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Cathy Fosnot Math Workshop
Transcript of Cathy Fosnot Math Workshop
Celebrate each accomplishment and then move the goal posts. Focus on generating strategies rather than applying memorized formulas. Looking through a new lens -Dos and Don'ts Young children will never discover math patterns unless they count large amounts. “Hierarchical inclusion” means numbers nest inside other numbers.ex.
1 - *
2 - **
3 - *** (It is not the third * that is 3 it is *** that is 3) Babies are hardwired to see 1, 2 and 3. - babies looked at dots till bored or till dots changed - did the same experiment with sounds, ex. 2 beeps
Seeing small units as a whole is known as subitizing. It takes about 5 years for a child to count to 7 and understand that it means he has 7 items.
- there is a difference between counting and understanding total amounts.
So, DON'T tell children to count. Instead, ask how many there are. Do not post a number line. Instead, have students create their own tape measures. This will lead to number line concepts that are understood rather than memorizing a posted number line. Concepts taught through patterns on a 100's chart do not transfer into number sense. You are better off to use a 100's chart by cutting apart rows and creating a number line together. Use it for some skip counting etc. But do not leave it posted in the class as a tool to use. Base ten blocks and 100's charts are manipulatives only. They help in reading answers but do not transfer into working number sense. Mini lessons: - great to do at the opening of a lesson when the group meets together. Post a Math Strategy Wall – similar to a Word Wall except instead of vocabulary you post ex. “Caroline's Strategy for Adding Doubles”.
Remind your students that = means balance. So, 152 + 149 = 150 + 150 + 2 – 1 It's fun to try different strategies BUT.....“A mathematician always says to himself – What's the fastest, most efficient way to solve this problem.” The best mathematicians look to the numbers first before choosing a strategy. Certain strategies are better at certain times depending on numbers. Number line is a model NOT a strategy. Use the number line for skip counting in patterning. Songs are helpful too for skip counting .ex. Count by 3's (We Will Rock You) count by 4's (Twinkle, Twinkle – each last note of a phrase)
Always use an open number line instead of a real number line so they will do operations instead of counting. To truly understand a number line you need to understand measurement. Always give students reflection time through partner talk. Thumb up when ready to respond. Who had a helpful partner who made things clearer for them? Do NOT put your attainment target on the board because it will influence thinking. The important thing is the Strategy Wall. Subtraction – emphasize that this means to find the difference.
That could mean take away
could add on
just find the difference – helpful to place it on a number line
DO NOT call subtraction take away. It doesn't have to be. *** the teacher is always the one to represent strategies on the number line
1. teacher develops models for students' lives
2. students see you use the number line to represent their strategies
3. learners will eventually use the number line tool during their investigations
Calculating Rack (Rekenrak)
develops sub units of 5
children need to realize the total will always be 5
always keep red beads on class left
always read beads left to right (like an equation)
then extend reading to 10
* this takes months of daily practise to develop
try the passing billboard drill
-use doubling and halving as a strategy
-ex. 24 X 15 = 12 X 30 = 6 X 60
-arrays – each square in an array is in a row and column simultaneously
-STUDENTS MUST clearly understand arrays before they can understand area.
-Use graph paper to model arrays.
Do not call multiplication repeated addition. It isn't.ex. 1/2 X 2/3 = For a Gallery Walk, not all solutions need to be complete. It is alright to show your thoughts and investigations so far. Gallery Walk Gallery Walk is to support development of proof-making. Don't confuse proof making with telling everything you did. Questioning:
ask students questions that mentor them to think mathematically ex. What model could you use?
Do you know what mathematicians do when they are stuck? They step back for a moment and say is there some kind of model that might help illustrate this problem? Let's try it and see what happens. Do not tell students what to do.
Model what a professional mathematician might do.
“So, are you saying, that doubling and halving is alright to do because of the associative property?” Before the Gallery Walk stage, teacher should walk around the classroom, celebrating with students and then upping the ante. Get on the students level. Say things like, “I wonder about that too.” This gives students the confidence to articulate their thoughts. While “kid-watching”, gather evidence only. Do not infer.
“I think what you're doing is _______________. Is that right?”
Math Congress is not a share time.
Teacher picks 2 or 3 pieces that will allow some interesting insights to the math community
during the Congress the teacher should be part of the audience.
Students need to convince the math community not just the teacher.
Start a Math Editing Book with your students. After a Math Congress, they can revise their math problems in this book and keep it as a source for strategies that have proved to be effective.
Have students clarify solutions – NOT the teacher.
Piaget said that every time you show children your way of doing something you have robbed them of investigation and empowerment. FIRST CLASS – find math continuum “Fosnot Curriculum Connections”
Go to SSS (School Support Services) then Curriculum Services, Elementary Math, Curriculum, Fosnot Connections. Assessment :
photocopy and reduce the size of student work
add it to student portfolio along with comments
photocopy a landscape page and highlight the accomplishments
Give a test of ten questions including related questions
ex. 10 X 13 =
5 X 13 =
2 X 13 =
10 X 13 =
Have students use pen for the test.
Write in red pen for the first two minutes.
Switch to blue pen and take five minutes more.
This will tell you what the student finds easiest.
It will also show you how they are progressing across the landscape. SOMETIMES use pen for open response questions so you can see ALL strategies that the students use. Do not post a number line.
Instead, have students create their own tape measures. This will lead to number line concepts that are understood rather than memorizing a posted number line.
demonstrate cube number line with alternating groups of 5 (ex. 5 white, 5 orange) – Place this high to force visual groupings rather than counting ones.
50 + 50 = 100 show on open number line
49 + 51 =
300 + 300 = 600
298 + 303 =
Strategy: use a friendly number or double that you already know and then branch out from there.