Excerpt from Teaching Numeracy
If a student(s) is stuck, do not tell her what is wrong, as hard as that may be. Arthur Hyde says that interventions are best done with questions that help a student or group rethink and repair on their own. When students rethink and repair themselves, they own the learning. It is actually the essence of deep understanding.
Help build good estimating sense with these measurement ideas:
Measurement Ideas: Early Elementary
What’s as Big as Me?
Take the height of each child using string. Cut the string and have the students use it to discover what else is as big as them in the classroom.
Have students categorize things that are smaller, the same, and larger than themselves.
Oh, That’ll Take About a Minute
Start a stop watch and ask students to clap when they think a minute is up. Talk about how close they were. Now do jumping jacks and ask students to stop when they think a minute is up. Continue with different ways to help students make sense of a minute.
Oh, That’ll Take About an Hour
Have students talk with family members to figure out activities that take about an hour to do. They should draw illustrations of two examples and bring them to class to share. Have groups talk about their findings and then identify different times during the day that an hour has passed. Discuss all the things that happened during that hour.
Measurement Ideas: Upper Elementary/Middle School
Measurement Scavenger Hunt
Students love a good measurement scavenger hunt. For your Ignition (see Component 2 for more on Ignition), place some “mystery measures” on the board and have them guess what in the room is that length, perimeter, area, volume, and so on. For example, you can ask, What in this room is 50 cm long? The next day ask, What is 20 cm long and weighs 10 grams? Then, how about, What is 20 cm long, weighs 10 grams, and is 5 cm in height? Make sure that what you are asking them to find are different objects. Students will then be challenged to consider the relationship among length, weight, and height. You will learn very quickly who has a good sense of measurement and who does not.
Mystery Staff Member
Another really fun idea is to have a “mystery staff member.” Put the mystery person’s measurements (height, arm span, shoulder span, whatever) out in the hall and make it a school challenge. Don’t forget to switch up the unit of measure. Challenge the students to “think outside the inches box.” It is a blast. Don’t forget to include the custodial or cafeteria staff or principal. It’s endless! This activity will really help build number sense, which really helps in monitoring and repairing math understanding.
“Size ’em Up”: Measuring Angles
Students often get confused about how to use a protractor properly. Alleviate this problem by having the students first “size-up” the angle before using the protractor. Use an “under 90°, over 90°” estimate. Then model using the protractor. Your estimate will lead you to the right measurement without any confusion. This method is very meaningful compared to some of the “catch phrases” used to teach measuring angles (e.g., “come right in or be left out”). Catch phrases are quick fixes without the reasoning. An estimate will automatically bring in reasoning and meaning.
Select a single object such as a box, a watermelon, or a jar (or you can select a person like the principal). Each day, choose a different attribute or dimension to estimate: length, weight, volume, surface area, and so on (Van de Walle & Lovin, 2006, p. 235).
Create Math Sense Moments
- A million drops of water equal how many liters? Discuss this idea in small groups. Draw and/or explain the process of discovery.
- Name a fraction that is close to 1 but not more than 1. Explain how you know this fraction is close to 1. Now write another fraction that is closer to 1 than the first one you picked. Explain how you know this fraction is even closer to one. Try one more. Do you see a pattern? Explain your findings in writing. Turn and talk to a partner and share your explanation. Would you like to revise your answers? If so, what did you change? Why?
- Make a case for why each of a series of numbers does not belong (e.g., 24, 12, 2, 11). Do this as an Ignition when the students come in. At the end of the class, as a Debrief, put up the series again and challenge the students to include the following concepts in their explanations: prime, composite, divisible, multiple, factor, whatever you are teaching at the time.
- Chips-Ahoy cookies claim to have 1,000 chips per box of cookies. How could your determine whether this claim is true without counting the chips in every cookie (Schuster & Canavan Anderson, 2005)? Identify the pattern so that you can discover the number of chocolate chips in “X” boxes.
Encourage Students to Monitor and Repair Their Understanding by Providing a Working Answer Key
Working Answer Keys help alleviate the meaningless nightmare of checking homework during valuable class time. Now hear this: We are certainly not saying that you should remove homework from the mathematics classroom. However, homework must induce deep thinking and not devour half your math class the following day and not allow you to provide meaningful feedback to your students. Tall order, right? Not really. Here’s how Working Answer Keys were born.
My son’s brilliant pre-calculus teacher provided the answer to a problem I was struggling with as a teacher. My son would come home with five problems to complete each night. Attached to his assignment was an answer key. Now, the answer key did not simply have the answers, it included the process as well. I hate to admit it, but my first thought was, “Wow, that was dumb to give to the kids. He’s just going to copy the problems and be done with it.” However, that is not what happened at all. I did realize that, without the proper setup by the teacher, it probably would not have worked, but this teacher had obviously taken the time to empower his students to value their own learning. My son wanted to discover the process and to succeed. I watched him try a problem and then check his work against each step on the answer key. I heard him wonder aloud why he was wrong. Then he’d discover where his mistake was and what he needed to do to be correct. It was amazing. He said he wished he had another classmate there to discuss the process. Watching my son sparked an idea about how to solve my homework nightmare!
