SIG=1mbfl54if/EXP=1270952227/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Derb%252Btesting%2526b%253D22%2526ni%253D21%2526ei%253Dutf-8%2526y%253DSearch%2526xargs%253D0%2526pstart%253D1%26w=135%26h=66%26imgurl=www.severnschool.com%252Fftpimages%252F45%252Fmisc%252Fmisc_26841.jpg%26rurl=http%253A%252F%252Fwww.severnschool.com%252Fcommoninc%252Fpushpage%252Fsevern_pushpage3.asp%253Fsend_id%253D%2526volume_id%253D6802%2526user_id%253D%2526mode%253Dpreview%2526news_id%253D258293%26size=20k%26name=misc%2b26841%2bjpg%26p=erb%2btesting%26oid=7bd9bda27f7346f6%26fr2=%26no=36%26tt=37%26b=22%26ni=21%26sigr=140e1h6fm%26sigi=11l4r2ur5%26sigb=138q3ka1ghttp://rds.yahoo.com/_ylt=A0WTefdr379LoXkAW4.jzbkF/SIG=126qeb11j/EXP=1270952171/**http%3a//apsiatpebblebeach.com/images/cb_logo_08.jpghttp://rds.yahoo.com/_ylt=A0WTefcr379L1HsAf_6jzbkF/SIG=12v0hhokq/EXP=1270952107/**http%3a//www.touchlocal.com/media/business/2759210/act_logo1.jpg%3f1250278235http://rds.yahoo.com/_ylt=A0WTefbj3r9Lo38ACY2JzbkF;_ylu=X3oDMTBpc2ozM2gzBHBvcwM0BHNlYwNzcgR2dGlkAw--/SIG=1if98k6pn/EXP=1270952035/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Deducational%252Btesting%252Bservices%252Blogo%2526sado%253D1%2526ei%253Dutf-8%2526fr2%253Dsg-gac%26w=121%26h=103%26imgurl=urnet.uah.edu%252Fnews%252FnewsImages%252Fets_logo.jpg%26rurl=http%253A%252F%252Furnet.uah.edu%252Fnews%252Fbusinessread.asp%253FnewsID%253D553%26size=12k%26name=ets%2blogo%2bjpg%26p=educational%2btesting%2bservices%2blogo%26oid=4bef7ed8691fb94e%26fr2=sg-gac%26no=4%26tt=25%26sigr=11l5aig9d%26sigi=11arv8lje%26sigb=13b3ichbc8/10/2019 Math So Hard Redux
32/77
What should we assess?
What we value
Learning
http://rds.yahoo.com/_ylt=A0WTefVZ479LhEcAwnejzbkF/SIG=12adflo6f/EXP=1270953177/**http%3a//www.unt.edu/identity/images/overview-sat-lg.jpg8/10/2019 Math So Hard Redux
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!
http://images.google.com/imgres?imgurl=http://www.hawaii.edu/lruby/art400/THINKER.GIF&imgrefurl=http://www.hawaii.edu/lruby/art400/sculpt.htm&h=600&w=428&sz=291&tbnid=On-qO243al4J:&tbnh=132&tbnw=94&prev=/images%3Fq%3Dthinker%26hl%3Den%26lr%3D&oi=imagesr&start=38/10/2019 Math So Hard Redux
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!
8/10/2019 Math So Hard Redux
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.JPG8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
39/77
Teaching mathematics is so hard because:
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.jpg8/10/2019 Math So Hard Redux
40/77
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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!
