Here you will see students as young as 4 and 5 years old doing algebra and "advanced" math, without ever knowing it's supposed to be hard. You are invited to learn how to use this method...
a² + b² = c² Is easy to understand when you can see it. I need a new white board this one has gotten quite a bit of use...anyhow the vid at the bottom of the page shows how to find the height of a triangle when you only know the sides.
Here is a young bright Honor student who has all "A's" in everything but math. All students (and a lot parents) blame the teacher first when they fail or don't understand. Sometimes this is true and sometimes the teacher is doing the best they can with the tools they have. Other times the student is a little brat and it wouldn't matter who the teacher was the kid would be failing or getting "C's" or what have you on their own account, not because of the teacher. There is also the personality factor, I don't get along with everybody, I don't know about you but I figured out a long time ago you can't please everybody and you will be unhappy if you try.
I always spend a little time getting to know the student first before I take them on, sometimes it's better to refer them elsewhere or say no than yes. It's pretty rare, but it happens. But I digress.
This student is smart, quick minded and when whatever it is, is explained she usually gets it right away or takes meticulous notes so she can study it later if it's iffy. This way we cover the most ground possible. This student does; however, have huge gaps in her math skill sets that need to be addressed but we usually spend all our time studying for the tests instead of increasing her conceptual knowledge base and brushing up on her computation. As with most students I meet, multiplication tables are not mastered, nor are fractions or the basics of algebra specifically hero zero and no fun get back to one.
Happily Pythagorean Theorem (⇐ click that to go to go to CRHOM's page on it) made sense to her because it is pretty straight forward and my explanation using blocks is visually obvious as the Montessori folk like to say.
Here she is being introduced to Pythagorean Theorem the Crewton Ramone House Of Math way. Note this student is advanced and we are pressed for time so this isn't the best presentation for little kids, go to the house of math for that.
I want to stress this is a sharp high school student. (A little freaked out by math but still very quick minded.) This is NOT how you would present it to a young student or a 6th grader or even a not so sharp junior. More examples and don't jump to subtraction after just one problem. You cab see the confusion this causes, happily I am there to clear things up, but I am not always there. And most often things don't get cleared up. Note how I use easy problems that have answers that are whole numbers that don't require a calculator. She can easily see 10 as an answer as compared to a problem with a long decimal for the answer. This also adds to the confusion. Once you understand the concepts all they can do is change the numbers...
After a very short time she is getting it. But three passes do not a quarterback make. PRACTICE is required. Also note how for the most part she told me the formulas, when I told her the formulas she had already got them she just had a little trouble verbalizing them. Contrast this with just telling them to memorize the formula.
Not the best lesson but a good lesson overall. In my defense we were rushing because we had a lot of ground to cover, she had an upcoming test, and she was flustered. Now, this would normally only be on a password protected page, but in the spirit of Christmas, the Flying Spaghetti Monster appeared to me in a dream while driving a fast car, and told me specifically to make this video available to anyone who could find it. In a few weeks this post will be buried on this blog like many other great posts I've made over the years and you will have to search to find it. I have 100's of videos on youtube now. Finding the one you want is made easier with a password because on my website they are grouped by subject. Also many of the vids are unlisted so like the one below you have to know where it is to find it, it won't show up on a search.
It is also on a couple of the password protected pages like advanced algebra, because the problems shown are classic algebra problems that appear in every textbook ever written about algebra...or just about. Now you might have to be in Algebra II but it's basic algebra application as far as I'm concerned. Very few kids figure out how to do this on their own and over the years many kids have come to me failing to understand how to do these kinds of problems...
Find h.
Students and teachers who have spent time at the house of math have very little trouble understanding this problem because they understand the basic concepts behind each step and the computation doesn't pose a problem. (Get it? I crack myself up.)
You can also see for sure that the drawing on her paper is "not to scale."
I have been told by more than one person the vid above is actually fun to watch because they "get it." A lot of that goes on at the house of math.
I did not write this or gather this information, I just found theis PDF and put on the web. Please feel free to cut and paste this elsewhere. Soon I will have post about a certain Danish study analyzing their graphical data from a purely mathematical point of view. ~Crewton Ramone
Bichel, "Post-vaccinial Lymphadenitis Developing into Hodgkin’s Disease", Acta Med Scand, 1976, Vol 199, p523-525.
Stewart, AM, et al, "Aetiology of Childhood Leukaemia", Lancet, 16 Oct, 1965, 2:789-790. [Listed under Vaccine Adverse Reactions.]
Glathe, H et al, "Evidence of Tumorigenic Activity of Candidate Cell Substrate in Vaccine Production by the Use of Anti-Lymphocyte Serum", Development Biol Std, 1977, 34:145-148.
Bolognesi, DP, "Potential Leukemia Virus Subunit Vaccines: Discussion", Can Research, Feb 1976, 36(2 pt 2):655-656.
Colon, VF, et al, "Vaccinia Necrosum as a Clue to Lymphatic Lymphoma", Geriatrics, Dec 1968, 23:81- 82.
Park-Dincsoy, H et al, "Lymphoid Depletion in a case of Vaccinia Gangrenosa", Laval Med, Jan 1968, 39:24-26.
Hugoson, G et al, "The Occurrence of Bovine Leukosis Following the Introduction of Babesiosis Vaccination", Bibl Haemat, 1968, 30:157-161.
Hartstock, , ""Post-vaccinial Lymphadenitis: Hyperplasia of Lymphoid Tissue That Simulates Malignant Lymphomas", Apr 1968, Cancer, 21(4):632-649.
Allerberger, F, "An Outbreak of Suppurative Lymphadenitis Connected with BCG Vaccination in Austria- 1990/1991," Am Rev Respir Disorder, Aug 1991, 144(2) 469.
Omokoku B, Castells S, "Post-DPT inoculation cervical lymphadenitis in children." N Y State J Med 1981 Oct;81(11):1667-1668.
Vaccines and Chromosome Changes Leading to Mutations:
Knuutila, S et al, "An Increased Frequency of Chromosomal Changes and SCE’s in Cultured Lymphocytes of 12 Subjects Vaccinated Against Smallpox," Hum Genet, 1978 Feb 23; 41(1):89-96.
Cherkeziia, SE, et al, "Disorders in the Murine Chromosome Apparatus Induced By Immunization with a Complex of Anti-viral Vaccines," Vopr Virusol, 1979 Sept Oct, (5):547-550.
[Note: SCE means sister chromatid exchange and is an indication that genetic mutations are occurring, which could possibly lead to cancer-causing mutations.]
Vaccines and Auto-immunity Citations:
Romanov, V A, et al, "Role of Auto-immune Processes in the Pathogenesis of Post-Vaccinal Lesions of the Nervous System", Oct 1977, Zh Mikrobiol Epidemiol Immunobiol, 10:80-83.
