What does ‘Mastery’ Mean? Gumby vs. Weeble Learning in Mathematics

Posted on June 30, 2014 by Janet Moeller

If you’re a Gen-X teacher, perhaps you remember a childhood where you played with Weebles and Gumby.

Weebles had a weighted bottom so that, no matter how long you held their heads to the ground, they always popped back up to their original positions. Gumby, on the other hand, was flexible, adapting to any imaginary setting in which he was placed.

I’ve been thinking about ‘Mastery Learning’ as I read The Smartest Kids in the World and How They Got That Way and comb through back issues of Educational Leadership. I realise that I was a compliant student who learned to perform tasks but not master material. I engaged in Weeble Learning rather than Gumby Learning.

Weeble Learning

I grew up as a Weeble learner of mathematics. I memorised the multiplication tables. Having some strange, innate desire to please my teachers and parents, I practiced pages of arithmetic facts, pushing myself for better speed and mastery of tables and algorithms. If we played “Around the World” games whereby I could compete against others in speed and accuracy of maths facts, I rocked.

When I got to worded problems, I hunted for the numbers and for little word clues that helped me decide whether to add, subtract, multiply or divide. Geometry involved memorising a formula into which numbers would go that I would again add, subtract, multiply, or divide. In High School, I looked for every formula in the chapter and figured out which one would best fit the numerals in the paragraph.

I didn’t realise that maths involved more than the four basic operations. Like a Weeble that can only tip and then return upright to the same position, maths problems could only be tackled with the four basic operations.

If my teachers had asked me about my strategies, I would have said ‘Guess and Check’. I guessed which operation was the correct one and checked to see whether or not the teacher marked it wrong.

Gumby Learning

Had I been less of a Weeble and more of a Gumby, I would have realised that there were multiple ways to tackle a mathematical problem and prove (to myself!) that answers were correct.

Had I been a Gumby mathematician, I would have been able to think of addition flexibly. I’d have realised that addition sometimes involved putting a whole bunch of things into one pile and sometimes related to jumping distances, comparing bars on a graph, or measuring perimeter. Multiplication might have been columns and rows or groups of items. I would connect factors and multiples in order to determine divisibility.

In the December/January issue of Educational Leadership, Grant Wiggins proposed the following definition of Mastery:

[The] effective transfer of learning in authentic and worthy performance. Students have mastered a subject when they are fluent, even creative, in using their knowledge, skills, and understanding in key performance challenges and contexts at the heart of that subject.

Wiggins’ definition aligned with Amanda Ripley’s use of the word ‘mastery’ in The Smartest Kids in the World and How They Got That Way. Ripley’s description of schools both in Finland and Poland was that “Schools existed to help students master complex academic material” [emphasis mine].

I have come to believe that rigour is the insistence that students be able to use both knowledge and skilils fluently, flexibly and creatively in the context of complex academic material.

Gumby mathematicians can solve ‘standardised test-type’ maths problems and open-ended maths problems because they have the flexibility to think beyond the basic four operations and beyond formulas. They mentally recognise that the difference between 1000 and 995 is 5 without having to write the numbers down and ‘borrow’. They are not flustered when faced with problems such as this one (from Math Olympiads, December 2005):

They are motivated by authentic tasks such as those listed in Victoria Teacher Resources for Assessing Multiplicative Thinking.

So how is your maths teaching? Do your choices of instruction materials, methods and assessments develop Weeble mathematicians or Gumby mathematicians? What is your evidence?

Teaching Before Christmas: Lesson Plans

Posted on December 3, 2012 by Janet Moeller

It’s time to get out the lesson plan book and plan for December. Below are ideas for math, reading, writing, and social studies.

Ideally, December lessons follow four criteria. They…

fit curriculum standards,
make students smile,
take very little prep time to pull together, and
require very little marking.

You can do holiday activities that address standards such as the Common Core or the Australian National Standards.

So what kinds of activities are holiday-like and fun and academic?

