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Mathematicians estimate before they calculate.

Walking through classrooms I see so many opportunities for students to engage in authentic mathematics by way of estimation. Estimation is such an important yet underused skill that students can use to their advantage if they know when and how to do so. This week’s mini-lesson “Mathematicians estimate before they calculate can ensure that students can make a reasonable answers and discover efficient calculation strategies. In fact, Standard for Mathematical Practice #5 “Use appropriate tools strategically” states, “They [students] detect possible errors by strategically using estimation and other mathematical knowledge.”

We can be of tremendous service to students if we instill the purpose for and the ways to to estimate easily before calculating. When we put the emphasis on correct answers only we send students a clear message (whether we know it or not) that getting it right quickly is all that matters. Students learn that good guesses and approximations are to be avoided and have little or no value. This is extremely unrealistic when one is in the process of learning something new. Summative assessments after a period of trying and learning are where we should expect to find exact answers. These exact answers come about as a result of learning new processes and testing out new strategies. Elon Musk, Thomas Edison, Marie Curie, and Elon Musk had to be wrong many times on the way to being right. Making estimates and good guesses are great places to start.

Estimation is a strategy that always works no matter what the problem may be. Estimation is a worthwhile strategy that students will be able to use throughout their lifetimes; it is well worth the time and effort of consistent instruction. In comparison to the tape diagram, while useful, it is not always efficient and may lead to misconceptions and confusion on the part of students. What do you think is a more valuable use of your instructional time?

Last week, I watched fourth graders in the process of dividing 432 by 5.  Many were making groups of  5 and 10 and using those amounts to count up until they got to 432. Oh, it was painful to sit back and watch them struggle. It was slow, full of opportunities for inaccuracies, and it was creating a group of students who had given up on the problem, some were dutifully making circle after circle, and many others were trying to figure out how to use base ten blocks to make groups of 5. Many students were doing all this work just to get the answer wrong, with no visible pathway to efficiency or success. Finally, I stopped the students and I asked them to consider estimation. “If you had $432 and you had to divide it equally between 4 people, about how much would each person get? (context always helps). Could they each get $10, $50, $100, or $200? I don’t need an exact amount but approximately how much would each person get? Estimate in your head or on paper before you calculate.” We went through this process, which was like pulling teeth initially but eventually students began to see the value of just thinking about estimates before calculating.Many books teach complex steps for calculating that can be take longer and be more confusing than the actual calculation. This can turn kids away from using and developing this skill. By orally asking students to calculate reasonableness in their heads given boundaries, students see the skill as more accessible and purposeful than doing two sets of calculations.

 Mini-lesson #11
“Mathematicians estimate before they calculate.”

Give students a problem (it can be a word problem or an equation). Write the equation on the board, label the equation with context, (see below). Ask them about how much do you think it will be? Provide some boundaries, not too close and not too far away but enough to get students to start thinking about reasonableness. Estimate what would be a reasonable amount. Record the possible estimates, then calculate. Repeat.

Examples of Problems

Separate Result Unknown (SRU)
25students – 9left to take pictures = how many are in the classroom
Teacher says, “There are 25 students in our classroom.  9 students left to take make-up pictures. About how many students are in our classroom right now?
About how many will there be, more than 20 or less than 20?” Provide quiet time to think before sharing out any answers. Record the estimates then calculate.

Grouping Problem (Number of Groups Unknown)
42/ ___ = 6 books in each basket =how many books total

42pencils /7in each bag = how many bags
Teacher says, “There are 42 pencils. I want to put 7 in each bag. About how many bags do you think you will need? Will 2 bags be enough, 10 bags?” Provide quiet time to think before sharing out any answers. Record the estimates then calculate.

Join Result Unknown (JRU)
982 books in the library + 450 books in our classroom= how many books

Teacher says, ” There are 982 books in the school library. I have 450 books in this classroom. How many books are there in our classroom and the school library combined?” About how much will it be? Will it be more than a 1,000 or less than 1,000? Will it be more than 2,000 or less than 2,000? Provide quiet time to think before sharing out any answers. Record the estimates then calculate.

 245 / 6 =

I have $245 to divide equally between 6 people. About how much do you think each person will get? Provide quiet time to think before sharing out any answers. Record the estimates then calculate.

*Note – one of two things tend to happen when you begin to use this mini-lesson with your students: 1) students will make completely unreasonable estimates, don’t worry this gets better with consistent practice, 2) you may realize that your students have very weak number sense. If they do not have good number sense then it is crucial that you engage them in this opportunity to develop their number sense. If students have number sense they can master grade level mathematics, without it they are more likely to fail. It is worthwhile to develop this skill in your students now as opposed to dealing with this weakness next week or next month. Don’t panic. Don’t give up. Remember, a culture of kindness, acceptance, and the analyzing of work to further the growth of the classroom must be a priority in a Cognitively Guided Instruction (CGI) classroom or any classroom for that matter. Show your students that you value their contributions by giving all students the opportunity to share their thinking and growth over time.