Grachev, V P, et al, "Formation of Auto-antibodies in Laboratory Animals After Inoculation of Viruses With Different Virulence. I. Results of Studies ..., July 1973, Acta Virol (Praha), 17:319-326.
Movsesiants, AA, et al, "Experimental Study of the Ability of Different Strains of Vaccinia Virus to Induce Auto-Antibody Formation", Vopr Virusol, May-Jun 1975; (3):297-302.
Negina, IuP, "Comparative Study of Auto-antibody Formation Following Immunization With Different Types of Typhoid Vaccines", Zh Mikrobiol Epidemiol Immunobiol, May 1980; (5):69-72.
Vaccinations and Diabetes Citations:
Sinaniotis, et al, "Diabetes Mellitus after Mumps Vaccination", Arc Dis Child, 1975, 50:749.66
Polster, H, "Diabetes insipidus after Smallpox vaccination", Z Aerztl Fortbild (Jena), 1 Apr 1966, 60:429-432.
Patan, "Postvaccinal Severe Diabetes Mellitus", Ter Arkh, Jul 1968, 40:117-118.
Classen, JB, MD, "The Timing of Immunization Affects The Development of Diabetes in Rodents", Autoimmunity, 1996, 24:137-145.
Classen JB, "The diabetes epidemic and the hepatitis B vaccines," N Z Med J, 109(1030):366 1996 Sep 27. [letter]
Classen JB, "Childhood immunisation and diabetes mellitus," N Z Med J, 109(1022):195 1996 May 24 [letter]
Poutasi K, " Immunisation and diabetes," N Z Med J 1996 Jul 26;109(1026):283. [letter; comment]
Other Articles Linking Diabetes to Vaccines:
Dokheel, T M, "An Epidemic of Childhood Diabetes in the United States? Evidence from ....", Diabetes Care, 1993, 16:1606-1611.
Parent ME, et al, "Bacille Calmette-Guerin vaccination and incidence of IDDM in Montreal, Canada," Diabetes Care 1997 May; 20(5):767-772.
House DV, Winter WE, "Autoimmune diabetes. The role of auto-antibody markers in the prediction and prevention of insulin-dependent diabetes mellitus," Clin Lab Med 1997 Sep; 17(3):499-545.
Zeigler, M et al , "[Autoantibodies in type 1 diabetes mellitus]" Z Arztl Fortbild (Jena). 1994 Aug; 88(7- 8):561-5
Vaccines and Nervous System Changes:
Bondarev, VN et al, "The Changes of the Nervous System in Children After Vaccination", Pediatria, Jun 1969; 48:20-24.
Ehrengut W, "Central nervous sequelae of vaccinations," Lancet 1986 May 31;1(8492):1275-1276.
Provvidenza, G et al, [On a Case of Benign Acute Cerebellar Ataxia in Childhood], Arch Ital Sci Med Trop, 43:189-194, Apr 1962.
Katsilambros, L, "[The Phenomenom of Apathy in Man and Animals After the Injection of Viruses in Very High Doses. Clinical Data]", Rev Med Moyen Orient, 20:539-546, Nov – Dec 1963.
Vaccinations and Autism Citations:
Eggers, C, "Autistic Syndrome (Kanner) And Vaccinations against Smallpox", Klin Paediatr, Mar 1976, 188(2):172-180.
Kiln MR, "Autism, inflammatory bowel disease, and MMR vaccine." Lancet 1998 May 2;351(9112):1358.
Selway, "MMR vaccination and autism 1998. Medical practitioners need to give more than reassurance." BMJ 1998 Jun 13;316(7147):1824.
Nicoll A, Elliman D, Ross E, "MMR vaccination and autism 1998," MJ 1998 Mar 7;316(7133):715-716.
Lindley K J, Milla PJ, "Autism, inflammatory bowel disease, and MMR vaccine."Lancet 1998 Mar 21;351(9106):907-908.
Bedford H, et al, "Autism, inflammatory bowel disease, and MMR vaccine." Lancet 1998 Mar 21;351(9106):907.
Vijendra K. Singh, Sheren X. Lin, and Victor C. Yang, "Serological Association of Measles Virus and Human Herpesvirus-6 with Brain Autoantibodies in Autism," Clinical Immunology and Immunopathology, Oct 1998, Vol. 89, No. 1, p 105-108. ["None of the autistic children in the study had measles in the past, but all had the MMR" stated David Whalgren.
Vaccines and Demyelination Citations:
Herroelen, L et al, "Central-Nervous-System Demyelination After Immunization with Recombinant Hepatitis B Vaccine", Lancet, Nov 9, 1991, 338(8776):1174-1175.
Kaplanski G, Retornaz F, Durand J, Soubeyrand J, "Central nervous system demyelination after vaccination against hepatitis B and HLA haplotype." J Neurol Neurosurg Psychiatry 1995 Jun; 58(6):758-759.
Matyszak MK, Perry VH, "Demyelination in the central nervous system following a delayed-type hypersensitivity response to bacillus Calmette-Guerin." Neuroscience 1995 Feb;64(4):967-977.
Tornatore CS, Richert JR, "CNS demyelination associated with diploid cell rabies vaccine." Lancet 1990 Jun 2;335(8701):1346-1347.
Adams, JM et al, "Neuromyelitis Optica: Severe Demyelination Occurring Years After Primary Smallpox Vaccinations", Rev Roum Neurol, 1973, 10:227-231.
In 1988, Dietrich used MRI to show that developmentally delayed children had alterations in their myelin. Coulter described that central nervous system damage can be exhibited as abnormal behavior of the child. In 1935, Thomas Rivers, experimental allergic encephalitis (EAE) can be the result of a viral or bacterial infection of the nervous system. "The fact of the matter is that it is a matter of record that it was known that vaccination produced encephalitis since 1926." The authors stated, "In regions in which there is no organized vaccination of the population, general paralysis is rare. ... It is impossible to deny a connection between vaccinations and the encephalitis (brain damage) which follows it."
Vaccines have been linked to seizures, convulsions and epilepsy.
Vaccinations and Seizures:
Hirtz DG, Nelson KB, Ellenberg J H, "Seizures following childhood immunizations", Pediatr 1983 Jan; 102(1):14-18.
Cherry JD, Holtzman AE, Shields WD, Buch D, Nielsen, "Pertussis immunization and characteristics related to first seizures in infants and children,"J Pediatr 1993 Jun;122(6):900-903.
Coplan J, "Seizures following immunizations," J Pediatr 1983 Sep;103(3):496.
Griffin MR, et al, "Risk of seizures after measles-mumps-rubella immunization," Pediatrics 1991 Nov;88(5):881-885.
Griffin MR, et al, "Risk of seizures and encephalopathy after immunization with the diphtheria- tetanus-pertussis vaccine," JAMA 1990 Mar 23-30;263(12):1641-1645.