8/10/2019 Math So Hard Redux
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.jpg8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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.jpg8/10/2019 Math So Hard Redux
50/77
Teaching mathematics is so hard because:
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-88/10/2019 Math So Hard Redux
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=12d87663e8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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-88/10/2019 Math So Hard Redux
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-88/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
56/77
Now it may be time to change this
8/10/2019 Math So Hard Redux
57/77
Imaginethe SchoolPad 2 12
http://www.amazon.com/gp/reader/0768924693/ref=sib_dp_pthttp://www.pearsonschool.com/index.cfm?locator=PSZ16d&PMDBSUBCATEGORYID=&PMDBSITEID=2781&PMDBSUBSOLUTIONID=&PMDBSOLUTIONID=6724&PMDBSUBJECTAREAID=&PMDBCATEGORYID=806&PMDbProgramID=66784http://rds.yahoo.com/_ylt=A0WTb_v6s3RLVDYADQeJzbkF;_ylu=X3oDMTBpcWpidGtpBHBvcwM4BHNlYwNzcgR2dGlkAw--/SIG=1i1i17juu/EXP=1266025850/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dpadlock%252Bicon%2526rs%253D0%2526fr%253Dslv1-adbe%26w=128%26h=128%26imgurl=www.iconarchive.com%252Ficons%252Fgakuseisean%252Fradium-neue%252FPadlock-128x128.png%26rurl=http%253A%252F%252Fwww.iconarchive.com%252Fshow%252Fradium-neue-icons-by-gakuseisean%252FPadlock-icon.html%26size=9k%26name=Padlock%2b128x128%2b...%26p=padlock%2bicon%26oid=73db6720b8c8373e%26fr2=%26no=8%26tt=947%26sigr=12ivsiff7%26sigi=1258q7hfu%26sigb=12d8psabvhttp://rds.yahoo.com/_ylt=A0WTb_0Ds3RLBX8AdkSJzbkF;_ylu=X3oDMTBqamdoM3Q5BHBvcwMxMgRzZWMDc3IEdnRpZAM-/SIG=1fsa06rcd/EXP=1266025603/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dti-connect%252Bsoftware%2526ei%253Dutf-8%2526fr%253Dslv1-adbe%26w=48%26h=48%26imgurl=img.informer.com%252Ficons%252Fpng%252F48%252F30%252F30091.png%26rurl=http%253A%252F%252Fti-connect.software.informer.com%252F%26size=2k%26name=30091%2bpng%26p=ti-connect%2bsoftware%26oid=d9e41e7715d95f48%26fr2=%26no=12%26tt=38%26sigr=118knhnld%26sigi=11ar0atp2%26sigb=12ojfim58http://rds.yahoo.com/_ylt=A0WTb_nrsHRLKjIApGuJzbkF;_ylu=X3oDMTBrMnYxMjV2BHBvcwMxMzEEc2VjA3NyBHZ0aWQD/SIG=1jc65apia/EXP=1266025067/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dhigh%252Bschool%252Blogo%2526b%253D127%2526ni%253D21%2526ei%253Dutf-8%2526xargs%253D0%2526pstart%253D1%2526fr%253Dyfp-t-701%26w=125%26h=124%26imgurl=sites.epals.com%252Fcleveland2%252Ffiles%252Fcleveland2%252Fimages%252F_clevhigh%252520logo.JPG%26rurl=http%253A%252F%252Fsites.epals.com%252Fcleveland2%26size=8k%26name=School%2bLogo%26p=high%2bschool%2blogo%26oid=1caa86db30337dee%26fr2=%26no=131%26tt=126282%26b=127%26ni=21%26sigr=1111cgpmi%26sigi=127497gbs%26sigb=13iet67v8http://rds.yahoo.com/_ylt=A0WTb_jarHRLZ1wAq0WJzbkF;_ylu=X3oDMTBpZTByOGFiBHBvcwMyBHNlYwNzcgR2dGlkAw--/SIG=1g6kb68fk/EXP=1266024026/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Ditunes%252Blogo%2526ei%253Dutf-8%2526fr%253Dyfp-t-701%26w=128%26h=128%26imgurl=upsetthesetup.wordpress.com%252Ffiles%252F2008%252F09%252Fitunes-blue-logo.jpg%26rurl=http%253A%252F%252Fnewgoldenera.com%252F%253Fp%253D99%26size=8k%26name=itunes%2bblue%2blogo%26p=itunes%2blogo%26oid=51efe12f50df077c%26fr2=%26no=2%26tt=34413%26sigr=10ti5pjpd%26sigi=11uu624r1%26sigb=12gf1rvn8http://rds.yahoo.com/_ylt=A0WTb_w1q3RLHHUArkeJzbkF;_ylu=X3oDMTBpY2Y5NXNiBHBvcwM2BHNlYwNzcgR2dGlkAw--/SIG=1inmalvm3/EXP=1266023605/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgrand%252Btheft%252Bauto%2526ei%253Dutf-8%2526fr%253Dyfp-t-701%26w=520%26h=598%26imgurl=www.xgn.nl%252Fimages%252Fboxarts%252Fps3%252Fbig%252Fgrand-theft-auto-iv.jpg%26rurl=http%253A%252F%252Fwww.xgn.nl%252Fps3%252Fnieuws%252F11636%252Fgrand-theft-auto-krijgt-de-schuld-van-dierenmoord%26size=211k%26name=grand%2btheft%2bauto...