We began today with a good discussion about the fact that we feel there are certain benchmarks that help us in teaching reading.  There was a question about whether there has been research on something like this for mathematical knowledge.  I did some research today and found some articles that were interesting but of course I have not been able to dig very deeply.  The one that fascinated me the most was https://link.springer.com/content/pdf/10.1007%2FBF03217400.pdf  and had some interesting ideas about the structure of math knowledge according to a psychological study referred to as the Steffe model whose key components were:

Building addition and subtraction through counting by ones;
• Building addition and subtraction through grouping;
• Building multiplication and division through equal counting
and grouping;
• Building place value through grouping;
• Forward number word sequences;
• Backward number word sequences;
• Number word sequences by 10s and 100s; and
• Numeral identification.

A couple of other sources I found were:

https://www.merga.net.au/documents/RR_treacy.pdf

https://books.google.com/books?id=JBmOAgAAQBAJ&printsec=frontcover#v=onepage&q&f=false

And Fran Arbaugh from University Park recommended the books:

https://www.amazon.com/Cognition-Based-Assessment-Teaching-Place-Value/dp/0325043434/ref=sr_1_1?ie=UTF8&qid=1502380549&sr=8-1&keywords=battista+mathematics

We had some great discussion about linear relationships that were both proportional and not proportional and non-linear relationships.  We followed up with some challenging questions from Block 4.

The end of our day consisted of a discussion about “Smarter than We Think” and a video about Cena whose understanding of place value might be a bit more fragile than her teacher thought.  Here are the ideas I shared about inverse and number sense that you could use in a few extra minutes of class.

“Tell me another way”

• I ask for two number whose product is 40 (or whose sum is 8).
• Then ask to find two other ones.
• After listing several look for a pattern.
• This can even lead in to negative numbers or fractions.

• Functions
• 6→∎→10 what could have happened in the box?
• 6→∎→10→∎→6 what could have happened in the boxes?

□→black box that multiplies by 2 and adds 4→  △  fill in the square and triangle with values.

• “What’s the process? – function of 2 variables”
• I put 2 and 3 in the black box and get out 5, what’s the black box doing?
• You can build on that with all sorts of processes.
• You can then have another box and have it undo the process of the first.

• Other ideas:
• I am starting with the number 7. I add 3 and then divide by 2 to get ______. How do I get back to the number 7 using the same number of steps I used?
• I am starting with the number 100, subtract 50, divide by 2, take square root. I end up with ______. How do I get back to 100 using the same number of steps I used?