From Chalkbeat:

The event, to be held in March at the SXSW EDU conference in Austin, Texas, will build on live-format shows that celebrate the hidden craftsmanship in other professions — think Top Chef, Project Runway, and The Voice — minus the competition. You can read more about the Teach-Off here.

The goal of this event is to highlight teaching as an intellectual activity and to make visible the invisible decisions that teachers constantly make when they teach.

If you wanted insight into teaching decision-making, who would choose? Which of these pairs of teachers would you like to learn more about their teaching?

Check out these really reflective teachers and help decide who will get their decision-making made visible for the world!

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The goal of this work was to take a repetitive task that almost all NY State public school math teachers do, which is to merge and typeset items from Regents exams, and greatly reduce how long this task takes, thus saving teachers time to do other more important tasks. We can easily typeset Regents questions centrally at a fairly low cost, and then a tool like Quiz Banker makes it easier for teachers to work with those typeset questions.

During the summer we asked teachers how long it would normally take them to take all of the Regents questions associated with a particular domain of mathematics and typeset them into the same document. Answers from teachers ranged from 5 minutes to 8 hours with most teachers estimating about an hour. When we demoed Quiz Banker, it took 2 minutes to accomplish this task, including the time spent installing the add-on.

During the summer suggestion I told teachers, “If it used to take you 40 minutes to create a quiz and now it takes 2 minutes, use those 38 minutes you saved to make sure that quiz assesses what you want it to assess.” As Patrick Honner notes frequently on his blog, not every Regents question is of equal quality.

Having a question typeset also means you can easily modify a multiple choice question into a more open-ended question, modify the language of a question to get a slightly different mathematical idea, or just increase the font-size so that students with differences in visual processing are able to read the question.

QuizBanker also includes meta-data like what Common Core Domain, Cluster, and Standard to which each question is aligned as well as alignment to the Units and Big Ideas in the New Visions’ Math Curriculum. This further reduces how long it takes teachers to aggregate those questions usefully.

More broadly I believe that if teachers are going to work on changing their teaching, this takes time, but time is a fixed quantity. The cheapest way to give teachers more time to work on improving their teaching is to take repetitive and time-consuming tasks they do and change the amount of time spent on these tasks either by eliminating those tasks altogether or by reducing how long the task takes to do.

What other tasks do you see math teachers doing frequently that could save time if there was a tool that made that task easier and faster to do?

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In learning teaching however, since every practice is connected to other teaching practices, it can be extremely challenging and potentially unhelpful to isolate individual teaching practices. For example, you cannot really get better at the 5 Practices (summary, book) without considering how those 5 Practices interact with each other. If you anticipate student ideas for an upcoming lesson, you will only get feedback on that anticipation if you also monitor what students do.

One strategy to reduce the complexity of learning to teach is to approximate teaching practice in various ways. Instead of teaching a whole class of students, one can teach at one table. Instead of teaching five classes of students a day, one can teach one class. Instead of teaching on one’s own, one can co-teach with a mentor teacher.

Another approximation of teaching we have found helpful in our work is the use of an instructional rehearsal, which is where one teacher (or perhaps a pair of co-teachers) leads the group in a teaching experience with everyone else playing the role of students. At either strategic instances or on request of the teacher(s) leading the experience, the action stops and everyone considers teaching practice either together or in small groups.

It is helpful for one person to act as the facilitator or coach for this experience, and for the rest of the participants to switch between playing the role of students while the teaching experience is in action, and to discuss the teaching as teachers when it is not. If each rehearsal has a different focus, then one can learn different elements of teaching over time, while still maintaining the complexity of teaching. The goal is for the core practices of teaching to become integrated rather than overly isolated. The Teacher Education by Design website has more details and resources for instructional rehearsals here.

A further design element of instructional rehearsals is that the activity to be rehearsed should be fairly clear for participants. We use instructional routines as the frame for our rehearsals because they constraint the scope of potential decisions to be made and subsequently discussed but are still complex enough examples of teaching to allow for different foci or teaching practices to be discussed in different rehearsals. We typically model an instructional routine a few times for teachers, unpack it collaboratively, then teachers plan around a task for the instructional routine, and then rehearse the instructional routine one or two times as a whole group.

