Tag Archives: math

Equity and Mathematics

17 May

Dan Meyer put together this talk from Uri Treisman at the NCTM conference in Denver.  Treisman puts together a series of statistics showing the US’s math performance relative to other countries, but he goes on to disaggregate the numbers by poverty level, and then by poverty level and state.   His message: poverty is important, and we don’t control poverty, but access to learning is also important, and we can influence that.  His talk is a call to math educators, and educators everywhere, to focus on what we can control.

Treisman manages to point out the incredible growth the US has made in mathematics education, and at the same time sound a call to arms to take control and do more.  Along the way he slips in stories of Boeing and the NYC police department, quips about Texas vs. California, and more.  It’s a great talk.

Building Perimeter

5 May

Area and perimeter have always been a doozy to teach, not because the concepts are so difficult but because students tend to mix them up.  When we get into problems like “keep the perimeter the same, but create the smallest and largest areas you can,” their heads really spin.

Enter the popsicle sticks.  It turns out that popsicle sticks are the perfect perimeter building material.  They could take the number of sticks they needed to represent a given perimeter (like 12) and create different rectangles.  Then they would count the squares inside and find the area.  You know an activity is a success when your strongest mathematicians want to build, and your struggling students are successful.

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Afterwards, students transferred their work to graph paper and wrote about which perimeter made the largest area.

 

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We had a discussion afterwards about what kinds of shapes made the largest areas (fat ones–the closer to a square, the better)and what kinds of shapes created the smallest area (long and skinny.)  Some students also came up with mathematical ways to create create shapes with a certain perimeter besides guess and check.  They discovered that for a perimeter of 14, for example, you need to create a shape with two sides that add up to 7, which then doubles for the opposite 2 sides.  Not all students were ready for that logic, but it spread through about half of the class.  Area and perimeter really lend themselves to building, we’ll have to do more in the future!

Do You Speak Math?

4 Feb

You can always count on Calvin and Hobbes to illustrate  how mystifying some schoolwork can be to children.  Take this one on math:

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Math truly is another language for children.  If your kids are English learners, it can feel like a double whammy–new math words, explained in a difficult language.

My first few years of teaching, I dealt with math vocabulary by ignoring it.  Denominators became “the bottom number” of fractions, the word “quotient” was nonexistent.  I thought I was making math accessible, but I was really denying my students any chance of becoming mathematically literate.

Now, I make sure I use the official terms in every lesson.  My class is comfortable using official terms when they talk about math.  I feel pretty confident in how I’m incorporating vocabulary into the subject.

Sort of.

But math talks need some more work in my room, as illustrated by a recent test on fractions, when multiple students weren’t sure what “equivalent fraction” meant.  We’d been using the word in lessons, it had been an official “word of the week,” and I thought it was one of the clearer terms (it has the word equal in it!).

What I was reminded of, is that me saying the word, or even having students repeat the word and use it in a sentence, isn’t enough for full comprehension.  Children need to say, read, and write words in context to help them solidify their understanding.  Even better, they need to make connections between words.

There are some graphic organizers and activities that can help children to make these connections.  Some of these I was already familiar with, and some I found after some research.

The Frayer model is one organizer I’ve mentioned before:

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You can also use a Venn Diagram for making connections in math:

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Or a web:

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Or concept circles:
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These organizer examples are available at this PDF site, along with blackline masters that you can print.

These organizers all help children to understand how vocabulary words are related and organized.  You can also get a lot of bang for your buck just by asking children to write how they solved a problem, using vocabulary words and some friendly sentence frames.  For example, I asked students to write how they knew that one fraction was larger than another during a compare and contrast lesson.  They could pick from a word bank of words, “greater than, less than, numerator, denominator, half.”  It was a challenge for many, but when they wrote, “I know 3/5 is greater than 2/8, because 3/5 is more than half, and 2/8 is less than half” I could tell they had a firm understanding of fraction size.

Shel Silverstein and Math

23 Jan

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Shel Silverstein poems are always fun, and many of his works can be used during math instruction.  A few years ago I asked the class how much money the boy lost in the poem “Smart.”  One student painstakingly worked out the exchanges of dollars to quarters to dimes to nickels to pennies, adding the cents together and then subtracting the decimals with agonized groans.  At least that’s how I remember it.

