Which problem is "just right" for you?

Today on my Twitter-stream there was a great deal of discussion about how we use homework in math class. Being a university math educator, I have more freedom than most K12 teachers when it comes to assigning homework but that doesn't mean that I am any less concerned about this topic. Homework is a important memory for many math learners and we teachers need to consider carefully how we use it.

If our schools require us to assign homework from the text but we have some freedom in what it looks like, it is time for us to work our magic as problem solvers. John Golden and I team-taught a course for preservice teachers a few years ago and modeled some ideas of adding to pre-existing items from a text. We called them "just right" problems in reference to the NCTM article Vygotsky and the Three Bears. Here's what the original items looked like:

from Scott Foresman – Addison Wesley Math [5^th Grade]

And here is how we tried to make them more thinker friendly:

Using these existing textbook items to demonstrate the processes associated with doing math. Pick one or two of the following to explore:

• Do either the odds or the evens – your choice. Why did you pick the evens (odds) to work on?
• Look over all the items. Which five do you consider the easiest? What makes them easy? Which five do you consider the hardest? What makes them hard?
• Pick an addition item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
• Pick a subtraction item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
• Pick one item to solve and write a metacognitive memoir that describes your cognitive efforts.
• The answer to one of the items is nineteen-twenty-fourths; which item is it? (Be sure to keep a record of your thinking)
• Which answer is closest to one? How can you be sure? (Be sure to keep a record of your thinking)
• Put the items in order based on their answers from least to greatest. (Be sure to keep a record of your thinking)
• Write your own ‘just right’ problem related to this review.

Which problem is "just right" for you?

Is there another problem we could add to this list?

Are you ready for "something completely different?"

My wife gave me The Complete Monty Python's Flying Circus for my 48th birthday. I just got around to watching the first episode and saw the following sketch. It demonstrates perfectly what happens when we fail to use formative assessment to inform planning and instruction.

It is important to understand that formative assessment does not need to be in the form of a test. We can gather data simply by observing. Imagine how this Italian class might have looked if the teacher had really seen the students during the lesson and adjusted his instruction accordingly. In the Design Thinking methodology, this is a part of the first phase - Empathy.

This reminds me of an article by Dr. Rheta Rubenstein titled, "The Learning of Mathematics: Perspectives of Different Researchers." (It comes from our text, Teaching and Learning Middle Grades Mathematics.) Under the section on Vygotsky, she includes the following diagram attributed to Dr. Lauren Resnick.

Vygotsky in Venn Diagram Form

As Terry Jones's character discovers, when teachers use scripted lessons that do not take learners' current level of understanding into account very little learning actually occurs.