Rehearsals can be places to discuss planning processes and protocols that might be necessary pre-steps to improve the enactment of a performative teaching practice. For example, while considering how to annotate a student strategy during a rehearsal, participants will likely realize that practicing different annotation strategies in advance of a lesson would be helpful and that in order to do this, one should first anticipate the student strategies that are likely to emerge for a particular.

We have found rehearsals to be helpful for teachers at all stages of their career, since all teachers have room to grow and to learn. The foci of the rehearsals for pre-service, early career, mid-career, and late career teachers may be different but the overall process is the same.

One other key idea of rehearsals: the goal is rarely to give the teacher leading the rehearsal feedback although that often happens but to collaborate together to consider teaching. The goal is to collectively improve teaching practice not individual teachers.

Rehearsal is not a replacement for working with a mentor teacher over time to learn ways to communicate with parents and other critical aspects of the role of a teacher. Some elements of teaching practice are hard or potentially impossible to rehearse. However the performance aspect of teaching is where most teachers will spend at least half of their time, and rehearsals are a good strategy for developing performative teaching practice.

**References:**

Kazemi, E., Franke, M., & Lampert, M. (2009). Developing pedagogies in teacher education to support novice teachers’ ability to enact ambitious instruction. In Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 12-30).

Knowing Teaching from the Inside Out: Implications of Inquiry in Practice for Teacher Education. (1999). In G. A. Griffin (Ed.), The education of teachers (pp. 167-184). Chicago, IL: NSSE.

Lampert, M. (1990). When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching. American Educational Research Journal, 27(1), 29-63. doi:10.3102/00028312027001029

Lampert, M. (2009). Learning Teaching in, from, and for Practice: What Do We Mean? Journal of Teacher Education, 61(1-2), 21-34. doi:10.1177/0022487109347321

Lampert, M., Franke, M. L., Kazemi, E., Ghousseini, H., Turrou, A. C., Beasley, H., . . . Crowe, K. (2013). Keeping It Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching. Journal of Teacher Education, 64(3), 226-243. doi:10.1177/0022487112473837

Lampert, M., & Graziani, F. (2009). Instructional Activities as a Tool for Teachers’ and Teacher Educators’ Learning. The Elementary School Journal, 109(5), 491-509. doi:10.1086/596998

Mcdonald, M., Kazemi, E., & Kavanagh, S. S. (2013). Core Practices and Pedagogies of Teacher Education: A Call for a Common Language and Collective Activity. Journal of Teacher Education, 64(5), 378-386. doi:10.1177/0022487113493807

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Here is a strategy that may help when you want students to use an interactive tool but either have limited access to devices or do not want to waste a bunch of classroom time handing out devices.

**1. Introduce the goal of using the technology to students.**

We introduce the goal first so that students have some sense of what they are trying to accomplish. The goal can be somewhat vague so that it doesn’t take any of the magic out of the lesson, but ideally upon reflection students should be able to see either how they reached or did not reach the goal.

**2. Have a pair of students come up to use the tool and demonstrate in front of the class.**

The pair of students will problem solve with direct access to the computer. They can manipulate sliders, drag things around, etc… and use the interactive tool as designed. Everyone else works with a partner to do the same thinking and discussion about the interactive tool but without the direct access that the one pair has. The rest of the class is relying on the pair at the front of the room to manipulate the interactive tool in ways that are useful to their own learning.

This also frees the teacher up to circulate around the room and listen in on the conversations students have. This will give you some formative data on what students are thinking about during their discussions.

Note that it is best if the pair at the front of the room takes enough time to finish using the tool so that everyone in the room has sufficient time to notice relationships (if that is the goal). Therefore the pair that comes to the front of the room should be a pair of students that you can rely on to move deliberately enough that everyone has access to the range of possible things noticeable via the tool.

**3. Have someone else describe their thinking.**

Have someone, other than the pair of students already at the front of the room, describe relationships they noticed while the pair already at the computer manipulates the interactive tool under their direction in order to demonstrate the thinking being described. At this stage, it may be helpful to annotate or otherwise record a representation of what is being discussed so that it is clear for all students. You may decide to ask a few different students to present and you may want to select which students to present based on circulating around the room earlier.

During these presentations, you may want to use a variety of strategies to make the thinking of the pair of students clear for everyone.

**4. Have students reflect on or apply what they learned.**

Students should have opportunities to reflect on experiences that they have, either by trying to apply the ideas to other problems or by writing about those experiences.