Cizewska S, Huber Z, Sluzewski W, "[Prophylactic inoculations and seizure activity in the EEG]," Neurol Neurochir Pol 1981 Sep-Dec;15(5-6):553-557. [Article in Polish]
Huttenlocher PR, Hapke RJ, "A follow-up study of intractable seizures in childhood." Ann Neurol 1990 Nov; 28(5):699-705.
Blumberg DA, "Severe reactions associated with diphtheria-tetanus-pertussis vaccine: detailed study of children with seizures, hypotonic-hypo-responsive episodes, high fevers, and persistent crying."Pediatrics 1993 Jun; 91(6):1158-1165.
Vaccinations and Convulsions Citations:
Prensky AL, et al, "History of convulsions and use of pertussis vaccine," J Pediatr 1985 Aug; 107(2):244-255.
Baraff LJ, "Infants and children with convulsions and hypotonic-hypo-responsive episodes following diphtheria-tetanus-pertussis immunization: follow-up evaluation," Pediatrics 1988 Jun; 81(6):789-794.
Jacobson V, "Relationship of pertussis immunization to the onset of epilepsy, febrile convulsions and central nervous system infections: a retrospective epidemiologic study," Tokai J Exp Clin Med 1988;13 Suppl: 137-142.
Cupic V,et al, "[Role of DTP vaccine in the convulsive syndromes in children]," Lijec Vjesn 1978 Jun; 100(6):345-348. [Article in Serbo-Croatian (Roman)]
Pokrovskaia NIa, "[Convulsive syndrome in DPT vaccination (a clinico-experimental study)]," Pediatriia 1983 May;(5):37-39. [Article in Russian]
Vaccinations and Epilepsy Citations:
Ballerini, Ricci, B, et al, "On Neurological Complications of Vaccination, With Special Reference to Epileptic Syndromes," Riv Neurol, Jul-Aug 1973, 43:254-258.
Wolf SM, Forsythe A, "Epilepsy and mental retardation following febrile seizures in childhood," Acta Paediatr Scand 1989 Mar;78(2):291-295.
Vaccines and Brain Swelling:
Iwasa, S et al, "Swelling of the Brain in Mice Caused by Pertussis ... Quantitative Determination and the Responsibility of the Vaccine", Jpn J Med Sci Biol, 1985 , 38(2):53-65.
Mathur R, Kumari S, "Bulging fontanel following triple vaccine." Indian Pediatr 1981 Jun;18(6):417- 418.
Barry W, Lenney W, Hatcher G, "Bulging fontanelles in infants without meningitis." Arch Dis Child 1989 Apr;64(4):635-636.
Shendurnikar N, "Bulging fontanel following DPT" Indian Pediatr 1986 Nov;23(11):960.
Gross TP, Milstien JB, Kuritsky JN, "Bulging fontanelle after immunization with diphtheria-tetanus- pertussis vaccine and diphtheria-tetanus vaccine." J Pediatr 1989 Mar;114(3):423-425.
Jacob J, Mannino F, "Increased intracranial pressure after diphtheria, tetanus, and pertussis immunization." Am J Dis Child 1979 Feb;133(2):217-218.
Dugmore, WN, "Bilateral Oedema at the Posterior Pole. Hypersensitivity Reaction to Alavac P injection." Br J Ophthalmol, Dec 1972, 55:848-849.
Vaccines and Neurological Damage:
Nedar P R, and Warren, R J, "Reported Neurological Disorders Following Live Measles Vaccine", 1968, Ped, 41:997-1001.
Paradiso, G et al, "Multifocal Demyelinating Neuropathy after Tetanus Vaccine", Medicina (B Aires), 1990, 50(1):52-54.
Landrigan, PJ, Whitte, J, "Neurologic Disorders Following Live Measles-virus Vaccination", JAMA, Mar 26, 1973, v223(13):1459-1462.
Turnbull, H M, "Encephalomyelitis Following Vaccination", Brit Jour Exper Path, 7:181, 1926.
Kulenkampff, M et al, "Neurological Complications of Pertussis Inoculation", Arch Dis Child, 1974, 49:46.
Strom, J, "Further Experience of Reactions, Especially of a Cerebral Nature in Conjunction with Triple Vaccination", Brit Med Jour, 1967, 4:320-323.
Berg, J M, "Neurological Complications of Pertussis Immunization," Brit Med Jour, July 5,1958; p 24.
Bondarev, VN et al, "The Changes of the Nervous System in Children After Vaccination", Pediatria, Jun 1969; 48:20-24.
Badalian, LO, "Vaccinal Lesions of the Nervous System in Children," Vop Okhr Materin Dets, Dec 1959, 13:54-59
Lorentz, IT, et al, "Post-Vaccinal Sensory Polyneuropathy with Myoclonus", Proc Aust Ass Neurol, 1969, 6:81-86.
Trump, R C, White, T R, "Cerebellar Ataxia Presumed Due To Live Attenuated Measles Virus Vaccine," JAMA, 1967, 199:165-166.
Allerdist, H, "Neurological Complications Following Measles Vaccination", Inter Symp, Brussels, 1978, Development Biol Std, Vol 43, 259-264.
Finley, K H, "Pathogenesis of Encephalitis Occurring With Vaccination, Variola and Measles, Arch Neur and Psychologist, 1938; 39:1047-1054.
Froissart, M et al, "Acute Meningoencephalitis Immediately after an Influenza Vaccination", Lille Med, Oct 1978, 23(8):548-551.
Pokrovskaia, Nia, et al, "Neurological Complications in Children From Smallpox Vaccination", Pediatriia, Dec 1978, (12):45-49.
Allerdist, H, "Neurological Complications Following Measles Virus Vaccination. Evaluation of the Cases seen Between 1971-1977'', Monatsschr Kinderheilkd, Jan 1979, 127(1): 23-28.
Ehrengut, W et al, "On Convulsive Reactions Following Oral vaccination Against Polio", Klin Paediatr, May 1979, 191(3):261-270.
Naumova, R P, et al, "Encephalitis Developing After Vaccination without a Local Skin Reaction", Vrach Delo, Jul 1979, (7):114-115.
Goswamy, BM, "Neurological Complications After Smallpox Vaccination", J Ass Phys India, Jan 1969, 17:41-43.
Schchelkunov, SN et al, "The Role of Viruses in the Induction of Allergic Encephalomyelitis," Dokl Akad Nauk SSSR, 1990,315(1):252-255. [Vaccines contain viruses, too]
Walker AM, "Neurologic events following diphtheria-tetanus-pertussis immunization," Pediatrics 1988 Mar;81(3):345-349.
Shields WD, et al, "Relationship of pertussis immunization to the onset of neurologic disorders: a retrospective epidemiologic study," J Pediatr 1988 Nov; 113(5):801-805.