%26p=grand%2btheft%2bauto%26oid=acfc55364a6240a6%26fr2=%26no=6%26tt=1518471%26sigr=12kg61b28%26sigi=11pcc0lhr%26sigb=12lg0itdmhttp://rds.yahoo.com/_ylt=A0WTb_qMrHRLnzIATj2JzbkF;_ylu=X3oDMTBqczRiNXN2BHBvcwM0MgRzZWMDc3IEdnRpZAM-/SIG=1j181te82/EXP=1266023948/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dpowerpoint%252Blogo%2526b%253D22%2526ni%253D21%2526ei%253Dutf-8%2526xargs%253D0%2526pstart%253D1%2526fr%253Dyfp-t-701%26w=50%26h=50%26imgurl=home.euphonynet.be%252Fbibamus%252Fimages%252Fpowerpoint_logo.gif%26rurl=http%253A%252F%252Fhome.euphonynet.be%252Fbibamus%252Fimages%253FS%253DA%26size=3k%26name=powerpoint%2blogo%2b...%26p=powerpoint%2blogo%26oid=c8a80dcdaf98162a%26fr2=%26no=42%26tt=28892%26b=22%26ni=21%26sigr=11ckfj9hn%26sigi=11lutb7ue%26sigb=13gq7l861http://www.amazon.com/Red-Badge-Courage-Evergreen-Classics/dp/0486434222/ref=sr_1_6?ie=UTF8&s=books&qid=1265936808&sr=1-6http://www.amazon.com/Romeo-Juliet-Shakespeare-Made-Easy/dp/0812035720/ref=sr_1_3?ie=UTF8&s=books&qid=1265936689&sr=1-3http://rds.yahoo.com/_ylt=A0WTefPVqHRLdi0AN1GJzbkF;_ylu=X3oDMTBpdnJhMHUzBHBvcwMxBHNlYwNzcgR2dGlkAw--/SIG=1gduo6rhv/EXP=1266022997/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dmicrosoft%252Bexcel%252Blogo%2526ei%253Dutf-8%2526fr%253Dslv1-adbe%26w=64%26h=64%26imgurl=www.i-community.ch%252Fimages%252Fpics%252Fexcel2003.jpg%26rurl=http%253A%252F%252Fwww.i-community.ch%252Fschulungen%252Fexcel2003%26size=1k%26name=excel2003%2bjpg%26p=microsoft%2bexcel%2blogo%26oid=d24a4a25946caa0a%26fr2=%26no=1%26tt=2184%26sigr=11e348143%26sigi=11c13ps0c%26sigb=12p2depefhttp://rds.yahoo.com/_ylt=A0WTefOkqHRLNi0A.nWJzbkF;_ylu=X3oDMTBqZDFlYmxzBHBvcwMxNgRzZWMDc3IEdnRpZAM-/SIG=1ha5o689n/EXP=1266022948/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253F_adv_prop%253Dimage%2526va%253Dmicrosoft%252Bword%252Blogo%2526fr%253Dslv1-adbe%26w=55%26h=55%26imgurl=www.palmettoshagclub.com%252Fimages%252Fmicrosoft_word_logo_55x55.png%26rurl=http%253A%252F%252Fwww.palmettoshagclub.com%252Fimages%253FM%253DD%26size=36k%26name=microsoft%2bword%2bl...%26p=microsoft%2bword%2blogo%26oid=5db33380fc6638c8%26fr2=%26no=16%26tt=7415%26sigr=11agrdocd%26sigi=11thhkl7u%26sigb=130mr9vl0http://rds.yahoo.com/_ylt=A0geu9tbqHRL7skAZhVXNyoA;_ylu=X3oDMTB0dDM1djdlBHNlYwNzYwRjb2xvA2FjMgR2dGlkA0g1MDRfMTQw/SIG=1iaq71pd3/EXP=1266022875/**http%3a//images.search.yahoo.com/images/view%3fback=http%253A%252F%252Fsearch.yahoo.com%252Fsearch%253Fei%253DUTF-8%2526p%253Dinternet%252Bexplorer%252Blogo%26w=200%26h=203%26imgurl=www.downloadsquad.com%252Fimages%252F2005%252F08%252Fie7logo.jpg%26size=15.4kB%26name=ie7logo%2bjpg%26rcurl=http%253A%252F%252Fwww.ithaber.com%252F%253Fp%253D30%26rurl=http%253A%252F%252Fwww.ithaber.com%252F%253Fp%253D30%26p=internet%2bexplorer%2blogo%26type=jpeg%26no=1%26tt=20%252C966%26oid=d26d7dbb1f5a9d7e%26tit=internet%2bexplorer%26sigr=10sbo9j5h%26sigi=11goi653q%26sigb=120ia4dq9http://www.pearsonschool.com/index.cfm?locator=PSZ16o&filter_161=&filter_422=&filter_423=6732&filter_424=&filter_281=&filter_425=&programFilterTypeList=161%2C422%2C423%2C424%2C281%2C425&PMDbSiteid=2781&PMDbSolutionid=6724&PMDbSubSolutionid=&PMDbCategoryid=815&&PMDbProgramID=60921http://www.pearsonschool.com/index.cfm?locator=PSZ16