Here is a good point from David – if you have a document camera you can use this same principle with anything that you have.

Yeah, provided you have some way for everyone to be able to clearly see the thing. Maybe could use with manipulatives underneath a document camera as well?

I should add this point to the blog post.

— David Wees (@davidwees) December 20, 2017

Take a look at this video of an interactive tool being manipulated so that you can experience what it is like to watch something being changed without being able to change it yourself.

(For those who are interested, the tool in the video is available here).

The student has clearly made a mistake. Why did they do it? I asked on Twitter and here are some theories:

Using stacked subtraction, how could a student subtract 43 from 180 and get 77 as answer? What are they thinking?

— David Wees (@davidwees) January 12, 2015

@davidwees My guess: they brought down the 7 (70) from after regrouping and forgot their hundred and their 40.

— Andy Mitchell (@gadfly1974) January 12, 2015

@davidwees Got the ones column right. Then crossed out the 8 and wrote 7 in the tens place instead of above. Then said "I'm done."?

— Josh Fisher (@sineof1) January 12, 2015

@davidwees Crossed out the 8 to borrow the 1 but wrote down a 1 next to the 8 instead?

— Michael Jacobs (@msbjacobs) January 12, 2015

@davidwees I think this sort of thing happens when the calculation ends up being demanding and we just default to associations w/numbers.

— Michael Pershan (@mpershan) January 12, 2015

@davidwees So 10-3=7 and then 7 is present in their memory for whatever reason (because they were just thinking about it?) so 77.

— Michael Pershan (@mpershan) January 12, 2015

@mpershan @davidwees If they "borrowed" from the 8 there's a 7 sitting there?

— Chris Robinson (@MrCBRobinson) January 12, 2015

—

@davidwees Maybe as they borrowed 1 from the 8, they wrote 1, so subtracted 4 from 11?

— Chen Mingi (@chen_mingi) January 12, 2015

@davidwees when the stu borrowed 1 from the 8, the 7 s/he wrote looked like a 1. So sloppiness.

— Steven Francis (@SFrancismath) January 15, 2015

I think all of these are possible answers. Only one of them is possibly correct for this particular student but any of them could potentially be reasons a student might do this calculation incorrectly.

But do each of them have the same instructional response? I’m not clear on that. I think we need to know more than what students did right or wrong, I think we need to know what thinking students were doing. And I think we need to know what students are thinking whether a student has done a problem correctly or incorrectly. I also think we should focus on learning, not just individual performances.

If all we know is if students go an answer right or wrong, the best instructional approach we can imagine is to essentially repeat our prior instruction, except maybe slower and louder than before. If we know more about student thinking, then we can focus on experiences that will change the information students are using to make decisions, which I think is far superior feedback to students than x’s or checks on a piece of paper or the digital equivalent via a computer.

If my theory that our responses could vary depending on different student thinking, then adaptive computerized systems have a long way to go before they are really going to meet student need. None of them currently has any hope in gaining better insight into student thinking than a teacher asking the simple question, “Can you explain to me what you did here?”

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Here is one example from the Khan Academy (apparently at least one of the flaws outlined below no longer applies to the Khan Academy. But that same flaw still applies to IXL. And Prodigy Math. And a thousand other practice apps out there.)

**Feedback is terrible or nonexistent**

How does everyone feel about those online maths practice things that only let you progress if you get a streak of several completely correct in a row? #MTBoS

— David Butler (@DavidKButlerUoA) November 27, 2017

Many programs will, as David points out above, only allow a student to progress after they have gotten a certain number in a row correct. But if a student is struggling to complete an activity and the feedback to the student is terrible, how exactly are students meant to achieve the streaks necessary in order to advance?

Note: Watching a video of a concept isn’t feedback if the learner has already watched that video before. That’s information the learner already has.

**Impossible to see patterns**

One way that people learn math is by observing patterns in their work or solution strategies as they work on a set of problems in a row.For example, what pattern do you see if you try the following exercise:

5 x 3 = ?

4 x 3 = ?

3 x 3 = ?

2 x 3 = ?

1 x 3 = ?

0 x 3 = ?

-1 x 3 = ?

-2 x 3 = ?

-3 x 3 = ?