Wilson J, "Proceedings: Neurological complications of DPT inoculation in infancy," Arch Dis Child 1973 Oct; 48(10):829-830.
Iakunin IuA, "[Nervous system complications in children after preventive vaccinations]," Pediatriia 1968 Nov; 47(11):19-26. [Article in Russian]
Greco D, et al, "Case-control study on encephalopathy associated with diphtheria-tetanus immunization in Campania, Italy," Bull World Health Organ 1985;63(5):919-925.
Ehrengut W at Institute of Vaccinology and Virology, Hamburg, Germany states, "Bias in the evaluation of CNS complications following pertussis immunization are the following: 1) Notifications of post-immunization adverse events, 2) Publications by vaccine producers on the frequency of adverse reactions, 3) Comparison of permanent brain damage after DPT and DT immunization, 4) Pro- immunization, 5) Immunization associated viral encephalitis, 6) Accuracy of statistics, 7) Personal. A review of these points indicates an underestimation of CNS complications after pertussis immunization."
Reference: Ehrengut W, "Bias in evaluating CNS complications following pertussis immunization." Acta Paediatr Jpn, 1991 Aug; 33(4):421-427.
Vaccinations and Unexplained Diseases:
Hiner, E E, Frasch, C E, "Spectrum of Disease Due to Haemophilus Influenza Type B Occurring in Vaccinated Children", J Infect Disorder, 1988 Aug; 158(2): 343-348.
Olin P, Romanus, V, Storsaeter, J, "Invasive Bacterial Infections During an Efficiacy Trial of Acellular Pertussis Vaccines — Implications For Future Surveilance In Pertussis Vaccine Programmes", Tokai J Exp Clin Med, 1988; 13 Suppl: 143-144.
Storsaeter, J, et al, "Mortality and Morbidity From Invasive Bacterial Infections During a Clinical Trial of Acellular Pertussis Vaccines in Sweden", Pediatr Infect Disorder J, 1988 Sept; 7(9):637-645.
Vadheim, CM, et al, "Effectiveness and Safety of an Haemophilus Influenzae type b Conjugate Vaccine (PRP-T) in Young Infants. Kaiser-UCLA Vaccine Study Group," Pediartics, 1993 Aug; 92(2):272-279. [The vaccines caused fevers, irritability, crying, and seizures, but were declared to be "safe and ... effective ... ".]
Stickl, H, "Estimation of Vaccination Damage", Med Welt, Oct 14, 1972, 23:1495-1497.
Waters, VV, et al, "Risk Factors for Measles in a Vaccinated Population", JAMA, Mar 27, 1991, 265(12): 1527.
Stickl, H, "Iatrogenic Immuno-suppression as a Result of Vaccination", Fortschr Med, Mar 5, 1981, 99(9);289-292.
Vaccine Citations Linking the Vaccine to the "prevented" Disease:
Nkowane, et al, "Vaccine-Associated Paralytic Poliomyelitis, US 1973 through 1984, JAMA, 1987, Vol 257:1335-1340.
Quast, et al, "Vaccine Induced Mumps-like Diseases", nd, Int Symp on Immun, Development Bio Stand, Vol 43, p269-272.
Green, C et al, "A Case of Hepatitis Related to Etretinate Therapy and Hepatitis B Vaccine", Dermatologica, 1991, 182(2):119-120.
Shasby, DM, et al, "Epidemic Measles in Highly Vaccinated Population", NEJM, Mar 1977, 296(11): 585-589.
Tesovic, G et al, "Aseptic Meningitis after Measles, Mumps and Rubella Vaccine", Lancet, Jun 12, 1993, 341(8859):1541.
Johnson, RH, et al, "Nosocomial Vaccinia Infection", West J Med, Oct 1976, 125(4):266-270.
Malengreau, M, "Reappearance of Post-Vaccination Infection of Measles, Rubella, and Mumps. Should Adolescents be re-vaccinated?" Pedaitric, 1992;47(9):597-601 (25 ref)
Basa, SN, "Paralytic Poliomyelitis Following Inoculation With Combined DTP Prophylactic. A review of Sixteen cases with Special Reference to Immunization Schedules in Infancy", J Indian Med Assoc, Feb 1, 1973, 60:97-99.
Landrigan, PJ et al, "Measles in Previously Vaccinated Children in Illinois", Ill Med J, Arp 1974, 141:367-372.
NA, "Vaccine-Associated Poliomyelitis", Med J Aust, Oct 1973, 2:795-796.
Vaccine Failures Citations:
Hardy, GE, Jr, et al, "The Failure of a School Immunization Campaign to Terminate an Urban Epidemic of Measles," Amer J Epidem, Mar 1970; 91:286-293.
Cherry, JD, et al, "A Clinical and Serologic Study of 103 Children With Measles Vaccine Failure", J Pediatr, May 1973; 82:801-808.
Jilg, W, et al, "Inoculation Failure Following Hepatitis B Vaccination", Dtsch Med wochenschr, 1990 Oct 12; 115(41):1514-1548.
Plotkin, SA, "Failures of Protection by Measles Vaccine," J Pediatr, May 1973; 82:798-801.
Bolotovskii, V, et al, "Measles Incidence Among Children Properly Vaccinated Against This Infection", ZH Mikrobiol Epidemiol Immunobiol, 1974; 00(5):32-35.
Landrigan, PJ, et al, "Measles in Previously Vaccinated Children in Illinois", Ill Med J, Apr 1974; 141:367-372.
Strebel, P et al, "An Outbreak of Whooping Cough in a Highly Vaccinated Urban Community", J Trop Pediatr, Mar 1991, 37(2): 71-76.
Forrest, JM, et al, "Failure of Rubella Vaccination to Prevent Congenital Rubella,"Med J Aust, 1977 Jan 15; 1(3): 77.
Jilg, W, "Unsuccessful Vaccination against Hepatitis B", Dtsch Med Wochenschr, Nov 16, 1990, 115(46):1773.
Coles, FB, et al, "An Outbreak of Influenza A (H3N2) in a Well-Immunized Nursing home Population," J Am ger Sociologist, Jun 1992, 40(6):589-592.
Jilg, W, et al, "Inoculation Failure following Hepatitis B Vaccination," Dtsch Med Wochenschr, Oct 12, 1990, 115(41):1545-1548.
Hartmann, G et al, "Unsuccessful Inoculation against Hepatitis B," Dtsch Med Wochenschr, May 17, 1991, 116(20): 797.
Buddle, BM et al, "Contagious Ecthyma Virus-Vaccination Failures", Am J Vet Research, Feb 1984, 45(2):263-266.
Mathias, R G, "Whooping Cough In Spite of Immunization", Can J Pub Health, 1978 Mar/Apr; 69(2):130-132.