That night, I worked out the problems I had assigned my students for the next day. I learned something valuable by doing the assignment myself: Assigning 10 thoughtful problems was more meaningful than assigning 20 or 30 repetitive problems. The next day, I placed what I dubbed Working Answer Keys in a labeled bin and started teaching my students to take responsibility for their own learning. I explained that they were to pick up an answer key, place it alongside their homework, and check the process as well as the answer. For the first few days, I modeled what using a Working Answer Key tool looked and sounded like and what it did not look or sound like. I tried to infuse humor, and it definitely got the point across. Because my students were in groups of four, I told them to use each other for clarification along the way. They discussed and shared their thinking behind the homework with one another. If all four students didn’t understand a problem, we would bring it up as a whole class. It took some time and patience and an occasional call to the home of any student who basically copied my work verbatim. “Copiers” quickly learned from their group that copying wouldn’t be tolerated. Many times I’d hear students say something like, “What’s the point of copying? You’re not learning anything.” By the second month, my classroom looked very different during homework check. I could quickly walk around and see who did or didn’t complete the assignment, and the students were doing all the work. Homework was now meaningful, providing opportunities for deep understanding and meaningful feedback, and taking half the time to check. They shared ideas, discussed problems, and celebrated successes. The journey was worth taking. They had learned how to monitor and repair their own math understanding.
Increase Mental Math Ability
“A good calculator does not need artificial aids” (Lao Tze, 531 BC). This quote gains in applicability as you provide opportunities for your students to think fast, think hard, and think deep. If you really want your students to have mathematical sense when it comes to problem solving, invest time in improving their mental math skills using quick reasoning in practical situations. Mental math does not mean only rote memorization. However, when it comes to multiplication tables, rote memorization is important. We are referring to quick reasoning in practical situations.
- Realizing how much we use mental math in real life: Teach the use of compensation and the distributive property when using mental math. Ask, “Can you multiply 29 × 7 in your head? I bet you can. Try it!” Let them wrestle with it, think about it, discuss it. Explain, “What your brain actually did was multiply (30 × 7) – (1 × 7) = 203. Now try this one. 42 × 6. Did anyone’s mind naturally multiply (40 × 6) + (2 × 6) = 252?” Some of your students’ minds don’t naturally do that. You can open your mathematical thinking to them in a think-aloud moment. They will greatly appreciate it. We like to call it “taking the mental out of the mental math.”
- Using “friendly” or compatible numbers to make meaning: When problem solving, change the problem by rounding the numbers into numbers you are more familiar or comfortable with. Sometimes, if you simply round the numbers, you can see how to solve the problem. This is a great way to monitor your understanding before, during, and after solving a problem. This little trick will open your mind to how to solve the problem.
- Model compatible numbers: Put the following on the overhead and let the students mentally add the numbers in 10 seconds. Let them in on how their brains search for “friendly numbers”—ones that “your brain just seems to clump together naturally.”
25 + 73 + 75 + 40 + 30 + 60 + 70
- Allow your students to have the experience of their brains chunking together the compatible numbers. It is amazing how many students can do the calculations in their heads. Now try others! There is no limit to building a sense of compatible numbers. This practice should continue to be encouraged through high school. Include percentages, calculating a tip in a restaurant, and estimating time and distance when taking trips.
- Making the implicit explicit: Model how to solve a problem by doing a think-aloud; in other words, demonstrate to students how to think through a problem. Opening up a teacher’s critical thinking habits to the students shows the process of problem solving. Many students don’t realize that the teacher uses the same critical thinking habits they are being asked to use. They simply think the teacher is smarter. The benefits of this method are amazing. It levels the playing field for those students who never knew “the smart kids” are actually processing, too. If you are a risk taker, try doing this “cold.” It is scary, because you might not know how to work the problem correctly, but isn’t that a powerful lesson as well? How will you make it work? How do you remain positive when things are getting tough? What monitoring and repairing strategies do you apply? Have your students notice what you are doing as a mathematician to stay in the problem. Then, let them discuss how this thinking helped you succeed as a problem solver. Now let them give it a go with a partner a few times. This works best in mixed ability groups. Students are empowered to learn from each other. It gives the slower processor a chance to hear strategic thinking happening during problem solving.
- Monitoring and repairing to maintain mental stamina, courtesy of professional baseball: The best way for students to keep focused through standardized testing is to have them monitor and repair their comprehension. Try this! Find two short video clips of two pitchers during a World Series. One clip should present a pitcher who is beginning to fall apart on the mound, whereas the other clip is of a pitcher who seems flawless. Show the failing pitcher first. Have the students fill in their “thoughts.” No matter what is going on with the pitcher, he still has to remain focused, somehow. Let them figure out how. Then show the “superstar.” What do they think he is saying to himself to keep focused? Okay, now comes the powerful part: Ask the students how this situation is like taking a test. Push the thinking such that the students realize that monitoring and repairing your comprehension gives you the stamina to remain focused through any problem.