## Saturday, September 20, 2014

Here are two estimation problems.

The first is from TERC Investigations:

The second is from Andrew Stadel's Estimation180:

I have some questions about the relationship of these two problems to each other:

1. Do these two problems help students get better at the same thing? Or do these two problems help students improve at different skills?
2. What sort of knowledge is needed to successfully tackle the first estimation task? The second?
3. What does the first problem seem to mean by "number sense"? What does the "second"?
4. Can we infer why estimation is valuable to the writers of the first problem? The second?
Thoughts?

1. The first task is focused on simplifying calculations to turn a calculator problem into a mental math problem, allowing you to approximate an answer. The second task is focused on simplifying a measurement task to make it a in-your-head problem. It requires the application of reference numbers to a new scenario so you can make a guess that makes sense near common benchmarks (typical male height, fence height, shrub height). Both are trying to get you to take a time-consuming task and make a reasonable first order approximation. This might serve as a sufficient answer or simply a reference to compare to for silly errors (unit conversion error with measurement or typo with calculation). Neat thought.

1. I think Andy's explanation is well said and answers most of your questions, Michael. Here are a few other thoughts.
1. Do these two problems help students get better at the same thing? Or do these two problems help students improve at different skills?
Both. I could see both tasks as helping students contextualize and decontextualize tasks. Students could draw a picture to contextualize the first task: say they draw 30 groups of 60 granola bars. This would be a waste of time to draw about 1,800 granola bars, but it's possible. Secondly, a student could simply draw three vertical lines, corresponding to the height of the shrub, fence, and me. They could attach numbers to the height of each vertical line.
Either way, a strength to each task is encouraging both teachers and students to contextualize or decontextualize the task.

2. What sort of knowledge is needed to successfully tackle the first estimation task? The second?
The student needs a basic understanding of multiplication for the first task, whereas the student needs a basic understanding of some type of measurement for a man for the second. The first task is bounded by the answer choices which is unfortunate in my opinion. The second task is bounded by physical limitations of the human body and height. I think bounds are important here and students are robbed of creating them in the first task.

3. What does the first problem seem to mean by "number sense"? What does the "second"?
Number sense in the first task seems more like a relationship with being able to project a range of answers from facts about multiplication of specific double digit numbers. This is a broad sense of number sense in my opinion. In that general definition, I just created a range of answers from 100 to about 10,000. Therefore, the task demands more specificity with which group of tens is being multiplied together. Oh, a group of tens in the 50’s to 60’s is being multiplied by a group of tens in the 30’s.
Number sense in second task is applicable to the context of the task. We're dealing with height and references. Therefore, students tap into at least two areas of number sense: height and unit of measurement. We should be asking what is the most useful unit of measurement for the height of an adult? This depends on where you live and your customary unit of measurement for height. Furthermore, number sense continues to get more detailed here: should we use all inches or feet AND inches? How do we write that? DO we use a decimal point or apostrophes, etc.? Should we use centimeters or meters or a combination again? Do we use a decimal point here. Why do we use a decimal point for one and not the other customary measurement?
These are vital questions for students and teachers to ask of each other.

2. 4. Can we infer why estimation is valuable to the writers of the first problem? The second?
As for the first task, I use this type of estimation when performing operations in problem-solving tasks. Once the situation has been decontextualized and I'm performing operations on the numbers, it's extremely helpful for me to project my answers as I work through a problem. As for the second, estimation is valuable because many daily interactions involve height. This could range from not sitting behind someone at a movie because they're too tall, to asking someone to grab something from a top shelf, to identifying a suspect's height in a crime. On a personal note, there are times when knowing my height affects what I'm doing and that can range from watching I don't hit my head on something to fitting in a small car (or plane), to riding (or not riding) a roller coaster. The value for me is simply paying attention to things around us that we can use a metric to function in our daily lives. The metric doesn’t have to be labeled numerically, but the number association can help with precision and communication via a common language of mathematics. I find that thrilling, but I’m weird like that, right?

