So we had our monthly math meeting for our district and one of the items on the agenda was the “final report.”
The focus was; “How do the programs we are piloting fit in with the recommendations of the N.A.M.P. report?” There was talk about the U.S. mathematic textbooks being too long and how that is in response to the state standards. We had some nice discussion about how countries in Asia are doing less but more (depth vs. breadth) but how they are turning to the U.S. for how they solve problems. Although there was a general consensus that we aren’t necessarily great problem solvers (I do not think that a problem masquerading as an algorithm constitutes problem solving. For example-There are 32 rows of corn in a field. Each row has 22 stalks. How many stalks are there in all?).
As I have read through this report several times, I realized that this report is a front for pushing an agenda. It seems as if the panel members are in line to receive federal funding ($260million) to create a research based program. Roger Schank has a great article about the problems arising from the report and the people involved. As well, they basically slam “real-world” problem solving, saying that it only measures how students solve real-world problems. And what is wrong with that? Are students going to work in the real world and solve problems that occur over two days (short answer day 1 and long answer day 2)?
Another point that I can to think about was, why is Algebra the end all be all. The blog dy/dan has an interesting post about the need for Algebra for the masses. The comments are great food for thought as well. Maybe some thought should be given to having students pursue rigorous math courses that are of interest to them.
Let me start off this post by saying that I do not think that this is necessary for all students but it may be necessary for some. After seeing a couple of students who still cannot not carry out the traditional division algorithm (including writing down the acronym D.M.S.B.-divide, multiply, subtract, bring down), I feel it is necessary to present to these students a different way of looking at it. Now I believe that these students are beyond the repeated subtraction phase of division but have yet to grasp the underpinnings of the traditional algorithm (which is suppose to be faster).
So let us take a look at the traditional algorithm, using the problem 3672 divided by 8.
As you can see the first thing we would do is to see how many times 8 goes into 3 (which is really 3,000) and it doesn’t so you move to the next digit and figure out how many time 8 goes into 36. We determine 4 times, which is 32. You then subtract 32 from 36 and are left with 4. We are then taught to bring down the next digit (the B in “D. M. S. B.”) which is a 7. We then begin the process again-determine how many times 8 goes into 47, which is 5, 8 times 5 is 40, and 47-40 is 7. We bring down the last digit, 2, and repeat the process one more time, which leaves us with no remainder. This is fine as long as the student understands what is going on.
For example, I would want them to realize that when we are talking about 36 divided by 8 we are really talking about 3,600 and the choice of 4 is 400. When we bring down the next digit, that is what it is not a number. According to Merriam-Webster OnLine a digit is; “a: any of the Arabic numerals 1 to 9 and usually the symbol 0 b: one of the elements that combine to form numbers in a system other than the decimal system” and a number is; “ a sum of units : total c (1): a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication.” For more information on digits and numerals you can go to Wikipedia.
Let’s take a look at the friendly number method. As you can see this method helps build on the strengths students already posses (rounding, multiplying by powers of 10, basic facts). It also looks similar to the traditional algorithm which would make the transition from the friendly number method to the traditional algorithm easier.
As point of emphasis, when working with struggling students, or using this for the first time, I would begin with 10 instead of 100. Students are usually strong with the 10-facts and they could then see how cumbersome and time-consuming it would be to use 10 over and over again. You could move them from 10 to 100, using their knowledge of powers of 10, then to any basic fact like in the second step (300 x 8 because they know 300 is 3 x 100 and therefore they can multiply 3 x 8 and 100 x 8).
As they become more comfortable and use this method they will begin to see the connection between the rounding and the numbers they choose to solve. At first they will use the rounding to see if their answer is “in the ballpark.” You can get them to see that they can use the estimate to help them right away. By that I mean that instead of starting with 100, they could choose 400 because they already know that 500 x 8 is 4,000 (inverse relationship between multiplication and division). They would therefore be as efficient as they can be and only need three steps to solve the problem as opposed to the 4 (above) or more.
Once they have became efficient and effective with this method you could transition them the friendly number to the traditional, if needed. The students can also compare and contrast the two methods to discuss the similarities and differences.
As you can see when dividing 36 by 8 you are really dividing 8 into 3,600. The same applies to the other two steps. The students can see how the 0’s have been eliminated in the traditional algorithm to further the comparison.
I think that for students who do not understand the traditional algorithm, this strategy can be beneficial.
* Division pictures written on ArtRage 2.5 Starter Edition
In “Blogging your vacation keeps loved ones in touch” by Beth Whitehouse, the author describes her use of logs to keep others in the loop of their vacations. In the article, Beth quotes Dane Atkinson chief executive of Squarespace.com who said; “If you do blog, you can craft a story about your trip.” What a great idea. Instead of just posting pictures and video you can focus on certain aspects of the trip. Maybe it was a hilarious experince, a great dinner, or a map of what you had done (and maybe calculating the amount of miles walked in a day at a theme park. I could see kids using skitch to show the map and point at starting points, ending points, the “trail” they took around the park with a key to determine calculations).
Another piece of the article that I found interesting was some of the suggestions the author made. One suggestions was to “respect your child’s voice.” Allow your child to create their story about their vacation without the parent telling what and how to write. We as educators should capitalize on this, with our upcoming spring vacation, four days away. This accomplishes not only the obvious, getting kids to write authentically, but introduces blogs to parents, creates another home-school connection and augments the goal of creating 21st century learners.