p&filter_161=&filter_423=6732&filter_424=&filter_425=&filter_281=&programFilterTypeList=161%2C423%2C424%2C425%2C281&PMDbSiteid=2781&PMDbSolutionid=6724&PMDbSubSolutionid=&PMDbCategoryid=813&&PMDbProgramID=64941http://www.pearsonschool.com/index.cfm?locator=PSZ16e&PMDBSUBCATEGORYID=&PMDBSITEID=2781&PMDBSUBSOLUTIONID=&PMDBSOLUTIONID=6724&PMDBSUBJECTAREAID=&PMDBCATEGORYID=3289&PMDbProgramID=56484http://www.pearsonschool.com/index.cfm?locator=PSZ16f&filter_161=&filter_423=&filter_422=&filter_424=&filter_281=&filter_425=&programFilterTypeList=161%2C423%2C422%2C424%2C281%2C425&PMDbSiteid=2781&PMDbSolutionid=6724&PMDbSubSolutionid=&PMDbCategoryid=814&PMDbSubcategoryid=24824&&PMDbProgramID=578218/10/2019 Math So Hard Redux
58/77
Teaching mathematics is so hard because:
There is so much more to teachingmathematics than just teaching
mathematics.
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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.
8/10/2019 Math So Hard Redux
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.shtml8/10/2019 Math So Hard Redux
65/77
Graphing calculators have made word
problems more accessible to students. The
emphasis has shifted much more towardmodeling.
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
67/77
After modeling the problem, there are two
easy methods of solving it:
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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
8/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A9G_Rtnk0P5Foo8A0gqJzbkF;_ylu=X3oDMTBjcXBoZjEwBHBvcwMzBHNlYwNzcg--/SIG=1hskb75ed/EXP=1174413924/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dprofessor%2526fr%253Dyfp-t-501%2526toggle%253D1%2526cop%253Dmss%2526ei%253DUTF-8%26w=391%26h=500%26imgurl=greywolf.critter.net%252Fimages%252Fdeadlands%252Fweirdwars%252Forient%252Fprofessor.jpg%26rurl=http%253A%252F%252Fgreywolf.critter.net%252Fimages%252Fdeadlands%252Fweirdwars%252Forient%253FM%253DA%26size=23.6kB%26name=professor.jpg%26p=professor%26type=jpeg%26no=3%26tt=1,580,394%26oid=4b46b1901fbcfdce%26ei=UTF-88/10/2019 Math So Hard Redux
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
http://rds.yahoo.com/_ylt=A9G_Rqz01f5FLbgAFh6JzbkF;_ylu=X3oDMTBjMHZkMjZyBHBvcwMxBHNlYwNzcg--/SIG=1jq0a4ppj/EXP=1174415220/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dstudent%252Braising%252Bhand%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=152%26h=225%26imgurl=www.free-graphics.com%252Fclipart%252FEducation%252FClassroom_And_Activities%252FStudent_Raising_Hand.jpg%26rurl=http%253A%252F%252Fwww.free-graphics.com%252Fclipart%252FEducation%252FClassroom_And_Activities%252FStudent_Raising_Hand.shtml%26size=10.0kB%26name=Student_Raising_Hand.jpg%26p=student%2Braising%2Bhand%26type=jpeg%26no=1%26tt=116%26oid=665c1d683aa31360%26ei=UTF-88/10/2019 Math So Hard Redux
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
http://rds.yahoo.com/_ylt=A9G_Rq7_1_5FK_wAqxqJzbkF;_ylu=X3oDMTBjYzZubXM2BHBvcwM4BHNlYwNzcg--/SIG=1hoocj3ti/EXP=1174415743/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dcartoon%252Bteacher%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=111%26h=115%26imgurl=www.substituteteachertoday.freewebspace.com%252Fimages%252Fcartoon_teacher_%2528man%2529.jpg%26rurl=http%253A%252F%252Fwww.substituteteachertoday.freewebspace.com%252Fcustom4.html%26size=5.6kB%26name=cartoon_teacher_%2528man%2529.jpg%26p=cartoon%2Bteacher%26type=jpeg%26no=8%26tt=1,767%26oid=fb3aeb8b84542486%26ei=UTF-88/10/2019 Math So Hard Redux
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.
