Making a Difference in the Math Classroom Marian Small November 2016
We will talk about… u
The renewed math strategy
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The importance of making sense of the math
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Critical thinking
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The importance of engagement
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The importance of a collaborative learning environment
Renewed math strategy u
There has been a focus on math for almost 10 years now.
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But we are still not where we need to be.
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So now there is intention to move schools, not individual teachers.
Some shifts u
Lots of attention to professional learning and building capacity, including that of administrators
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Particular attention to students with learning disabilities
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Staffing of grades 7 and 8 with those with more math background
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More math time in some cases
The big take-away u
Math matters and we need to do it right.
Some of you u
are with a particular group of students for a day or a few days
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Others are their teacher for a significant period of time
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Clearly, some things about those roles will be different but we will address our overall goals
Math has to make sense u
So if I said to you :
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Choose a number.
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Double it.
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Add the number to the double.
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What are some answers you could get?
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What are some answers you cannot get?
It makes sense u
That the number could be 3, 6, 9, … but not other numbers if you started with a whole number.
Or… u
Choose a 2-digit or 3-digit number.
u
You subtract a number from it, so that your answer is a LITTLE more than you subtracted.
u
What would you subtract?
It makes sense u
That you subtracted a little less than half
Pattern blocks You might ask: u If yellow is worth 6, make a design worth 20. u (or if a yellow is worth ½, make a design worth5/3) u
Could there have been more than one colour? u Did there have to have been? Why? u What is the most number of yellows you could have used? u
What comes next? u
What is the 20th number? How do you know? Are you sure?
u
14, 17, 20, 23, 26,….
What comes next? u
What is the 100th number? How do you know? Are you sure?
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14, 17, 20, 23, 26,….
Or a problem like this u
The perimeter of a rectangle is 3 times as much as its length.
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Use LINKING CUBES to show what the dimensions could be.
Some great class starters u
Are based on the notion of Which One Doesn’t Belong
For example u
Which one do you think does not belong?
u
5
8
9
15
For example u
Which one do you think does not belong?
u
4+8
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14 – 2
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9+3
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6+4+2
For example u
Which one do you think does not belong?
u
2/3
3/4
7/9
8/7
For example u
Which one do you think does not belong?
u
2x + 15
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30 – 5x
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5x
u
25/x
Number talks u
NO PENCILS OR CALCULTORS!
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How could you figure out 38 + 49?
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How about 42 – 17?
Number talks u
NO PENCILS OR CALCULTORS!
u
How could you figure out 0.5 x 284?
u
4.242 ÷ 0.21
Dot talks u
How many dots are there? How do you know?
Web sites like Estimation 180 u
www.estimation180.com
Solve me website
Critical thinking u
Critical and creative thinking are essential in a 21st century education.
It couldBbe u
A third shape is more like shape A than shape B.
u
What might it look like?
A
B
It couldBbe u
There is a plate with more than 200, but less than 300, cookies.
u
You think it is VERY EASY to share them equally onto either 3 plates or 5 plates.
u
How many cookies might there have been?
It couldBbe u
You show a number with a LOT more ten rods than one cubes.
u
What could the number be?
It couldBbe u
An amount you can show with 6 coins is added to an amount you can show with 3 coins.
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How many coins might you NEED to show the sum?
It couldBbe u
An amount you can show with 6 algebra tiles is added to an amount you can show with 3 algebra tiles
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How many tiles might you NEED to show the sum?
It couldBbe u
Which is a better description of how far away your birthday is?
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Is it 50 days?
u
Is it about 7 weeks?
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Is it almost 2 months?
u
Why is it better?
It couldBbe u
How could 1000 be a lot?
u
How could it be a little?
Engagement u
About how many text messages have you sent this year?
Engagement u
About how many steps would it take you to get across the room?
Engagement u
The McNuggets problem
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If McNuggets came only in 6s, 9s and 20s (they used to in Britain), what is the greatest number of McNuggets you could not get?
Engagement u
Compare the records of the Blue Jays and the Red Sox over the past 10 years. Which team is better? More consistent?
Debates u
A very engaging format is a debate.
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You put out a hypothesis.
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Students choose whether they agree or disagree.
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Representatives “debate” their perspectives.
Possible debate topics u
Jason says that you can’t get a little answer when you subtract two big numbers.
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Lia says you can.
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With whom do you agree? Why?
Possible debate topics u
A 3-D figure can have more vertices than edges, sometimes.
Possible debate topics u
Kyle says that when the numerator and a denominator of one fraction are closer together than the numerator and denominator of another, it is greater.
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Lia says that this might not be true.
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With whom do you agree? Why?
Possible debate topics u
When a line is steeper, it has a greater slope.
What might you suggest u
As a debate topic
Another great format u
2 truths and a lie
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Which is the lie?
Which is the lie? 1.
A number that takes 4 words to say can be greater than a number that takes 7 words to say.
2.
A number with more digits is always greater.
3.
There are only 90 numbers between 100 and 999 that are of the form ___hundreds + ___ ones if single digit whole numbers go in the blanks.
Which is the lie? 1.
When you subtract a fraction with a denominator of 3 from a fraction with a denominator of 4, your answer could have a denominator of 2.
2.
When you multiply two fractions, the answer could have a denominator of 1.
3.
When you divide a proper fraction by an improper fraction, the answer could be either more or less than 1.
What might you suggest u
For 2 truths and a lie
Another way to get engagement is to ensure questions are appropriate for ALL u
Open questions are one vehicle
For example… u
The answer is 100.
u
What might the question have been?
For example… u
You multiply two decimal numbers and the answer is close to 16. What might they be?
For example… u
The volume of a cylinder is a little less than 100 cm3. What could the dimensions be?
For example… u
Choose two of these points. What is the equation of the line joining them?
Collaboration u
Students need to learn to work together.
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I am going to give you a problem.
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The rule is:
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Work with 2 other people.
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Each of you has to contribute to the solution.
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Each of you has to ask at least one question of the others.
Problem 1 u
You add two numbers.
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You subtract the same two numbers.
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The answers are 24 apart.
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What might your numbers have been?
Problem 2 u
An item that was on sale at 40% off costs the same as one that was on sale at 20% off.
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How were the original prices related?
Problem 3 u
The maximum value of a quadratic occurs at 10.
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What could the graph look like?
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What could the equation be?
I have not talked about today… u
Something extremely important.
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The notion of choosing learning goals that focus on ideas rather than only skills and teaching to those ideas.
For example… u
Rather than considering my learning goal simply that kids can subtract 2-digit numbers:
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I can describe problems I would subtract in different ways and why I would do that.
For example… u
Rather than considering my learning goal simply that kids can use formulas to measure, my goal might be:
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I can describe which measures of a shape matter in finding its area and which don’t and why.
For example… u
Rather than considering my learning goal simply that kids can use the exponent laws.
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I can show how and when simplifying powers can make calculations easier.
The previous work u
Is probably the most important work the longterm teacher of a class (whether regular teacher or LTO) can do.
It is tied to u
The consolidation part of the 3-part lesson, i.e. helping students at the end of working on a problem see how the big idea emerges from the problem.
Most important u
An environment where kids want to learn
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an environment where kids are made successful
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An environment where math makes sense
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An environment where connections are repeatedly made
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