Osterholm, MT, et al, "Lack of Efficacy of Haemophilus b Polysacharide Vaccine in Minnesota", JAMA, 1988 Sept 9; 260(10:1423-1428.
Johnson, RH, et al, "Nosocomial Vaccinia Infection", West J Med, Oct 1976, 125(4):266-270.
Vaccines Causing Another Vaccinal Disease:
Basa, SN, "Paralytic Poliomyelitis Following Inoculation With Combined DTP Prophylactic. A review of Sixteen cases with Special Reference to Immunization Schedules in Infancy", J Indian Med Assoc, Feb 1, 1973, 60:97-99.
Pathel, JC, et al, "Tetanus Following Vaccination Against Small-pox", J Pediatr, Jul 1960; 27:251-263.
Favez, G, "Tuberculous Superinfection Following a Smallpox Re-Vaccination", Praxis, July 21, 1960; 49:698-699.
Quast, Ute, and Hennessen, "Vaccine-Induced Mumps-like Diseases", Intern Symp on Immunizations , Development Bio Stand, Vol 43, p 269-272.
Forrest, J M, et al, "Clinical Rubella Eleven months after Vaccination," Lancet, Aug 26, 1972, 2:399-400.
Dittman, S, "Atypical Measles after Vaccination", Beitr Hyg Epidemiol, 19891, 25:1-274 (939 ref) Sen S, et al, "Poliomyelitis in Vaccinated Children", Indian Pediatr, May 1989, 26(5): 423-429.
Arya, SC, "Putative Failure of Recombinant DNA Hepatitis B Vaccines", Vaccine, Apr 1989, 7(2): 164- 165.
Lawrence, R et al, "The Risk of Zoster after Varicella Vaccination in Children with Leukemia", NEJM, Mar 3, 1988, 318(9): 543-548.
Vaccination Citations and Death:
Na, "DPT Vaccination and Sudden Infant Death – Tennessee, US Dept HEW, MMWR Report, Mar 23, 1979, vol 28(11): 132.
Arevalo, "Vaccinia Necrosum. Report on a Fatal Case", Bol Ofoc Sanit Panamer, Aug 1967, 63:106-110.
Connolly, J H, Dick, G W, Field, CM, "A Case of Fatal Progressive Vaccinia", Brit Med Jour, 12 May 1962; 5288:1315-1317.
Aragona, F, "Fatal Acute Adrenal Insufficiency Caused by Bilateral Apoplexy of the Adrenal Glands (WFS) following Anti-poliomyelitis Vaccination", Minerva Medicolegale, Aug 1960; 80:167-173.
Moblus, G et al, "Pathological-Anatomical Findings in Cases of Death Following Poliomyelitis and DPT Vaccination", Dtsch Gesundheitsw, Jul 20, 1972, 27:1382-1386.
NA, "Immunizations and Cot Deaths", Lancet, Sept 25, 1982, np.
Goetzeler, A, "Fatal Encephalitis after Poliomyelitis Vaccination", 22 Jun 1961, Muenchen Med Wschr, 102:1419-1422.
Fulginiti, V, "Sudden Infant Death Syndrome, Diphtheria-Tetanus Toxoid-Pertussis Vaccination and Visits to the Doctor: Chance Association or Cause and Effect?", Pediatr Infect Disorder, Jan-Feb 1983, 2(1): 7-11.
Baraff, LJ, et al, "Possible Temporal Association Between Diphtheria-tetanus toxoid-Pertussis Vaccination and Sudden Infant Death Syndrome", Pediatr Infect Disorder, Jan-Feb 1983, 2(1): 5-6.
Reynolds, E, "Fatal Outcome of a Case of Eczema Vaccinatum", Lancet, 24 Sept 1960, 2:684-686.
Apostolov. et al, "Death of an Infant in Hyperthermia After Vaccination", J Clin Path, Mar 1961, 14:196-197.
Bouvier-Colle, MH, "Sex-Specific Differences in Mortality After High-Titre Measles Vaccination", Rev Epidemiol Sante Publique, 1995; 43(1): 97.
Stewart GT, "Deaths of infants after triple vaccine.", Lancet 1979 Aug 18;2(8138):354-355.
Flahault A, "Sudden infant death syndrome and diphtheria/tetanus toxoid/pertussis/poliomyelitis immunisation.", Lancet 1988 Mar 12;1(8585):582-583.
Larbre, F et al, "Fatal Acute Myocarditis After Smallpox Vaccination", Pediatrie, Apr-May 1966, 21:345- 350.
Mortimer EA Jr, "DTP and SIDS: when data differ", Am J Public Health 1987 Aug; 77(8):925-926.
Vaccines and Metabolism Citations:
Deutsch J, " [Temperature changes after triple-immunization in infant age]," Padiatr Grenzgeb 1976;15(1):3-6.[Article in German]
NA, "[Temperature changes after triple immunization in childhood]," Padiatr Grenzgeb 1976;15(1):7- 10. [Article in German]
[Considering that the thyroid controls our Basal Metabolism, it would appear that vaccines altered (depressed) thyroid activity.]
Vaccines Altering Resistance to Disease:
Burmistrova AL, "[Change in the non-specific resistance of the body to influenza and acute respiratory diseases following immunization diphtheria-tetanus vaccine]," Zh Mikrobiol Epidemiol Immunobiol 1976; (3):89-91. [Article in Russian]
Vaccinations and Deafness Citations:
So I did a background check to see if there was any scientific evidence linking vaccines to deafness and hearing loss. Here are some of the articles I found:
Kaga, "Unilateral Total Loss of Auditory and Vestibular Function as a Complication of Mumps Vaccination", Int J Ped Oto, Feb 1998, 43(1):73-73
Nabe-Nielsen, Walter, "Unilateral Total Deafness as a Complication of the Measles- Mumps- Rubella Vaccination", Scan Audio Suppl, 1988, 30:69-70
Hulbert, et al, "Bilateral Hearing Loss after Measles and Rubella Vaccination in an Adult", NEJM, 1991 July, 11;325(2):134
Healy, "Mumps Vaccine and Nerve Deafness", Am J Disorder Child, 1972 Jun; 123(6):612
Jayarajan, Sedler, "Hearing Loss Following Measles Vaccination", J Infect, 1995 Mar; 30(2):184-185
Pialoux, P et al, "Vaccinations and Deafness", Ann Otolaryng (Paris), Dec 1963, 80:1012-1013.
Angerstein, W, et al, "Solitary Hearing and Equilibrium Damage After Vaccinations", Gesundheitswesen, May 1995, 57(5): 264-268.
