An off-the-cuff math question, “How much do you suppose they’re making off this ridiculous spectacle?” turned an excruciatingly dull evening into a creative endeavor. Yep, a math question. The other chaperones and I were hostages to a truly ghastly theatrical performance.…
A few years ago, two of the school districts I work with adopted new mathematics curricula–they each chose the one the other was abandoning. Golly, they could have saved a lot of money by simply boxing up all the teacher and student editions and swapping!
When student performance falls short of expectations, most educators seem to go hunting for new tools such as the ideal curriculum, the ideal personalized intervention program, the ideal iPad app, and so on.
But take a look at my post from last week, The Right Kind of Lazy for Math. What depth of math knowledge do teachers need to guide students in the deep number sense required for this kind of mathematical agility?
What if students approached math problem sets as if they were fascinating puzzles rather than boring worksheets? And did great work while being lazy? Sound crazy? It’s not–and there’s a neuroscience connection!
Quick–multiply 19 x 24. Are you reaching for paper and pencil? Or a calculator? That’s what students (and adults) trained in algorithms do. My colleague, Dr. Dario Nardi, has gathered EEG images of dozens of college students solving such problems. Those trained in algorithms use just a couple areas in their brains for calculations. The good news is that they solve these problems quickly and accurately. The bad news? They don’t access the part of the brain used for skillful working of unusual problems.
Instead of reaching for paper or calculator, did you say to yourself, Well, I can make this easier and do it in my head…20 x 24 – 24. If so, you used more parts of your brain, connected with understanding the steps in a process, rotating images, and viewing things holistically. And, if you learned to use strategies such as “friendly numbers” in collaborative groups, you also activated the speaking and listening areas of your brain. You did less work and fired more areas of the brain, creating new synapses for later, even more complex mathematical thinking!
This is the right kind of “lazy” for math.
A blog this week asked us to guess the grade level for which this math problem was written:
Kristen has four flowers. She gives some to a friend. Now Kristen has two flowers. How many did Kristen give her friend? Draw pictures to help you solve the problem.
It’s listed as a kindergarten homework problem.
If you teach math, you know this problem includes some of the biggest arithmetic concepts there are and you’re not deceived by the use of small numbers.
- Students need to understand hierarchical inclusion–that 4 includes 2
- They need to understand conservation–that the number of objects remains the same, no matter how they are arranged
- And, they need to understand cardinality, that the name of a number relates to a specific quantity–including the huge idea that “two” isn’t the second object, but a set of two objects. This is a major leap in knowledge, often hindered by memorizing names of numbers. Too often, students learn to count to 30 or 100 but don’t understand the concepts involved.
Last week I voiced my concerns about decreasing the emphasis on narrative writing in schools, with more effort going to evidence-based argument and other skills that business and colleges say are essential.
Could we think beyond the needs of business for a moment?
My first reason for wanting to do so: Could we think of other cultures? And the high value placed on story as one conveys values, truths, societal norms, warnings, rules, wisdom and so much more through narrative? I’ve done some work with schools for Native American children, for example. In a powerful book, Our Stories Remember, James Bruchak says this about the use of narratives.
What is the place and purpose of stories? What is their proper use? Stories were never “just a story,” in the sense of being merely entertainment. They were and remain a powerful tool for teaching. Lesson stories were used by every American Indian nation as a way of socializing the young and strengthening the values of their tribal nation for both young and old (p. 35)
It’s official. Here’s the research: having students analyze how math mistakes were made is more powerful than having them solve equation after equation.
I often show math professional learning communities a great film that Lucy West provided to me. Picture an 8th grade math class in an urban school. One girl explains the equation she developed to describe the number of tiles that surround the edge of a swimming pool. Her equation is correct, but instead of saying, “Right!” the teacher says, “Thank you for sharing. Who has a different answer?”
Another boy comes to the front, places his equation and diagram on the overhead projector and says, “After hearing her answer, I think I’m wrong, but I can’t figure out my mistake. Who can help me?” And the other students, not the teacher, analyze the two threads of thinking and determine not just which is correct, but how the second student got off track. The teacher intervenes only to help the students make their diagrams clearer and ask for reasoning.
Powerful, isn’t it.
What will be measured drives what will be taught. Any more reason needed for paying attention to the assessments being designed to align with the Common Core State Standards (CCSS)?
And there is hope. Dan Meyer (follow him on Twitter) posted this morning The Smarter Balanced Assessment Items which has a great example of an item that a) requires thinking b) shows the real-world usefulness of the concept being assessed and c) uses technology appropriately. Here’s the item:
Five swimmers compete in the 50-meter race. The finish time for each swimmer is shown in the video. Explain how the results of the race would change if the race used a clock that rounded to the nearest tenth.
Way better than a set of practice problems on rounding, isn’t it?
A few weeks ago I wrote about similarities in coaching myself as a cyclist and the strategies of differentiated coaching. A key component is knowing when to help teachers develop a routine or ritual to overcome a persistent struggle.
My problem with biking was remembering to unclip from pedals soon enough to avoid tipping over at stop signs. I needed a routine. To develop it, I identified when to clip in and out, practiced to determine how far in advance I needed to start the process (way sooner than other bikers), and also experimented with whether twisting my feet together or separately was the better way (separately so that I could then flip each pedal and not accidentally re-clip). These routines have so far kept me fall-free.
Here are a few examples of routines that helped teachers find flow in their classrooms:
In my last blog, I discussed how my experiences with coaching myself as a cyclist parallel my most effective strategies for coaching teachers.
Setting individualized goals is a key component. When I first tried out my biking shoes, I aimed for my husband’s goal: clip in as much as possible. After the second fall, I changed my goal: clip in wherever it is safe. I unclip on any stretch of road or trail that might bring surprises.
Often, new teachers—or their mentors—assume that a strategy needs to be implemented just as other teachers use it. They end up either biting off more than they can chew at first, or struggling inordinately with something that looked so easy when modeled by a teacher with very different strengths. Think for example about ways to develop student responsibility for materials. A new middle school teacher wanted to use a colleague’s chart for tracking which class had the highest percentage of students who came each day with pencil, notebook, and homework.
Before you read this, read Bill Ferriter’s great recent blog, Are We Asking the Right Questions? He is spot on about the dangers of focusing too much on “right answers” when asking questions about what is and isn’t working in schools. “Why did this student answer this question wrong?” is a far different question than “Can this student use what we’ve taught to innovate in some way?” Bill points out,
Phrases like “what would happen if” and “why should we believe in” that play a regular role in the language of innovators and entrepreneurs are replaced with phrases like “do you know how to” and “what do you remember about” which do nothing more than emphasize the skills required to find the right answers to someone else’s questions.
As an outside consultant, I’m often struck by how reluctant or even afraid professional learning communities can be to raise, and then seek answers to, questions involving anything but instruction and testing these days, even when finding answers might improve student outcomes. !