I’ve tried this with students and most of them notice that they are subtracting each time to find the next product, and so then they make a leap and decide that -1 x 3 must be -3.

But if an online practice program only ever shows one question at a time and the numbers for these questions are selected randomly, there will be very limited opportunity for students to notice and subsequently use any patterns that emerge.

**Blocked practice**

Except for a handful of studies, there is a lot of research that suggests that for most people, if the goal is to remember some mathematical idea, practicing topics in blocks will take longer than if different topics are interleaved together. Almost all of the programs out there focus on students practicing discrete topics. Caveat: I did read a study recently that suggested that for students entering a course with weak prior performance, while interleaved instruction was beneficial, interleaved practice was less effective for these students than blocked practice. Further caveat: I cannot find the link to this study.

**Too easy or too hard**

For some students the exercises are too easy. Sometimes this is because kids select easier problems for themselves, sometimes this is because students already know a bunch of mathematics and do not need this particular practice activity. Either way, needing to work through a streak of 5 or 10 problems just to be able to move on is ineffective for these students.

For other students the exercises are too hard. A student who really doesn’t know a particular area of mathematics doesn’t benefit from practice in that area – they need teaching or access to information.

**Inappropriate medium**

For many, many math problems, the best choice of a medium to work on the problem is a piece of paper. Or maybe the best choice for working on a particular problem is a programming language.

These online systems offer neither. This means students are often working in a possibly unfamiliar medium without the most useful tools available for them to work.

This also restricts the people who design questions for the system as they end up likely severely restricted as to what kinds of questions they can ask if they need the answer to the problem assessed by a computer.

**It obscures information from teachers**

If you are a teacher and you are using one of these online systems for your students to practice, there is usually a dashboard you can look at to see how well your students are doing with a particular exercise. But these dashboards truncate an enormous amount of information about the progress of learning and actually make it harder for you as a teacher to gather the information you need to be able to act to improve your students’ learning.

They also are likely to lead to teachers looking for students making mistakes instead of looking for student conceptions, which promotes a deficit view of students instead of treating students as sense-makers.

**It can lead to bad practice**

Virginia Tech has an online remedial math program where students go to sit at a terminal and watch videos on math and then take quizzes on what they learned, over and over again. There is a Facebook post where almost all of the students complain about how much they hate this mathematics class. If the online practice programs did not exist, neither would this course.

Teach to One uses a computer online practice program to inform teachers when small group instruction should occur. But in this middle to upper class neighbourhood, parents revolted and the program was scrapped. But what about districts where parents have less power?

Dan Meyer outlines the many problems with Rocketship Learning Labs, another personalized learning model, in this post.

**It isn’t really mathematics**

If you ask a mathematician or anyone who uses mathematics regularly what mathematics is, literally none of them will answer “it is a series of multiple choice questions or short response questions asked and answered on a computer screen.” While practicing mathematics is a decent way to get better at what you know how to do, it isn’t really the goal of teaching students mathematics.

If answering a series of problems is the only experience of math that students have, they are likely to end up with a very limited definition of what mathematics actually is.

Note, I’m not opposed to students practicing math at all. This is obviously an example of good practice and there is plenty of research to support this claim. I’m opposed to this being the *primary* experience of math that students have.

**Conclusion**

If you can possibly avoid it, don’t use these programs. Or at least try the program yourself for a couple of hours to see what the experience is really like for students. And if you are a designer of one of these online computer practice systems, for the love of God please do a better job than the industry currently is.

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My bold prediction: none of these 10 will disrupt education. None. https://t.co/pUAtmUOK7E

— Daniel Willingham (@DTWillingham) October 16, 2017

In no particular order, here are ten things that might actually disrupt US education.

**Teachers being afforded respect as a profession by policymakers and others**

You do not enact law like No Child Left Behind if you fundamentally believe that teaching is a profession. You know who primarily regulates lawyers, engineers, and doctors – that’s right, in many countries they do that themselves.

**Teachers, especially elementary school teachers, having adequate time to plan**

In some states elementary school teachers teach all subject areas and have a total of 45 minutes to plan AND are paid so little that many of them need second jobs.

**Provide curated resources to teachers**

Although increased planning time may reduce this tendency, designing ambitious curriculum is difficult and extremely time-consuming, so most teachers would benefit from curated resources that they can modify and adapt using their professional judgement. Surprisingly, many teachers have to use Pinterest and/or Google to find resources for their classroom because of a lack of curriculum resources aligned to their new state standards.