Math

Graphing, Probability, Statistics: Younger students can have fun graphing fun Christmas foods. Older students can use the same items to investigate sample sizes or play with probability.

3D Shapes: Make these ornaments and refer to them during class while naming them, finding surface area, etc.

Problem-Solving: You’ve probably sung The 12 Days of Christmas. One partridge was given on the first day. On the second day, there were 2 turtle doves and a partridge – making a total of 3 gifts. If this pattern continues, how many total gifts would be given on the 12th day? How many total over the full 12 days?

Build the excitement with a picture book based on the song – or the song might be in you iTunes playlist. My favorite version:

Writing

Stories: So what would life be like for the person who received all 12 days of gifts? Discuss. If inspired, write the story. Create a character who gives and one who receives. Put them in a setting. What is the motivation for the gifts? What are the daily consequences?

Poetry: In Ralph Fletcher’s book Poetry Matters, he shares that there was a year he wrote poems as gifts for each member of his family. Poems were written based on interests and hobbies of the person who would receive the poem. Poems can be decorated or framed – something that is good to do the final hours of the final day.

Persuasive Writing: Jen from Runde’s Room suggests students write letters to Santa from the point of view of a literary villain. She posts some examples that students can use as mentor texts.

Vocabulary, Grammar, and Word Study: Students guess the identities of obfuscated Christmas carols. Younger students can go to Visuwords for help. My favorites:

The Slight Percussionist Lad (Little Drummer Boy)
Quiescent Nocturnal Period (O Holy Night)

Christmas Mad Libs can help students review parts of speech.

The Resourceful Teacher has a great elementary writing activity using holiday stamps.

Those in Christian religious schools can delve into all the metaphors used to describe the coming Messiah. What does it mean that Jesus would be the Lion of Judah? Prince of Peace? What about other metaphorical references to Jesus as the Good Shepherd? The Door? Others?

Reading/Writing Connections:

For an ESL reading/writing connection, students can talk through various pages from Peter Spier’s Christmas, a wordless picture book. Pick pages to talk through each day and label items like mistletoe with post-its. Students can later either write out words for the page or they can compare Peter Spier’s Christmas to Christmas in their country of heritage.

The Polar Express website includes an interactive calendar of short writing activities – one activity for each day in December. The North Pole has additional activities.

Reading

Summarizing/Retelling: Review story mapping using Chris Van Allsburg’s The Polar Express.

Fluency: If you have students who are still struggling with reading fluency, plays are available online. Just glancing through the first page of playscripts, I found Babushka, The Gingerbread Boy, The Three Trees, A Christmas Carol, and The Nativity. A list of Christmas plays can also be found on Little Fun, LIttle Learning (be sure to look at the links in the comments section too!) and Dramatrix. Spread plays over a number of days and adding a tech twist where students practice lines on Garage Band or other recording tool.

Analyzing/Inferring: If your library has a good stock of Christmas picture books (or Hanakkuh), bring some back to the classroom. Have pairs of students read a book and come to consensus on the most important line in each book. The pairs write the most important line on a sheet of butcher. Rotate the books so that each pair gets a different story for each of the 3-5 days. Do the ‘most important lines’ agree? if not, can groups come to consensus? Wrap it up by asking students what they believe the author believes is the meaning of Christmas. Do students agree or disagree? Defend. Ruth Morgan has put together books on Hanukkah.

Those who do not have a large stock of Christmas books can listen to stories on Storynory or The North Pole. Again, have students create story mountains, summarize, or find the message behind a story or two.

History/Social Studies/Geography

History: Many students play soccer (football, for my Commonwealth friends) at recess. They might be interested to know that soldiers in WWI played soccer during periods of truce. In fact, both German and English soldiers played on Christmas.

The story is told to music by John McCutcheon. Commonwealth friends may want to preview and decide if the line “He’s singing bloody well, you know!” is appropriate for your age/classroom/school. The lyrics are online, if you care to print and distribute.

