modeling for middle school connecting context with math
TRANSCRIPT
Modeling for Middle SchoolConnecting Context with Math
Who am I?
Started teaching 7 years ago (Orosi, CA)
Intern = lowest on the totem pole
1st year “Algebra for ALL” (8th grade)
Math = NOT a priority
How do I reach these kids???
“Access” prior knowledge… Go WAY back
Teach Algebra by connecting it to primary math
“Math is so much easier now”
Now?
Started new position this year in Sanger, CA (8th grade)
Personal goal = Improve every day so that I can help make math learning a positive experience for as many students as possible
Offer my learning and experiences – spark thought, creativity, collaboration
Not an expert (…yet!) – just a teacher who tries stuff!
What about Y
OU
? Name
From (Location)
Grade(s)/Role
Hope to learn/do/accomplish in this session?
What is Modeling?
Modeling in Middle School
The Common Core Standards focus heavily on pushing students towards abstract mathematical models (algebra) in middle school and examples with “un-friendly” (rational) numbers
Many students see algebraic models as something “new” – no connection
Our focus as educators should be on finding ways to connect concrete examples (that make sense to students) with the abstract mathematical representations of those models.
Concrete Abstract
The problem is making the connection between each of these representations (they all mean the SAME thing)
Example: Adding Fractions
Common Errors?Why do students make these errors?
*Comprehension
Let’s Try: Adding Fractions
Questions Probing Comprehension
• What does ___ mean?
• Could you give an example?
• How could we show/represent/draw ___?
• How could you use your model/picture to demonstrate what happens?
• How can you prove/justify your answer using your model/picture?
• What if….?
• …other ideas???
Got Comprehension? NOW what?
CONNECT the understanding of concept with a mathematical representation:
“How can we represent our work with the model/picture just with math?”
Let’s go back: Adding Fractions
The REAL point of my time with you today…
Build comprehension through concrete and pictorial representations
Use these models to demonstrate and find the mathematical representation.
If students can “discover” the mathematical “short cut” (abstract) through a model they understand, it will make a lot more sense (and will carry over to more complex mathematics in the future)
CONTINUE ASKING the same questions – have students explain, justify, demonstrate, and connect the math to the original conceptual models
Something to Think About…
My personal belief: It isn’t so much about the models themselves as it is about students’ sense-making process.
If students have not used a particular strategy before, you will have to teach both the strategy itself, AND the math content relative to your grade level (and you thought time was an issue BEFORE!)
Sometimes it is more effective to leave it open ended and see what students come up with
Common modeling strategies
Area models: Nice connection with real life (area of a rectangle) and many different concepts that continue into higher levels of mathematics (multiplication, division, distribution, factoring, polynomials, estimating square roots, completing the square…)
Bar Models/Tape Diagrams: I like these best for word problems and representing numbers (fractions, percentages), but can be used for almost any type of problem. *Algebra tiles
*Number lines: Can be very helpful, but not innately understood by most students – have to build up understanding (this is already an abstraction in most situations)
Some resources for modeling strategies:
Grade level progression documents (a little more work, but show suggested strategies from lower grades that carry over): http://math.arizona.edu/~ime/progressions/
Tool for Bar Modeling with lots of video demos (apps that go along with this site are called “Thinking Blocks” but might be over-scaffolded for many students) http://www.mathplayground.com/ThinkingBlocks/thinking_blocks_start.html
National Library of Digital Manipulatives: http://nlvm.usu.edu/en/nav/grade_g_3.html
Honestly… I google a lot… Use vetted resources (NCTM, illustrativemathematics, illuminations, etc.)
What concepts cause the most confusion for your students?
We will focus on models and strategies that support YOUR requests/needs
Shortcut to the Payoff: (what was my point)
1st #: Speaker was well-prepared and knowledgeable (0-3)
2nd #: Speaker was engaging and an effective presenter (0-3)
3rd #: Session matched title and description in program book (0-3)