Loading in 2 Seconds...
Loading in 2 Seconds...
Middle School Performance Tasks and Student Thinking for Mathematics. CFN 609 Professional Development | March 29, 2012 RONALD SCHWARZ Math Specialist, America’s Choice,| Pearson School Achievement Services. Got change?.
CFN 609Professional Development | March 29, 2012
RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services
Try to figure out a way to make change for a dollar that uses exactly 50 coins. Is there more than one way?
Have you tried out any of the strategies, tasks or ideas from our previous sessions, and if so, what were the results?
What are one or two ways that US math instruction differs from that in higher-performing countries?
Describe and list some of the Standards for Math Practice.
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision.
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.
Development of expertise from novice to apprentice to expert
The Content of their mathematical Character
How are levels of cognitive demand used in looking at math tasks?
What do we mean by formative assessment and what are some strategies involved in it?
A train one mile long travels at a rate of one mile per minute through a tunnel that is one mile long. How long will it take the train to pass completely through the tunnel?
You have a 3-liter jar and a 5-liter jar. Neither of them have any markings and you do not have any extra jars. You can easily measure out exactly 3, 5 and 8 liters. Is it possible to measure out exactly 1 liter? If so, how? What about 4 liters? 6 liters? 7 liters?
A tree doubled in height each year until it reached its maximum height in 20 years. How many years did it take this tree to reach half its maximum height?
1_+ 1_+ 1_+ 1_+ 1_+ 1_
1×2 2×3 3×4 4×5 5×6 6×7
A tennis tournament has 50 contestants, with these rules: no tie games and the loser of each game is eliminated, the winner goes on to play in the next round. How many games are needed to determine a champion?
Properties of operations: their role in arithmetic and algebra
Mental math and (algebra vs. algorithms)
Units and unitizing
Operations and the problems they solve
Understanding the arithmetic of fractions draws upon four prior progressions that inform the CCSS:
Just nine: foundation for arithmetic
Exact same properties work for whole numbers, fractions, negative numbers, rational numbers, letters, expressions.
Same properties in 3rd grade and in calculus
Not just learning them, but learning to use them
Distributive property of multiplication over addition
a × (b+c) = (a×b) + (a×c)
a(b+c) = ab + ac
There are many patterns in the multiplication table, most of them are consequences of the properties of operations:
Find patterns and explain how they come from the properties
Find the distributive property patterns
Why you think it’s true and why you think it makes sense.
Saying “distributive property isn’t enough, you have to show how the distributive property applies to the problem.
In your head
Composing and decomposing
Place value in base 10
Factor x2 +4x + 4 in your head
If three bananas are worth two oranges, how many oranges are 24 bananas worth?
How they arise and how to deal with them
“Ours is not to question why,
just invert and multiply.”
What fraction of an hour is that? How many tubs could hose A fill in one hour?
The relationship between the time it takes for a hose to fill a tub, and the number of tubs that can be filled in one hour is called a reciprocal relationship. Each is also called the multiplicative inverse of the other.
Compare the relationship between a number and its multiplicative inverse with the relationship between a number and its additive inverse. What is similar?
Hose A takes 30 minutes to fill a tub with water. Hose B can do the same in 45 minutes. If you use both hoses, how long will it take to fill a tub?
When you add or subtract, line the numbers up on the right, like this:
Not like this
3.24 + 2.1 = ?
If you “Line the numbers up on the right “ like you spent all last year learning, you get this:
You get the wrong answer doing what you learned last year. You don’t know why.
Teach: line up decimal point.
Continue developing place value concepts
Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalization that the pupil has made. It may in fact be a natural stage of development.
Lean and clean lessons that are simple and focused on the math to be learned
Rituals and routines that maximize student interaction with the mathematics
Emphasis on students, student work, and student discourse
Teaches and motivates how to be a good math student
Assessment that is ongoing and instrumental in promoting student learning
Goal is to surface and make students aware of their misconceptions
Begin with a problem or activity that surfaces the various ways students may think about the math.
Engage in reflective discussion (challenging for teachers but research shows that it develops long-term learning)
Reference: Bell, A. Principles for the Design of Teaching Educational Studies in Mathematics. 24: 5-34, 1993
RONALD SCHWARZ, facilitator