Math Professional Development day with Peter Liljedahl: Two sessions

1. Exploring Producibles: Getting the Genie Back in the Bottle.

2. Assessing Numeracy: How do you Photograph the Wind?

One of my goals for this year is to make Professional Development Days more useful. Too often I gather great information from a session and then ‘file’ it away never to be used… despite its usefulness! I have decided that I will always pick one thing from a session and make sure that I implement it immediately into my practice. I will take the problem below into my Math class on Monday! Problem solving is something I have always valued and this session was a good synthesis of many ideas that I find important in teaching Math. I will be taking a lot more than one thing from this session…

Problem:

You have two glass orbs of equal strength and a 40 story building.

Your task is to determine the highest floor from which you can drop an orb without it breaking.

What is the least number of drops required to do this?

Both orbs may be broken in order to determine your answer.

Problematic problems:

Solving ‘True problems’ rather than traditional ‘word problems’. “With word problems the problem is in the words, with true problems the problem is in the problem”… word problems are about the ‘right’ answer, “A train leaves Toronto at…”. True ‘Problem Solving’ is about your path to the answer, it has ‘better’ answers, not necessarily a right answer… (It may indeed have a ‘right’ answer, but when you get it, you can’t necessarily guarantee that it really is the best answer.) The question above is about problem solving! (Tell the problem in a story rather than putting it in print.)

Producibles:(anything that students produce – not necessarily to ‘hand-in’ -List below created in our session.)

• Presentation – Solution/ Think-Pair-Share

• Case Study – give them 3 floors… try it, did your theory hold up?

• Create a similar problem

• Pictures/Drawings (with explanation)

• Self Assessment

• Skit/Video/Audio

Aspects of Problem solving: Communication (talking, listening, sharing, debating, working together, questioning, helping, taking roles, co-operation), Brainstorming, creating, interpreting, testing, hypothesizing/predicting/guessing, organizing, risk-taking, getting stuck/frustration/disengagement, comfortable frustration, self-correcting, messy/loud, getting stuck, brainstorming, perseverance, using multiple strategies, thinking!

Which aspects do we value… and want to develop in our students?

What to we want to assess?

“A hundred years ago we used Grimm fairy tales to keep kids in line, the fear of a witches or big-bad-wolf. Now we use assessment.”

We need to make thinking valuable in class… if we just access 20 questions at the end of a unit/lesson etc. then what are we telling students is important? “Show your work” is really ‘show my work’. How good are students at truly showing their thinking? Your thinking is chronological not logical.”

Here are some producibles that can be used to look at ‘The Process’:

Reflective Journaling- Does not disturb the process, but creates a producible that is reflective of the process- can journal on many things- Tell me how working in a group contributed to your answer. Convince me of how hard you worked on this problem. (Moving the thing that you value into your evaluation)… but you must teach students what you expect from journaling!

Black/Whiteboards- Put everyone on a board… teacher in the center of the room.

Digital Photos- print and ask, “What were you doing at this point?” “How were you contributing to your group when this photo was taken?” –can be ‘in-action’ photos or after they are done.

Poster- lots of variety here… not just what is your answer… look at process.

Pictures- Draw a 3 (or 4) panel cartoon that shows me how you got to the answer that you did.

More producible strategies: Show your work; Show me how you know; Quiz/test; Problem solving journal; Graphic organizers; Narrative/story; Placemat; Time line organizer; Presentation/sharing; Self and peer evaluation.

*Pair these together: Digital photos used to spur a reflective journal entry.

Here are some Cooperative Learning Strategies.

When you assess process you need to:

1. Let students know what you are assessing, (Today I will be looking at your perseverance or team work etc.- Pick only one!)

2. Mark them/give them feedback on what you are assessing, (don’t be afraid to be tough on them… and give them a chance to improve the next time)- call them on it!

-Show that you value the things that you value! (Problem solving aspects or skills that are important should be both announced and assessed.)

Kinds of Problems:

1. Planning Problem: “How many chocolate bars/drinks should you buy for the school dance?”

2. Fair Share Problem: Goody bags- you need 10 bags and you have 6 suckers, 8 chocolate bars, 12 glow-sticks, 3 party horns… “How do you ‘fairly’ make 10 goody bags?”

3. Estimating a large number of variables: “How many candy canes does the Santa at the mall hand out in a day?”

4. Modeling problem: “How do you organize classes so that you can get every other Friday off… without losing overall teaching time, and keeping a fair distribution of class times?”

Possible Producibles for All Students:

• Right/Wrong -sometimes you can just say, “No, that’s not right”. (If the class culture makes this ok.)

• Step by Step Rubrics/Part Marks

• Content Rubrics

• Performance Standard Rubrics

• Holistic Rubrics

Assessing through Observation (Some students per day- not all)

• A holistic or specific focus observation of selected students on a given day… (Announce and Assess).

Here is a very useful Holistic Marking Rubric .

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There is a problem with our fascination with “Explain your thinking” as a tool to asses. I am reminded of an analogy in Blink: The Power of Thinking Without Thinking by Malcolm Gladwell, on:

Unconscious Intuitive Thin-slicing: People are often unaware of how they make intuitive judgments.

Imagine that you have to solve this problem: two ropes are hanging vertically from the ceiling, too far apart for you to reach both at once, and you have to find ways to tie them together. One solution, which very few people come up with on their own, is to swing one rope, grab the other rope, and then catch the swinging rope. In one study, people get a subtle hint: the experimenter casually walks across the room in a way that involves brushing against one rope and making it swing slightly. Most of the people were able to pick up on the hint and identify the rope-swinging solution. However, only 1 of these people realized that they got the idea from the experimenter’s brush with the rope. The rest came up with unrelated explanations of their inspiration. They had no idea of the process that led to their (successful) intuition. [Taken from Blargh Blog]

…Students don’t always know where their ideas come from. Our school goal of ‘Articulate Your Thinking’ grew out of an e-mail that I wrote to our Math Learning Team that we had almost thee years ago. I still think it is important for students to articulate their thinking, and I have found this session interesting in the way that it challenged some of my thinking. Getting students to record their ideas chronologically can be a very insightful process that I will explore further.

This was an excellent opportunity to look at what is really important in Math. Just as I have been focusing on 21st Century Skills and the use of technology for learning, it was great to also revisit the parallels to Problem Solving in my Math Class.

It is interesting to note that Peter is fully aware that it is difficult to get all students engaged in problem solving, and he noted several times that this is true for other subjects too, and yet we still work at it… (Can you say that all your students were engaged with the Water Cycle in Science or with the Square Dancing unit in Gym?). You still have to use your skills as a teacher to engage students in the problems. Also, as you work on this ‘sometimes messy’ math, it is important to let students know what specifically you are assessing them for, and then you need to ‘raise the bar’ on expected producibles as students: get used to you assessing ‘different’ skills; and, as they improve their thinking skills!

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Resource: NRICH –need to filter, but some really good problems can be found here. It has a good internal search: by topic/grade(UK-so actually ‘level’)/difficulty.

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My ‘best answer’, so far, for the initial problem… with a 40 story building, the maximum number of orb drops I would need to determine the highest floor from which you can drop an orb without it breaking is ____ (I’ve got it in less than 10… so far). Feel free to post your best answer, but don’t ruin it for others with an explanation here!

I am starting a collection of good numeracy tasks on my ‘Practic-All’ blog. Do you have a Numeracy Task you would like to share? (Click here)

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Images: ‘Sphere_2720’ by doviende, ‘Producibles’ chart by Peter Liljedahl, ‘Swimming Hole Rope’ by Cindy Seigle.