**Paying teachers enough that they do not need second jobs and can afford to live in the communities that they work in**

One way to make getting into teaching competitive would be to pay people enough that it makes teaching an attractive choice. It would also mean fewer people leaving the profession to find more lucrative careers and leaving vacancies, especially in harder to fill content areas.

**Policies intended to improve teaching not teachers**

As Jim Stigler and notes in the Teaching Gap, much of US policy is engineered at supporting individual teachers at getting better and that as soon as these teachers retire or quit, their professional knowledge leaves with them and the profession of teaching in the US remains relatively unchanged. It’s a good thing for individual teachers to get better at their practice, it is better that the professional benefits from what they learn.

**Equitable funding across US schools**

In some school districts, schools spend $9000 per student while a few miles away in a suburban district, schools spend $26,000 a student. While this inequity exists, resources are unevenly distributed across US education and in most cases the students who need the most support to be successful receive the least amount of funding.

**Equitable access to teachers across US schools**

In almost all large urban areas, it pays better to work in the suburbs than it does to work in the city. This results in teachers leaving the cities for high paying, lower stress jobs outside of the cities and in uneven amounts of teacher experience across the schools in the city.

**Design school structures which are coherent and communicate across all levels of education**

Imagine a system where the person who teaches teachers never sets foot in a school, the person who runs a school has no time to read research or even see their teachers teach, the person who runs research has never taught, the instructional coach who supports a teacher has their own idiosyncratic teaching style, and a teacher who has to listen to all of these people give them different advice on teaching. This is considered normal for teachers across the United States. But it does not have to be that way! It is possible to design systems where all of these people work collectively rather than individually.

**End economic inequality in the United States**

Income inequality in the United States is increasing and given that we know already that there exists a relationship between income and educational achievement, any shift toward more economically equitable society is likely to result in improvements in education for most students.

**The end of systematic oppression of people of colour**

The United States has a long dark history of oppressing people of color in various ways. One way this occurs in the US school system is that the schools attended by children of color are much more likely to be closed and/or labelled as failing than other schools. Ending this systematic oppression would transform the United States educational landscape.

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So we contacted the organizers of the two NCTM regional conferences and proposed a possible solution. Instead of running one session, we will run 4. Instead of 4 separate sessions, we will plan those 4 sessions to connect together. Given how closely my colleagues and I work, we were each able to be the lead speaker on a different proposal. Both the NCTM Orlando and NCTM Chicago conference organizing teams agreed to this proposal and scheduled our sessions both so they do not overlap and also sequentially as requested.

So although we have 4 separate workshops listed in the program guide, these sessions are actually one of our day-long workshops divided into four sessions. Our hope is that some participants will experience one workshop and be no worse off than before – they will still learn something even if it is not the complete picture – but that participants who attend multiple sessions will have more insight and ability to use our work.

Here is a video of Contemplate then Calculate in action, with Kaitlin Ruggiero as the teacher and some teachers from one of our courses playing the role of students.

If this teaser intrigues you, our four sessions are:

**Experiencing Instructional Routines:**In this session participants will experience the same instructional routine three times with three different tasks to consider what elements of the teaching that occurs are part of the routine and what elements probably depend on the task and the students.

**Unpacking Instructional Routines:**Next, participants will experience the routine again (this will give access to people for whom this is their first session) and name the parts of the routine, why those parts are helpful, and what questions they have about the routine.

**Planning and Preparing Instructional Routines:**There is good evidence that a new teaching idea sticks better for participants if they have an opportunity to incorporate it into their existing teaching by planning and preparing to use the idea, so that will be the primary focus of this session. This will also connect the planning process for the instructional routines to the 5 practices for orchestrating productive mathematical discussions.

**Rehearsing Instructional Routines:**There are two goals of this session. First that some participants will have an opportunity to apply what they have learned and actually practice using the instructional routine before trying it out with students. Second, rehearsal of teaching is a useful way to norm around teaching practice and to try things out in teaching in a lower pressure situation than with a group of students.

Here are the times and locations for these sessions in Orlando:

And here are the times and locations for our sessions in Chicago:

Further reading about instructional routines:

- Why instructional routines?
- Instructional Routines as Formative Assessment
- Ambitious Mathematics Curriculum

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• Does this curriculum assume that children will forget ideas over time?

