Week 1: Chapters 1 - 3, Question #5 Conrad Wolfram’s four stages of working on mathematics are…

Conrad Wolfram’s four stages of working on mathematics are…
a. Posing a question
b. Going from the real world to a mathematical model
c. Performing a calculation
d. Going from the model back to the real world, to see if the original question was answered.

Challenge: Reframe a familiar lesson to follow Wolfram’s four stages of working on Mathematics. Write a simple lesson plan and share it in the discussion board.

Comments

I was able to read the first three chapters and the whole concept of mistakes being beneficial to brain learning is interesting and very antithetical to the way most of us are taught.  However, I am trying to wrap my head around how to implement the above when dealing with adult learners who are still trying to master the very basics (fractions, decimals, etc.), The above four stages makes sense if you are looking at a scientific problem or perhaps analyzing correlations between data points.

In my case, I am helping a student master the kinds of things that are found on the TASC high school equivalency test.  We are proceeding very slowly so that the basics can sink in before we move on to complicated things like Geometry and Algebra.  And brain science seems to confirm that speed does not always equal proficiency.  However, the TASC test is timed and that can be daunting for someone who is not yet confident (and when I took it myself, I would have liked more time to finish).

We've been playing around with a lesson structure called a Three-Act Task. It can be used at any level of mathematics. Teachers are using it at all levels, including elementary and kindergarten. A Three-Act Task consists of the following parts:

an engaging and perplexing Act One (usually an image or video)

an information and solution seeking Act Two

a solution discussion and solution revealing Act Three

(https://hcpss.instructure.com/courses/9414/pages/three-act-tasks)

Here's an example of a Three-Act using percentages:

What's the first question that pops into your mind? 

In a class, you would write down the answer, then talk to a partner, then share with the group, generating lots of questions, including a question the teacher is hoping for...

Which coupon should I use? Or Which is the best deal?

In most three-acts, the next step would be to ask students what information they need in order to answer the question and then give them the information we have prepared if they ask for it. In this three-act, the teacher creates stations with different items such as...

Students move around to each station and make calculations to decide for each item whether $20 or 20% off would be a better deal. At some point, the teacher will want to ask if there is a faster way to figure this out than to do these calculations every time. Something like, At what price is 20% off better than $20 off?

I think the idea of modeling with mathematics is not about the level of the mathematics, but about using a process that allows students to determine the question, decide what information they need, do calculations and then see whether their calculations make sense with the original situation.

For information on how to use three-act math tasks in adult education, I recommend this post by Mark Trushkowsky on CollectEdNY.org.

 

 

I think Eric said it really well when he wrote, "I think the idea of modeling with mathematics is not about the level of the mathematics, but about using a process that allows students to determine the question, decide what information they need, do calculations and then see whether their calculations make sense with the original situation."

As he described, the three-act math structure is a great one for including students at any level into the problem-posing aspects of mathematics. One of the resources I mention in my CollectEdNY review is 101qs.com. At 101 questions you can find videos and photographs for inspiring perplexity and problem-posing in students. The site has hundreds of 3-act lessons, which aren't lessons in the usual sense, but offer teachers the materials they need to pull off the different stages with students. 101qs also has a search feature that allows you to target specific grade levels - check it out here.

Another suggestion I would make is Notice/Wonder. It doesn't give students the same opportunity to have to decide what information they need, which is one of the things I really like about the 3-act math tasks. On the other hand, notice/wonder is incredibly simple to do right away.

Here are the basic steps:

1. Find a mathematical situation or graph.

2. Remove the question.

3. Ask students what they notice.

4. Ask students what they wonder.

For example:

Tyler and Rachel go to the movies with their three nephews, aged 7, 8 and 12. At the movie theater, children's tickets are half the price of the adult tickets. They paid a total of $43.75 for all the tickets.

What do you notice?

What do you wonder?

Asking students what they notice does a lot of things. First and foremost, it emphasizes the importance of taking time to make sense of the situation. If there was a questions, too often students move to answer it immediately, without fully understanding the situation or the conditions. It also gives them time to make personal connections. Hearing what they notice gives teachers insight into what they understand and it allows them to share and build their collective understanding with each other. 

The first thing asking students what they wonder does is engage student curiosity. It also allows them to ask clarifying questions. Finally, instead of one problem to solve, you get many. For example, what is the price of one adult ticket? What is the price of one kids ticket? How much did they pay for the 3 kids? How much was Rachel and Tyler's share of the total? What if the 12 year old has to have an adult ticket? You can use the questions students generate in different ways: you can have them choose one, you can direct them all to the one you want them to answer (this is similar to the three-act approach), you can have different groups work on each of the questions, etc.

Here's another one:

What do you notice?

What do you wonder?

I've done this one with students without the Percentage Cost column, but I can't figure out how to post an image on LINCS that is not attached to a website. It works with that column too. I wanted to share this one because I've had a lot of success with using notice/wonder with charts, infographics and graphs.