Religious Studies: Those of you that work in religious institutions, can label the prophesies on a timeline and compare them to the birth of Jesus. Simple and short, but powerful.

Research/History/Geography: Students pick a country that celebrates Christmas and research the celebrations in that country. Have students look for references to a Santa-like figure, other saints, and references to the Magi (if the Magi bring gifts in that particular country). Students can place references on Google maps using the same process as was described in the post about Haikus on Google maps. The research and presentation could last at least a week.

Move from the North Pole to the South pole and explore Antarctica street view. The landscapes are stunning.

Quizzes/Trivia/Research:Turn these pre-made Christmas quizzes into a bit of research fun.

Character Development: The story of Rudolph can be turned into an anti-bullying lesson.

Still haven’t planned for all of December? Over 2,500 Christmas activities are shared by teaching professionals on BetterLesson.com and ShareMyLesson.com. Check out some of Ian Byrd’s ideas or Shelly Terrell’s ideas – they’re great.

Have a wonderful time planning for Christmas lessons. Please add any of your favorite activities in the comments box. Any ideas for science?

If you find this post valuable, please consider doing one or more of the things in the storyboard below…

Create a Copy

photo credit: LinksmanJD via photopin cc

Why the Problem Solving Process Doesn’t Work: Problem Solving Series #4

Posted on November 21, 2012 by Janet Moeller

You have good materials. Students have unpacked the question and know what they are looking for. Students freeze.

Now what?

Most problem-solving materials will tell you to use the four steps:

Find out what the problem is about.
Choose a strategy.
Solve it.
Look back.

I suspect the people who ‘invented’ this process were rock star problem solvers in school. Then there are the rest of us…

So what is wrong with the process? How can we help students think like mathematicians and be less intimidated by the process of problem solving?

Don’t ask students to ‘Choose a Strategy’.

The ‘Choose a strategy’ step of the problem-solving process implies that there is one, correct strategy – students just need to find it. The ‘you-need-to-choose-the-correct-strategy’ is especially logical to children when you consider that, in other strands of mathematics, students find the one right way to get the one right answer. In students’ minds ‘picking’ may be something like drawing straws. They hope they pick the right one.

Often, multiple strategies can (and should!) be used. A student might…

draw a diagram of the initial story,
put the numbers into the diagram, and
logically discover the missing piece(s).

In another problem, students might…

draw a diagram of the initial story,
label and organize numbers,
find patterns in the table, and
logically decide how the pattern in the table would continue.

In each of these scenarios, students are using multiple strategies. No need to pick or name the strategies.

If you want students to name things, have them name the math vocabulary words that match the particular problem. Larger conceptual understandings form when students connect the problem with the concepts and vocabulary of other mathematical strands.

‘Solve it!’ discourages experimentation.

‘Solve it.’ What do you think when you hear those words? How do those words compare to ‘Play with the numbers” or “Find patterns”?

For me, ‘Solve it’ translates into ‘Just get there already…’

The problem becomes all about the destination.

The process should be as important as the product. Students understand this better when they are allowed or encouraged to play with numbers and number patterns.

The Abercrombie Process for Problem Solving (or APPS):

Find out what the problem is about.
Label the numbers.
Find, describe, and continue any patterns you see.
Relate the patterns or ideas to previous mathematics lessons.
Look back at the exact question to locate the answer.
State the answer in a complete sentence.

Find out what the problem is about. See the second part of this series.

Label the numbers.

Part of finding patterns is labeling the numbers. Numbers are adjectives. They never stand alone. Replay scenes of The Count from Sesame Street (if you have to) to reinforce this point.

Find, describe, and continue any patterns you see.

Mathematicians don’t just compute. They find patterns. They continue patterns. They make predictions based on those patterns.

Mathematicians look at number patterns like artists look at sunsets.

Allow students to play with the numbers. Encourage them to label their numbers on post-its and move them around the desk. Perhaps students want to move labeled numbers around on an Interactive White Board or on Wallwisher.

