# Week 2: Chapters 4 - 5, Question #5 Look at the six questions that Boaler uses to create rich mathematical tasks.

Submitted by Amy Vickers on April 23, 2017 - 8:27pm

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Look at the six questions that Boaler uses to create rich mathematical tasks. Which question appeals to you? Why? Which question is the most challenging for you? Why?

*Challenge: Use at least one of the questions to revise or create an activity for your class. Share the activity in the discussion board.*

## Comments

## THose different achievers

S Jones April 24, 2017 - 5:15pm 0 Likes

One of the most common utterances of my students is, "So I am making this harder than it should be." Yes, these are students who don't use number sense.

The overwhelming majority of my students have long abandoned number sense. Some of them never developed it. Others have it -- but don't see how it connects to This Math Stuff (and there's a ton of research that supports this -- students believe their natural thinking of numbers is basically unrelated to Math Course Problems). Others drive me nuts because it's there, they're using it ... and then anxiety kicks in and it flies out the window...

I wish our lower courses infused number sense. I think we tend to conflate it with "knowing the facts." I'm thinking it would make sense to spend time with number sense and use it to build more fluency.

Just some starting thoughts... I remember Sebastian Thrun for claiming that MOOCs were going to bridge the equity gap in math, and then deciding that no, poor people had too many other problems for that to happen ...

## Amen, sister!

donnawparrish April 30, 2017 - 4:30pm 0 Likes

I am reminded of the student that walked into our learning center for required remediation after having failed Math 20 three times. He was very secure with the idea that all he had to do was memorize and retain long enough to pass a test. I almost bit a hole in my tongue to keep from saying, "I can't help you, then." Well, I couldn't help him. He used up all his financial aid and had no college course to his credit.

Number sense is often missing from the learners we deal with. It takes a long time to teach that to students who have struggled with math for years, but I agree with you that it is vitally important. For students to memorize rule after rule without having connections is a waste of time.

## The Staircase Example

Marti Reppetto May 15, 2017 - 12:45pm 1 Like

## The formula after much struggle

Marti Reppetto May 15, 2017 - 12:46pm 0 Likes

^{N (squared) - (N-1)N}^{ 2}## negative staircase

ecappleton April 28, 2017 - 8:13am 0 Likes

Marti,

I like the expression you came up with! You were using a negative space way of looking at the figure, right? I'm imagining the n^2 as the enclosing square formed by squaring the base. (n-1)n/2 is the triangle that is being taken away from that square.

This is the task, right?

I wonder what equivalent expressions there could be based on different ways of seeing the growth. For example, I see it growing from the bottom, so the total squares are:

1

1 + 2

1 + 2 + 3

1 + 2 + 3 + 4

...

## Yes. In fact I ordered the

Marti Reppetto April 28, 2017 - 1:21pm 0 Likes

Yes. In fact I ordered the numbers the way you did which made me think of trying to subtract 1 from a set of squares and try to average them out. I was trying all sorts of things to the point where I am not exactly sure how I stumbled on the formula. I just know that the three sided square added up to a total of 6 and the 4 sided square added up to 10...like so.

4 + 1

3 + 2

2 + 3

1 This being the extra leftover.

## Six Questions to Create Rich Mathematical Tasks

Mary Jo Chmielewski April 28, 2017 - 1:30am 2 Likes

I personally think all six questions are appealing and challenging all at the same time. While I see and understand the need for conceptual understanding, multiple representations, inquiry based...... lessons, the bottom line in my class is that they must do well on the Accuplacer test which does not support any of these findings as far as what good math teaching looks like. I know it's our reality, but it makes creating this balance a challenge. I have used a lot of Robert Kaplinsky's ideas and lessons. http://staging.robertkaplinsky.com/lessons/ These lessons create a culture of inquiry and also encourages a productive struggle. For instance, the lesson on the 100 x 100 In and Out Burger - I teach this lesson before introducing anything about slope or y-intercept. Once the students have figured out various ways of demonstrating their solutions, we see if it would be possible to graph our findings. This leads naturally to the idea of how come the line can't start at zero? Or why are we increasing by 90cents each time we move up on the graph. The concepts suddenly have meaning not to mention it comes with a lot of excitement from the students. I would highly recommend Robert Kaplinsky's website as he also goes into why it's so important to anticipate students' questions beforehand and even has a handout for students to keep track of their work as they are attempting different strategies to solve a problem.

On another note, I was thrilled to hear that Jo Boaler would write her daughter's teachers a note after they had proved they knew the concept of any math problem. I have done the same thing for my children and it actually motivated my children to teach me their own understanding of a concept so they could move onto something else and not spend hours laboriously doing 30 addition problems.

## Meaningful tasks?

Patricia Helmuth April 29, 2017 - 12:08pm 0 Likes

I found Mary Jo's comment about homework interesting, and applaud her (and Jo Boaler) for monitoring their children's conceptual understanding of math concepts and supporting them by sending notes to school that "opt-out" of pages of tedious and redundant math problems.

I admit, though, that I do send home self-study guides with my adult students to support or extend what we do in class, or for students that ask for additional practice that they can do at home. I give them similar advice on moving through those booklets: do a few problems and when you're sure you understand it, move onto the next page. So, this leads me to a few questions:

I find that most students, even those who ask for those booklets, spend little (if any) time working through them.

## Number Sense and Conceptual Understanding

Sarah Sirois April 30, 2017 - 7:12pm 0 Likes

HI Mary Jo,

I enjoyed reading your post. I too like the work of Robert Kaplinsky and use some of his tasks with my students. I understand your frustration with the Accuplacer test. I believe however that if we help students develop number sense and conceptual understanding, they will be able to make better sense of the problems they encounter on the Accuplacer and other standardized tests and thus develop greater confidence in solving them. Teaching strategies that are useful in any problem is key. Sometimes I think we worry too much about teaching to a test and should focus instead on helping students develop mathematical thinking and the mindset that Boaler speaks of.

Just my thoughts,

Sarah

## Six Questions

Duane Dorion May 2, 2017 - 8:23pm 0 Likes

The question that appeals to me the most is asking problem before teaching the method. I can think of a few of my classes where I can use this concept where it should work extremely well. A few of them are In geometry and another would be with exponent rules. When you have two like bases and your multiplying exponents you add the exponents. This is why we come up with feet squared. Or when you multiply three numbers it will become the cube of something. This all stems from the formula for exponents where when you have like bases and your are multiplying you are adding the exponents. In geometry, I can see where this can be utilized for finding area, perimeter, and volume for sure. Or showing why a triangle has the formula of 1/2BH.