Brodsky, Stanievich, "Sensorineural Hearing Loss Following Live Measles Virus Vaccination", Int J Ped Oto, 1985 Nov; 10(2):159-163
Koga, et al, "Bilateral Acute Profound Deafness After MMR Vaccination- Report of a Case", Nippon Jibiin Gakkai Kai, 1991 Aug;94(8):1142-5
Seiferth, LB, "Deafness after Oral Poliomyelitis Vaccination – a Case Report and Review", HNO, 1977 Aug; 25(8): 297-300
Pantazopoulos, PE, "Perceptive Deafness Following Prophylactic use of Tetanus anittoxin", Laryngoscope, Dec 1965, 75:1832-1836.
Zimmerman, W, "Observation of a case of Acute Bilateral Hearing Impairment Following Preventive Poliomyelitis Vaccination (type 3)", Arch Ohr Nas Kehlkopfheilk, 1965, 185:723-725.
Vaccinations and Kidney Disorders Citations:
Jacquot, C et al, "Renal Risk in Vaccination", Nouv Presse Med, Nov 6, 1982, 11(44):3237-3238.
Giudicelli, et al, "Renal Risk in Vaccination", Presse Med, Jun 11, 1982, 12(25):1587-1590.
Tan, SY, et al, "Vaccine Related Glomerulonephritis", BMJ, Jan 23, 1993, 306(6872):248.
Pillai, JJ, et al, "Renal Involvement in Association with Post-vaccination Varicella", Clin Infect Disorder, Dec 1993, 17(6): 1079-1080.
Eisinger, AJ et al, "Acute Renal Failure after TAB and Cholera Vaccination", B Med J, Feb 10, 1979, 1(6160):381-382.
Silina, ZM, et al, "Causes of Postvaccinal Complications in the Kidneys in Young Infants", Pediatria, Dec 1978, (12):59-61.
Na, "Albuminurias", Concours Med, Mar 1964, 85:5095-5098.[vaccination adverse reactions]
Oyrl, A, et al, "Can Vaccinations Harm the Kidney?", Clin Nephrol, 1975, 3(5):204-205.
Mel’man Nia, "[Renal lesions after use of vaccines and sera]." Vrach Delo 1978 Oct;(10):67-9, [Article in Russian]
Silina ZM, Galaktionova TIa, Shabunina NR, "[Causes of postvaccinal complications in the kidneys in young infants]." Pediatriia 1978 Dec;(12):59-61, [Article in Russian]
Silina EM, et al, "[Some diseases of the kidneys in children during the 1st year of life, following primary smallpox vaccination and administration of pertusis-diphtheria-tetanus vaccine]." Vopr Okhr Materin Det 1968 Mar; 13(3):79-80, [Article in Russian]
Lupton GP, "Discoid lupus erythematosus occurring in a smallpox vaccination scar," J Am Acad Dermatol, 1987 Oct; 17(4):688-690.
Kompier, A J, "Some Skin Diseases caused by Vaccinia Virus [Smallpox]," Ned Milt Geneesk T, 15:149- 157, May 1962.
Weber, G et al, "Skin Lesions Following Vaccinations," Deutsch Med Wschr, 88:1878-1886, S7 Sept 1963.
Copeman, P W, "Skin Complications of Smallpox Vaccination," Practitioner, 197:793-800, Dec 1966.
Denning, DW, et al, "Skin Rashes After Triple Vaccine," Arch Disorder Child, May 1987, 62(5): 510-511.
Vaccinations and Abcesses:
Sterler, HC, et al, "Outbreaks of Group A Steptococcal Abcesses Following DTP Vaccination", Pediatrics, Feb 1985, 75(2):299-303.
DiPiramo, D, et al, "Abcess Formation at the Site of Inoculation of Calmette-Guerin Bacillus (BCG)," Riv Med Aeronaut Spaz, Jul-Dec 1981, 46(3-4):190-199.
Vaccinations and Shock:
Caileba, A et al, "Shock associated with Disseminated Intravascular Coagulation Syndrome following Injection of DT.TAB Vaccine, Prese Med, Sept 15, 1984, 13(3):1900.
Vaccines: The Weird, The Wild and The Hilarious Citations:
Sometimes there are articles published about the strangest facts related to vaccines that defies our imagination and ability to understand them. They were written seriously by well-meaning scientific persons, but their titles can be seen differently. Some are funny, some are sad and some are purely scientific folly. See if you can figure these out:
Pathel, JC, et al, "Tetanus Following Vaccination Against Small-pox", J Pediatr, Jul 1960; 27:251-263. [Now you need a tetanus vaccination!]
Favez, G, "Tuberculous Superinfection Following a Smallpox Re-Vaccination", Praxis, July 21, 1960; 49:698-699. [Super means large/big/great!]
Bonifacio, A et al, "Traffic Accidents as an expression of "Iatrogenic damage", Minerva Med, Feb 24, 1971, 62:735-740. [But officer I was just vaccinated!]
Baker, J et al, "Accidental Vaccinia: Primary Inoculation of a Scrotum", Clin Pediatr (Phila), Apr 1972, 11:244-245. [Ooops, the needle slipped.]
Edwards, K, "Danger of Sunburn Following Vaccination", Papua New Guinea Med J, Dec 1977, 20(4):203. [Are vaccines phototoxic?]
Stroder, J, "Incorrect Therapy in Children", Folia Clin Int (Barc), Feb 1966, 16:82-90. [Agreed.]
Wehrle PF, "Injury associated with the use of vaccines," Clin Ther 1985;7(3):282-284. [Dah!]
Alberts ME, "When and where will it stop", Iowa Med 1986 Sep; 76(9):424. [When!]
Breiman RF, Zanca JA, "Of floors and ceilings — defining, assuring, and communicating vaccine safety", Am J Public Health 1997 Dec;87(12):1919-1920. [What is in between floors and ceilings?]
Stewart, AM, et al, "Aetiology of Childhood Leukaemia", Lancet, 16 Oct, 1965, 2:789-790.
Nelson, ST, "John Hutchinson On Vaccination Syphilis (Hutchinson, J)", Arch Derm, (Chic), May 1969, 99:529-535. [Vaccinations and STDs!]
Mather, C, "Cotton Mather Anguishes Over the Consequences of His Son’s Inoculation Against Smallpox", Pediatrics, May 1974; 53:756. [Is it for or against?]
Thoman M, "The Toxic Shot Syndrome", Vet Hum Toxicol, Apr 1986, 28(2):163-166. [Animals are not exempt from vaccination damage either!]
Johnson, RH, et al, "Nosocomial Vaccinia Infection", West J Med, Oct 1976, 125(4):266-270. [Nosocomial means a disease acquired in a doctor's office or hospital.]
Heed, JR, "Human Immunization With Rabies Vaccine in Suckling Mice Brain," Salud Publica, May-Jun 1974, 16(3): 469-480. [Have you had your suckling mice brains today?]
Tesovic, G et al, "Aseptic Meningitis after Measles, Mumps and Rubella Vaccine", Lancet, Jun 12, 1993, 341(8859):1541. [AM has same symptoms as poliomyelitis!]
