Archive for December, 2010
There are many different types of scaffolds. We should be sure to use a variety of scaffolds to accommodate our students (Alibali, 2006). For example, we use scaffolds to accommodate the readiness levels of learners, novice, apprentice, practitioner and expert. But, we also use scaffolds to accommodate the progression of knowledge among learners, so the apprentice might start with one type of scaffold and then move on to a more complex one later. Hartman (2002) states that scaffolds may include models, cues, prompts, hints, partial solutions, think aloud modeling and direct instruction.
A procedural facilitator can be a hint, cue card or partially completed example (cited in Van der Stuyf, 2002). For example, when teaching rounding, the graphic organizer with the hills on it and the numbers tell students what to do (GOOGLE ANTHILL NUMBER ROUNDOFF/ http://www.learnnc.org/lp/pages/3420 ). Another example for rounding is a colored number grid http://www.superteacherworksheets.com/paper/hundredschart-rounding2.pdf Here is another example of a graphic organizer being used as a scaffold for rounding. This one uses the number line http://www.superteacherworksheets.com/rounding/rounding-nearest-hundred-d.pdf Also, poems like the rounding poem http://www.proteacher.org/a/79954_Rounding_Rap.html can be considered a scaffold (a type of cue card) because they give prompts to what should happen. Here is another example of a cue card type of scaffold. Notice the prompts at the top of the sheet. [you have to copy and insert url into your browser directly to get to site] http://math.about.com/library/5roundinga.pdf
Hartman, H. (2002). Scaffolding & Cooperative Learning. Human Learning and Instruction (pp. 23-69). New York: City College of City University of New York.
Google: Scaffolding as a Teaching Strategy (Van der Stuyf, 2002)Read Full Post | Make a Comment ( 1 so far )
What is Scaffolded Instruction?
Scaffolding is helping students become successful through a series of guided steps. Bruner (1975) coined the term, based on the work of Vygotsky. The main ideas is that children can become successful doing things that they can’t do on their own yet, with a little help from both their teacher and friends. In the beginning a great deal of support is given and then gradually the support is decreased until the student can successfully do it own their own. Remember when you learned to ride a bike? Those extra back set of trainer wheels were one level of scaffolding. Then, when they came off, whoever push started you and followed close behind was another level. Finally, you were off, down the street, doing it on your own, grinning all the way!
Hogan and Pressley (1997) found 8 essential elements to scaffold instruction:
1. Pre-engagement with the student and the curriculum
2. A shared goal
3. Ongoing Assessment- Pre/During/End
4. Tailored assistance – This may include cueing or prompting, questioning, modeling, telling, or discussing.
5. Ongoing Goal Setting
6. Specific Feedback
7. Attention to student disposition/mental and emotional engagement
8. Internalization, independence, and generalization to other contexts –
(adapted from citation in Larkin, 2002)
Larkin (2001) found that teachers who scaffold also
- Meet students where they are/ focus on what they can do
- Scaffold success quickly so that the “cycle of failure” is broken
- Help students to “be” like everyone else
- Know when it is time to stop – “Less is more”
- Foster Independence
The open number line is a powerful tool to scaffold the development of mathematical proficiency, specifically conceptual understanding, procedural fluency and strategic competence. It helps to show the magnitude of distances on a line, equivalent quantities and proximity of numbers to landmark numbers (Dreambox, 2010). The open numberline provides the opportunity for children to practice efficient strategies such as jumps of ten, splitting numbers and compensation.
Different from a “regular” numberline with the counting numbers written on it, the “open” number line is a line that children draw to show/record their thinking as they solve addition and subtraction problems. It is allows them to illustrate their strategies. Students only write the numbers that they are using in the problem. They record these number as jumps along the number line.
MORE OPEN NUMBERLINE VIDEOS
Read this Vignette of a Small Guided Math Group in Action:Read Full Post | Make a Comment ( None so far )
The open number line is a power tool — one that promotes powerful mathematical thinking. It helps children to show and explain their invented strategies, builds flexibility with numbers and scaffolds the mental representation of number and number operations to support mental arithmetic strategies (Fosnot,2007 ; Beishuizen, 1993 & Gravemeijer, 1991).
The open number line gives students a model for representing their thinking. It requires that they be actively engaged in their explanations. It is more cognitively demanding than either base ten blocks or the hundred chart according to Klein, Beishuizen and Treffers, 2002 cited in Fosnot,2007).
It is great to use this strategy with the whole group, but if you really want to have in-depth conversations then you should do it in small guided math groups so children can have the time to explain their thinking. In a guided math group, the teacher would model use of the open number line and then give the students the opportunity to work on a problem together with the open number line. Finally, the teacher would give each student an opportunity to solve a problem using the open number line as the model while explaining their thinking to the group.
Here are some great videos of the open number line in action!
BE SURE TO READ PART 2:)
Dreambox has a virtual numberline that we can use for free!!! Yeah, we love good free stuffJ http://www.dreambox.com/blog/the-latest-free-dreambox-teacher-tool-open-number-line-developing-number-sense%E2%84%A2
Fosnot, C.T. (2007). Measuring for the art show. Portsmouth NH: Heinemann or Fosnot, C.T. and Dolk, M (2001) Young mathematicians at work: Constructing early number sense, addition and subtraction. Portsmouth, NH: Heinemann.