http://www.pearsonschool.com/index.cfm?locator=PSZ16d&PMDBSUBCATEGORYID=&PMDBSITEID=2781&PMDBSUBSOLUTIONID=&PMDBSOLUTIONID=6724&PMDBSUBJECTAREAID=&PMDBCATEGORYID=806&PMDbProgramID=66784http://www.pearsonschool.com/index.cfm?locator=PSZ16d&PMDBSUBCATEGORYID=&PMDBSITEID=2781&PMDBSUBSOLUTIONID=&PMDBSOLUTIONID=6724&PMDBSUBJECTAREAID=&PMDBCATEGORYID=806&PMDbProgramID=667848/10/2019 Math So Hard Redux
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.
http://rds.yahoo.com/_ylt=A9gnMiZkT_9FPyIBmjyJzbkF;_ylu=X3oDMTBjYzZubXM2BHBvcwM4BHNlYwNzcg--/SIG=1hcm1lkig/EXP=1174446308/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dleibniz%2526fr%253Dyfp-t-501%2526toggle%253D1%2526ei%253DUTF-8%2526fr2%253Dtab-web%26w=252%26h=400%26imgurl=www.nlb-hannover.de%252FLeibniz%252FGesellschaft%252FDokumente%252FLeibniz.png%26rurl=http%253A%252F%252Fwww.nlb-hannover.de%252FLeibniz%252FGesellschaft%252FVeranstaltungen%252FKongress%26size=184.5kB%26name=Leibniz.png%26p=leibniz%26type=png%26no=8%26tt=27,175%26oid=4f22000002251e62%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMik3Tv9Fus4AWEyJzbkF;_ylu=X3oDMTBjMHZkMjZyBHBvcwMxBHNlYwNzcg--/SIG=1fr7r393d/EXP=1174446007/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dpythagoras%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=570%26h=728%26imgurl=mason.gmu.edu%252F%257Ejmartin6%252Fhowe%252FImages%252Fpythagoras.jpg%26rurl=http%253A%252F%252Fmason.gmu.edu%252F%257Ejmartin6%252Fpythagpythag.html%26size=130.4kB%26name=pythagoras.jpg%26p=pythagoras%26type=jpeg%26no=1%26tt=20,799%26oid=88b32d1e0373c904%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMinuTf9FXM4AIFGJzbkF;_ylu=X3oDMTBjdmNoOTVjBHBvcwMyBHNlYwNzcg--/SIG=1er5d8g3a/EXP=1174445934/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgalileo%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=244%26h=326%26imgurl=www.mathematik.ch%252Fmathematiker%252FGalileo.jpg%26rurl=http%253A%252F%252Fwww.mathematik.ch%252Fmathematiker%252Fgalileo.php%26size=12.1kB%26name=Galileo.jpg%26p=galileo%26type=jpeg%26no=2%26tt=167,794%26oid=67081d8e501e6dbe%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiChTf9FaCgBg4GJzbkF;_ylu=X3oDMTBjdmNoOTVjBHBvcwMyBHNlYwNzcg--/SIG=1fcqvg6i5/EXP=1174445857/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Deuclid%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=248%26h=295%26imgurl=www.nndb.com%252Fpeople%252F724%252F000087463%252Feuclid-1-sized.jpg%26rurl=http%253A%252F%252Fwww.nndb.com%252Fpeople%252F724%252F000087463%26size=35.8kB%26name=euclid-1-sized.jpg%26p=euclid%26type=jpeg%26no=2%26tt=40,537%26oid=040a1fd214bb2fd6%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiSFTf9F4yIB7yOJzbkF;_ylu=X3oDMTBjMHZkMjZyBHBvcwMxBHNlYwNzcg--/SIG=1f6lfmc5d/EXP=1174445829/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Ddescartes%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=256%26h=273%26imgurl=www.myweblog.com%252Fmisc%252Fdescartes.jpg%26rurl=http%253A%252F%252Fwww.myweblog.com%252Fapollyon%252Fcategory%252F17%252FPhilosophy%26size=4.9kB%26name=descartes.jpg%26p=descartes%26type=jpeg%26no=1%26tt=32,098%