Buddle, BM et al, "Contagious Ecthyma Virus-Vaccination Failures", Am J Vet Research, Feb 1984, 45(2):263-266.
Freter, R et al, "Oral Immunization And Production of Coproantibody in Human Volunteers", J Immunol, Dec 1963, 91:724-729. [Guess what copro- means .... Feces.]
NA, "Vaccination, For and Against", 1964, Belg T Geneesk, 20:125-130. [Is it for or against?]
Sahadevan, MG et al, "Post-vaccinal Myelitis", J Indian Med Ass, Feb 16, 1966, 46:205-206. [Did I mention myelitis?]
Castan, P et al, "Coma Revealing an acute Leukosis in a child, 15 days after an Oral Anti-poliomyelitis Vaccination," Acta Neurol Bekg, May 1965, 65:349-367. [Coma from vaccines!]
Stickl, H, et al, "Purulent [pus] meningitides Following Smallpox Vaccination. On the Problem of Post- Vaccinal Decrease of Resistance", Deutsch Med Wschr, Jul 22, 1966, 91:1307-1310.
[Vaccines are the injection of viruses cultured from pus ... ]
Mercury:
You will want to watch these several times they are PACKED with info:
Packed with info but he never does tell you how to detox mercury...
Here is this:
10 biggest vaccine news stories of 2011
(NaturalNews) A lot happened in the vaccine industry in 2011 -- a year of accelerating change and increased awareness among the people. One of the big realizations that hit home in 2011 is that vaccines are far more dangerous than most people previously thought, and even conventional scientists are starting to question the over-vaccination of infants and children.
Here are the 10 biggest vaccine news stories of 2011, reported on NaturalNews.com:
We cover all of these in the longer 20 minute video. Here is the short version, but you can get to the longer vid from here if you like:
In order to get here we played with basic concepts for months. With a password you can have access to hours and hours of video,sample lessons, PDF's and more and you can see step by step how a 6 year old got here. People are quite amazed at just how much you can get for 3 bucks or 12 bucks for an annual pass. Those who buy annual passes before the end of the year are in for some pleasant surprises next year, along with those of you who have already bought a annual pass to show your support...prices will be going up and there are more products being worked on as we speak that will be available at good discounts for those who have already gotten an annual pass. Also there are going to be separate passwords for various topics but right now one password does it all. For those who just want to take a peak 3 bucks gets you a good 60 days worth...but don't expect any special treatment.
[Update: prices have indeed gone up. Look at the side bar for details.]
For a more complete blog post on with these very same videos go to Math Genius Making.
Of course be sure to find us on FaceBook, and go to the House Of Math for much more on the best use of base ten blocks and manipulatives for math enrichment so your child or students can be doing math like this with no problem.
Stay tuned like I say on my website, "I'm just getting started."
Lately there seems to be a bit of a resurgence in understanding that those big wooden blocks had more purpose than meets the eye.
There are also articles appearing on blogs like this one and others. People seem to be rediscovering a very simple idea. Early is better when it comes to language, and math is a language. Further, the more senses you use the more learning takes place and the easier it is to understand. The brain likes multiple inputs.
Here is a simple wall made with the big blocks, also a lesson on proportion, wholes, halves and quarters as well as addition and multiplication...just by asking simple questions. What if all the blocks were little ones...well them the bottom blocks would take four...in fact each row would be four...and we can sing a little song about fours as we figure out how many fours it would take if the wall was made of the small blocks, what if we used the halves? Then we'd be counting by twos... For more advanced students (and teachers) we could do a quick lesson on bases...the most important part is to PLAY. These kinds of lessons have to be quick and easy and as natural as possible so the kids learn without knowing they are learning...we're just playing using our imagination and asking silly questions as we build stuff. It's not a formal lesson per se.
Here for sure we are playing. It's a fun preschool math activity AND they learn things that are not so obvious to the casual observer...as Doug Clements explains above.
We talked about the words symmetry, symmetric, symmetrical...
Bust mostly we had fun building a castle.
Then of course there was a little formal math where we practiced all 45 addends...this is a math activity that will be repeated MANY times before the recall is instant and mastery is achieved. Some of the slower teachers will drill these addends over and over again on a daily basis until the students have them down and have acquired a distaste for math in the process. The more enlightened teachers will make this one of a myriad of math activities that keep preschoolers, kindergartners, and first graders engaged for hours over many days and weeks. Kids never get tired of playing.
Here is another fast easy lesson on finding the area of a rectangle where the students tell you the formula not the other way around. They will forget this in a few weeks, and you can teach it again as a fresh lesson. Seriously. After a a few times they will remember it...internalize it, understand it and not because they looked at these symbols and memorized a formula:
1/2bh = A
The lesson with the blocks above is SO much more powerful than even this lesson:
However, I posted that video several places and the comments about it were how powerful and easy a drawing can be to illustrate the simple concept. I find that presentation creates HUGE "AH-HA" moments in students and teachers alike. Picture worth a thousand words as they say. Kinesthetic experience with blocks worth exponentially more than that...
Short explanation of a symbol in geometry, with a very lame explanation of how to use a negative inverse to make a perpendicular line.... After further review we did a much more thorough lesson on how to make perpendicular lines with slope intercept form. Anyhow math really is no fun if you don't know what the symbols mean.
Remembering complimentary angles and supplementary angles can be tricky. It also causes some people to think all math is, is memorization.
I always hear questions like, "can this do algebra, can this do geometry, can this do calculus?"...with "this" meaning Mortensen Math and the blocks. How do you apply manipulatives to Geometry or Trig? The basic concepts remain the same throughout the mathematics.
In cases like this all we are doing is learning vocabulary:
So there is a fast way to remember vocabulary. The problem is when you confuse vocabulary with understanding concepts or where you confuse memorization with conceptualization and problem solving. Some seem to think there is no difference between memorizing vocabulary words and memorizing formulae or addends or multiplication tables. Nothing could be further from the truth. We can "figure out" or use an algorithm that requires some thinking and doing to derive the quadratic, or even πr2 or something simple like how to remember 9's times tables rather than just commit a bunch of facts to memory but with definitions there is no "figuring out" per se you either know it or you don't.
Although with reasoning and logic you may use induction or deduction to figure out a word's meaning it's not quite the same as knowing enough algebra to solve for x when given a second degree polynomial in standard form set equal to zero.
ax2 + bx + c = 0 and coming up with the quadratic formula...and just memorizing the quadratic formula. With complimentary angles and supplementary angles it is very difficult to deduce which one means two angles which when added add up to 90 degrees and which one means two angles which when added add up to 180 degrees, unless you know quite a bit more English than the average highschooler has at their command...you either know what "table" means or you don't. Same with complimentary angles and supplementary angles. Don't confuse memorizing definitions of words or terms with wrote memorization of formula.
