mathematics

The Truth About Math Curriculum

question markA 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? 

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The Right Kind of Lazy for Math

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Is math a chore or a fun puzzle?

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.

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What’s Involved in this Math Problem?

k problemA 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.
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If the Common Core Devalues Narrative Writing, Who Are We Devaluing?

L Superior 2011-08-20 199Last 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)

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What Kind of “Complex Text”?

My “To Read” Shelf

Saturday morning, I read a book cover to cover. I laughed, I pondered, I underlined, I got ideas for my own writing, I…loved every word of it.

It was a book on mathematics.

Honest.

Now, honestly, have you run across a mathematics textbook that would glue you to your chair on a Saturday morning? Or a science textbook? Or history? Or is your experience much like mine—finding that the texts are boring or outdated or, worse, irrelevant to the real work of the discipline.

A history teacher told me, “I never assign more than five pages of reading because the students get too bogged down. It’s too complex. Oh, and 10/20 on a quiz is a C because the information is so difficult.” I took a good look at the book. Folks, it was an encyclopedia of names and dates, not “complex text” that explored the fascinating ideas and motivations that drive history. The ideas he was testing weren’t difficult, either—they just required hours of memorization. I will say that the publisher had added lots of pictures of torture and brothels, I assume in an attempt to keep students turning the pages.

The Common Core State Standards emphasize complex text:

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Aren’t Math Mistakes Beautiful?

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.

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Math Assessment Hope

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?

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