At the end of class, as everyone shared their work, he heard a fellow student say, “all I really had to do was stop and think that he started with a dollar and ended with 5 pennies, which is a difference of 95 cents.”  When he realized the difference in their strategies, I thought his head would explode.  But he did learn a good lesson: stopping to think before you jump in to calculating can save you a lot of pain in the end. As a teacher I can’t help but think, “that was a lot of good math practice he had, too!”

Here are some questions you can use with poems to spark some great math thinking.

Shel Silverstein Math and Poetry

These problems vary a lot in difficulty.  Some would be fine for upper-elementary students, some are probably more at the middle school level.  You can modify them as you see fit, but they give you an idea of the kinds of questions you can ask.

I would have students work on these problems in pairs or small groups (and I would probably have done some lessons with one of the poems whole class, to model some strategies for attacking the problems.)  I can also imagine this being an activity that would look great as  a presentation, with the poem on a poster and the students showing with pictures, numbers, and words, how they solved the problems.

The Illuminations website also has a lesson relating Silverstein’s “Shape” poem to a math lesson.

Real Life Math Problems

22 Nov

I stumbled upon this post by Matt Ives about Real Life Math Problems and I knew it was something I wanted to try.   Matt’s problems are typical word problems, but with all of the numbers removed.  For example, instead of, “You want to buy a cookie for $2 and some milk for .50 cents.  If you have $5 to spend, how much money will you have left over?” the problem might read, “You want to buy a cookie and some milk.  How much money will you have left?”  Students then need to figure out what information they need to solve the problem, and ask the teacher, “How much money do I have to start with?  How much money is a cookie?  How much does the milk cost?”  If children ask a relevant question, the teacher gives them the information needed.  If students ask an irrelevant question, the teacher asks them to rethink about what’s important and what information would help them.

There are so many advantages to this method of problem solving.  It avoids the classic problem of children grasping on to the numbers in a problem and immediately and randomly trying out different operations.  How many times have we seen students read a word problem and then look at us and say, “I multiply right?  Divide?  Add?  Subtract?” without any clear understanding of why they would be using a particular operation.   They’re desperate to plug and chug.  Without numbers, they have to consider what the problem is actually asking before they dive into finding a solution.

This method is also highly engaging.  Students are talking and brainstorming what possible information would be helpful.  My room was loud, but it was on-task loud, with students trying to figure out what information they needed and then what they could do with it.

Students also remain engaged with these problems for more time.  One group was stuck on the first problem for almost ten minutes.  Ten minutes is an eternity for a nine year old, but they kept trying to find the necessary information and failing, trying and failing, until eventually  they succeeded.  Their exuberance when they nailed it was that much sweeter because they’d been working at it for so long.

What Worked:

Students were placed in groups of three.  I find that 3-4 students per group tends to maximize participation of each person while providing for a lot of different ideas to be heard.   If students struggle to collaborate, I might drop it down to pairs.

The math problems grow in order of difficulty, so students began at problem one and worked down through problem 5.  They wrote what information they needed to gather, and then showed the work they did, with numbers, pictures, or words, to get the answer. Students worked on large whiteboards so they could all see and access their work (and I could see what they were doing at a glance.)  I made these whiteboards by buying showerboard at Home Depot.  They come in 8 x 4 foot sheets, which I had Home Depot cut to 2 x 3 foot boards, and then I wrapped the edges in duct tape.

One norm we’ve established is the idea that all students must “share the pen” meaning everyone gets a chance to record.  For this session, I gave each group one pen, but sometimes I give each member a different color pen, with the expectation that I should see three or four colors on the board at the end of the math period.

I required different members of the group to ask me the questions about additional information, so everyone practiced asking clear questions.  Some students had to go back to their group to clarify what they were asking me, so they got even more practice in listening and speaking to others.

At the end of the session we put the boards on a table and had a “gallery walk” of every team’s work.  Then we debriefed in a circle and talked about what went well about working together on these problems, and what was hard.

Next Steps:

I’d like to incorporate this kind of problem solving about once a week in my math class.  My class does a nice job overall of listening to each other and working together respectfully, but we probably need some more instruction in helping each other with wait time and inviting everyone to take participate, even those shy or more struggling.

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