Help students be specific when they describe the patterns they see. Instead of saying “This column is ‘plus two'”, teach them to identify the numbers in a column as being two more than numbers in another column. What if this pattern continued? Can you predict what would happen if the table continued to…?

This is the place where students begin to connect the problem to other mathematical concepts. Are the numbers in one column all even? square?

Relate the patterns or ideas to previous mathematics lessons.

The NCTM process standards now include Connections. Student should connect the problem to mathematical ideas they have previously encountered.

While the idea of making connections to past lessons seems obvious to us, the idea is not obvious to most elementary students.

Create a classroom mantra: “Never forget a math lesson.” Let students look back at previous work to find the rules of divisibility or the formula for circumference if the concepts relate to the larger problem.

Do the number patterns remind you of other mathematics lessons? Do the patterns result in square numbers? Do rules of divisibility apply? Area and perimeter? Rectangular arrays?

Look back at the question to locate the answer.

How does the actual question relate to the patterns you have found? Previous math lessons? Make it like a treasure map. The answer is somewhere in the pattern…but where?

State the answer in a complete sentence.

Not only do students need to locate the answer, they need to see whether or not the answer they have located actually answers the question. They can do that by putting the answer in a complete sentence.

Notice that the complete sentence is the only complete sentence I ask them to write. Do students really need to write, “First I…Then I…Next I…” to effectively communicate their thinking?

The next post will show you why such sentences are unnecessary. It will also include a newly-created problem-solving rubric you can use for assessment.

To be one of the first to receive this rubric, sign up to receive Expat Educator by email. Also, please consider passing this post on to colleagues or friends.

photo credit: Old Shoe Woman via photopin cc

Problem Solving Series #3: What Teaching Resources Should I Buy?

Posted on November 12, 2012 by Janet Moeller

So far, this series has included…

Classroom procedures that help students explore and construct problem-solving strategies.
Ways to make sure your low readers and second language students are not at a disadvantage.

Is there a perfect Problem Solving teaching resource?

Bad news: Instructional materials are only as good as the instructor. More bad news: Materials from the major textbook companies will probably not be adequate – even if their representatives tell you otherwise.

In 2003 and 2007, two representatives of a major textbook company tried to convince me that the problem solving activities attached to the summative assessments were adequate in helping students develop problem-solving skills. My first issue: One problem solver per unit means that students get 12 opportunities during the year to build these skills. Students need more than that. My second issue: Students have had no scaffolding throughout the unit that would make them successful.

Students turned in their assessments with looks of shame and defeat.

One representative claimed there was research defending the position that problem solving strategies need not be explicitly taught. Given enough time, students will develop strategies on their own.

I have scoured the research. Can’t find it. The only research I can find post-millenium states that students with disabilities benefit from explicit, repeated instruction .

Look for materials that explicitly teach strategies.

If students are going to transfer problem solving skills to real-world problems in a different context, Grant Wiggins suggests students must make four cognitive moves:

independently realize what the question is asking and think about which answers/approaches make sense;
infer the most relevant prior learning from plausible alternatives;
try out an approach, making adjustments as needed given the context or wording; and
adapt their answer, perhaps, in the face of a somewhat novel or odd setting

Students must have a mental portfolio of plausible, alternative approaches. Without a mental portfolio of possible strategies, elementary students will tend to do one of the following:

randomly add, subtract, multiply, or divide numbers – hoping they pick the right operation.
tell you they are using the “guess and check” method. Their paper will full of random computational guesses. One of the guesses will be circled.

If you spend at least a few lessons each year explicitly working with and repeating strategies, students have a mental portfolio of approaches from which they can draw.

Give them a chance to explore one type of problem using the procedures I outlined in the first part of this series. Give them a similar problem the next day. Then give them a homework assignment using that strategy. The fourth time, almost all students can independently use the strategy.

…but don’t name the strategies.