Fosnot, C.T. and Uittenbogaard, W. (2007). Minilessons for extending addition and subtraction. Portsmouth NH: Heinemann.
Lucky 8 and Lucky 9 refer to the math strategy of compensation – a strategy that fiddles with the numbers in order to make the problem easier to solve. Both the Common Core first grade and second grade Operations and Algebraic Domain, specifically refer to teaching a variety of mental math strategies.
When practicing Lucky 8 and Lucky 9 students are adjusting these numbers to become 10 so that they can work with them more efficiently. Remember that when we are teaching math we always want to think in the frame of concrete, pictorial and abstract.
I use the double ten frame to teach this so students can see what is happening. Look at an example of that here http://www.teachervision.fen.com/addition/graphic-organizers/44540.html I do this in small groups and give each child a double ten frame and counters. They get to manipulate the numbers. So take the problem, 8 + 7. Students see if we have “lucky 8” that there are two empty squares in the first frame where we built 8. So they take two from the second frame and move them into the first frame to make a ten and then we add 10 + 5. We practice this many times so that students get a conceptual understanding of the model. Then we do the same thing with “lucky 9”.
At this site they can practice compensation on the Illuminations site under the Add game http://illuminations.nctm.org/activitydetail.aspx?id=75 On this great site that talks about fact families, you can see a visual of how this works at http://www.montgomeryschoolsmd.org/schools/oaklandes/mathstudentworkpages/gr1un2page6.html
I also do number grid activities to teach this. See post on number grid
Here are two poems that I made up that I use with the students when teaching this strategy.
You’re So Great!
When I see you
I know what to do
Go to the other number
And take two!
You’re So Fine!
Glad to see you everytime
It’s just too much fun
To go to the other number
And take one!
Songsforteaching.com has some great poems as well. They have one called 9 Be My Friend http://www.songsforteaching.com/carlsherrill/9bemyfriend.htm …the first line goes “9 be my friend, let me turn you into a ten…” These are great for talking about the problem and having an abstract mnemonic to remember the strategy. They have another poem called the 9 Rule http://www.songsforteaching.com/jeffschroeder/9rule.htm which also talks about compensation.
Google this “Math 4 – Act. 01: Mental Math: Addition and Subtraction.” It is a UEN lesson plan with a great graphic organizer.
Happy Mathing and Happy Holidays!
Dr. NickiRead Full Post | Make a Comment ( None so far )
Here are some interactive websites for teaching time to your students. Many of these can be done on an interactive board. You can also pull your students into small groups and play these in the group. These are also great sites to set up at the computer center so students can practice. Remember we teach digital natives:)
http://www.youtube.com/watch?v=fXtnLoxGtGo (these cost but they are good)
Dr. NickiRead Full Post | Make a Comment ( 1 so far )
Subitizing provides a basis for early addition skills. Different arrangements help children to see and discuss different ways to name a number. Ten frames are another tool to use to build subitizing skills. Subitizing is a fundamental skill in the development of students’ understanding of number (Baroody, 1987). Clements (1999) writes that students can use subitizing to develop understanding of number, conservation and compensation.
Here are some game ideas. There are several others listed in the links below. When you show your students the cards you should ask “What do you See?” This is different from “How many? because it frames the question in a way that they think about different number arrangements. More question examples http://naturalmaths.com.au/numblocks/mr_subitization.htm
1. Quick image games where the teacher flashes dot cards and the students tell how many.
2. Give children a set of cards with equivalent names but one that doesn’t belong. Have them select the one that doesn’t belong.
3. Have students match the number with the dot card.
4. Have the students play matching games with cards that show equivalent names.
Great Resources for Subitizing:
There is a new video out by Marilyn Burns called Number Talks that shows a teacher using dot cards and ten frames.
ReferencesRead Full Post | Make a Comment ( 1 so far )
Subitizing is being able to look at a number and know how many without counting. It refers to “rapid, accurate and confident judgments performed for small numbers of items. It comes from the Latin adjective subitus (meaning “sudden”)” and has to do with immediately knowing how many items one sees for a small set of numbers. When we are talking about larger sets we often estimate or count.
There are two types of subitizing. Perceptual subitizing is looking at the number and knowing how many without any mathematical processes taking place. Conceptual subitizing is based in mathematical processes, such as looking at the parts and the whole. An example of this iswhen you see the eight domino and you know that it is five on one side and three on the other and that makes eight. Spatial patterns are just one kind. Other patterns include kinesthetic ones such as finger patterns, rhythmic and spatial-auditory (Clements, 1999).
We usually teach students to subitize up to ten. See http://www.poweroften.ca/index.php?option=com_content&view=article&id=37&Itemid=103
Spatial arrangements make subitizing easy or hard. Rectangular arrangements seem to be the easiest followed by linear, circular and scrambled arrangements increasing in difficulty (cited Clements, 1999). You can use dot cards, domino cards, number-cube cards and ten frames to teach subitizing. You can also use bingo chips and interlocking cubes.
Subitizing is a fundamental skill in the development of students’ understanding of number (Baroody, 1987). Clements (1999) writes that students can use subitizing to develop understanding of number, conservation and compensation. In part 2 we will discuss this further.
ReferencesRead Full Post | Make a Comment ( 2 so far )
Here are some great videos to introduce addition. I use them as part of the mini lesson and then facilitate a discussion about what it means to add something. These are great ways to build conceptual understanding. I then pull small guided math groups to further that discussion.
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