\doc\web\99\12\average2.txt Exemplars was supposedly based on national standards, but every math standard I've ever seen does not introduce this concept until 4th or 5th grade. 2nd graders typically are lucky if they get integer multiplication, decimal division is typically after 4th grade. Division with fraction exact answers is also 6th grade fraction math. In my opinion, average and median don't belong until after 5th grade. This problem would be difficult for most 7th or 8th graders, assuming no direct instruction in how to solve the problem. This is very typical of "high standards". The rule is to demand "high standards" but not bother to teach kids how to get there. As I recall, the 4th grade 1964 book had a page asking kids to figure out out the forumula sum/ number but only the teachers edition gave the correct formula! In other words, if you didn't figure it out, the textbook didn't tell you the answer. Did you have a chance to discuss this with the teacher, and ask where she got the idea that these skills are taught in 2nd grade math? Of course considering my 1st grade teacher expected my kid to read "estivation" and write sentences his first week homework, I shouldn't be surprised. Mean = Average = sum / number 2nd Exemplars 4th MCAS assesment 4th Vermont (begin simple concepts) 4th Addison Wesley 1991 --------- assessment appropriate ------- 5th Addison Wesley 1964 5th Holt 1970 5-8 Vermont - appropriately use From: Nanny714@aol.com Date sent: Tue, 17 Aug 1999 14:45:35 EDT Subject: [education-consumers] Math Problem Solving/Exemplars To: "ClearingHouse" Send reply to: Nanny714@aol.com ===================================================================== Here is one of the "so-called" problem solving tasks that was "developed" in my daughter's first grade classroom. Keep in mind that the majority of the children had not mastered addition and subtration and received little, if any direct instruction. Our K-6 school (Marion Cross School, Norwich, VT) has been extremely progressive for years, heralds constructivism, frowns on direct instruction, promotes political ideology, and doesn't use textbooks. This one put a hundred bucks in the teachers pocket and is now sold by Exemplars and is linked by Exemplars to some of your state standards. (I have others which are children were exposed to which the teacher was not so fortunate in earning money from). I have posted the teacher comments exactly as they appeared relative to why individual students were scored in a particular group. Question: How do we "solve" the "problem" of our children being subjected to such claptrap?? Your comments will be MOST welcome and appreciated! : Grade K-2 Task Average Number of letters in a group of five names. As a group, discuss strategies to figure out how to find the average number of letters in a group of five names. Remember that there will be five groups in the end because you started with five names. Some of you will have even groups of five and others will have uneven groups and will have to decide how to solve the problem. Context We have been working on averages in several areas including graphing averages in a science unit on bubbles. We have discussed and practiced averages of lower numbers and small groups of objects. this gave my students the opportunity to try averages in a different context. What this task accomplishes This problem gives the students the opportunity to apply their knowledge of making even groups out of objects they can see and move around. For kids with uneven groups, they have a chance to physically divide whole numbers and fractions. What the student will do Several kids who have had experience with averages will have suggestions like using a list of the names and/or cutting out the letters. Provide them with both so they have options to use their own strategies. * Make 5 even groups with cut out letters and split the leftover letters up into equal pieces for each of the five groups (halves, quarters, fifths, tenths) and call the average number rough but close. * Make 5 even groups and split the leftover letters into the same number as the groups so that each group gets a different size piece of the letter and call all pieces one. * Make 5 even groups of letters (if they have a number divisble by 5). * Make six or seven groups so that the numbers divide up evenly. * Make 5 groups of 3 or 4 and then split the last letters and put them in groups even though splitting up wasn't necessary. * Cross off letters on a list and rewrite them in blank spaces until the groups are even. * Cross off letters on a list and rewrite some in other spaces. Concepts to be assessed and skills to be developed Division Grouping Counting Problem Solving Reasoning Fractions Organization/Planning Rubrics and Benchmarks Novice (left 22 letters unused) Student: "I put the y next to the