Turns out this stuff was super easy after all. We also covered distance formula which as just slightly advanced Pythagorean Theorem and a few or math terms and definitions that come with geometry like parallel and perpendicular, and then the names of angles, and poof, a few problems and the hour was over.
At my house this doesn't even qualify as "math time" anymore. It's just playing blocks. The fact that they are learning their addends and other math facts as well as refining their motor skills is an added bonus.
When we did do math we basically fooled around with factoring quadratics and distributive theory.
They multiplied problems like 5(3x + 2) = 15x + 10.
And factored problems like x2 + 12x + 27...
They learned LOTS of math doing these and more problems like them. We laffed, we cried, we spent some time learning addends and multiplication and factoring via the algebra. Most places you are supposed to know your math facts BEFORE you get to algebra. If you want to watch the lesson you will need a password so you can go to the sample lessons page. If you click that link you will go directly to it or click the picture below...it will ask you for your password and should redirect you to the vid, or just click on the URL in the address window of your browser and hit enter, if not just scroll down until you see the video titled More Algebra for Basic Ops. Also check out the vids right ABOVE it.
Here is a super short sample of the 36 minute video.
Passwords are cheap and easy to get and don't change near as often as they should. # bucks will get you in for at leaste 60 days, and it will take you about that long to watch all the stuff I have up.
Basically all we did was play with our food. By the time I thought about getting the camera out we were down to about 20 grapes...we had already eaten more than half. We practiced the all important concept "numbers are made of other numbers".
I was mostly focusing on addition and subtraction and multiplication he threw in division on his own at the end...so we covered all the basic operations. We could have easily gone down the fractions road...but we didn't. Main thing is to keep it simple and fun. Talking about math concepts as you manipulate real world objects helps them understand and add meaning to the abstract symbols they see.
Also the theme of late is math rich environment. That doesn't mean you have to be rich and spend a lot of money on manipulatives you can create a math rich environment pretty easily...just takes a little imagination and an idea of what you want to impart that day. Everyday math activities should be relatively short simple and fun.
You can't go wrong just play math. This is a great preschool math activity...or depending on where your child is and how late you started it could be a first grade math activity, or even second grade. I teach math concepts early using this method compared to public schools. By the end of second grade this boy will know his 45 addends and multiplication tables thru 12. Very little of it will come from drills with pencil and paper or math worksheets. Most of it will come from play. The worksheets and paper and pencil will reinforce what he already knows it won't be the primary tool for learning these math facts and concepts. He will also have tools and algorithms for figuring out problems he hasn't been exposed to and will quickly find there are quite a few patterns and rules that are immutable. A basic concept for teachers to understand is all math is IS COUNTING. So you can do adding or algebra, counting or calculus from day one. It's all just counting.
Seriously. Which way would YOU have rather learned?
For new comers this may be a little hard to digest. Search this blog and my website you will see how we actually use algebra to teach counting and other basic concepts instead of the mindset where these concepts have to be mastered BEFORE we introduce algebra...
You have seen pictures and descriptions of addends towers and other games at Crewton Ramone's House of Math since pretty much day one, but here it is on video for the first time. This is a staple of Mortensen Math. You HAVE to set the rules for the lesson: they have to build the levels with addends not just four 7's or whatever.
These two little girls are a ton of fun and a bit of a challenge when it comes to behavior management. This was part of a lesson that engaged them the whole hour. At the end of it we did algebra but we barely got started and it was time to go...it felt like they were there for 5 or 10 minutes. Thus proving my own axiom to myself: "make it fun."
With these two I just have to get a little more creative and keep things moving so te have little or no time to quarrel with each other...and certainly no time to get bored. I fully understand in a classroom situation you can't make it exciting every single minute, that would be exhausting for everybody; however, on the other hand some classrooms are so boring the kids get diagnosed with ADD or ADHD, they're not attention deficient, they bored to tears. Solution: drugs.
I have a different solution. Engage the student with MRE's. Math Rich Environments. Make sure those MRE's are activity rich too. Book work is boring. Necessary, but boring for little kids. As they get older you will be moving to symbols anyway but at the younger ages they need hands on activities. Think of it the same way you think of teaching reading. In the beginning the books have a lot more pictures than words and gradually move to words, but by 7 there is still lots more page devoted to pictures than to words.
Same for math...remember start in the concrete, then move to pictures and lastly to symbols. If you do it right the symbols might even be a reward and something they look forward to...drawing with pens on the board is fun, and along with the pictures move them to making symbols...
And as always don't just use these powerful tools to teach addition and place value. You can use them to teach counting through calculus...and one day you will see that as I use algebra to teach counting you can use calculus to teach counting and multiplication too...so literally teach counting through calculus.
The best way to start any lesson on Math Symbols is with a three period lesson. You may need to repeat it once or twice...it's important that they hear you read the symbols and that you add meaning as you go.
Here is a synopsis of about 30 minutes of a lesson on some math symbols. We had also done this several times before. This student is Autistic which doesn't mean can't learn, it means takes more time and impressions to get info into the part of the brain where it can be recalled. This one is often missed by students who have made the rule hard and fast in their heads two negatives equal a positive: -|-3| because they don't understand the concept of absolute value and fail to read the math symbols correctly.
When doing absolute value I often just talk about the block itself doesn't matter what side it's on a four block is "four" long. In her case though this explanation didn't sit as well because she wanted to be sure the block was on the positive side since all absolute values are positive. Other students seem to get the idea of "it's four from here to here" no matter which way you go. This also explains why the difference of two negative numbers is still positive.
Bright young students might inquire why it is that -22 - -28 = 6 where all the numbers are negative and we're even subtracting! The answer somehow ends up positive.
The difference can be explained in terms of direction on a number line, but as far as I have seen number lines and direction help confuse things, not clarify them. It's not about direction, it's about distance. The amount of "space" between the numbers, delineated by numbers themselves. Most kids don't ask. Might be wise not to bring it up especially with students who have poor self confidence in math. For the inquisitive ones and older students it can quickly devolve into a discussion on philosophy. How do you know the distance from here to here is four...? Keep it simple instead.
Explain what the math symbols mean and use blocks to bring the points home. Integers are no problem if presented properly. Look for more lessons on integers here on this blog and at the house of math where Sarah has her own page. The way I see it if you can explain it to Autistic students or little kids in a way they can understand then adults and older students as well as "gifted" students should be duck soup. Most of the time this is true. Often you have to get through math anxiety and other problems first with the older students.
Bottom line keep it simple.
“If you can't explain it simply, you don't understand it well enough”. ~Albert Einstein
I have made 100's of dollars re-explaining integers and reading symbols to all manner of students and have almost universally heard that using blocks makes it easier. Of late I have run into a lot of students who just want something to memorize rather than wanting to understand...understanding concepts is better.