Some resources to explore Notice/Wonder

 

Also, on a completely different note, Eric and I are in New York which is also a TASC state. In February we did a webinar where we brought representatives from the NYS Dept of Education and the Test Development Program Manager from Data Recognition Corporation/CTB to answer some questions about the TASC, especially around the high-emphasis content and the changes on the new GHI forms of the TASC. If you're interested, you can watch it here: http://www.collectedny.org/2017/02/cuny-webinar-teaching-high-emphasishigh-utility-math-content-for-the-tasc/   

 

Thanks, I plan to check out the links provided here.  I am also in New York and I have been following along with the McGraw Hill TASC preparation materials.  However, I feel I have been relying a little too much on "boring" practice drills provided in the books and not enough real world application (although I have been using pie charts etc. for fractions).  This should help.

This one (and many more) we learned during the ANI Institutes:The million dollar question.

Pose a question: What would you rather have, a million dollars or all the one dollar bills needed to cover a football field?

Real world to mathematical problem: Find areas of dollar bill and football field, ideally in comparable units.

Perform calculations: unit conversions, division

Back to the real world: Should anyone ever offer, definitely take the million dollars!

You can do this to many of the other math activities the ANI  program touched on: Input/output rules, finding Pi (measure circumference and diameter of different size circles), number of the day, number line (fractions, decimals, percents, pictorial representations) from 0 to 1, comparing different size/shape areas, the popcorn cylinder volume experiment, etc. I have tried most of these in a remedial community college course offered by our adult education program free of charge; they work, they definitely get students engaged.

What a cool idea - the million dollar question - I love it!  I teach at a community college in NH and I can see this task as something I can give my students in the developmental class as well as in an applied math class that I teach.  Thanks!

Marti,

Along with some of the great resources that have already been mentioned is this thread is the CUNY HSE Mathematics Curriclum Framework that focuses on Problem Solving in Functions in Algebra.(Did someone mention it yet? If so, I missed it).

I use this with all students at various ability levels. Sometimes, I think our students get boxed into a certain "math zone", either by their own perception of what they think they can't do or by program curriculum that is set up to to sequence the learning of mathematics as linear: addition, subtraction, multiplication, division, fractions, decimals, percents, algebra, and then geometry. 

The CUNY curriculum provides scaffolding that enables students to work through Wolfram's four stages in regards to functions even though they haven't yet mastered the aforementioned linear sequence of mathematics instruction. Students can understand functions especially when they see how it applies to real life situations.

Patricia

 

 

Hi Marti,

Your response " I am helping a student master the kinds of things that are found on the TASC high school equivalency test.  We are proceeding very slowly so that the basics can sink in before we move on to complicated things like Geometry and Algebra.  And brain science seems to confirm that speed does not always equal proficiency." shows a great deal of understanding about the complexities of dealing with emerging learners.  TASC or no TASC, our students rarely plan to spend years gaining skills in math or any other subject, so the time element is exceedingly important. What are the basics that you are teaching your students? Are you using area models to help them learn multiplication of all sorts? Are they observing patterns and verbalizing them as they look at, for example, multiplication of whole numbers? There is a valuable intertwining of the math strands that allows the learning in one strand to be applied to others. Now THAT is a beautiful thing.

Lacking deep familiarity with TASC and perhaps showing ignorance here, calculator use is permitted, right? Here's a great TED Talk Wolfram video that argues for moving learning from arithmetic to "real math".  I am way behind in this course so maybe someone else suggested it. https://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers

Congratulations on your ponderings!

Half of the TASC test allows the use of a scientific calculator.  I imagine that is more for square roots and complex formulas (I am still a little fuzzy on that myself).  For now, I am working with basic computation involving fractions and decimals (long division etc.).  I try to illustrate this using pies (fractions) and graphical illustrations to show what .1,.01 and .001 actually looks like.

I teach adult basic ed in Michigan (but for the GED). I have recently started using something from New York and I have been amazed at the results so far! I usually started like you did and tried to work the students through the basics first and found that I would never get past that. This time I tried the curriculum from CUNY. It is discovery based. I can't beleive how many concepts the students know when they are presented in a different format. We worked two days on functions. The students more deeply understood functions after two days using this method than they would have in two weeks of the methods I used to teach.

I do know that I will have to go back and address some of the holes students have in their knowledge and some of the basic skills, but I figure if I can get them past the math phobia and show them that they can do real math they will realize that they can do the basics also. For now I let them use a calculator.

The materials I used are found here:

http://www2.cuny.edu/academics/academic-programs/model-programs/cuny-college-transition-programs/adult-literacy/cuny-hse-curriculum-framework/

 

I go over a scenario in my Math 2 class on good vs bad credit. I ask the question, "is it better to pay your bills on time and have good credit or to not pay your bills on time and have terrible credit."   I use a car and a house with good credit at 5%, mediocre credit at 10% and terrible credit at 15% for a house and 20% for a car. I show the difference in monthly, yearly and the students calculate total payments of these two scenarios.  The students then have to calculate the difference of total payments on the loans for the three different interest rates.  At the end I ask the students, "does it pays off to pay your bills on time or just pay them when you want to.".  All the students see this as an eye opening experience to see how much more they will pay for a car and a house just because they do not have good credit.  They feel it was a valuable lesson to learn going through these scenarios. Most students will tell me that paying their bills on time will be something they will make sure happens.  Some students also say that is not what their parents currently do. They also see why their parents spend so much each month on their car payments.