h from Brittany, the d from Clifford next to the y. I put the t from Catherine next to the d, r came from Brittany. I think the a came from Brittany (The student didn't check to make sure she had used all the letters with this list. Instead she used this list. First, she put some letters on and folded about five of them when she moved them to blamk spaces. When I asked her what the reason was, she said "to move them to another group.") Teacher: "Are all your groups the same size?" Student: "I guess" Teacher: "What is the average number of letters in a group?" Student: "8" This student had limited awareness of the problem. His/her organization of the problem was random and weak. S/he applied inappropriate concepts and inappropriate procedures therefore no solution was reached. S/he put a few cut letters on the list (her name and a few random letters), folded five others and placed them in blank spaces and left about 22 letters off the list. This student didn't have a clear or connected explanation of his/her solution. S/he was unable to explain his/her strategy. Apprentice This student made 5 even groups of five. Then he tried seven groups but had an extra. He put them back into 5 groups. He placed 3 letters in groups. He divided the other half and a half in quarters. He placed 5 halves and 2 quarters in each group and called the average 7 because there was seven pieces in each group. This student was able to use the correct mathematical procedure to solve this problem but couldn't carry out the procedure to find a correct solution. His/her strategy was partially useful. S/he lined up the different size pieces vertically in seven columns and came to the conclusion that the average was seven. This student has a basic understanding of the problem but a lack of understanding of fractions and was unable to get a solution. Practitioner The student was frustrated when he reached 5 goups of four with 3 left over. He cut two in half and placed 4 halves in four groups and 1 whole in the 5th group. I asked him if the groups were even. He said they weren't and tried 6 groups of four but was short one to be even. (He forgot that 5 groups was what he needed to keep). He went back to his first try and decided to go with it. Some groups had 4 1/2 and one had five. He said 4 3/4 was the average and wrote 4 1/3 (as 4 3/4). This student has a broad understaning of the problem. S/he was able to come close to the correct solution using fractions. S/he had three extra pieces to put in five groups. S/he placed a whole in a group and split the other two in half and placed them in the remaining groups. S/he labeled each group with either 4 and 1/2 or 5 and called the average 4 and 3/4 written as 4 and 1/3. S/he has a solution that reflects effective mathematical reasoning because s/he came up with a roughly correct average. Expert The student made 5 even groups of four whole pieces. She asked if she could cut the pieces. With the ok she cut the three into halves and put 5 of them in the 5 groups. She told me she had an average of 4 1/2 with a little extra piece that fit into 5 groups. She cut it into 1/10 slivers and asked how much it was. I told her to. She wrote her average as 4 and a half and one tenth. This student has a deep understanding of the problem. S/he divided up the six cut halves into the five groups and was able to express the amount in each group using correct terminology. S/he referred to the leftover half as "a little extra in each group." S/he proceded to cut the last half into five equal pieces. S/he concluded that the average was 4 and 1/2 and 1/10 which is true. S/he applied his/her understanding of fractions and the procedures necessary to find a correct solution. His/her explanation was clear. S/he generalized from previous mathematics experience. Author CINDY PIERCE graduated from the Upper Valley Teacher Training Program, an alternative certification program that matches teaching interns with strong, innovative teachers. She is in her third year of teaching ------------------------------------------------------------------------------ --------------------------------------------- ===================================================================== EDUCATION CONSUMERS CLEARINGHOUSE networking and information for parents and taxpayers on the internet Website & Archives: http://education-consumers.com You are currently subscribed to education-consumers as: arthurhu@halcyon.com TO UNSUBSCRIBE: Send a blank email to leave-education-consumers-989462S@lists.dundee.net ===================================================================== For less mail, use the following link and choose 1) a daily digest, 2) a daily list of subjects, or 3) no mail (read postings on Web) http://lists.dundee.net/scripts/lyris.pl?enter=education-consumers For more help & info: http://www.lyris.com/help or