In the next entry, I’ll get on my proverbial soapbox about the “pick a strategy” step in the frequently-published ‘problem solving process.’ Rather than say, ‘This kind of problem is solved by [name the strategy]’, you still want students to construct a strategy or two that works.

When students find something that works and defend their strategy, then you want them to solidify the conceptual understanding in different (but similar) problems.

Who cares what they call it? If students are going to name things, ask them to connect vocabulary from other math strands to the patterns they are finding.

Collect the strategies in a journal, notebook, or portfolio.

Math journals help students keep a record of previous lessons. Some of the best journals I’ve seen are found at Runde’s Room.

During a lesson where a new strategy is introduced, expect the journal pages will be rather messy. The students will start well, including sketches and bullet points showing their understanding of the scenario and the question. Then, to construct meaning, students need to try different things, collaborate with classmates, change approaches, then collaborate some more. They will cross things out. They will erase until the page tears (although I discourage erasing). They will repair pages with scotch tape. They will use white-out strips. They will circle things and use arrows when explaining work to others. This is a good thing.

The second day, students will get to an answer more directly because they can refer to the procedures that worked the day before and apply them in the new and different situation.

The homework that follows might be given as a worksheet. Rather than collecting the homework for grading, have students take the work out at the beginning of the lesson, compare their answers in small groups, and come to consensus.

If changes need to be made, students can make the changes in a different color pencil. Then, students write a short “things to remember” note and paste the homework in their journal.

Wean students off of the repetition.

At some point, the types of problems should spiral more than repeat. Questions for students: Does this problem resemble any you’ve seen before? How does it relate to prior knowledge in other areas of mathematics?

Some will benefit from looking back at the journal for ideas. Others will not need that step.

Remember that not all the students will need repetitions of strategies.

Once a student demonstrates the ability to independently use a strategy, there is no reason to give him or her more of the same. You will have students that catch on the first time and immediately apply the strategy to new situations. These students will be held back if you require them to have them do the same problems as the rest of the class.

Try giving these students a problem that is at least a grade level higher. Assuming the student has no trouble with that, move him/her on to a project.

What materials help teach problem solving strategies?

I’ve had the best luck with The Problem Solver – with reservations that I will explain in the fourth part of this series.

The real power of the Problem Solver comes when teachers can match Problem Solving strategies with the conceptual ideas of a mainstream curriculum math unit. Some examples:

Students need to find all the factors of numbers. They learn the Rules of Divisibility and continually ask themselves Is 1 a factor? Is 2 a factor? Is 3 a factor? This unit provides a great opportunity for the strategy ‘Make an organized list’. In both situations, students need to think about and organize numbers in a more systematic way.
You are teaching a unit on fractions. Fractions combine well with the ‘Working
backward’ strategy.
If students are learning to graph coordinates on a plane, they might also practice ‘Make a picture or diagram’.

Mathematical strands that tend to match Problem Solving strategies.

Think of problem solving as an umbrella that covers all the mathematical strands.

No hard and fast rules exist to match problem solving with other mathematical strands – only experience will help you make the matches. Also, problem-solving strategies overlap. Many students begin a problem with a picture or diagram. Why stop there? An organized list might lead to a pattern that can be graphed. Celebrate when students find the overlaps!

If you’re new to a mathematics series or to problem solving instruction, here are some general guidelines:

Conclusion

Find a set of materials that explicitly teach problem solving strategies. Teach the strategies for a decent chunk of the year. Don’t rely on teaching strategies the whole year, but give students enough background knowledge and confidence that they can approach a scenario with a few tried and true options.

Do not force repetition on all your students because some will not need it. An initial problem or a pre-assessment might be the same but the follow-up problems need not be.

Connect the strategies with other mathematical strands.

Have you found any research on the pros and cons of teaching strategies? What materials have you found that help teach problem solving? Please share!

If you like what you read, please subscribe to Expat Educator. You’ll have instant access to subsequent posts in this series. The next post: What the Problem Solving Process is Missing.

photo credit: Marquette La via photopin cc