This has been fun, Michael. I think the biggest take away for me in working with students and teachers, is the context of estimation and number sense. In the math class, estimation can start with estimation jars in primary grades, and unfortunately stay there until students start seeing worksheets like the one you posted. I feel there's a strong disconnect between estimation and number sense in and out of the classroom.This is a jumbled thought, so I will clear it up and eventually blog about it. However, long story short, estimation gets a bad wrap because of the worksheet above, and yet that task has value. Step outside of math class and I guarantee we'd all be surprised how many times we estimate the things around us each day. If we ask the (unfair) question, which task is more important? I think it all depends on the context. Therefore, they're both important.

3. Thanks for the extensive thoughts!

Here's what I'm hearing: you're saying that there's a disconnect between contextualized estimation and decontextualized estimation in school math. This is a problem because you'd like to kids to be able to use their estimation skills during their daily lives.

You also point out that there are very different skills needed for these two tasks. I agree, and I'd add that being great at estimating products doesn't help much for estimating heights. They're both estimation tasks, but they use different sets of skills.

Say that we want students to spontaneously use their product-estimating skills in other contexts. I'm convinced by you that if we only use them in decontextualized situations, it's unlikely that students will bust those skills out unprompted. I'm also convinced that giving kids contextualized estimation tasks is part of the solution. What I'm wondering, though, is would we need to give students chances to estimate products in context?

To clarify and sum up: Kids are unlikely to spontaneously estimate products if they are only asked to do it on estimation worksheets. But will asking kids to estimate height make them more likely to spontaneously estimate products? Or do we need to give kids chances to estimate products in context?

4. Asking students to estimate the area of a room without a calculator would be a dual task -- first use your contextual estimation skills to get two lengths, then round them off to numbers you can reasonably multiply in your head to get a product. Both of the simpler problems you posted would be good scaffolds for a question like this.

5. I think that's a great suggestion, Andy. We could also ask kids to estimate the number of objects in some enormous array.

6. Michael, a thought provoking set of questions. Here’s my experience, and I’ve taught both: In the first one, we’d tell a kid, “Round each number to the nearest ten. Multiply the ‘basic fact’ you see (and here we’d underline the 6 and the 3) and add two zeros. Your answer should be about 1,800.” Don’t ask what happens when they have to multiply numbers and get a multiple of 10 (i.e. 60 x 50). This will often come out to 300. In this scenario I doubt many kids understand what’s happening, place-value wise (there are teachers who don’t, either). They’re just following a procedure. This gets mixed up with the procedure we have them do when they add, i.e. 57 + 32… 60 + 30…here we add 6 + 3 and only add one zero! Anyway, many kids have trouble understanding why they just wouldn’t add or multiply the original numbers and just be done with it. As both you and Andrew have pointed out, devoid of context, it’s meaningless. We might say something like, “Well, when you’re shopping and you want to estimate how much something costs…or if we wanted to put carpet down on the classroom floor.." like 9 year-olds find themselves in these situations on a regular basis. I suppose what we are doing there is building a foundation for multiplying using partial products, maybe hoping the kids will use the strategy to check the reasonableness of their answer.
The second task is totally different. Do they have to estimate how tall people are on a regular basis? It doesn’t matter. The task itself is interesting. And I have a special place in my heart for that one. It was the first task we did last year when we made the decision to abandon our traditional “do now” (which ironically might have been something akin to the first task) and replace it with estimation180. Here’s what came out of it: After choosing an appropriate unit (which was a lesson in itself), finding clues, placing too lows, too highs, and just rights number lines (both vertical and horizontal), the task evolved into a discussion of just what 6’4” meant. Where would that be placed on a number line? Between 6’ and 7’? Is that the same as 6.4 feet, as many kids argued? What is halfway between 6’ and 7’ and how tall is Mr. Stadel in relation to that benchmark? Totally unplanned. Everyone engaged. And for the following several days, as they worked through the “how tall” series, all these concepts were explored and we could see understanding unfold before our eyes.
The second task is estimation. The first one might be used to estimate something, but until we find that convincing, meaningful “something”, not a pseudo-context something, it’s just another problem on another page in another workbook.