26oid=647b7c2261f7b6b4%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiBTTf9F1icBdGGJzbkF;_ylu=X3oDMTBjZmpxdmw3BHBvcwM3BHNlYwNzcg--/SIG=1e9c0lieg/EXP=1174445779/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Deinstein%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=259%26h=342%26imgurl=www.tarrworks.com%252Fimages%252Feinstein.jpg%26rurl=http%253A%252F%252Fwww.tarrworks.com%252Fwraqt1a.html%26size=29.9kB%26name=einstein.jpg%26p=einstein%26type=jpeg%26no=7%26tt=282,946%26oid=bd40c6eb2eb32c8e%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiYeTf9FXx8B52eJzbkF;_ylu=X3oDMTBjZGM1ZGE1BHBvcwM1BHNlYwNzcg--/SIG=1fvnaj037/EXP=1174445726/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Deuler%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=344%26h=400%26imgurl=www.csc.liv.ac.uk%252F%257Efrans%252FProgramming%252FleonhardEuler.jpeg%26rurl=http%253A%252F%252Fwww.csc.liv.ac.uk%252F%257Efrans%252FProgramming%252Feuler.html%26size=16.7kB%26name=leonhardEuler.jpeg%26p=euler%26type=jpeg%26no=5%26tt=33,477%26oid=4721ca233ed06f34%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMinwTP9F084AoQ2JzbkF;_ylu=X3oDMTBkYTNuNGk0BHBvcwMxMARzZWMDc3I-/SIG=1fm9c8l64/EXP=1174445680/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Dgauss%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=271%26h=347%26imgurl=www.knowledgerush.com%252Fwiki_image%252F2%252F22%252FGauss.jpg%26rurl=http%253A%252F%252Fwww.knowledgerush.com%252Fkr%252Fencyclopedia%252FKarl_Friedrich_Gauss%26size=53.5kB%26name=Gauss.jpg%26p=gauss%26type=jpeg%26no=10%26tt=54,845%26oid=2ff472380ebfd992%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMiK7TP9FjawAMXGJzbkF;_ylu=X3oDMTBjcXBoZjEwBHBvcwMzBHNlYwNzcg--/SIG=1fpo05o7p/EXP=1174445627/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Disaac%252Bnewton%2526ei%253DUTF-8%2526fr%253Dyfp-t-501%2526x%253Dwrt%26w=340%26h=307%26imgurl=www.biografiasyvidas.com%252Fbiografia%252Fn%252Ffotos%252Fnewton.jpg%26rurl=http%253A%252F%252Fwww.lunchboxing.com%252F2005_07_24_archive.shtml%26size=8.2kB%26name=newton.jpg%26p=isaac%2Bnewton%26type=jpeg%26no=3%26tt=20,415%26oid=7820e33d9a1eee2c%26ei=UTF-8http://rds.yahoo.com/_ylt=A9gnMicBTP9FLWEA9GqJzbkF;_ylu=X3oDMTBkYTNwY3VnBHBvcwMxNgRzZWMDc3I-/SIG=1gtnia8l9/EXP=1174445441/**http%3A//images.search.yahoo.com/search/images/view%3Fback=http%253A%252F%252Fimages.search.yahoo.com%252Fsearch%252Fimages%253Fp%253Darchimedes%2526fr%253Dyfp-t-501%2526toggle%253D1%2526cop%253Dmss%2526ei%253DUTF-8%26w=360%26h=400%26imgurl=quest.arc.nasa.gov%252Faero%252Fplanetary%252Fimages%252FArchimedes_7.jpeg%26rurl=http%253A%252F%252Fquest.arc.nasa.gov%252Faero%252Fplanetary%252Farchimedes.html%26size=23.8kB%26name=Archimedes_7.jpeg%26p=archimedes%26type=jpeg%26no=16%26tt=27,634%26oid=669c0a48bb6ae2a6%26ei=UTF-88/10/2019 Math So Hard Redux
75/77
MATHEMATICS
Culture
Quantitative
Literacy
Technology
Mental
Discipline
Research
College
Prep)
But whatever you do have fun!
8/10/2019 Math So Hard Redux
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!
8/10/2019 Math So Hard Redux
77/77
dkennedy@baylorschool.org