• Does this curriculum provide instructional supports that increase the odds that all children have access to it?

• Does the curriculum assume all students are capable of learning and doing interesting mathematics?

• Are the connections between different mathematical ideas made explicit, both for me as a teacher, and for students who will experience the curriculum?

• Is it possible, based on the license and format of the materials, for me to extend / adapt / modify the curriculum based on student need?

• Does the material make it easier or harder for me to use formative assessment practices each day?

If the answer to all of these questions is not yes, I don’t want to use that curriculum. A curriculum which is no more than a collection of tasks is no more useful to me than my ability to search for resources in Google.

What other questions do you ask yourself when reviewing curriculum?

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The issue is that mathematical representations are not intrinsically meaningful on their own. Some mathematical representations are completely arbitrary and for others it can be challenging to determine to what elements of the representation to pay attention.

Here is an example intended to highlight how some mathematical representations, even ones that are very familiar, are somewhat arbitrary. Check out the diagram below and ask yourself, “What is meant by each of these models for the less than, equals to, and greater than signs?”

The less than, equals to, and greater than signs are arbitrary. They are symbols to which we denote meaning and which otherwise do not contain any mathematical information without that meaning assigned.

Another issue is that students do not always attend to the critical features of a mathematical representation. For example, I have often seen a shape and a formula for calculating the area of that shape introduced together, possibly like it is shown below with a calculation of area alongside the visual.

But to what exactly in this representation do we expect students to attend? The most obvious features of the diagram of the rectangle that correspond to the area formula are the 5 and the 3. These refer to the quantities of length and width. But what is meant by the multiplication of those two quantities? How is this multiplication represented in the diagram? There is no special reason from diagrams like these that children will attend to the space occupied by the rectangle and match that to the area of the rectangle, so we need to find ways to draw their attention to this element of rectangles.

Mathematical representations have potential power to subtly introduce ideas to students as well. The number line is a good example of a representation that is often introduced early and may lead to some powerful questions by students.

Each of these questions has a mathematical answer and the number line can again be used to represent this answer (warning: but not always very well).

I worked recently with a group of teachers, and we looked for shortcuts to solving the equation x + x = 116 – 84.

Here are some of their shortcuts.

Strategy 1 | Strategy 2 | Strategy 3 |
---|---|---|

“I combined the x’s together and I subtracted the 84 from the 116, which gave me 32. I could do this quickly because I knew that 11 – 8 = 3 and 6 – 4 = 2. This gave me 2x = 32, so then I divided both sides by 2 to get x = 16.” |
“I saw the x + x and changed it to 2x. Then I decided to divide everything by 2 to make the calculation simpler, and got x = 58 – 42. Since 58 – 42 is 16, this means x = 16.” |
“I noticed that 116 and 84 are both 16 away from 100. So I can rewrite this as x + x = 16 + 16 and therefore x = 16.” |

But what if we tried to represent Strategy 2 and 3 on a number line? Here are a couple of different visualizations of these strategies. What information is captured differently by the different visualizations?

In our math curriculum, when introducing an area model for factoring and completing the square, we first introduce the representation itself before we do any other mathematical work using it.

Try out this applet and ask yourself, *“What relationships between the visual and the expression do you notice as you change the value of a?”*

In a classroom setting, we could ask students to share their answers to this prompt with a partner and then we could ask some students to share their answers with the entire class. After this, if necessary, we could add an observation from another class, so that students know to what elements of this representation to attend.

Next, after students practice writing out expressions for unaltered diagrams, we could ask students to write expression for the following altered diagrams:

In my experience, this geometric approach to completing the square results in more students in having access to the algebraic approach, and makes the name of the algebraic strategy more obvious.

Mathematical representations can offer explicit ways for students to make connections across different mathematical topics. In our Algebra I curriculum, we do not have a unit on graphing. Instead interpreting and using graphs is part of all seven units, increasing the odds students make connections in and between those units and also that students remember key ideas from the course.

**To summarize:**

- Don’t assume that mathematical understanding is transmitted by the representation.
- Some mathematical ideas are easier to introduce using some representations rather than others.
- Reusing a specific mathematical representation over and over again will both help students make mathematical connections and remember key concepts from the year.

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