Long Island’s wireless internet project appears to have hit a snag. The project, which is endorsed by Suffolk and Nassau County Executives, Steve Levy and Tom Suozzi and is suppose to be built by E-Path Communications (based in Florida), has past its initial target date. The project is suppose to be built without tax-payer money. The N.Y. Times recently ran an article about the failures nationwide regarding the implementation of wi-fi. There are projects out there that I believe are having some success (Minneapolis, St. Cloud and Providence) but have used a lot of public financing and pledges from local governments to purchase access.
A company called Meraki sells low-cost equipment that can be placed in your home to broadcast a signal. Another piece of hardware, called a repeater, can be purchased and placed on roofs, walls, etc. to capture the original signal. In San Francisco there are approximately 70,000 users.
There seems like there are models out there that can be copied. One can only hope that people deliver on their promises.
I am reading a book by Victor Canto entitled Cocktail Economics-Discovering Investment Truths from Everyday Conversations. I came across these couple of lines that I keep coming back to in my head. I don’t know if I have an “oceanic issue” with my writing but I keep thinking about whether or not education and POB in particular are elastic enough to provide an appropriate 21st century education.
“By elasticity, I very simply mean the ability of an industry to adjust to economic shocks. An elastic industry shifts and alters and transforms when an economic “tsunami” envelops it, thereby helping to guarantee its own survival” (Canto 50).
What if instead it said “adjust to 21st century literacy shocks” and instead of economic “tsunami” it says “21st century literacy tsunami.” What are we doing district-wide to ensure we are shifting, altering, and transforming as opposed to being devastated by the 21st century literacy “tsunami.” In terms of acquiring technology, there is some promise as we hope to see more projectors and and finally a number of smartboards. As well we will have a district management system. The Kindergarten Center will finally have a computer lab. This is all based on the budget passing. Although I understand the tools are not the answer by themselves although how many classrooms have you seen that have 2 dusty Dells sitting there looking quite lonely?
What about best practice? The pursuit of understanding digital literacy (even though I am not exactly sire what this entails yet) and the tools associated with it, for me, is what it is about. Students of this generation are digital natives. Technology is a part of who they are. It is very disappointing to here people in our district say things like we need to coax parents to restrict computer access, kindergarten students don’t need computers, and technology is just a waste of time. We cannot be pushing the teaching of digital literacies out of the schools. Adults in our school should not be saying I am taking that Ipod away, we should be figuring out how to best incorporate that tool into our day.
We are at an interesting juncture at POB, a fight, as I see it, for 20th century vs. 21st century learning and teaching. Do we want to be a district that has good test scores but students who cannot think, synthesize, analyze, create, collaborate, and converse? Personally, if we have to live with the state exams, so be it (although they are not measuring what we want or expect from our students). But if the answer is a worksheet for homework that has 70 problems on it, I can’t be a part of that. If a student knows the first 10 and the next 60 are the same thing (with no rhyme or reason) what is the point? That a student persevered? This would seem to be quite inelastic.
I am hopeful. More teachers are attending conferences related to technology. There have been more informal conversations about the use of technology and its role in best practice. Wikis have been created, Google Docs is being used regularly, VoiceThreads have been used as lessons, and websites have been developed to build communities of learners. That seems elastic.
There is more to be done. Are we going to adapt and change or be smothered by the tsunami?
What do you think?
Canto, Victor A. Cocktail Economics-Discovering Investment Truths from Everyday Conversations. Upper Saddle River: Financial Times Press, 2007.
One technique or tool that I have been using this year with the students I educate, is the mini-lesson. Usually 10 minutes in length, they are designed to develop efficient mental math strategies. They are also designed to highlight certain strategies such as the distributive property.
Mini-lessons are generally done with the entire class. There are usually 5-8 problems and they are displayed one at a time. As each problem is revealed, students are asked to solve them with their hands (they can use pencil and paper to keep track or use a model). We want the students to examine the numbers and use them to efficiently solve problems. Each problem is related in that they are helping to promote an understanding of a mathematical idea (i.e.-commutative property, associative property, etc.).
I would like to take a look at an example of a number string that we used this year with a fifth grade class. We had some students who still were not fluent with the basic multiplication facts. Facts such as 6 x 8 were not be fluent and were not automatic. As I began to use the number strings, I noticed students were able to use what they knew and apply it to what they didn’t and many have a better recall of the basic facts.
One string we issued this year involved the following number facts (remember I would only put one problem up at a time, then students would solve and discuss):
3 x 4
3 x 8
6 x 8
12 x 4
24 x 2
48 x 1
3 x 16
As you can see, the first three problems are doubles of the previous problem. Students knew that 3 x 4 was 12 and that 3 x 8 was 24. Some did not recall 6 x 8 and had to resort to a counting-on strategy. There was not much discussion here as most had recall of these facts, so I would model next to the problem an array that depicted the problem.
The next three problems help promote the doubling/halving strategy. They all have the same answer, which helped begin a discussion about equivalence. The students I was working with did not see the doubling/halving relationship between the 6 x 8 and 12 x 4. They either knew automatically or used another strategy to solve the problems. When I put the 24 x 2 on the board to solve this is when the students began to make the connection. They could see that 24 was double 12 and 2 was half of 4 and each problem had the same product. A discussion then ensued about why 6 x 8 also had the same product. Students could see either through the numbers or the arrays that 6 was half of 12 and 8 was double 4 and that they had the same products.
I had facilitated the discussion about the arrays to help them see what was happening visually to what they were explaining to the class. As you can see, in the 12 x4 and 24 x 2 problems the 12 was doubled to 24 doubling the width of the array and the 4 was halved cutting the length of the array in half as well.
The last problem was no longer following the doubling/halving strategy but tripling and thirds which led to some good discussions.
Using these types of number strings has really helped students develop fluency and efficiency when computing.